Subgroups — C₆₇×#60568 #30739819

Order 17152.

2138 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#540×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3126	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3122	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3119	C₂	1	8	yes	maximal, complemented
6	C₆₇×#3094	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3080	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3038	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3006	C₂	1	2	yes	maximal, complemented
10	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	—
14	#540×C₆₇	—	2	2	no	—
15	#540×C₆₇	—	2	2	no	—
16	#540×C₆₇	C₂²	1	4	yes	—
17	#540×C₆₇	C₂²	1	4	yes	complemented
18	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂×#123×C₆₇	—	2	2	no	—
21	C₂×#121×C₆₇	—	2	2	no	—
22	C₂×#121×C₆₇	C₂²	1	2	yes	—
23	#520×C₆₇	—	2	2	no	—
24	#520×C₆₇	C₂²	1	4	yes	complemented
25	#519×C₆₇	C₂²	1	2	yes	complemented
26	#518×C₆₇	—	2	2	no	—
27	#518×C₆₇	C₂²	1	2	yes	complemented
28	C₄×#41×C₆₇	—	2	2	no	—
29	C₄×#41×C₆₇	C₂²	1	2	yes	—
30	C₄×#41×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
32	C₂×#117×C₆₇	—	2	2	no	—
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
35	#482×C₆₇	C₂²	1	8	yes	complemented
36	#481×C₆₇	C₂²	1	4	yes	complemented
37	#480×C₆₇	C₂²	1	4	yes	complemented
38	#479×C₆₇	C₂²	1	8	yes	complemented
39	#478×C₆₇	C₂²	1	8	yes	complemented
40	#460×C₆₇	C₂²	1	8	yes	complemented
41	#459×C₆₇	C₂²	1	8	yes	complemented
42	#449×C₆₇	C₂²	1	8	yes	complemented
43	#448×C₆₇	C₂²	1	4	yes	complemented
44	#447×C₆₇	C₂²	1	4	yes	complemented
45	#442×C₆₇	C₂²	1	8	yes	complemented
46	#440×C₆₇	C₂²	1	8	yes	complemented
47	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
48	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
49	#433×C₆₇	C₂²	1	4	yes	complemented
50	#433×C₆₇	C₂²	1	4	yes	complemented
51	#428×C₆₇	C₂²	1	4	yes	complemented
52	#423×C₆₇	C₂²	1	2	yes	complemented
53	#418×C₆₇	C₂²	1	4	yes	complemented
54	#417×C₆₇	C₂²	1	4	yes	—
55	#407×C₆₇	C₂²	1	2	yes	—
56	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
57	C₂×#41×C₆₇	—	2	8	no	—
58	C₂×#41×C₆₇	—	2	1	no	—
59	C₂×#41×C₆₇	—	2	1	no	—
60	C₂×#41×C₆₇	C₂³	1	2	yes	—
61	C₂×#41×C₆₇	C₂³	1	2	yes	—
62	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	—	2	2	no	—
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
67	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	—	2	1	no	—
69	C₂³×C₄×C₆₇	—	2	2	no	—
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
72	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
73	C₂×D₈×C₆₇	C₂³	1	8	yes	—
74	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
75	#127×C₆₇	—	2	2	no	—
76	#127×C₆₇	C₂³	1	4	yes	—
77	#125×C₆₇	—	2	2	no	—
78	#125×C₆₇	C₂³	1	4	yes	—
79	#123×C₆₇	—	2	8	no	—
80	#123×C₆₇	—	2	8	no	—
81	#121×C₆₇	—	2	2	no	—
82	#121×C₆₇	—	2	2	no	—
83	#121×C₆₇	—	2	8	no	—
84	#121×C₆₇	—	2	8	no	—
85	#121×C₆₇	C₂³	1	4	yes	—
86	#121×C₆₇	C₂³	1	4	yes	—
87	C₄×Q₈×C₆₇	—	2	4	no	—
88	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
89	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
90	C₄×D₄×C₆₇	—	2	4	no	—
91	C₄×D₄×C₆₇	—	2	4	no	—
92	C₄×D₄×C₆₇	—	2	4	no	—
93	C₄×D₄×C₆₇	—	2	4	no	—
94	C₄×D₄×C₆₇	—	2	4	no	—
95	C₄×D₄×C₆₇	—	2	4	no	—
96	C₄×D₄×C₆₇	C₂³	1	4	yes	—
97	C₄×D₄×C₆₇	C₂³	1	8	yes	—
98	C₄×D₄×C₆₇	C₂³	1	4	yes	—
99	#117×C₆₇	—	2	1	no	—
100	#117×C₆₇	—	2	1	no	—
101	#117×C₆₇	—	2	4	no	—
102	#117×C₆₇	—	2	4	no	—
103	#117×C₆₇	—	2	2	no	—
104	#117×C₆₇	—	2	2	no	—
105	#117×C₆₇	D₄	1	2	yes	—
106	#117×C₆₇	D₄	1	2	yes	—
107	#117×C₆₇	C₂³	1	2	yes	—
108	#117×C₆₇	C₂³	1	2	yes	—
109	#117×C₆₇	C₂³	1	2	yes	—
110	#117×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
120	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
121	C₂×C₄²×C₆₇	—	2	1	no	—
122	C₂×C₄²×C₆₇	—	2	2	no	—
123	C₂×C₄²×C₆₇	—	2	2	no	—
124	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	#107×C₆₇	C₂³	1	8	yes	—
127	#106×C₆₇	C₂³	1	8	yes	—
128	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
129	#103×C₆₇	C₂³	1	8	yes	—
130	#102×C₆₇	C₂³	1	8	yes	—
131	#102×C₆₇	C₂³	1	8	yes	—
132	#102×C₆₇	C₂³	1	8	yes	—
133	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
134	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
135	#97×C₆₇	C₂³	1	4	yes	—
136	C₄×C₈×C₆₇	C₂³	1	4	yes	—
137	C₂⁴×C₆₇	—	2	1	no	—
138	C₂⁴×C₆₇	—	2	2	no	—
139	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
140	#41×C₆₇	—	2	8	no	—
141	#41×C₆₇	—	2	8	no	—
142	#41×C₆₇	—	2	4	no	—
143	#41×C₆₇	—	2	4	no	—
144	#41×C₆₇	—	2	4	no	—
145	#41×C₆₇	#41	1	8	yes	—
146	C₂×Q₈×C₆₇	—	2	4	no	—
147	C₂×Q₈×C₆₇	—	2	1	no	—
148	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
149	C₂×D₄×C₆₇	—	2	8	no	—
150	C₂×D₄×C₆₇	—	2	2	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	1	no	—
156	C₂×D₄×C₆₇	—	2	2	no	—
157	C₂×D₄×C₆₇	—	2	8	no	—
158	C₂×D₄×C₆₇	—	2	8	no	—
159	C₂×D₄×C₆₇	—	8	4	no	—
160	C₂×D₄×C₆₇	—	8	4	no	—
161	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
162	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₆₇	—	2	8	no	—
164	C₂²×C₄×C₆₇	—	2	1	no	—
165	C₂²×C₄×C₆₇	—	2	2	no	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	—	2	4	no	—
168	C₂²×C₄×C₆₇	—	2	2	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	—	2	4	no	—
172	C₂²×C₄×C₆₇	—	2	8	no	—
173	C₂²×C₄×C₆₇	—	8	4	no	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	SD₁₆×C₆₇	—	2	16	no	—
183	D₈×C₆₇	—	2	16	no	—
184	M₁₆×C₆₇	—	2	8	no	—
185	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	8	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
192	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
195	C₂²⋊C₄×C₆₇	—	2	2	no	—
196	C₂²⋊C₄×C₆₇	—	2	2	no	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
203	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
204	C₄²×C₆₇	—	2	2	no	—
205	C₄²×C₆₇	—	2	4	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	2	no	—
208	C₄²×C₆₇	C₂×D₄	1	2	yes	—
209	C₄²×C₆₇	C₂×D₄	1	2	yes	—
210	C₄²×C₆₇	C₂⁴	1	2	yes	—
211	C₄²×C₆₇	C₂⁴	1	2	yes	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	2	2	no	—
215	C₂³×C₆₇	—	2	4	no	—
216	C₂³×C₆₇	—	4	2	no	—
217	C₂³×C₆₇	—	4	4	no	—
218	C₂³×C₆₇	—	4	4	no	—
219	C₂³×C₆₇	—	4	4	no	—
220	C₂³×C₆₇	—	8	2	no	—
221	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₇	—	2	4	no	—
225	Q₈×C₆₇	—	2	2	no	—
226	Q₈×C₆₇	C₂×#41	1	4	yes	—
227	D₄×C₆₇	—	2	8	no	—
228	D₄×C₆₇	—	2	4	no	—
229	D₄×C₆₇	—	2	8	no	—
230	D₄×C₆₇	—	2	2	no	—
231	D₄×C₆₇	—	2	8	no	—
232	D₄×C₆₇	—	4	4	no	—
233	D₄×C₆₇	—	8	8	no	—
234	D₄×C₆₇	—	8	4	no	—
235	D₄×C₆₇	—	8	4	no	—
236	D₄×C₆₇	C₂×#41	1	4	yes	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	8	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	8	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	4	2	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	—	8	4	no	—
252	C₂×C₄×C₆₇	—	8	4	no	—
253	C₂×C₄×C₆₇	—	8	8	no	—
254	C₂×C₄×C₆₇	—	8	2	no	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
257	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
258	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
261	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
262	C₈×C₆₇	—	2	8	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	1	no	—
267	C₂²×C₆₇	—	2	1	no	—
268	C₂²×C₆₇	—	2	4	no	—
269	C₂²×C₆₇	—	4	4	no	—
270	C₂²×C₆₇	—	4	4	no	—
271	C₂²×C₆₇	—	4	4	no	—
272	C₂²×C₆₇	—	4	4	no	—
273	C₂²×C₆₇	—	8	2	no	—
274	C₂²×C₆₇	—	8	8	no	—
275	C₂²×C₆₇	—	8	4	no	—
276	C₂²×C₆₇	#525	1	2	yes	—
277	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
278	C₄×C₆₇	—	2	4	no	—
279	C₄×C₆₇	—	2	4	no	—
280	C₄×C₆₇	—	2	1	no	—
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	2	1	no	—
283	C₄×C₆₇	—	4	4	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	#525	1	2	yes	—
286	C₄×C₆₇	C₂²×#41	1	2	yes	—
287	#60568	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
288	C₂×C₆₇	—	2	2	no	—
289	C₂×C₆₇	—	4	1	no	—
290	C₂×C₆₇	—	4	1	no	—
291	C₂×C₆₇	—	4	4	no	—
292	C₂×C₆₇	—	8	4	no	—
293	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
294	C₂×C₆₇	#3585	1	2	yes	—
295	C₂×#540	C₂×C₆₇	1	1	yes	characteristic, complemented
296	#3428	C₂×C₆₇	1	1	yes	characteristic, complemented
297	#3126	C₂×C₆₇	1	4	yes	complemented
298	#3122	C₂×C₆₇	1	4	yes	complemented
299	#3119	C₂×C₆₇	1	8	yes	complemented
300	#3094	C₂×C₆₇	1	2	yes	complemented
301	#3080	C₂×C₆₇	1	2	yes	complemented
302	#3038	C₂×C₆₇	1	2	yes	complemented
303	#3006	C₂×C₆₇	1	2	yes	complemented
304	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	4	yes	complemented
305	#2921	C₂×C₆₇	1	1	yes	characteristic, complemented
306	C₆₇	#60568	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	—
309	#540	—	2	2	no	—
310	#540	—	2	2	no	—
311	#540	C₂²×C₆₇	1	4	yes	—
312	#540	C₂²×C₆₇	1	4	yes	complemented
313	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic, complemented
314	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
315	C₂×#123	—	2	2	no	—
316	C₂×#121	—	2	2	no	—
317	C₂×#121	C₂²×C₆₇	1	2	yes	—
318	#520	—	2	2	no	—
319	#520	C₂²×C₆₇	1	4	yes	complemented
320	#519	C₂²×C₆₇	1	2	yes	complemented
321	#518	—	2	2	no	—
322	#518	C₂²×C₆₇	1	2	yes	complemented
323	C₄×#41	—	2	2	no	—
324	C₄×#41	C₂²×C₆₇	1	2	yes	—
325	C₄×#41	C₂²×C₆₇	1	2	yes	—
326	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
327	C₂×#117	—	2	2	no	—
328	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
330	#482	C₂²×C₆₇	1	8	yes	complemented
331	#481	C₂²×C₆₇	1	4	yes	complemented
332	#480	C₂²×C₆₇	1	4	yes	complemented
333	#479	C₂²×C₆₇	1	8	yes	complemented
334	#478	C₂²×C₆₇	1	8	yes	complemented
335	#460	C₂²×C₆₇	1	8	yes	complemented
336	#459	C₂²×C₆₇	1	8	yes	complemented
337	#449	C₂²×C₆₇	1	8	yes	complemented
338	#448	C₂²×C₆₇	1	4	yes	complemented
339	#447	C₂²×C₆₇	1	4	yes	complemented
340	#442	C₂²×C₆₇	1	8	yes	complemented
341	#440	C₂²×C₆₇	1	8	yes	complemented
342	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
343	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
344	#433	C₂²×C₆₇	1	4	yes	complemented
345	#433	C₂²×C₆₇	1	4	yes	complemented
346	#428	C₂²×C₆₇	1	4	yes	complemented
347	#423	C₂²×C₆₇	1	2	yes	complemented
348	#418	C₂²×C₆₇	1	4	yes	complemented
349	#417	C₂²×C₆₇	1	4	yes	—
350	#407	C₂²×C₆₇	1	2	yes	—
351	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
352	C₂×#41	—	2	8	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	—	2	1	no	—
355	C₂×#41	C₂³×C₆₇	1	2	yes	—
356	C₂×#41	C₂³×C₆₇	1	2	yes	—
357	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
358	C₂²×D₄	—	2	1	no	—
359	C₂²×D₄	—	2	2	no	—
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
362	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
363	C₂³×C₄	—	2	1	no	—
364	C₂³×C₄	—	2	2	no	—
365	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
366	AGL(1,ℤ/8)	—	2	16	no	—
367	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
368	C₂×D₈	C₂³×C₆₇	1	8	yes	—
369	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
370	#127	—	2	2	no	—
371	#127	C₂³×C₆₇	1	4	yes	—
372	#125	—	2	2	no	—
373	#125	C₂³×C₆₇	1	4	yes	—
374	#123	—	2	8	no	—
375	#123	—	2	8	no	—
376	#121	—	2	2	no	—
377	#121	—	2	2	no	—
378	#121	—	2	8	no	—
379	#121	—	2	8	no	—
380	#121	C₂³×C₆₇	1	4	yes	—
381	#121	C₂³×C₆₇	1	4	yes	—
382	C₄×Q₈	—	2	4	no	—
383	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
384	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
385	C₄×D₄	—	2	4	no	—
386	C₄×D₄	—	2	4	no	—
387	C₄×D₄	—	2	4	no	—
388	C₄×D₄	—	2	4	no	—
389	C₄×D₄	—	2	4	no	—
390	C₄×D₄	—	2	4	no	—
391	C₄×D₄	C₂³×C₆₇	1	8	yes	—
392	C₄×D₄	C₂³×C₆₇	1	4	yes	—
393	C₄×D₄	C₂³×C₆₇	1	4	yes	—
394	#117	—	2	1	no	—
395	#117	—	2	1	no	—
396	#117	—	2	4	no	—
397	#117	—	2	4	no	—
398	#117	—	2	2	no	—
399	#117	—	2	2	no	—
400	#117	D₄×C₆₇	1	2	yes	—
401	#117	D₄×C₆₇	1	2	yes	—
402	#117	C₂³×C₆₇	1	2	yes	—
403	#117	C₂³×C₆₇	1	2	yes	—
404	#117	C₂³×C₆₇	1	2	yes	—
405	#117	C₂³×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₄	—	2	2	no	—
407	C₂×C₄⋊C₄	—	2	4	no	—
408	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₂²⋊C₄	—	2	2	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	—	2	4	no	—
415	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
416	C₂×C₄²	—	2	1	no	—
417	C₂×C₄²	—	2	2	no	—
418	C₂×C₄²	—	2	2	no	—
419	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
421	#107	C₂³×C₆₇	1	8	yes	—
422	#106	C₂³×C₆₇	1	8	yes	—
423	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
424	#103	C₂³×C₆₇	1	8	yes	—
425	#102	C₂³×C₆₇	1	8	yes	—
426	#102	C₂³×C₆₇	1	8	yes	—
427	#102	C₂³×C₆₇	1	8	yes	—
428	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
429	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
430	#97	C₂³×C₆₇	1	4	yes	—
431	C₄×C₈	C₂³×C₆₇	1	4	yes	—
432	C₂⁴	—	2	1	no	—
433	C₂⁴	—	2	2	no	—
434	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
435	#41	—	2	8	no	—
436	#41	—	2	8	no	—
437	#41	—	2	4	no	—
438	#41	—	2	4	no	—
439	#41	—	2	4	no	—
440	#41	#41×C₆₇	1	8	yes	—
441	C₂×Q₈	—	2	4	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
444	C₂×D₄	—	2	8	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	8	4	no	—
455	C₂×D₄	—	8	4	no	—
456	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
457	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
458	C₂²×C₄	—	2	8	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	8	no	—
468	C₂²×C₄	—	8	4	no	—
469	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
473	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
475	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
477	SD₁₆	—	2	16	no	—
478	D₈	—	2	16	no	—
479	M₁₆	—	2	8	no	—
480	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	8	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
487	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
488	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
489	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
498	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
499	C₄²	—	2	2	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	2	4	no	—
502	C₄²	—	2	2	no	—
503	C₄²	C₂×D₄×C₆₇	1	2	yes	—
504	C₄²	C₂×D₄×C₆₇	1	2	yes	—
505	C₄²	C₂⁴×C₆₇	1	2	yes	—
506	C₄²	C₂⁴×C₆₇	1	2	yes	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	4	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	4	no	—
513	C₂³	—	4	4	no	—
514	C₂³	—	4	4	no	—
515	C₂³	—	8	2	no	—
516	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
517	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
518	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
519	Q₈	—	2	4	no	—
520	Q₈	—	2	2	no	—
521	Q₈	C₂×#41×C₆₇	1	4	yes	—
522	D₄	—	2	8	no	—
523	D₄	—	2	4	no	—
524	D₄	—	2	8	no	—
525	D₄	—	2	2	no	—
526	D₄	—	2	8	no	—
527	D₄	—	4	4	no	—
528	D₄	—	8	8	no	—
529	D₄	—	8	4	no	—
530	D₄	—	8	4	no	—
531	D₄	C₂×#41×C₆₇	1	4	yes	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	8	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	8	8	no	—
547	C₂×C₄	—	8	4	no	—
548	C₂×C₄	—	8	4	no	—
549	C₂×C₄	—	8	2	no	—
550	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
552	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
553	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
555	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
556	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
557	C₈	—	2	8	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	1	no	—
562	C₂²	—	2	1	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	8	2	no	—
569	C₂²	—	8	8	no	—
570	C₂²	—	8	4	no	—
571	C₂²	#525×C₆₇	1	2	yes	—
572	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
573	C₄	—	2	4	no	—
574	C₄	—	2	4	no	—
575	C₄	—	2	1	no	—
576	C₄	—	2	4	no	—
577	C₄	—	2	1	no	—
578	C₄	—	4	4	no	—
579	C₄	—	4	4	no	—
580	C₄	#525×C₆₇	1	2	yes	—
581	C₄	C₂²×#41×C₆₇	1	2	yes	—
582	C₂	—	2	2	no	—
583	C₂	—	4	1	no	—
584	C₂	—	4	1	no	—
585	C₂	—	4	4	no	—
586	C₂	—	8	4	no	—
587	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
588	C₂	C₆₇×#3585	1	2	yes	—

	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×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#540×C₆₇	C₂	semidirect
2. C₆₇×#3428	C₂	semidirect
3. C₆₇×#3126	C₂	semidirect
4. C₆₇×#3122	C₂	semidirect
5. C₆₇×#3119	C₂	semidirect
6. C₆₇×#3094	C₂	semidirect
7. C₆₇×#3080	C₂	semidirect
8. C₆₇×#3038	C₂	semidirect
9. C₆₇×#3006	C₂	semidirect
10. C₄×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
11. C₆₇×#2921	C₂	semidirect
17. #540×C₆₇	C₂²	semidirect
18. C₂×#127×C₆₇	C₂²	semidirect
19. C₂×#125×C₆₇	C₂²	semidirect
24. #520×C₆₇	C₂²	semidirect
25. #519×C₆₇	C₂²	semidirect
27. #518×C₆₇	C₂²	semidirect
31. C₂×C₄×D₄×C₆₇	C₂²	semidirect
35. #482×C₆₇	C₂²	semidirect
36. #481×C₆₇	C₂²	semidirect
37. #480×C₆₇	C₂²	semidirect
38. #479×C₆₇	C₂²	semidirect
39. #478×C₆₇	C₂²	semidirect
40. #460×C₆₇	C₂²	semidirect
41. #459×C₆₇	C₂²	semidirect
42. #449×C₆₇	C₂²	semidirect
43. #448×C₆₇	C₂²	semidirect
44. #447×C₆₇	C₂²	semidirect
45. #442×C₆₇	C₂²	semidirect
46. #440×C₆₇	C₂²	semidirect
47. C₄×SD₁₆×C₆₇	C₂²	semidirect
48. C₄×D₈×C₆₇	C₂²	semidirect
49. #433×C₆₇	C₂²	semidirect
50. #433×C₆₇	C₂²	semidirect
51. #428×C₆₇	C₂²	semidirect
52. #423×C₆₇	C₂²	semidirect
53. #418×C₆₇	C₂²	semidirect
56. C₄×M₁₆×C₆₇	C₂²	semidirect
287. #60568	C₆₇	direct (N × H)
295. C₂×#540	C₂×C₆₇	semidirect
296. #3428	C₂×C₆₇	semidirect
297. #3126	C₂×C₆₇	semidirect
298. #3122	C₂×C₆₇	semidirect
299. #3119	C₂×C₆₇	semidirect
300. #3094	C₂×C₆₇	semidirect
301. #3080	C₂×C₆₇	semidirect
302. #3038	C₂×C₆₇	semidirect
303. #3006	C₂×C₆₇	semidirect
304. C₄×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
305. #2921	C₂×C₆₇	semidirect
306. C₆₇	#60568	direct (N × H)
312. #540	C₂²×C₆₇	semidirect
313. C₂×#127	C₂²×C₆₇	semidirect
314. C₂×#125	C₂²×C₆₇	semidirect
319. #520	C₂²×C₆₇	semidirect
320. #519	C₂²×C₆₇	semidirect
322. #518	C₂²×C₆₇	semidirect
326. C₂×C₄×D₄	C₂²×C₆₇	semidirect
330. #482	C₂²×C₆₇	semidirect
331. #481	C₂²×C₆₇	semidirect
332. #480	C₂²×C₆₇	semidirect
333. #479	C₂²×C₆₇	semidirect
334. #478	C₂²×C₆₇	semidirect
335. #460	C₂²×C₆₇	semidirect
336. #459	C₂²×C₆₇	semidirect
337. #449	C₂²×C₆₇	semidirect
338. #448	C₂²×C₆₇	semidirect
339. #447	C₂²×C₆₇	semidirect
340. #442	C₂²×C₆₇	semidirect
341. #440	C₂²×C₆₇	semidirect
342. C₄×SD₁₆	C₂²×C₆₇	semidirect
343. C₄×D₈	C₂²×C₆₇	semidirect
344. #433	C₂²×C₆₇	semidirect
345. #433	C₂²×C₆₇	semidirect
346. #428	C₂²×C₆₇	semidirect
347. #423	C₂²×C₆₇	semidirect
348. #418	C₂²×C₆₇	semidirect
351. C₄×M₁₆	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#540×C₆₇	2
C₆₇×#3428	2
C₆₇×#3126	2
C₆₇×#3122	2
C₆₇×#3119	2
C₆₇×#3094	2
C₆₇×#3080	2
C₆₇×#3038	2
C₆₇×#3006	2
C₄×AGL(1,ℤ/8)×C₆₇	2
C₆₇×#2921	2
#60568	67

Subgroups — C67×#60568 #30739819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60568 #30739819