Subgroups — C₆₇×#20937 #30700188

Order 17152.

1814 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1880	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
10	C₂×#340×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#135×C₆₇	—	2	4	no	—
15	C₂×#135×C₆₇	—	2	4	no	—
16	C₂×#135×C₆₇	C₄	1	8	yes	complemented
17	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#520×C₆₇	—	2	4	no	—
24	#520×C₆₇	—	2	4	no	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	—	2	2	no	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×#117×C₆₇	—	2	1	no	—
29	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	#417×C₆₇	—	2	4	no	—
34	#417×C₆₇	—	2	4	no	—
35	C₂×#103×C₆₇	C₂²	1	2	yes	—
36	C₂×#103×C₆₇	C₂²	1	2	yes	—
37	C₂×#102×C₆₇	C₂²	1	2	yes	—
38	C₂×#102×C₆₇	C₂²	1	2	yes	—
39	#391×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#386×C₆₇	—	2	2	no	—
42	#380×C₆₇	—	2	1	no	—
43	#380×C₆₇	—	2	1	no	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	#340×C₆₇	C₂²	1	2	yes	—
46	#336×C₆₇	C₂²	1	2	yes	—
47	#336×C₆₇	C₂²	1	2	yes	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	C₂×#41×C₆₇	—	2	2	no	—
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	—	2	2	no	—
55	C₂×#41×C₆₇	—	2	2	no	—
56	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
57	C₂×#41×C₆₇	C₂×C₄	1	4	yes	complemented
58	C₂×#41×C₆₇	D₄	1	4	yes	—
59	C₂×#41×C₆₇	D₄	1	4	yes	—
60	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	—	2	1	no	—
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	—	2	1	no	—
67	C₂³×C₄×C₆₇	—	2	1	no	—
68	C₂³×C₄×C₆₇	—	2	1	no	—
69	C₂³×C₄×C₆₇	—	2	2	no	—
70	C₂³×C₄×C₆₇	—	2	1	no	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	#135×C₆₇	—	2	32	no	—
73	C₂×Q₁₆×C₆₇	—	2	2	no	—
74	C₂×Q₁₆×C₆₇	—	2	2	no	—
75	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
76	C₂×SD₁₆×C₆₇	—	2	2	no	—
77	C₂×SD₁₆×C₆₇	—	2	2	no	—
78	C₂×SD₁₆×C₆₇	—	2	2	no	—
79	C₂×SD₁₆×C₆₇	—	2	2	no	—
80	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
81	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
82	C₂×D₈×C₆₇	—	2	2	no	—
83	C₂×D₈×C₆₇	—	2	2	no	—
84	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
85	C₂×M₁₆×C₆₇	—	2	2	no	—
86	C₂×M₁₆×C₆₇	—	2	2	no	—
87	C₂×M₁₆×C₆₇	—	2	2	no	—
88	C₂²×C₈×C₆₇	—	2	2	no	—
89	C₂²×C₈×C₆₇	—	2	2	no	—
90	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
91	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
92	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×Q₈×C₆₇	—	2	4	no	—
95	C₄×Q₈×C₆₇	—	2	4	no	—
96	C₄×D₄×C₆₇	—	2	16	no	—
97	C₄×D₄×C₆₇	—	2	4	no	—
98	C₄×D₄×C₆₇	—	2	4	no	—
99	#117×C₆₇	—	2	8	no	—
100	#117×C₆₇	—	2	2	no	—
101	#117×C₆₇	—	2	2	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
109	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
118	C₂×C₄²×C₆₇	—	2	2	no	—
119	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
120	#103×C₆₇	—	2	4	no	—
121	#103×C₆₇	—	2	4	no	—
122	#102×C₆₇	—	2	4	no	—
123	#102×C₆₇	—	2	4	no	—
124	#95×C₆₇	—	2	2	no	—
125	#95×C₆₇	—	2	2	no	—
126	C₂⁴×C₆₇	—	2	1	no	—
127	C₂⁴×C₆₇	—	2	1	no	—
128	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
129	#41×C₆₇	—	2	8	no	—
130	#41×C₆₇	—	2	8	no	—
131	#41×C₆₇	—	2	8	no	—
132	#41×C₆₇	—	2	8	no	—
133	#41×C₆₇	—	2	8	no	—
134	#41×C₆₇	—	4	4	no	—
135	#41×C₆₇	—	4	4	no	—
136	#41×C₆₇	C₂²⋊C₄	1	16	yes	complemented
137	C₂×Q₈×C₆₇	—	2	1	no	—
138	C₂×Q₈×C₆₇	—	2	1	no	—
139	C₂×Q₈×C₆₇	—	2	1	no	—
140	C₂×Q₈×C₆₇	—	2	1	no	—
141	C₂×Q₈×C₆₇	—	2	1	no	—
142	C₂×Q₈×C₆₇	—	2	1	no	—
143	C₂×Q₈×C₆₇	—	2	1	no	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	4	no	—
151	C₂×D₄×C₆₇	—	2	4	no	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	1	no	—
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	—	2	1	no	—
156	C₂×D₄×C₆₇	—	2	1	no	—
157	C₂×D₄×C₆₇	—	2	1	no	—
158	C₂×D₄×C₆₇	—	2	1	no	—
159	C₂×D₄×C₆₇	—	2	1	no	—
160	C₂×D₄×C₆₇	—	2	1	no	—
161	C₂×D₄×C₆₇	—	4	2	no	—
162	C₂×D₄×C₆₇	—	4	2	no	—
163	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
164	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
165	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	—	2	1	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	—	2	8	no	—
174	C₂²×C₄×C₆₇	—	2	4	no	—
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	—	2	2	no	—
177	C₂²×C₄×C₆₇	—	4	2	no	—
178	C₂²×C₄×C₆₇	—	4	2	no	—
179	C₂²×C₄×C₆₇	—	4	2	no	—
180	C₂²×C₄×C₆₇	—	8	2	no	—
181	C₂²×C₄×C₆₇	—	8	2	no	—
182	C₂²×C₄×C₆₇	—	8	1	no	—
183	C₂²×C₄×C₆₇	—	8	1	no	—
184	C₂²×C₄×C₆₇	—	8	1	no	—
185	C₂²×C₄×C₆₇	—	8	1	no	—
186	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
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	2	yes	—
190	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
193	C₂²×C₄×C₆₇	#41	1	2	yes	—
194	C₂²×C₄×C₆₇	#41	1	2	yes	—
195	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
196	Q₁₆×C₆₇	—	2	8	no	—
197	SD₁₆×C₆₇	—	2	8	no	—
198	SD₁₆×C₆₇	—	2	8	no	—
199	D₈×C₆₇	—	2	8	no	—
200	M₁₆×C₆₇	—	4	4	no	—
201	C₂×C₈×C₆₇	—	2	8	no	—
202	C₂×C₈×C₆₇	—	2	1	no	—
203	C₂×C₈×C₆₇	—	2	1	no	—
204	C₂×C₈×C₆₇	—	2	1	no	—
205	C₂×C₈×C₆₇	—	2	1	no	—
206	C₂×C₈×C₆₇	—	2	2	no	—
207	C₂×C₈×C₆₇	—	2	2	no	—
208	C₂×C₈×C₆₇	—	2	2	no	—
209	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
210	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
211	C₄⋊C₄×C₆₇	—	2	8	no	—
212	C₄⋊C₄×C₆₇	—	2	2	no	—
213	C₄⋊C₄×C₆₇	—	2	4	no	—
214	C₄⋊C₄×C₆₇	—	2	2	no	—
215	C₄⋊C₄×C₆₇	—	2	2	no	—
216	C₄⋊C₄×C₆₇	—	2	2	no	—
217	C₄⋊C₄×C₆₇	—	2	2	no	—
218	C₄⋊C₄×C₆₇	—	4	2	no	—
219	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
220	C₂²⋊C₄×C₆₇	—	2	8	no	—
221	C₂²⋊C₄×C₆₇	—	2	4	no	—
222	C₂²⋊C₄×C₆₇	—	8	2	no	—
223	C₂²⋊C₄×C₆₇	—	8	2	no	—
224	C₄²×C₆₇	—	2	8	no	—
225	C₄²×C₆₇	—	2	2	no	—
226	C₄²×C₆₇	—	2	2	no	—
227	C₂³×C₆₇	—	2	1	no	—
228	C₂³×C₆₇	—	2	1	no	—
229	C₂³×C₆₇	—	2	1	no	—
230	C₂³×C₆₇	—	2	1	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	2	2	no	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	2	2	no	—
235	C₂³×C₆₇	—	4	1	no	—
236	C₂³×C₆₇	—	4	1	no	—
237	C₂³×C₆₇	—	4	1	no	—
238	C₂³×C₆₇	—	4	1	no	—
239	C₂³×C₆₇	—	4	1	no	—
240	C₂³×C₆₇	—	4	1	no	—
241	C₂³×C₆₇	—	4	1	no	—
242	C₂³×C₆₇	—	4	1	no	—
243	C₂³×C₆₇	—	4	1	no	—
244	C₂³×C₆₇	—	4	1	no	—
245	C₂³×C₆₇	C₄×D₄	1	2	yes	—
246	C₂³×C₆₇	#120	1	2	yes	—
247	C₂³×C₆₇	#121	1	2	yes	—
248	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
249	Q₈×C₆₇	—	2	2	no	—
250	Q₈×C₆₇	—	2	2	no	—
251	Q₈×C₆₇	—	2	2	no	—
252	Q₈×C₆₇	—	2	2	no	—
253	Q₈×C₆₇	—	2	2	no	—
254	Q₈×C₆₇	—	4	1	no	—
255	Q₈×C₆₇	—	4	1	no	—
256	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
257	D₄×C₆₇	—	2	16	no	—
258	D₄×C₆₇	—	2	2	no	—
259	D₄×C₆₇	—	2	2	no	—
260	D₄×C₆₇	—	2	2	no	—
261	D₄×C₆₇	—	2	2	no	—
262	D₄×C₆₇	—	2	2	no	—
263	D₄×C₆₇	—	4	8	no	—
264	D₄×C₆₇	—	4	1	no	—
265	D₄×C₆₇	—	4	1	no	—
266	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
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	16	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	4	1	no	—
280	C₂×C₄×C₆₇	—	4	1	no	—
281	C₂×C₄×C₆₇	—	4	2	no	—
282	C₂×C₄×C₆₇	—	4	2	no	—
283	C₂×C₄×C₆₇	—	4	8	no	—
284	C₂×C₄×C₆₇	—	4	2	no	—
285	C₂×C₄×C₆₇	—	4	2	no	—
286	C₂×C₄×C₆₇	—	4	2	no	—
287	C₂×C₄×C₆₇	—	8	4	no	—
288	C₂×C₄×C₆₇	—	8	8	no	—
289	C₂×C₄×C₆₇	—	8	4	no	—
290	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
291	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
292	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
293	C₂×C₄×C₆₇	#120	1	2	yes	—
294	C₂×C₄×C₆₇	#121	1	4	yes	—
295	C₂×C₄×C₆₇	#121	1	2	yes	—
296	C₂×C₄×C₆₇	#123	1	4	yes	—
297	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
298	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
302	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
303	C₈×C₆₇	—	2	2	no	—
304	C₈×C₆₇	—	2	2	no	—
305	C₈×C₆₇	—	4	2	no	—
306	C₂²×C₆₇	—	2	2	no	—
307	C₂²×C₆₇	—	2	2	no	—
308	C₂²×C₆₇	—	2	2	no	—
309	C₂²×C₆₇	—	2	2	no	—
310	C₂²×C₆₇	—	2	2	no	—
311	C₂²×C₆₇	—	2	2	no	—
312	C₂²×C₆₇	—	2	4	no	—
313	C₂²×C₆₇	—	4	2	no	—
314	C₂²×C₆₇	—	4	2	no	—
315	C₂²×C₆₇	—	4	2	no	—
316	C₂²×C₆₇	—	4	2	no	—
317	C₂²×C₆₇	—	4	2	no	—
318	C₂²×C₆₇	—	4	2	no	—
319	C₂²×C₆₇	—	4	2	no	—
320	C₂²×C₆₇	—	4	2	no	—
321	C₂²×C₆₇	—	4	2	no	—
322	C₂²×C₆₇	—	8	1	no	—
323	C₂²×C₆₇	—	8	1	no	—
324	C₂²×C₆₇	—	8	1	no	—
325	C₂²×C₆₇	—	8	1	no	—
326	C₂²×C₆₇	#386	1	4	yes	—
327	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
328	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
329	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
330	C₂²×C₆₇	#576	1	1	yes	characteristic
331	C₂²×C₆₇	#577	1	1	yes	characteristic
332	C₂²×C₆₇	#577	1	1	yes	characteristic
333	C₂²×C₆₇	#578	1	1	yes	characteristic
334	C₄×C₆₇	—	2	4	no	—
335	C₄×C₆₇	—	4	2	no	—
336	C₄×C₆₇	—	4	4	no	—
337	C₄×C₆₇	—	4	2	no	—
338	C₄×C₆₇	—	4	2	no	—
339	C₄×C₆₇	—	8	2	no	—
340	C₄×C₆₇	#386	1	4	yes	—
341	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
342	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
343	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
344	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
345	#20937	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
346	C₂×C₆₇	—	2	4	no	—
347	C₂×C₆₇	—	4	4	no	—
348	C₂×C₆₇	—	8	2	no	—
349	C₂×C₆₇	C₂×#386	1	1	yes	characteristic
350	C₂×C₆₇	#2951	1	1	yes	characteristic
351	C₂×C₆₇	#2952	1	1	yes	characteristic
352	C₂×C₆₇	#3012	1	1	yes	characteristic
353	C₂×C₆₇	#3013	1	1	yes	characteristic
354	C₂×C₆₇	#3045	1	1	yes	characteristic
355	C₂×C₆₇	#3046	1	1	yes	characteristic
356	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
357	C₂×#520	C₂×C₆₇	1	1	yes	characteristic, complemented
358	C₂×#417	C₂×C₆₇	1	1	yes	characteristic, complemented
359	C₂×#417	C₂×C₆₇	1	1	yes	characteristic, complemented
360	#2311	C₂×C₆₇	1	1	yes	characteristic, complemented
361	#1880	C₂×C₆₇	1	2	yes	complemented
362	#1879	C₂×C₆₇	1	2	yes	complemented
363	#1878	C₂×C₆₇	1	2	yes	complemented
364	#1877	C₂×C₆₇	1	2	yes	complemented
365	C₂×#340	C₂×C₆₇	1	1	yes	characteristic, complemented
366	#1729	C₂×C₆₇	1	1	yes	characteristic, complemented
367	C₆₇	#20937	1	1	yes	characteristic, direct product factor, Sylow
368	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
369	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
370	C₂×#135	—	2	4	no	—
371	C₂×#135	—	2	4	no	—
372	C₂×#135	C₄×C₆₇	1	8	yes	complemented
373	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
374	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
375	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
376	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
377	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
378	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
379	#520	—	2	4	no	—
380	#520	—	2	4	no	—
381	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
382	C₂×C₄×D₄	—	2	2	no	—
383	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
384	C₂×#117	—	2	1	no	—
385	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
386	C₂²×C₄⋊C₄	—	2	1	no	—
387	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
388	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
389	#417	—	2	4	no	—
390	#417	—	2	4	no	—
391	C₂×#103	C₂²×C₆₇	1	2	yes	—
392	C₂×#103	C₂²×C₆₇	1	2	yes	—
393	C₂×#102	C₂²×C₆₇	1	2	yes	—
394	C₂×#102	C₂²×C₆₇	1	2	yes	—
395	#391	C₂²×C₆₇	1	2	yes	—
396	#390	C₂²×C₆₇	1	2	yes	—
397	#386	—	2	2	no	—
398	#380	—	2	1	no	—
399	#380	—	2	1	no	—
400	#340	C₂²×C₆₇	1	2	yes	—
401	#340	C₂²×C₆₇	1	2	yes	—
402	#336	C₂²×C₆₇	1	2	yes	—
403	#336	C₂²×C₆₇	1	2	yes	—
404	C₂×#41	—	2	2	no	—
405	C₂×#41	—	2	2	no	—
406	C₂×#41	—	2	2	no	—
407	C₂×#41	—	2	2	no	—
408	C₂×#41	—	2	2	no	—
409	C₂×#41	—	2	2	no	—
410	C₂×#41	—	2	2	no	—
411	C₂×#41	—	2	2	no	—
412	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
413	C₂×#41	C₂×C₄×C₆₇	1	4	yes	complemented
414	C₂×#41	D₄×C₆₇	1	4	yes	—
415	C₂×#41	D₄×C₆₇	1	4	yes	—
416	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
417	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
418	C₂²×D₄	—	2	1	no	—
419	C₂²×D₄	—	2	1	no	—
420	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
421	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
422	C₂³×C₄	—	2	1	no	—
423	C₂³×C₄	—	2	1	no	—
424	C₂³×C₄	—	2	1	no	—
425	C₂³×C₄	—	2	2	no	—
426	C₂³×C₄	—	2	1	no	—
427	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
428	#135	—	2	32	no	—
429	C₂×Q₁₆	—	2	2	no	—
430	C₂×Q₁₆	—	2	2	no	—
431	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
432	C₂×SD₁₆	—	2	2	no	—
433	C₂×SD₁₆	—	2	2	no	—
434	C₂×SD₁₆	—	2	2	no	—
435	C₂×SD₁₆	—	2	2	no	—
436	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
437	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
438	C₂×D₈	—	2	2	no	—
439	C₂×D₈	—	2	2	no	—
440	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
441	C₂×M₁₆	—	2	2	no	—
442	C₂×M₁₆	—	2	2	no	—
443	C₂×M₁₆	—	2	2	no	—
444	C₂²×C₈	—	2	2	no	—
445	C₂²×C₈	—	2	2	no	—
446	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
447	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
448	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
449	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
450	C₄×Q₈	—	2	4	no	—
451	C₄×Q₈	—	2	4	no	—
452	C₄×D₄	—	2	16	no	—
453	C₄×D₄	—	2	4	no	—
454	C₄×D₄	—	2	4	no	—
455	#117	—	2	8	no	—
456	#117	—	2	2	no	—
457	#117	—	2	2	no	—
458	C₂×C₄⋊C₄	—	2	8	no	—
459	C₂×C₄⋊C₄	—	2	1	no	—
460	C₂×C₄⋊C₄	—	2	2	no	—
461	C₂×C₄⋊C₄	—	2	2	no	—
462	C₂×C₄⋊C₄	—	2	2	no	—
463	C₂×C₄⋊C₄	—	2	2	no	—
464	C₂×C₄⋊C₄	—	2	1	no	—
465	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
466	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
467	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
468	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
469	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
470	C₂×C₂²⋊C₄	—	2	2	no	—
471	C₂×C₂²⋊C₄	—	2	2	no	—
472	C₂×C₂²⋊C₄	—	2	2	no	—
473	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
474	C₂×C₄²	—	2	2	no	—
475	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
476	#103	—	2	4	no	—
477	#103	—	2	4	no	—
478	#102	—	2	4	no	—
479	#102	—	2	4	no	—
480	#95	—	2	2	no	—
481	#95	—	2	2	no	—
482	C₂⁴	—	2	1	no	—
483	C₂⁴	—	2	1	no	—
484	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
485	#41	—	2	8	no	—
486	#41	—	2	8	no	—
487	#41	—	2	8	no	—
488	#41	—	2	8	no	—
489	#41	—	2	8	no	—
490	#41	—	4	4	no	—
491	#41	—	4	4	no	—
492	#41	C₂²⋊C₄×C₆₇	1	16	yes	complemented
493	C₂×Q₈	—	2	1	no	—
494	C₂×Q₈	—	2	1	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
502	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
503	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
504	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	2	4	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	2	1	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	1	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	4	2	no	—
518	C₂×D₄	—	4	2	no	—
519	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
520	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
521	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
522	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	—	2	4	no	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	8	no	—
530	C₂²×C₄	—	2	4	no	—
531	C₂²×C₄	—	2	4	no	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	4	2	no	—
534	C₂²×C₄	—	4	2	no	—
535	C₂²×C₄	—	4	2	no	—
536	C₂²×C₄	—	8	2	no	—
537	C₂²×C₄	—	8	2	no	—
538	C₂²×C₄	—	8	1	no	—
539	C₂²×C₄	—	8	1	no	—
540	C₂²×C₄	—	8	1	no	—
541	C₂²×C₄	—	8	1	no	—
542	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
543	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
544	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
545	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
546	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
547	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
548	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
549	C₂²×C₄	#41×C₆₇	1	2	yes	—
550	C₂²×C₄	#41×C₆₇	1	2	yes	—
551	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
552	Q₁₆	—	2	8	no	—
553	SD₁₆	—	2	8	no	—
554	SD₁₆	—	2	8	no	—
555	D₈	—	2	8	no	—
556	M₁₆	—	4	4	no	—
557	C₂×C₈	—	2	8	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	2	no	—
563	C₂×C₈	—	2	2	no	—
564	C₂×C₈	—	2	2	no	—
565	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
566	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
567	C₄⋊C₄	—	2	8	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	4	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	2	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	4	2	no	—
575	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
576	C₂²⋊C₄	—	2	8	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	8	2	no	—
579	C₂²⋊C₄	—	8	2	no	—
580	C₄²	—	2	8	no	—
581	C₄²	—	2	2	no	—
582	C₄²	—	2	2	no	—
583	C₂³	—	2	1	no	—
584	C₂³	—	2	1	no	—
585	C₂³	—	2	1	no	—
586	C₂³	—	2	1	no	—
587	C₂³	—	2	2	no	—
588	C₂³	—	2	2	no	—
589	C₂³	—	2	2	no	—
590	C₂³	—	2	2	no	—
591	C₂³	—	4	1	no	—
592	C₂³	—	4	1	no	—
593	C₂³	—	4	1	no	—
594	C₂³	—	4	1	no	—
595	C₂³	—	4	1	no	—
596	C₂³	—	4	1	no	—
597	C₂³	—	4	1	no	—
598	C₂³	—	4	1	no	—
599	C₂³	—	4	1	no	—
600	C₂³	—	4	1	no	—
601	C₂³	C₄×D₄×C₆₇	1	2	yes	—
602	C₂³	#120×C₆₇	1	2	yes	—
603	C₂³	#121×C₆₇	1	2	yes	—
604	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
605	Q₈	—	2	2	no	—
606	Q₈	—	2	2	no	—
607	Q₈	—	2	2	no	—
608	Q₈	—	2	2	no	—
609	Q₈	—	2	2	no	—
610	Q₈	—	4	1	no	—
611	Q₈	—	4	1	no	—
612	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
613	D₄	—	2	16	no	—
614	D₄	—	2	2	no	—
615	D₄	—	2	2	no	—
616	D₄	—	2	2	no	—
617	D₄	—	2	2	no	—
618	D₄	—	2	2	no	—
619	D₄	—	4	8	no	—
620	D₄	—	4	1	no	—
621	D₄	—	4	1	no	—
622	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	16	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	4	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	2	no	—
638	C₂×C₄	—	4	2	no	—
639	C₂×C₄	—	4	8	no	—
640	C₂×C₄	—	4	2	no	—
641	C₂×C₄	—	4	2	no	—
642	C₂×C₄	—	4	2	no	—
643	C₂×C₄	—	8	4	no	—
644	C₂×C₄	—	8	8	no	—
645	C₂×C₄	—	8	4	no	—
646	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
647	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
648	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
649	C₂×C₄	#120×C₆₇	1	2	yes	—
650	C₂×C₄	#121×C₆₇	1	4	yes	—
651	C₂×C₄	#121×C₆₇	1	2	yes	—
652	C₂×C₄	#123×C₆₇	1	4	yes	—
653	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
654	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
655	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
656	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
657	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
658	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
659	C₈	—	2	2	no	—
660	C₈	—	2	2	no	—
661	C₈	—	4	2	no	—
662	C₂²	—	2	2	no	—
663	C₂²	—	2	2	no	—
664	C₂²	—	2	2	no	—
665	C₂²	—	2	2	no	—
666	C₂²	—	2	2	no	—
667	C₂²	—	2	2	no	—
668	C₂²	—	2	4	no	—
669	C₂²	—	4	2	no	—
670	C₂²	—	4	2	no	—
671	C₂²	—	4	2	no	—
672	C₂²	—	4	2	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	4	2	no	—
677	C₂²	—	4	2	no	—
678	C₂²	—	8	1	no	—
679	C₂²	—	8	1	no	—
680	C₂²	—	8	1	no	—
681	C₂²	—	8	1	no	—
682	C₂²	#386×C₆₇	1	4	yes	—
683	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
684	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
685	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
686	C₂²	#576×C₆₇	1	1	yes	characteristic
687	C₂²	#577×C₆₇	1	1	yes	characteristic
688	C₂²	#577×C₆₇	1	1	yes	characteristic
689	C₂²	#578×C₆₇	1	1	yes	characteristic
690	C₄	—	2	4	no	—
691	C₄	—	4	2	no	—
692	C₄	—	4	4	no	—
693	C₄	—	4	2	no	—
694	C₄	—	4	2	no	—
695	C₄	—	8	2	no	—
696	C₄	#386×C₆₇	1	4	yes	—
697	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
698	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
699	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
700	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
701	C₂	—	2	4	no	—
702	C₂	—	4	4	no	—
703	C₂	—	8	2	no	—
704	C₂	C₂×#386×C₆₇	1	1	yes	characteristic
705	C₂	C₆₇×#2951	1	1	yes	characteristic
706	C₂	C₆₇×#2952	1	1	yes	characteristic
707	C₂	C₆₇×#3012	1	1	yes	characteristic
708	C₂	C₆₇×#3013	1	1	yes	characteristic
709	C₂	C₆₇×#3045	1	1	yes	characteristic
710	C₂	C₆₇×#3046	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#520×C₆₇	C₂	semidirect
3. C₂×#417×C₆₇	C₂	semidirect
4. C₂×#417×C₆₇	C₂	semidirect
5. C₆₇×#2311	C₂	semidirect
6. C₆₇×#1880	C₂	semidirect
7. C₆₇×#1879	C₂	semidirect
8. C₆₇×#1878	C₂	semidirect
9. C₆₇×#1877	C₂	semidirect
10. C₂×#340×C₆₇	C₂	semidirect
11. C₆₇×#1729	C₂	semidirect
16. C₂×#135×C₆₇	C₄	semidirect
57. C₂×#41×C₆₇	C₂×C₄	semidirect
75. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
80. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
81. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
84. C₂×D₈×C₆₇	C₂×C₄	semidirect
90. C₂²×C₈×C₆₇	C₂×C₄	semidirect
136. #41×C₆₇	C₂²⋊C₄	semidirect
345. #20937	C₆₇	direct (N × H)
357. C₂×#520	C₂×C₆₇	semidirect
358. C₂×#417	C₂×C₆₇	semidirect
359. C₂×#417	C₂×C₆₇	semidirect
360. #2311	C₂×C₆₇	semidirect
361. #1880	C₂×C₆₇	semidirect
362. #1879	C₂×C₆₇	semidirect
363. #1878	C₂×C₆₇	semidirect
364. #1877	C₂×C₆₇	semidirect
365. C₂×#340	C₂×C₆₇	semidirect
366. #1729	C₂×C₆₇	semidirect
367. C₆₇	#20937	direct (N × H)
372. C₂×#135	C₄×C₆₇	semidirect
413. C₂×#41	C₂×C₄×C₆₇	semidirect
431. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
436. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
437. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
440. C₂×D₈	C₂×C₄×C₆₇	semidirect
446. C₂²×C₈	C₂×C₄×C₆₇	semidirect
492. #41	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂×#520×C₆₇	2
C₂×#417×C₆₇	2
C₆₇×#2311	2
C₆₇×#1880	2
C₆₇×#1879	2
C₆₇×#1878	2
C₆₇×#1877	2
C₂×#340×C₆₇	2
C₆₇×#1729	2
#20937	67

Subgroups — C67×#20937 #30700188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20937 #30700188