Subgroups — C₆₇×#61704 #30740955

Order 17152.

2022 conjugacy classes of subgroups.

The subgroup lattice has 640 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
2	C₂×#552×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3343	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3310	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3234	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3193	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3187	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3112	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3094	C₂	1	2	yes	maximal, complemented
10	C₂×#480×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	#552×C₆₇	—	2	4	no	—
14	#552×C₆₇	C₂²	1	8	yes	complemented
15	#547×C₆₇	C₂²	1	4	yes	—
16	#546×C₆₇	—	2	4	no	—
17	D₄²×C₆₇	C₂²	1	4	yes	—
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	—	2	1	no	—
24	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
25	C₂×#123×C₆₇	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×#121×C₆₇	C₂²	1	2	yes	—
29	C₂×#121×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#117×C₆₇	—	2	1	no	—
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	#484×C₆₇	C₂²	1	4	yes	complemented
38	#483×C₆₇	C₂²	1	4	yes	complemented
39	#482×C₆₇	C₂²	1	8	yes	complemented
40	#481×C₆₇	C₂²	1	4	yes	complemented
41	#480×C₆₇	C₂²	1	8	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	#477×C₆₇	C₂²	1	4	yes	complemented
46	#468×C₆₇	C₂²	1	8	yes	complemented
47	#465×C₆₇	C₂²	1	8	yes	complemented
48	#460×C₆₇	C₂²	1	4	yes	complemented
49	#450×C₆₇	C₂²	1	4	yes	complemented
50	#450×C₆₇	C₂²	1	4	yes	complemented
51	#448×C₆₇	C₂²	1	4	yes	complemented
52	#440×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×#107×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×Q₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	—	2	2	no	—
67	C₂²×D₄×C₆₇	—	2	4	no	—
68	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
69	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
70	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
71	C₂³×C₄×C₆₇	—	2	1	no	—
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
77	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
78	#127×C₆₇	C₂³	1	2	yes	—
79	#126×C₆₇	—	2	8	no	—
80	#125×C₆₇	—	2	8	no	—
81	#125×C₆₇	—	2	2	no	—
82	#125×C₆₇	C₂³	1	4	yes	—
83	#124×C₆₇	—	2	2	no	—
84	#123×C₆₇	—	2	8	no	—
85	#123×C₆₇	—	2	4	no	—
86	#123×C₆₇	—	2	4	no	—
87	C₂²⋊Q₈×C₆₇	—	2	2	no	—
88	C₂²⋊Q₈×C₆₇	—	2	2	no	—
89	C₂²⋊Q₈×C₆₇	—	2	4	no	—
90	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
91	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
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	4	no	—
98	#121×C₆₇	C₂³	1	4	yes	—
99	#121×C₆₇	C₂³	1	4	yes	—
100	#121×C₆₇	C₂³	1	4	yes	—
101	#120×C₆₇	—	2	4	no	—
102	#120×C₆₇	—	2	4	no	—
103	C₄×Q₈×C₆₇	—	2	2	no	—
104	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
105	C₄×D₄×C₆₇	—	2	2	no	—
106	C₄×D₄×C₆₇	D₄	1	4	yes	—
107	C₄×D₄×C₆₇	C₂³	1	4	yes	—
108	C₄×D₄×C₆₇	C₂³	1	4	yes	—
109	#117×C₆₇	—	2	2	no	—
110	#117×C₆₇	—	2	8	no	—
111	#117×C₆₇	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
113	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₄⋊C₄×C₆₇	—	2	1	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	1	yes	characteristic
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	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	D₄	1	1	yes	characteristic
132	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
133	C₂×C₄²×C₆₇	—	2	1	no	—
134	C₂×C₄²×C₆₇	—	2	1	no	—
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	#107×C₆₇	C₂³	1	8	yes	—
137	#107×C₆₇	C₂³	1	8	yes	—
138	#106×C₆₇	C₂³	1	8	yes	—
139	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
140	#103×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	2	no	—
149	C₂⁴×C₆₇	—	2	2	no	—
150	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
152	#41×C₆₇	—	4	8	no	—
153	#41×C₆₇	—	4	16	no	—
154	C₂×Q₈×C₆₇	—	2	1	no	—
155	C₂×Q₈×C₆₇	—	2	2	no	—
156	C₂×Q₈×C₆₇	—	4	2	no	—
157	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
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	2	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	—	4	4	no	—
167	C₂×D₄×C₆₇	—	4	4	no	—
168	C₂×D₄×C₆₇	—	4	16	no	—
169	C₂×D₄×C₆₇	—	4	8	no	—
170	C₂×D₄×C₆₇	—	4	8	no	—
171	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
172	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₇	—	2	1	no	—
174	C₂²×C₄×C₆₇	—	2	1	no	—
175	C₂²×C₄×C₆₇	—	2	1	no	—
176	C₂²×C₄×C₆₇	—	2	1	no	—
177	C₂²×C₄×C₆₇	—	2	2	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	2	4	no	—
180	C₂²×C₄×C₆₇	—	2	4	no	—
181	C₂²×C₄×C₆₇	—	4	2	no	—
182	C₂²×C₄×C₆₇	—	4	2	no	—
183	C₂²×C₄×C₆₇	—	4	4	no	—
184	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
189	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
193	SD₁₆×C₆₇	—	4	8	no	—
194	M₁₆×C₆₇	—	4	4	no	—
195	C₂×C₈×C₆₇	—	4	4	no	—
196	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
197	C₄⋊C₄×C₆₇	—	2	1	no	—
198	C₄⋊C₄×C₆₇	—	2	1	no	—
199	C₄⋊C₄×C₆₇	—	2	4	no	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	4	no	—
202	C₄⋊C₄×C₆₇	—	2	4	no	—
203	C₄⋊C₄×C₆₇	—	2	4	no	—
204	C₄⋊C₄×C₆₇	—	2	4	no	—
205	C₄⋊C₄×C₆₇	—	2	1	no	—
206	C₄⋊C₄×C₆₇	—	2	2	no	—
207	C₄⋊C₄×C₆₇	—	2	2	no	—
208	C₄⋊C₄×C₆₇	—	2	2	no	—
209	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
210	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
211	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
212	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
213	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
214	C₂²⋊C₄×C₆₇	—	2	4	no	—
215	C₂²⋊C₄×C₆₇	—	2	4	no	—
216	C₂²⋊C₄×C₆₇	—	2	4	no	—
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	C₂×D₄	1	4	yes	—
222	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
223	C₄²×C₆₇	—	2	4	no	—
224	C₄²×C₆₇	—	2	4	no	—
225	C₄²×C₆₇	C₂×D₄	1	2	yes	—
226	C₄²×C₆₇	C₂⁴	1	2	yes	—
227	C₂³×C₆₇	—	2	2	no	—
228	C₂³×C₆₇	—	2	4	no	—
229	C₂³×C₆₇	—	4	1	no	—
230	C₂³×C₆₇	—	4	1	no	—
231	C₂³×C₆₇	—	4	1	no	—
232	C₂³×C₆₇	—	4	1	no	—
233	C₂³×C₆₇	—	4	4	no	—
234	C₂³×C₆₇	—	4	4	no	—
235	C₂³×C₆₇	—	8	2	no	—
236	C₂³×C₆₇	—	8	2	no	—
237	C₂³×C₆₇	—	8	2	no	—
238	C₂³×C₆₇	—	8	2	no	—
239	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
240	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
241	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
242	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	Q₈×C₆₇	—	2	4	no	—
244	Q₈×C₆₇	—	4	2	no	—
245	Q₈×C₆₇	—	4	4	no	—
246	D₄×C₆₇	—	2	8	no	—
247	D₄×C₆₇	—	2	8	no	—
248	D₄×C₆₇	—	4	4	no	—
249	D₄×C₆₇	—	4	8	no	—
250	D₄×C₆₇	—	4	8	no	—
251	D₄×C₆₇	—	4	8	no	—
252	D₄×C₆₇	—	8	2	no	—
253	D₄×C₆₇	—	8	2	no	—
254	D₄×C₆₇	—	8	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	4	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	4	2	no	—
265	C₂×C₄×C₆₇	—	4	2	no	—
266	C₂×C₄×C₆₇	—	4	1	no	—
267	C₂×C₄×C₆₇	—	4	1	no	—
268	C₂×C₄×C₆₇	—	4	1	no	—
269	C₂×C₄×C₆₇	—	4	8	no	—
270	C₂×C₄×C₆₇	—	4	2	no	—
271	C₂×C₄×C₆₇	—	4	4	no	—
272	C₂×C₄×C₆₇	—	4	2	no	—
273	C₂×C₄×C₆₇	—	4	2	no	—
274	C₂×C₄×C₆₇	—	4	2	no	—
275	C₂×C₄×C₆₇	—	4	8	no	—
276	C₂×C₄×C₆₇	—	4	4	no	—
277	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
283	C₈×C₆₇	—	2	8	no	—
284	C₂²×C₆₇	—	2	1	no	—
285	C₂²×C₆₇	—	2	2	no	—
286	C₂²×C₆₇	—	2	1	no	—
287	C₂²×C₆₇	—	2	2	no	—
288	C₂²×C₆₇	—	2	2	no	—
289	C₂²×C₆₇	—	2	2	no	—
290	C₂²×C₆₇	—	2	2	no	—
291	C₂²×C₆₇	—	4	4	no	—
292	C₂²×C₆₇	—	4	4	no	—
293	C₂²×C₆₇	—	4	4	no	—
294	C₂²×C₆₇	—	4	4	no	—
295	C₂²×C₆₇	—	8	2	no	—
296	C₂²×C₆₇	—	16	4	no	—
297	C₂²×C₆₇	—	16	4	no	—
298	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
299	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
300	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
301	C₄×C₆₇	—	2	2	no	—
302	C₄×C₆₇	—	2	1	no	—
303	C₄×C₆₇	—	2	2	no	—
304	C₄×C₆₇	—	2	2	no	—
305	C₄×C₆₇	—	2	2	no	—
306	C₄×C₆₇	—	4	2	no	—
307	C₄×C₆₇	—	4	2	no	—
308	C₄×C₆₇	—	4	2	no	—
309	C₄×C₆₇	—	4	2	no	—
310	C₄×C₆₇	—	4	4	no	—
311	C₄×C₆₇	#585	1	2	yes	—
312	#61704	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
313	C₂×C₆₇	—	2	2	no	—
314	C₂×C₆₇	—	2	4	no	—
315	C₂×C₆₇	—	4	2	no	—
316	C₂×C₆₇	—	8	4	no	—
317	C₂×C₆₇	#3485	1	1	yes	characteristic
318	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₂×C₆₇	#3586	1	1	yes	characteristic
320	#3496	C₂×C₆₇	1	1	yes	characteristic
321	C₂×#552	C₂×C₆₇	1	1	yes	characteristic, complemented
322	#3343	C₂×C₆₇	1	8	yes	complemented
323	#3310	C₂×C₆₇	1	8	yes	complemented
324	#3234	C₂×C₆₇	1	2	yes	complemented
325	#3193	C₂×C₆₇	1	2	yes	complemented
326	#3187	C₂×C₆₇	1	2	yes	complemented
327	#3112	C₂×C₆₇	1	2	yes	complemented
328	#3094	C₂×C₆₇	1	2	yes	complemented
329	C₂×#480	C₂×C₆₇	1	2	yes	complemented
330	#2973	C₂×C₆₇	1	1	yes	characteristic, complemented
331	C₆₇	#61704	1	1	yes	characteristic, direct product factor, Sylow
332	C₂×H(5,𝔽₂)	—	2	1	no	—
333	#552	—	2	4	no	—
334	#552	C₂²×C₆₇	1	8	yes	complemented
335	#547	C₂²×C₆₇	1	4	yes	—
336	#546	—	2	4	no	—
337	D₄²	C₂²×C₆₇	1	4	yes	—
338	#537	—	2	1	no	—
339	#534	—	2	1	no	—
340	#534	—	2	1	no	—
341	#532	C₂²×C₆₇	1	2	yes	—
342	C₂×#126	—	2	1	no	—
343	C₂×#125	—	2	1	no	—
344	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
345	C₂×#123	—	2	1	no	—
346	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
347	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
348	C₂×#121	C₂²×C₆₇	1	2	yes	—
349	C₂×#121	C₂²×C₆₇	1	2	yes	—
350	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
351	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
352	C₂×#117	—	2	1	no	—
353	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic, complemented
354	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
355	#501	C₂²×C₆₇	1	4	yes	complemented
356	#490	C₂²×C₆₇	1	4	yes	complemented
357	#484	C₂²×C₆₇	1	4	yes	complemented
358	#483	C₂²×C₆₇	1	4	yes	complemented
359	#482	C₂²×C₆₇	1	8	yes	complemented
360	#481	C₂²×C₆₇	1	4	yes	complemented
361	#480	C₂²×C₆₇	1	8	yes	complemented
362	#480	C₂²×C₆₇	1	4	yes	complemented
363	#480	C₂²×C₆₇	1	4	yes	complemented
364	#478	C₂²×C₆₇	1	8	yes	complemented
365	#477	C₂²×C₆₇	1	4	yes	complemented
366	#468	C₂²×C₆₇	1	8	yes	complemented
367	#465	C₂²×C₆₇	1	8	yes	complemented
368	#460	C₂²×C₆₇	1	4	yes	complemented
369	#450	C₂²×C₆₇	1	4	yes	complemented
370	#450	C₂²×C₆₇	1	4	yes	complemented
371	#448	C₂²×C₆₇	1	4	yes	complemented
372	#440	C₂²×C₆₇	1	8	yes	complemented
373	#435	C₂²×C₆₇	1	8	yes	complemented
374	#432	C₂²×C₆₇	1	2	yes	complemented
375	#428	C₂²×C₆₇	1	4	yes	complemented
376	C₂×#107	C₂²×C₆₇	1	4	yes	complemented
377	#418	C₂²×C₆₇	1	4	yes	complemented
378	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
379	#407	C₂²×C₆₇	1	2	yes	complemented
380	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
381	H(5,𝔽₂)	—	4	8	no	—
382	C₂×#41	—	2	4	no	—
383	C₂×#41	—	2	2	no	—
384	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
385	C₂²×D₄	—	2	2	no	—
386	C₂²×D₄	—	2	2	no	—
387	C₂²×D₄	—	2	4	no	—
388	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
389	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
390	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
391	C₂³×C₄	—	2	1	no	—
392	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
393	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
394	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
395	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
396	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
397	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
398	#127	C₂³×C₆₇	1	2	yes	—
399	#126	—	2	8	no	—
400	#125	—	2	8	no	—
401	#125	—	2	2	no	—
402	#125	C₂³×C₆₇	1	4	yes	—
403	#124	—	2	2	no	—
404	#123	—	2	8	no	—
405	#123	—	2	4	no	—
406	#123	—	2	4	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	2	no	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
411	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
412	#121	—	2	2	no	—
413	#121	—	2	2	no	—
414	#121	—	2	2	no	—
415	#121	—	2	2	no	—
416	#121	—	2	2	no	—
417	#121	—	2	4	no	—
418	#121	C₂³×C₆₇	1	4	yes	—
419	#121	C₂³×C₆₇	1	4	yes	—
420	#121	C₂³×C₆₇	1	4	yes	—
421	#120	—	2	4	no	—
422	#120	—	2	4	no	—
423	C₄×Q₈	—	2	2	no	—
424	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
425	C₄×D₄	—	2	2	no	—
426	C₄×D₄	D₄×C₆₇	1	4	yes	—
427	C₄×D₄	C₂³×C₆₇	1	4	yes	—
428	C₄×D₄	C₂³×C₆₇	1	4	yes	—
429	#117	—	2	2	no	—
430	#117	—	2	8	no	—
431	#117	C₂³×C₆₇	1	4	yes	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	2	2	no	—
434	C₂×C₄⋊C₄	—	2	1	no	—
435	C₂×C₄⋊C₄	—	2	1	no	—
436	C₂×C₄⋊C₄	—	2	1	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
442	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
443	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
445	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
446	C₂×C₂²⋊C₄	—	2	1	no	—
447	C₂×C₂²⋊C₄	—	2	1	no	—
448	C₂×C₂²⋊C₄	—	2	1	no	—
449	C₂×C₂²⋊C₄	—	2	2	no	—
450	C₂×C₂²⋊C₄	—	2	2	no	—
451	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
452	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
453	C₂×C₄²	—	2	1	no	—
454	C₂×C₄²	—	2	1	no	—
455	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
456	#107	C₂³×C₆₇	1	8	yes	—
457	#107	C₂³×C₆₇	1	8	yes	—
458	#106	C₂³×C₆₇	1	8	yes	—
459	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
460	#103	C₂³×C₆₇	1	8	yes	—
461	#102	C₂³×C₆₇	1	8	yes	—
462	#102	C₂³×C₆₇	1	8	yes	—
463	#102	C₂³×C₆₇	1	8	yes	—
464	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
465	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
466	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
467	#97	C₂³×C₆₇	1	4	yes	—
468	C₂⁴	—	2	2	no	—
469	C₂⁴	—	2	2	no	—
470	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
471	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
472	#41	—	4	8	no	—
473	#41	—	4	16	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	2	no	—
476	C₂×Q₈	—	4	2	no	—
477	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
478	C₂×D₄	—	2	4	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	2	no	—
484	C₂×D₄	—	2	2	no	—
485	C₂×D₄	—	2	2	no	—
486	C₂×D₄	—	4	4	no	—
487	C₂×D₄	—	4	4	no	—
488	C₂×D₄	—	4	16	no	—
489	C₂×D₄	—	4	8	no	—
490	C₂×D₄	—	4	8	no	—
491	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
492	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	4	2	no	—
502	C₂²×C₄	—	4	2	no	—
503	C₂²×C₄	—	4	4	no	—
504	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
505	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
508	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
509	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
510	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
513	SD₁₆	—	4	8	no	—
514	M₁₆	—	4	4	no	—
515	C₂×C₈	—	4	4	no	—
516	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	4	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
530	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
531	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
532	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
533	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
542	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
543	C₄²	—	2	4	no	—
544	C₄²	—	2	4	no	—
545	C₄²	C₂×D₄×C₆₇	1	2	yes	—
546	C₄²	C₂⁴×C₆₇	1	2	yes	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	4	no	—
549	C₂³	—	4	1	no	—
550	C₂³	—	4	1	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	4	1	no	—
553	C₂³	—	4	4	no	—
554	C₂³	—	4	4	no	—
555	C₂³	—	8	2	no	—
556	C₂³	—	8	2	no	—
557	C₂³	—	8	2	no	—
558	C₂³	—	8	2	no	—
559	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
560	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
561	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
562	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
563	Q₈	—	2	4	no	—
564	Q₈	—	4	2	no	—
565	Q₈	—	4	4	no	—
566	D₄	—	2	8	no	—
567	D₄	—	2	8	no	—
568	D₄	—	4	4	no	—
569	D₄	—	4	8	no	—
570	D₄	—	4	8	no	—
571	D₄	—	4	8	no	—
572	D₄	—	8	2	no	—
573	D₄	—	8	2	no	—
574	D₄	—	8	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	8	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	2	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	8	no	—
596	C₂×C₄	—	4	4	no	—
597	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
598	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
599	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
600	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
601	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
602	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
603	C₈	—	2	8	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	2	1	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	4	4	no	—
612	C₂²	—	4	4	no	—
613	C₂²	—	4	4	no	—
614	C₂²	—	4	4	no	—
615	C₂²	—	8	2	no	—
616	C₂²	—	16	4	no	—
617	C₂²	—	16	4	no	—
618	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
619	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
620	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	1	no	—
624	C₄	—	2	2	no	—
625	C₄	—	2	2	no	—
626	C₄	—	4	4	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	#585×C₆₇	1	2	yes	—
632	C₂	—	2	2	no	—
633	C₂	—	2	4	no	—
634	C₂	—	4	2	no	—
635	C₂	—	8	4	no	—
636	C₂	C₆₇×#3485	1	1	yes	characteristic
637	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
638	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3496	C₂
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
2. C₂×#552×C₆₇	C₂	semidirect
3. C₆₇×#3343	C₂	semidirect
4. C₆₇×#3310	C₂	semidirect
5. C₆₇×#3234	C₂	semidirect
6. C₆₇×#3193	C₂	semidirect
7. C₆₇×#3187	C₂	semidirect
8. C₆₇×#3112	C₂	semidirect
9. C₆₇×#3094	C₂	semidirect
10. C₂×#480×C₆₇	C₂	semidirect
11. C₆₇×#2973	C₂	semidirect
14. #552×C₆₇	C₂²	semidirect
24. C₂×#125×C₆₇	C₂²	semidirect
26. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
27. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
30. C₂×C₄×Q₈×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. #484×C₆₇	C₂²	semidirect
38. #483×C₆₇	C₂²	semidirect
39. #482×C₆₇	C₂²	semidirect
40. #481×C₆₇	C₂²	semidirect
41. #480×C₆₇	C₂²	semidirect
42. #480×C₆₇	C₂²	semidirect
43. #480×C₆₇	C₂²	semidirect
44. #478×C₆₇	C₂²	semidirect
45. #477×C₆₇	C₂²	semidirect
46. #468×C₆₇	C₂²	semidirect
47. #465×C₆₇	C₂²	semidirect
48. #460×C₆₇	C₂²	semidirect
49. #450×C₆₇	C₂²	semidirect
50. #450×C₆₇	C₂²	semidirect
51. #448×C₆₇	C₂²	semidirect
52. #440×C₆₇	C₂²	semidirect
53. #435×C₆₇	C₂²	semidirect
54. #432×C₆₇	C₂²	semidirect
55. #428×C₆₇	C₂²	semidirect
56. C₂×#107×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
312. #61704	C₆₇	direct (N × H)
321. C₂×#552	C₂×C₆₇	semidirect
322. #3343	C₂×C₆₇	semidirect
323. #3310	C₂×C₆₇	semidirect
324. #3234	C₂×C₆₇	semidirect
325. #3193	C₂×C₆₇	semidirect
326. #3187	C₂×C₆₇	semidirect
327. #3112	C₂×C₆₇	semidirect
328. #3094	C₂×C₆₇	semidirect
329. C₂×#480	C₂×C₆₇	semidirect
330. #2973	C₂×C₆₇	semidirect
331. C₆₇	#61704	direct (N × H)
334. #552	C₂²×C₆₇	semidirect
344. C₂×#125	C₂²×C₆₇	semidirect
346. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
347. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
350. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
353. C₂×#117	C₂²×C₆₇	semidirect
354. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
355. #501	C₂²×C₆₇	semidirect
356. #490	C₂²×C₆₇	semidirect
357. #484	C₂²×C₆₇	semidirect
358. #483	C₂²×C₆₇	semidirect
359. #482	C₂²×C₆₇	semidirect
360. #481	C₂²×C₆₇	semidirect
361. #480	C₂²×C₆₇	semidirect
362. #480	C₂²×C₆₇	semidirect
363. #480	C₂²×C₆₇	semidirect
364. #478	C₂²×C₆₇	semidirect
365. #477	C₂²×C₆₇	semidirect
366. #468	C₂²×C₆₇	semidirect
367. #465	C₂²×C₆₇	semidirect
368. #460	C₂²×C₆₇	semidirect
369. #450	C₂²×C₆₇	semidirect
370. #450	C₂²×C₆₇	semidirect
371. #448	C₂²×C₆₇	semidirect
372. #440	C₂²×C₆₇	semidirect
373. #435	C₂²×C₆₇	semidirect
374. #432	C₂²×C₆₇	semidirect
375. #428	C₂²×C₆₇	semidirect
376. C₂×#107	C₂²×C₆₇	semidirect
377. #418	C₂²×C₆₇	semidirect
378. C₂×#102	C₂²×C₆₇	semidirect
379. #407	C₂²×C₆₇	semidirect
380. 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×#552×C₆₇	2
C₆₇×#3343	2
C₆₇×#3310	2
C₆₇×#3234	2
C₆₇×#3193	2
C₆₇×#3187	2
C₆₇×#3112	2
C₆₇×#3094	2
C₂×#480×C₆₇	2
C₆₇×#2973	2
#61704	67

Subgroups — C67×#61704 #30740955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61704 #30740955