Subgroups — C₆₇×#61707 #30740958

Order 17152.

2086 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#553×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3345	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3318	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3234	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3192	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3186	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3109	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3094	C₂	1	2	yes	maximal, complemented
10	C₂×#481×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2973	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
13	#553×C₆₇	—	2	4	no	—
14	#553×C₆₇	C₂²	1	8	yes	complemented
15	#547×C₆₇	C₂²	1	4	yes	complemented
16	#546×C₆₇	—	2	4	no	—
17	D₄²×C₆₇	C₂²	1	4	yes	complemented
18	#537×C₆₇	—	2	1	no	—
19	#534×C₆₇	—	2	1	no	—
20	#534×C₆₇	—	2	1	no	—
21	#532×C₆₇	C₂²	1	2	yes	—
22	C₂×#126×C₆₇	—	2	1	no	—
23	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×#123×C₆₇	—	2	1	no	—
25	C₂×#121×C₆₇	—	2	1	no	—
26	C₂×#121×C₆₇	C₂²	1	2	yes	—
27	C₂×#121×C₆₇	C₂²	1	2	yes	—
28	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₄×D₄×C₆₇	—	2	1	no	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#501×C₆₇	C₂²	1	4	yes	complemented
36	#490×C₆₇	C₂²	1	4	yes	complemented
37	#482×C₆₇	C₂²	1	8	yes	complemented
38	#481×C₆₇	C₂²	1	8	yes	complemented
39	#481×C₆₇	C₂²	1	4	yes	complemented
40	#481×C₆₇	C₂²	1	4	yes	complemented
41	#481×C₆₇	C₂²	1	4	yes	complemented
42	#480×C₆₇	C₂²	1	4	yes	complemented
43	#480×C₆₇	C₂²	1	4	yes	complemented
44	#478×C₆₇	C₂²	1	8	yes	complemented
45	#476×C₆₇	C₂²	1	4	yes	complemented
46	#469×C₆₇	C₂²	1	8	yes	complemented
47	#466×C₆₇	C₂²	1	8	yes	complemented
48	#459×C₆₇	C₂²	1	4	yes	complemented
49	#447×C₆₇	C₂²	1	4	yes	complemented
50	#447×C₆₇	C₂²	1	4	yes	complemented
51	#447×C₆₇	C₂²	1	4	yes	complemented
52	#442×C₆₇	C₂²	1	8	yes	complemented
53	#435×C₆₇	C₂²	1	8	yes	complemented
54	#432×C₆₇	C₂²	1	2	yes	complemented
55	#428×C₆₇	C₂²	1	4	yes	complemented
56	C₂×#106×C₆₇	C₂²	1	4	yes	complemented
57	#418×C₆₇	C₂²	1	4	yes	complemented
58	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
59	#407×C₆₇	C₂²	1	2	yes	complemented
60	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
61	H(5,𝔽₂)×C₆₇	—	4	8	no	—
62	C₂×#41×C₆₇	—	2	4	no	—
63	C₂×#41×C₆₇	—	2	2	no	—
64	C₂²×D₄×C₆₇	—	2	1	no	—
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	—	2	2	no	—
67	C₂²×D₄×C₆₇	—	2	1	no	—
68	C₂²×D₄×C₆₇	—	2	4	no	—
69	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
70	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
72	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
73	C₂³×C₄×C₆₇	—	2	1	no	—
74	C₂³×C₄×C₆₇	—	2	1	no	—
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×D₈×C₆₇	C₂³	1	8	yes	—
78	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
80	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
81	#127×C₆₇	C₂³	1	2	yes	—
82	#126×C₆₇	—	2	8	no	—
83	#125×C₆₇	—	2	2	no	—
84	#125×C₆₇	C₂³	1	4	yes	—
85	#124×C₆₇	—	2	2	no	—
86	#123×C₆₇	—	2	8	no	—
87	#123×C₆₇	—	2	4	no	—
88	#123×C₆₇	—	2	4	no	—
89	C₂²⋊Q₈×C₆₇	—	2	4	no	—
90	#121×C₆₇	—	2	8	no	—
91	#121×C₆₇	—	2	2	no	—
92	#121×C₆₇	—	2	2	no	—
93	#121×C₆₇	—	2	2	no	—
94	#121×C₆₇	—	2	2	no	—
95	#121×C₆₇	—	2	2	no	—
96	#121×C₆₇	—	2	2	no	—
97	#121×C₆₇	—	2	2	no	—
98	#121×C₆₇	—	2	4	no	—
99	#121×C₆₇	C₂³	1	4	yes	—
100	#121×C₆₇	C₂³	1	4	yes	—
101	#121×C₆₇	C₂³	1	4	yes	—
102	#121×C₆₇	C₂³	1	4	yes	—
103	#121×C₆₇	C₂³	1	4	yes	—
104	#120×C₆₇	—	2	4	no	—
105	#120×C₆₇	—	2	4	no	—
106	C₄×D₄×C₆₇	—	2	8	no	—
107	C₄×D₄×C₆₇	—	2	2	no	—
108	C₄×D₄×C₆₇	—	2	2	no	—
109	C₄×D₄×C₆₇	D₄	1	4	yes	—
110	C₄×D₄×C₆₇	C₂³	1	4	yes	—
111	C₄×D₄×C₆₇	C₂³	1	4	yes	—
112	C₄×D₄×C₆₇	C₂³	1	4	yes	—
113	#117×C₆₇	—	2	2	no	—
114	#117×C₆₇	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
127	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
128	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
129	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
130	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
131	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
132	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
133	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	—	2	1	no	—
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	#107×C₆₇	C₂³	1	8	yes	—
137	#106×C₆₇	C₂³	1	8	yes	—
138	#106×C₆₇	C₂³	1	8	yes	—
139	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
140	#102×C₆₇	C₂³	1	8	yes	—
141	#102×C₆₇	C₂³	1	8	yes	—
142	#102×C₆₇	C₂³	1	8	yes	—
143	#102×C₆₇	C₂³	1	8	yes	—
144	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
145	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
146	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
147	#97×C₆₇	C₂³	1	4	yes	—
148	C₂⁴×C₆₇	—	2	1	no	—
149	C₂⁴×C₆₇	—	2	2	no	—
150	C₂⁴×C₆₇	—	2	2	no	—
151	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
153	#41×C₆₇	—	4	8	no	—
154	#41×C₆₇	—	4	16	no	—
155	C₂×Q₈×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	4	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	4	no	—
162	C₂×D₄×C₆₇	—	2	1	no	—
163	C₂×D₄×C₆₇	—	2	2	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	2	2	no	—
166	C₂×D₄×C₆₇	—	2	2	no	—
167	C₂×D₄×C₆₇	—	4	2	no	—
168	C₂×D₄×C₆₇	—	4	4	no	—
169	C₂×D₄×C₆₇	—	4	4	no	—
170	C₂×D₄×C₆₇	—	4	16	no	—
171	C₂×D₄×C₆₇	—	4	8	no	—
172	C₂×D₄×C₆₇	—	4	4	no	—
173	C₂×D₄×C₆₇	—	4	8	no	—
174	C₂×D₄×C₆₇	—	8	2	no	—
175	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
176	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
177	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	—	2	1	no	—
179	C₂²×C₄×C₆₇	—	2	1	no	—
180	C₂²×C₄×C₆₇	—	2	1	no	—
181	C₂²×C₄×C₆₇	—	2	2	no	—
182	C₂²×C₄×C₆₇	—	2	2	no	—
183	C₂²×C₄×C₆₇	—	2	4	no	—
184	C₂²×C₄×C₆₇	—	2	4	no	—
185	C₂²×C₄×C₆₇	—	2	4	no	—
186	C₂²×C₄×C₆₇	—	4	2	no	—
187	C₂²×C₄×C₆₇	—	4	2	no	—
188	C₂²×C₄×C₆₇	—	8	2	no	—
189	C₂²×C₄×C₆₇	—	8	2	no	—
190	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
199	D₈×C₆₇	—	4	8	no	—
200	M₁₆×C₆₇	—	4	4	no	—
201	C₂×C₈×C₆₇	—	4	4	no	—
202	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
203	C₄⋊C₄×C₆₇	—	2	1	no	—
204	C₄⋊C₄×C₆₇	—	2	1	no	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	4	no	—
207	C₄⋊C₄×C₆₇	—	2	4	no	—
208	C₄⋊C₄×C₆₇	—	2	1	no	—
209	C₄⋊C₄×C₆₇	—	2	2	no	—
210	C₄⋊C₄×C₆₇	—	2	2	no	—
211	C₄⋊C₄×C₆₇	—	2	2	no	—
212	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
213	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
214	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
215	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
216	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
217	C₂²⋊C₄×C₆₇	—	2	4	no	—
218	C₂²⋊C₄×C₆₇	—	2	4	no	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	—	2	4	no	—
221	C₂²⋊C₄×C₆₇	—	2	4	no	—
222	C₂²⋊C₄×C₆₇	—	2	4	no	—
223	C₂²⋊C₄×C₆₇	—	2	4	no	—
224	C₂²⋊C₄×C₆₇	—	2	4	no	—
225	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
226	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
227	C₄²×C₆₇	—	2	4	no	—
228	C₄²×C₆₇	C₂×D₄	1	2	yes	—
229	C₄²×C₆₇	C₂⁴	1	2	yes	—
230	C₂³×C₆₇	—	2	2	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	2	4	no	—
233	C₂³×C₆₇	—	4	1	no	—
234	C₂³×C₆₇	—	4	1	no	—
235	C₂³×C₆₇	—	4	1	no	—
236	C₂³×C₆₇	—	4	1	no	—
237	C₂³×C₆₇	—	4	2	no	—
238	C₂³×C₆₇	—	4	2	no	—
239	C₂³×C₆₇	—	4	4	no	—
240	C₂³×C₆₇	—	4	2	no	—
241	C₂³×C₆₇	—	4	4	no	—
242	C₂³×C₆₇	—	8	2	no	—
243	C₂³×C₆₇	—	8	2	no	—
244	C₂³×C₆₇	—	8	2	no	—
245	C₂³×C₆₇	—	8	2	no	—
246	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
247	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
248	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
249	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
250	Q₈×C₆₇	—	4	4	no	—
251	D₄×C₆₇	—	2	8	no	—
252	D₄×C₆₇	—	2	4	no	—
253	D₄×C₆₇	—	2	8	no	—
254	D₄×C₆₇	—	4	2	no	—
255	D₄×C₆₇	—	4	4	no	—
256	D₄×C₆₇	—	4	8	no	—
257	D₄×C₆₇	—	4	8	no	—
258	D₄×C₆₇	—	4	8	no	—
259	D₄×C₆₇	—	4	8	no	—
260	D₄×C₆₇	—	8	2	no	—
261	D₄×C₆₇	—	8	2	no	—
262	D₄×C₆₇	—	8	2	no	—
263	D₄×C₆₇	—	8	4	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	4	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	4	2	no	—
273	C₂×C₄×C₆₇	—	4	2	no	—
274	C₂×C₄×C₆₇	—	4	1	no	—
275	C₂×C₄×C₆₇	—	4	1	no	—
276	C₂×C₄×C₆₇	—	4	1	no	—
277	C₂×C₄×C₆₇	—	4	8	no	—
278	C₂×C₄×C₆₇	—	4	2	no	—
279	C₂×C₄×C₆₇	—	4	2	no	—
280	C₂×C₄×C₆₇	—	4	4	no	—
281	C₂×C₄×C₆₇	—	4	2	no	—
282	C₂×C₄×C₆₇	—	4	4	no	—
283	C₂×C₄×C₆₇	—	8	4	no	—
284	C₂×C₄×C₆₇	—	8	4	no	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
290	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
291	C₈×C₆₇	—	2	8	no	—
292	C₂²×C₆₇	—	2	1	no	—
293	C₂²×C₆₇	—	2	2	no	—
294	C₂²×C₆₇	—	2	1	no	—
295	C₂²×C₆₇	—	2	2	no	—
296	C₂²×C₆₇	—	2	2	no	—
297	C₂²×C₆₇	—	2	2	no	—
298	C₂²×C₆₇	—	2	2	no	—
299	C₂²×C₆₇	—	4	4	no	—
300	C₂²×C₆₇	—	4	2	no	—
301	C₂²×C₆₇	—	4	4	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	4	4	no	—
304	C₂²×C₆₇	—	4	4	no	—
305	C₂²×C₆₇	—	4	2	no	—
306	C₂²×C₆₇	—	8	2	no	—
307	C₂²×C₆₇	—	8	4	no	—
308	C₂²×C₆₇	—	16	4	no	—
309	C₂²×C₆₇	—	16	4	no	—
310	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
311	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
312	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
313	C₄×C₆₇	—	2	2	no	—
314	C₄×C₆₇	—	2	1	no	—
315	C₄×C₆₇	—	2	2	no	—
316	C₄×C₆₇	—	2	2	no	—
317	C₄×C₆₇	—	2	2	no	—
318	C₄×C₆₇	—	4	2	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	—	4	4	no	—
321	C₄×C₆₇	—	4	2	no	—
322	C₄×C₆₇	#585	1	2	yes	—
323	#61707	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
324	C₂×C₆₇	—	2	2	no	—
325	C₂×C₆₇	—	2	4	no	—
326	C₂×C₆₇	—	4	2	no	—
327	C₂×C₆₇	—	8	2	no	—
328	C₂×C₆₇	—	8	4	no	—
329	C₂×C₆₇	#3485	1	1	yes	characteristic
330	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
331	C₂×C₆₇	#3585	1	1	yes	characteristic
332	#3496	C₂×C₆₇	1	1	yes	characteristic, complemented
333	C₂×#553	C₂×C₆₇	1	1	yes	characteristic, complemented
334	#3345	C₂×C₆₇	1	8	yes	complemented
335	#3318	C₂×C₆₇	1	8	yes	complemented
336	#3234	C₂×C₆₇	1	2	yes	complemented
337	#3192	C₂×C₆₇	1	2	yes	complemented
338	#3186	C₂×C₆₇	1	2	yes	complemented
339	#3109	C₂×C₆₇	1	2	yes	complemented
340	#3094	C₂×C₆₇	1	2	yes	complemented
341	C₂×#481	C₂×C₆₇	1	2	yes	complemented
342	#2973	C₂×C₆₇	1	1	yes	characteristic, complemented
343	C₆₇	#61707	1	1	yes	characteristic, direct product factor, Sylow
344	C₂×H(5,𝔽₂)	—	2	1	no	—
345	#553	—	2	4	no	—
346	#553	C₂²×C₆₇	1	8	yes	complemented
347	#547	C₂²×C₆₇	1	4	yes	complemented
348	#546	—	2	4	no	—
349	D₄²	C₂²×C₆₇	1	4	yes	complemented
350	#537	—	2	1	no	—
351	#534	—	2	1	no	—
352	#534	—	2	1	no	—
353	#532	C₂²×C₆₇	1	2	yes	—
354	C₂×#126	—	2	1	no	—
355	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
356	C₂×#123	—	2	1	no	—
357	C₂×#121	—	2	1	no	—
358	C₂×#121	C₂²×C₆₇	1	2	yes	—
359	C₂×#121	C₂²×C₆₇	1	2	yes	—
360	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic, complemented
361	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic, complemented
362	C₂×C₄×D₄	—	2	1	no	—
363	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
364	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
365	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic, complemented
366	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
367	#501	C₂²×C₆₇	1	4	yes	complemented
368	#490	C₂²×C₆₇	1	4	yes	complemented
369	#482	C₂²×C₆₇	1	8	yes	complemented
370	#481	C₂²×C₆₇	1	8	yes	complemented
371	#481	C₂²×C₆₇	1	4	yes	complemented
372	#481	C₂²×C₆₇	1	4	yes	complemented
373	#481	C₂²×C₆₇	1	4	yes	complemented
374	#480	C₂²×C₆₇	1	4	yes	complemented
375	#480	C₂²×C₆₇	1	4	yes	complemented
376	#478	C₂²×C₆₇	1	8	yes	complemented
377	#476	C₂²×C₆₇	1	4	yes	complemented
378	#469	C₂²×C₆₇	1	8	yes	complemented
379	#466	C₂²×C₆₇	1	8	yes	complemented
380	#459	C₂²×C₆₇	1	4	yes	complemented
381	#447	C₂²×C₆₇	1	4	yes	complemented
382	#447	C₂²×C₆₇	1	4	yes	complemented
383	#447	C₂²×C₆₇	1	4	yes	complemented
384	#442	C₂²×C₆₇	1	8	yes	complemented
385	#435	C₂²×C₆₇	1	8	yes	complemented
386	#432	C₂²×C₆₇	1	2	yes	complemented
387	#428	C₂²×C₆₇	1	4	yes	complemented
388	C₂×#106	C₂²×C₆₇	1	4	yes	complemented
389	#418	C₂²×C₆₇	1	4	yes	complemented
390	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
391	#407	C₂²×C₆₇	1	2	yes	complemented
392	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
393	H(5,𝔽₂)	—	4	8	no	—
394	C₂×#41	—	2	4	no	—
395	C₂×#41	—	2	2	no	—
396	C₂²×D₄	—	2	1	no	—
397	C₂²×D₄	—	2	2	no	—
398	C₂²×D₄	—	2	2	no	—
399	C₂²×D₄	—	2	1	no	—
400	C₂²×D₄	—	2	4	no	—
401	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
402	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
403	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
404	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
405	C₂³×C₄	—	2	1	no	—
406	C₂³×C₄	—	2	1	no	—
407	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
409	C₂×D₈	C₂³×C₆₇	1	8	yes	—
410	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
411	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
412	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
413	#127	C₂³×C₆₇	1	2	yes	—
414	#126	—	2	8	no	—
415	#125	—	2	2	no	—
416	#125	C₂³×C₆₇	1	4	yes	—
417	#124	—	2	2	no	—
418	#123	—	2	8	no	—
419	#123	—	2	4	no	—
420	#123	—	2	4	no	—
421	C₂²⋊Q₈	—	2	4	no	—
422	#121	—	2	8	no	—
423	#121	—	2	2	no	—
424	#121	—	2	2	no	—
425	#121	—	2	2	no	—
426	#121	—	2	2	no	—
427	#121	—	2	2	no	—
428	#121	—	2	2	no	—
429	#121	—	2	2	no	—
430	#121	—	2	4	no	—
431	#121	C₂³×C₆₇	1	4	yes	—
432	#121	C₂³×C₆₇	1	4	yes	—
433	#121	C₂³×C₆₇	1	4	yes	—
434	#121	C₂³×C₆₇	1	4	yes	—
435	#121	C₂³×C₆₇	1	4	yes	—
436	#120	—	2	4	no	—
437	#120	—	2	4	no	—
438	C₄×D₄	—	2	8	no	—
439	C₄×D₄	—	2	2	no	—
440	C₄×D₄	—	2	2	no	—
441	C₄×D₄	D₄×C₆₇	1	4	yes	—
442	C₄×D₄	C₂³×C₆₇	1	4	yes	—
443	C₄×D₄	C₂³×C₆₇	1	4	yes	—
444	C₄×D₄	C₂³×C₆₇	1	4	yes	—
445	#117	—	2	2	no	—
446	#117	C₂³×C₆₇	1	4	yes	—
447	C₂×C₄⋊C₄	—	2	1	no	—
448	C₂×C₄⋊C₄	—	2	2	no	—
449	C₂×C₄⋊C₄	—	2	1	no	—
450	C₂×C₄⋊C₄	—	2	1	no	—
451	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
452	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
453	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
454	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
455	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
457	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₂²⋊C₄	—	2	1	no	—
460	C₂×C₂²⋊C₄	—	2	1	no	—
461	C₂×C₂²⋊C₄	—	2	2	no	—
462	C₂×C₂²⋊C₄	—	2	2	no	—
463	C₂×C₂²⋊C₄	—	2	1	no	—
464	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
465	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
466	C₂×C₄²	—	2	1	no	—
467	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
468	#107	C₂³×C₆₇	1	8	yes	—
469	#106	C₂³×C₆₇	1	8	yes	—
470	#106	C₂³×C₆₇	1	8	yes	—
471	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
472	#102	C₂³×C₆₇	1	8	yes	—
473	#102	C₂³×C₆₇	1	8	yes	—
474	#102	C₂³×C₆₇	1	8	yes	—
475	#102	C₂³×C₆₇	1	8	yes	—
476	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
477	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
478	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
479	#97	C₂³×C₆₇	1	4	yes	—
480	C₂⁴	—	2	1	no	—
481	C₂⁴	—	2	2	no	—
482	C₂⁴	—	2	2	no	—
483	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
484	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
485	#41	—	4	8	no	—
486	#41	—	4	16	no	—
487	C₂×Q₈	—	2	2	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	4	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	2	4	no	—
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	2	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	4	2	no	—
500	C₂×D₄	—	4	4	no	—
501	C₂×D₄	—	4	4	no	—
502	C₂×D₄	—	4	16	no	—
503	C₂×D₄	—	4	8	no	—
504	C₂×D₄	—	4	8	no	—
505	C₂×D₄	—	4	4	no	—
506	C₂×D₄	—	8	2	no	—
507	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
508	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
509	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	2	4	no	—
517	C₂²×C₄	—	2	4	no	—
518	C₂²×C₄	—	4	2	no	—
519	C₂²×C₄	—	4	2	no	—
520	C₂²×C₄	—	8	2	no	—
521	C₂²×C₄	—	8	2	no	—
522	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
523	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
525	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
526	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
527	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
528	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
530	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
531	D₈	—	4	8	no	—
532	M₁₆	—	4	4	no	—
533	C₂×C₈	—	4	4	no	—
534	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
545	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
546	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
547	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
548	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
558	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
559	C₄²	—	2	4	no	—
560	C₄²	C₂×D₄×C₆₇	1	2	yes	—
561	C₄²	C₂⁴×C₆₇	1	2	yes	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	4	no	—
565	C₂³	—	4	1	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	1	no	—
569	C₂³	—	4	2	no	—
570	C₂³	—	4	2	no	—
571	C₂³	—	4	4	no	—
572	C₂³	—	4	2	no	—
573	C₂³	—	4	4	no	—
574	C₂³	—	8	2	no	—
575	C₂³	—	8	2	no	—
576	C₂³	—	8	2	no	—
577	C₂³	—	8	2	no	—
578	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
579	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
580	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
581	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
582	Q₈	—	4	4	no	—
583	D₄	—	2	8	no	—
584	D₄	—	2	4	no	—
585	D₄	—	2	8	no	—
586	D₄	—	4	4	no	—
587	D₄	—	4	2	no	—
588	D₄	—	4	8	no	—
589	D₄	—	4	8	no	—
590	D₄	—	4	8	no	—
591	D₄	—	4	8	no	—
592	D₄	—	8	2	no	—
593	D₄	—	8	2	no	—
594	D₄	—	8	2	no	—
595	D₄	—	8	4	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	4	2	no	—
605	C₂×C₄	—	4	2	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	8	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	4	no	—
613	C₂×C₄	—	4	4	no	—
614	C₂×C₄	—	4	2	no	—
615	C₂×C₄	—	8	4	no	—
616	C₂×C₄	—	8	4	no	—
617	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
618	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
619	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
620	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
621	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
622	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
623	C₈	—	2	8	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	4	4	no	—
632	C₂²	—	4	4	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	4	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	4	no	—
638	C₂²	—	8	2	no	—
639	C₂²	—	8	4	no	—
640	C₂²	—	16	4	no	—
641	C₂²	—	16	4	no	—
642	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
643	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
644	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
645	C₄	—	2	2	no	—
646	C₄	—	2	2	no	—
647	C₄	—	2	1	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	4	4	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	2	no	—
654	C₄	#585×C₆₇	1	2	yes	—
655	C₂	—	2	2	no	—
656	C₂	—	2	4	no	—
657	C₂	—	4	2	no	—
658	C₂	—	8	2	no	—
659	C₂	—	8	4	no	—
660	C₂	C₆₇×#3485	1	1	yes	characteristic
661	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
662	C₂	C₆₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#3496	C₂	semidirect
2. C₂×#553×C₆₇	C₂	semidirect
3. C₆₇×#3345	C₂	semidirect
4. C₆₇×#3318	C₂	semidirect
5. C₆₇×#3234	C₂	semidirect
6. C₆₇×#3192	C₂	semidirect
7. C₆₇×#3186	C₂	semidirect
8. C₆₇×#3109	C₂	semidirect
9. C₆₇×#3094	C₂	semidirect
10. C₂×#481×C₆₇	C₂	semidirect
11. C₆₇×#2973	C₂	semidirect
14. #553×C₆₇	C₂²	semidirect
15. #547×C₆₇	C₂²	semidirect
17. D₄²×C₆₇	C₂²	semidirect
23. C₂×#125×C₆₇	C₂²	semidirect
28. C₂×#121×C₆₇	C₂²	semidirect
29. C₂×#121×C₆₇	C₂²	semidirect
32. C₂×C₄×D₄×C₆₇	C₂²	semidirect
33. C₂×#117×C₆₇	C₂²	semidirect
34. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
35. #501×C₆₇	C₂²	semidirect
36. #490×C₆₇	C₂²	semidirect
37. #482×C₆₇	C₂²	semidirect
38. #481×C₆₇	C₂²	semidirect
39. #481×C₆₇	C₂²	semidirect
40. #481×C₆₇	C₂²	semidirect
41. #481×C₆₇	C₂²	semidirect
42. #480×C₆₇	C₂²	semidirect
43. #480×C₆₇	C₂²	semidirect
44. #478×C₆₇	C₂²	semidirect
45. #476×C₆₇	C₂²	semidirect
46. #469×C₆₇	C₂²	semidirect
47. #466×C₆₇	C₂²	semidirect
48. #459×C₆₇	C₂²	semidirect
49. #447×C₆₇	C₂²	semidirect
50. #447×C₆₇	C₂²	semidirect
51. #447×C₆₇	C₂²	semidirect
52. #442×C₆₇	C₂²	semidirect
53. #435×C₆₇	C₂²	semidirect
54. #432×C₆₇	C₂²	semidirect
55. #428×C₆₇	C₂²	semidirect
56. C₂×#106×C₆₇	C₂²	semidirect
57. #418×C₆₇	C₂²	semidirect
58. C₂×#102×C₆₇	C₂²	semidirect
59. #407×C₆₇	C₂²	semidirect
60. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
323. #61707	C₆₇	direct (N × H)
332. #3496	C₂×C₆₇	semidirect
333. C₂×#553	C₂×C₆₇	semidirect
334. #3345	C₂×C₆₇	semidirect
335. #3318	C₂×C₆₇	semidirect
336. #3234	C₂×C₆₇	semidirect
337. #3192	C₂×C₆₇	semidirect
338. #3186	C₂×C₆₇	semidirect
339. #3109	C₂×C₆₇	semidirect
340. #3094	C₂×C₆₇	semidirect
341. C₂×#481	C₂×C₆₇	semidirect
342. #2973	C₂×C₆₇	semidirect
343. C₆₇	#61707	direct (N × H)
346. #553	C₂²×C₆₇	semidirect
347. #547	C₂²×C₆₇	semidirect
349. D₄²	C₂²×C₆₇	semidirect
355. C₂×#125	C₂²×C₆₇	semidirect
360. C₂×#121	C₂²×C₆₇	semidirect
361. C₂×#121	C₂²×C₆₇	semidirect
364. C₂×C₄×D₄	C₂²×C₆₇	semidirect
365. C₂×#117	C₂²×C₆₇	semidirect
366. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
367. #501	C₂²×C₆₇	semidirect
368. #490	C₂²×C₆₇	semidirect
369. #482	C₂²×C₆₇	semidirect
370. #481	C₂²×C₆₇	semidirect
371. #481	C₂²×C₆₇	semidirect
372. #481	C₂²×C₆₇	semidirect
373. #481	C₂²×C₆₇	semidirect
374. #480	C₂²×C₆₇	semidirect
375. #480	C₂²×C₆₇	semidirect
376. #478	C₂²×C₆₇	semidirect
377. #476	C₂²×C₆₇	semidirect
378. #469	C₂²×C₆₇	semidirect
379. #466	C₂²×C₆₇	semidirect
380. #459	C₂²×C₆₇	semidirect
381. #447	C₂²×C₆₇	semidirect
382. #447	C₂²×C₆₇	semidirect
383. #447	C₂²×C₆₇	semidirect
384. #442	C₂²×C₆₇	semidirect
385. #435	C₂²×C₆₇	semidirect
386. #432	C₂²×C₆₇	semidirect
387. #428	C₂²×C₆₇	semidirect
388. C₂×#106	C₂²×C₆₇	semidirect
389. #418	C₂²×C₆₇	semidirect
390. C₂×#102	C₂²×C₆₇	semidirect
391. #407	C₂²×C₆₇	semidirect
392. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3496	2
C₂×#553×C₆₇	2
C₆₇×#3345	2
C₆₇×#3318	2
C₆₇×#3234	2
C₆₇×#3192	2
C₆₇×#3186	2
C₆₇×#3109	2
C₆₇×#3094	2
C₂×#481×C₆₇	2
C₆₇×#2973	2
#61707	67

Subgroups — C67×#61707 #30740958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61707 #30740958