Subgroups — C₆₇×#60569 #30739820

Order 17152.

2010 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3428	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3127	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3123	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3120	C₂	1	8	yes	maximal, complemented
6	C₆₇×#3095	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3080	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3039	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3007	C₂	1	2	yes	maximal
10	C₄×#137×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×#137×C₆₇	C₂²	1	4	yes	—
14	#541×C₆₇	—	2	2	no	—
15	#541×C₆₇	—	2	2	no	—
16	#541×C₆₇	C₂²	1	4	yes	—
17	#541×C₆₇	C₂²	1	4	yes	complemented
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	C₂×#126×C₆₇	—	2	2	no	—
20	C₂×#125×C₆₇	—	2	2	no	—
21	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
23	#520×C₆₇	—	2	2	no	—
24	#520×C₆₇	C₂²	1	4	yes	—
25	#519×C₆₇	C₂²	1	2	yes	—
26	#518×C₆₇	—	2	2	no	—
27	#518×C₆₇	C₂²	1	2	yes	—
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×Q₈×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	#485×C₆₇	C₂²	1	8	yes	—
36	#484×C₆₇	C₂²	1	4	yes	complemented
37	#483×C₆₇	C₂²	1	4	yes	complemented
38	#479×C₆₇	C₂²	1	8	yes	complemented
39	#478×C₆₇	C₂²	1	8	yes	complemented
40	#462×C₆₇	C₂²	1	8	yes	complemented
41	#461×C₆₇	C₂²	1	8	yes	complemented
42	#452×C₆₇	C₂²	1	8	yes	—
43	#451×C₆₇	C₂²	1	4	yes	—
44	#450×C₆₇	C₂²	1	4	yes	—
45	#441×C₆₇	C₂²	1	8	yes	complemented
46	#440×C₆₇	C₂²	1	8	yes	complemented
47	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
48	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
49	#433×C₆₇	C₂²	1	4	yes	—
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	#419×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×Q₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×D₄×C₆₇	—	2	1	no	—
65	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	—	2	1	no	—
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	#137×C₆₇	—	2	16	no	—
69	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
70	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
72	C₄⋊Q₈×C₆₇	—	2	2	no	—
73	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
74	#126×C₆₇	—	2	8	no	—
75	#126×C₆₇	—	2	8	no	—
76	#125×C₆₇	—	2	8	no	—
77	#125×C₆₇	—	2	8	no	—
78	#125×C₆₇	—	2	2	no	—
79	#125×C₆₇	C₂³	1	4	yes	—
80	C₂²⋊Q₈×C₆₇	—	2	2	no	—
81	C₂²⋊Q₈×C₆₇	—	2	2	no	—
82	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
83	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
84	C₄×Q₈×C₆₇	—	2	4	no	—
85	C₄×Q₈×C₆₇	—	2	4	no	—
86	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
87	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
88	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
89	C₄×D₄×C₆₇	—	2	4	no	—
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	C₂³	1	4	yes	—
95	C₄×D₄×C₆₇	C₂³	1	4	yes	—
96	#117×C₆₇	—	2	1	no	—
97	#117×C₆₇	—	2	1	no	—
98	#117×C₆₇	—	2	4	no	—
99	#117×C₆₇	—	2	4	no	—
100	#117×C₆₇	—	2	2	no	—
101	#117×C₆₇	—	2	2	no	—
102	#117×C₆₇	D₄	1	2	yes	—
103	#117×C₆₇	D₄	1	2	yes	—
104	#117×C₆₇	C₂³	1	2	yes	—
105	#117×C₆₇	C₂³	1	2	yes	—
106	#117×C₆₇	C₂³	1	2	yes	—
107	#117×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
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	4	no	—
119	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
120	C₂×C₄²×C₆₇	—	2	1	no	—
121	C₂×C₄²×C₆₇	—	2	2	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	#103×C₆₇	C₂³	1	8	yes	—
131	#103×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	C₂×D₄	1	1	yes	characteristic
139	#41×C₆₇	—	2	8	no	—
140	#41×C₆₇	—	2	8	no	—
141	#41×C₆₇	—	2	4	no	—
142	#41×C₆₇	—	2	4	no	—
143	#41×C₆₇	—	2	4	no	—
144	#41×C₆₇	#41	1	8	yes	—
145	C₂×Q₈×C₆₇	—	2	4	no	—
146	C₂×Q₈×C₆₇	—	2	1	no	—
147	C₂×Q₈×C₆₇	—	2	2	no	—
148	C₂×Q₈×C₆₇	—	2	8	no	—
149	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
150	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
151	C₂×D₄×C₆₇	—	2	8	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	2	2	no	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	1	no	—
156	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₆₇	—	2	8	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	2	2	no	—
165	C₂²×C₄×C₆₇	—	2	4	no	—
166	C₂²×C₄×C₆₇	—	2	4	no	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	Q₁₆×C₆₇	—	2	16	no	—
176	SD₁₆×C₆₇	—	2	16	no	—
177	M₁₆×C₆₇	—	2	8	no	—
178	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
179	C₄⋊C₄×C₆₇	—	2	2	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	8	no	—
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	2	4	no	—
185	C₄⋊C₄×C₆₇	—	2	8	no	—
186	C₄⋊C₄×C₆₇	—	2	8	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
189	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
190	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
191	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
192	C₂²⋊C₄×C₆₇	—	2	2	no	—
193	C₂²⋊C₄×C₆₇	—	2	2	no	—
194	C₂²⋊C₄×C₆₇	—	2	4	no	—
195	C₂²⋊C₄×C₆₇	—	2	4	no	—
196	C₂²⋊C₄×C₆₇	—	2	4	no	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
199	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
200	C₄²×C₆₇	—	2	2	no	—
201	C₄²×C₆₇	—	2	4	no	—
202	C₄²×C₆₇	—	2	4	no	—
203	C₄²×C₆₇	—	2	2	no	—
204	C₄²×C₆₇	—	2	8	no	—
205	C₄²×C₆₇	C₂×D₄	1	2	yes	—
206	C₄²×C₆₇	C₂×D₄	1	2	yes	—
207	C₄²×C₆₇	C₂⁴	1	2	yes	—
208	C₄²×C₆₇	C₂⁴	1	2	yes	—
209	C₂³×C₆₇	—	2	4	no	—
210	C₂³×C₆₇	—	2	2	no	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	—	4	2	no	—
213	C₂³×C₆₇	—	8	2	no	—
214	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
215	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
216	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₆₇	—	2	4	no	—
218	Q₈×C₆₇	—	2	2	no	—
219	Q₈×C₆₇	—	2	8	no	—
220	Q₈×C₆₇	—	4	4	no	—
221	Q₈×C₆₇	C₂×#41	1	4	yes	—
222	D₄×C₆₇	—	2	8	no	—
223	D₄×C₆₇	—	2	4	no	—
224	D₄×C₆₇	—	2	8	no	—
225	D₄×C₆₇	—	2	2	no	—
226	D₄×C₆₇	C₂×#41	1	4	yes	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	8	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	8	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	4	2	no	—
240	C₂×C₄×C₆₇	—	4	2	no	—
241	C₂×C₄×C₆₇	—	4	4	no	—
242	C₂×C₄×C₆₇	—	4	4	no	—
243	C₂×C₄×C₆₇	—	8	4	no	—
244	C₂×C₄×C₆₇	—	8	4	no	—
245	C₂×C₄×C₆₇	—	8	2	no	—
246	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
247	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
249	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
252	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
253	C₈×C₆₇	—	2	8	no	—
254	C₂²×C₆₇	—	2	2	no	—
255	C₂²×C₆₇	—	2	2	no	—
256	C₂²×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	2	1	no	—
258	C₂²×C₆₇	—	2	1	no	—
259	C₂²×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	4	4	no	—
261	C₂²×C₆₇	—	8	2	no	—
262	C₂²×C₆₇	—	8	4	no	—
263	C₂²×C₆₇	#525	1	2	yes	—
264	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
265	C₄×C₆₇	—	2	4	no	—
266	C₄×C₆₇	—	2	4	no	—
267	C₄×C₆₇	—	2	1	no	—
268	C₄×C₆₇	—	2	4	no	—
269	C₄×C₆₇	—	2	1	no	—
270	C₄×C₆₇	—	4	4	no	—
271	C₄×C₆₇	—	4	4	no	—
272	C₄×C₆₇	—	4	4	no	—
273	C₄×C₆₇	#525	1	2	yes	—
274	C₄×C₆₇	C₂²×#41	1	2	yes	—
275	#60569	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂×C₆₇	—	2	2	no	—
277	C₂×C₆₇	—	4	1	no	—
278	C₂×C₆₇	—	4	1	no	—
279	C₂×C₆₇	—	4	4	no	—
280	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
281	C₂×C₆₇	#3586	1	2	yes	—
282	C₂×#541	C₂×C₆₇	1	1	yes	characteristic, complemented
283	#3428	C₂×C₆₇	1	1	yes	characteristic
284	#3127	C₂×C₆₇	1	4	yes	complemented
285	#3123	C₂×C₆₇	1	4	yes	complemented
286	#3120	C₂×C₆₇	1	8	yes	complemented
287	#3095	C₂×C₆₇	1	2	yes	complemented
288	#3080	C₂×C₆₇	1	2	yes	complemented
289	#3039	C₂×C₆₇	1	2	yes	complemented
290	#3007	C₂×C₆₇	1	2	yes	—
291	C₄×#137	C₂×C₆₇	1	4	yes	complemented
292	#2921	C₂×C₆₇	1	1	yes	characteristic, complemented
293	C₆₇	#60569	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
295	C₂×#137	C₂²×C₆₇	1	4	yes	—
296	#541	—	2	2	no	—
297	#541	—	2	2	no	—
298	#541	C₂²×C₆₇	1	4	yes	—
299	#541	C₂²×C₆₇	1	4	yes	complemented
300	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
301	C₂×#126	—	2	2	no	—
302	C₂×#125	—	2	2	no	—
303	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
304	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
305	#520	—	2	2	no	—
306	#520	C₂²×C₆₇	1	4	yes	—
307	#519	C₂²×C₆₇	1	2	yes	—
308	#518	—	2	2	no	—
309	#518	C₂²×C₆₇	1	2	yes	—
310	C₄×#41	—	2	2	no	—
311	C₄×#41	C₂²×C₆₇	1	2	yes	—
312	C₄×#41	C₂²×C₆₇	1	2	yes	—
313	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
314	C₂×#117	—	2	2	no	—
315	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
317	#485	C₂²×C₆₇	1	8	yes	—
318	#484	C₂²×C₆₇	1	4	yes	complemented
319	#483	C₂²×C₆₇	1	4	yes	complemented
320	#479	C₂²×C₆₇	1	8	yes	complemented
321	#478	C₂²×C₆₇	1	8	yes	complemented
322	#462	C₂²×C₆₇	1	8	yes	complemented
323	#461	C₂²×C₆₇	1	8	yes	complemented
324	#452	C₂²×C₆₇	1	8	yes	—
325	#451	C₂²×C₆₇	1	4	yes	—
326	#450	C₂²×C₆₇	1	4	yes	—
327	#441	C₂²×C₆₇	1	8	yes	complemented
328	#440	C₂²×C₆₇	1	8	yes	complemented
329	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
330	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
331	#433	C₂²×C₆₇	1	4	yes	—
332	#433	C₂²×C₆₇	1	4	yes	complemented
333	#428	C₂²×C₆₇	1	4	yes	complemented
334	#423	C₂²×C₆₇	1	2	yes	complemented
335	#419	C₂²×C₆₇	1	4	yes	complemented
336	#417	C₂²×C₆₇	1	4	yes	—
337	#407	C₂²×C₆₇	1	2	yes	—
338	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
339	C₂×#41	—	2	8	no	—
340	C₂×#41	—	2	1	no	—
341	C₂×#41	—	2	1	no	—
342	C₂×#41	C₂³×C₆₇	1	2	yes	—
343	C₂×#41	C₂³×C₆₇	1	2	yes	—
344	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
345	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
346	C₂²×D₄	—	2	1	no	—
347	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
348	C₂³×C₄	—	2	1	no	—
349	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
350	#137	—	2	16	no	—
351	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
352	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
353	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
354	C₄⋊Q₈	—	2	2	no	—
355	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
356	#126	—	2	8	no	—
357	#126	—	2	8	no	—
358	#125	—	2	8	no	—
359	#125	—	2	8	no	—
360	#125	—	2	2	no	—
361	#125	C₂³×C₆₇	1	4	yes	—
362	C₂²⋊Q₈	—	2	2	no	—
363	C₂²⋊Q₈	—	2	2	no	—
364	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
365	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
366	C₄×Q₈	—	2	4	no	—
367	C₄×Q₈	—	2	4	no	—
368	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
369	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
370	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	2	4	no	—
373	C₄×D₄	—	2	4	no	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	—	2	4	no	—
376	C₄×D₄	C₂³×C₆₇	1	4	yes	—
377	C₄×D₄	C₂³×C₆₇	1	4	yes	—
378	#117	—	2	1	no	—
379	#117	—	2	1	no	—
380	#117	—	2	4	no	—
381	#117	—	2	4	no	—
382	#117	—	2	2	no	—
383	#117	—	2	2	no	—
384	#117	D₄×C₆₇	1	2	yes	—
385	#117	D₄×C₆₇	1	2	yes	—
386	#117	C₂³×C₆₇	1	2	yes	—
387	#117	C₂³×C₆₇	1	2	yes	—
388	#117	C₂³×C₆₇	1	2	yes	—
389	#117	C₂³×C₆₇	1	2	yes	—
390	C₂×C₄⋊C₄	—	2	2	no	—
391	C₂×C₄⋊C₄	—	2	2	no	—
392	C₂×C₄⋊C₄	—	2	2	no	—
393	C₂×C₄⋊C₄	—	2	2	no	—
394	C₂×C₄⋊C₄	—	2	4	no	—
395	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	2	no	—
400	C₂×C₂²⋊C₄	—	2	4	no	—
401	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
402	C₂×C₄²	—	2	1	no	—
403	C₂×C₄²	—	2	2	no	—
404	C₂×C₄²	—	2	2	no	—
405	C₂×C₄²	—	2	2	no	—
406	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
408	#107	C₂³×C₆₇	1	8	yes	—
409	#106	C₂³×C₆₇	1	8	yes	—
410	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
411	#103	C₂³×C₆₇	1	8	yes	—
412	#103	C₂³×C₆₇	1	8	yes	—
413	#103	C₂³×C₆₇	1	8	yes	—
414	#102	C₂³×C₆₇	1	8	yes	—
415	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
416	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
417	#97	C₂³×C₆₇	1	4	yes	—
418	C₄×C₈	C₂³×C₆₇	1	4	yes	—
419	C₂⁴	—	2	1	no	—
420	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
421	#41	—	2	8	no	—
422	#41	—	2	8	no	—
423	#41	—	2	4	no	—
424	#41	—	2	4	no	—
425	#41	—	2	4	no	—
426	#41	#41×C₆₇	1	8	yes	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	2	8	no	—
431	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
432	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
439	C₂²×C₄	—	2	8	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
457	Q₁₆	—	2	16	no	—
458	SD₁₆	—	2	16	no	—
459	M₁₆	—	2	8	no	—
460	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	2	8	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
471	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
472	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
473	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	2	4	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
481	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
482	C₄²	—	2	2	no	—
483	C₄²	—	2	4	no	—
484	C₄²	—	2	4	no	—
485	C₄²	—	2	2	no	—
486	C₄²	—	2	8	no	—
487	C₄²	C₂×D₄×C₆₇	1	2	yes	—
488	C₄²	C₂×D₄×C₆₇	1	2	yes	—
489	C₄²	C₂⁴×C₆₇	1	2	yes	—
490	C₄²	C₂⁴×C₆₇	1	2	yes	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	8	2	no	—
496	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
497	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
498	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
499	Q₈	—	2	4	no	—
500	Q₈	—	2	2	no	—
501	Q₈	—	2	8	no	—
502	Q₈	—	4	4	no	—
503	Q₈	C₂×#41×C₆₇	1	4	yes	—
504	D₄	—	2	8	no	—
505	D₄	—	2	4	no	—
506	D₄	—	2	8	no	—
507	D₄	—	2	2	no	—
508	D₄	C₂×#41×C₆₇	1	4	yes	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	8	4	no	—
526	C₂×C₄	—	8	4	no	—
527	C₂×C₄	—	8	2	no	—
528	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
529	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
531	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
533	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
534	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
535	C₈	—	2	8	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	8	2	no	—
544	C₂²	—	8	4	no	—
545	C₂²	#525×C₆₇	1	2	yes	—
546	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
547	C₄	—	2	4	no	—
548	C₄	—	2	4	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	4	no	—
551	C₄	—	2	1	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₄	#525×C₆₇	1	2	yes	—
556	C₄	C₂²×#41×C₆₇	1	2	yes	—
557	C₂	—	2	2	no	—
558	C₂	—	4	1	no	—
559	C₂	—	4	1	no	—
560	C₂	—	4	4	no	—
561	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
562	C₂	C₆₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
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×#541×C₆₇	C₂	semidirect
3. C₆₇×#3127	C₂	semidirect
4. C₆₇×#3123	C₂	semidirect
5. C₆₇×#3120	C₂	semidirect
6. C₆₇×#3095	C₂	semidirect
7. C₆₇×#3080	C₂	semidirect
8. C₆₇×#3039	C₂	semidirect
10. C₄×#137×C₆₇	C₂	semidirect
11. C₆₇×#2921	C₂	semidirect
17. #541×C₆₇	C₂²	semidirect
18. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
21. C₂×#125×C₆₇	C₂²	semidirect
31. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
36. #484×C₆₇	C₂²	semidirect
37. #483×C₆₇	C₂²	semidirect
38. #479×C₆₇	C₂²	semidirect
39. #478×C₆₇	C₂²	semidirect
40. #462×C₆₇	C₂²	semidirect
41. #461×C₆₇	C₂²	semidirect
45. #441×C₆₇	C₂²	semidirect
46. #440×C₆₇	C₂²	semidirect
47. C₄×Q₁₆×C₆₇	C₂²	semidirect
48. C₄×SD₁₆×C₆₇	C₂²	semidirect
50. #433×C₆₇	C₂²	semidirect
51. #428×C₆₇	C₂²	semidirect
52. #423×C₆₇	C₂²	semidirect
53. #419×C₆₇	C₂²	semidirect
56. C₄×M₁₆×C₆₇	C₂²	semidirect
275. #60569	C₆₇	direct (N × H)
282. C₂×#541	C₂×C₆₇	semidirect
284. #3127	C₂×C₆₇	semidirect
285. #3123	C₂×C₆₇	semidirect
286. #3120	C₂×C₆₇	semidirect
287. #3095	C₂×C₆₇	semidirect
288. #3080	C₂×C₆₇	semidirect
289. #3039	C₂×C₆₇	semidirect
291. C₄×#137	C₂×C₆₇	semidirect
292. #2921	C₂×C₆₇	semidirect
293. C₆₇	#60569	direct (N × H)
299. #541	C₂²×C₆₇	semidirect
300. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
303. C₂×#125	C₂²×C₆₇	semidirect
313. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
318. #484	C₂²×C₆₇	semidirect
319. #483	C₂²×C₆₇	semidirect
320. #479	C₂²×C₆₇	semidirect
321. #478	C₂²×C₆₇	semidirect
322. #462	C₂²×C₆₇	semidirect
323. #461	C₂²×C₆₇	semidirect
327. #441	C₂²×C₆₇	semidirect
328. #440	C₂²×C₆₇	semidirect
329. C₄×Q₁₆	C₂²×C₆₇	semidirect
330. C₄×SD₁₆	C₂²×C₆₇	semidirect
332. #433	C₂²×C₆₇	semidirect
333. #428	C₂²×C₆₇	semidirect
334. #423	C₂²×C₆₇	semidirect
335. #419	C₂²×C₆₇	semidirect
338. 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×#541×C₆₇	2
C₆₇×#3428	2
C₆₇×#3127	2
C₆₇×#3123	2
C₆₇×#3120	2
C₆₇×#3095	2
C₆₇×#3080	2
C₆₇×#3039	2
C₆₇×#3007	2
C₄×#137×C₆₇	2
C₆₇×#2921	2
#60569	67

Subgroups — C67×#60569 #30739820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60569 #30739820