Subgroups — C₆₇×#34350 #30713601

Order 17152.

2106 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2383	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2382	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2379	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2378	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2377	C₂	1	2	yes	maximal
7	C₆₇×#2376	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2298	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2296	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2295	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2293	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2290	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2290	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
16	#529×C₆₇	C₄	1	16	yes	complemented
17	C₂×#126×C₆₇	C₂²	1	2	yes	—
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#123×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₇	C₂²	1	2	yes	—
29	C₂×#117×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#391×C₆₇	C₂²	1	4	yes	—
33	#390×C₆₇	C₂²	1	4	yes	—
34	#389×C₆₇	C₂²	1	4	yes	—
35	#389×C₆₇	C₂²	1	4	yes	—
36	#389×C₆₇	C₂²	1	4	yes	—
37	#389×C₆₇	C₂²	1	4	yes	—
38	#388×C₆₇	C₂²	1	4	yes	—
39	#388×C₆₇	C₂²	1	4	yes	—
40	#388×C₆₇	C₂²	1	4	yes	—
41	#388×C₆₇	C₂²	1	4	yes	—
42	#388×C₆₇	C₂²	1	4	yes	—
43	#388×C₆₇	C₂²	1	4	yes	—
44	#387×C₆₇	C₂²	1	4	yes	—
45	#387×C₆₇	C₂²	1	4	yes	—
46	#387×C₆₇	C₂²	1	4	yes	—
47	#387×C₆₇	C₂²	1	4	yes	—
48	#387×C₆₇	C₂²	1	4	yes	—
49	#387×C₆₇	C₂²	1	4	yes	—
50	#386×C₆₇	C₂²	1	4	yes	—
51	#386×C₆₇	C₂²	1	4	yes	—
52	#385×C₆₇	C₂²	1	4	yes	—
53	#385×C₆₇	C₂²	1	4	yes	—
54	#385×C₆₇	C₂²	1	4	yes	—
55	#385×C₆₇	C₂²	1	4	yes	—
56	#384×C₆₇	C₂²	1	2	yes	—
57	#383×C₆₇	C₂²	1	2	yes	—
58	#382×C₆₇	C₂²	1	2	yes	—
59	#382×C₆₇	C₂²	1	2	yes	—
60	#381×C₆₇	C₂²	1	2	yes	—
61	#380×C₆₇	C₂²	1	2	yes	—
62	#380×C₆₇	C₂²	1	2	yes	—
63	#380×C₆₇	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
65	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
66	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
67	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
68	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
70	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
71	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
72	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
73	#376×C₆₇	C₂²	1	2	yes	—
74	#376×C₆₇	C₂²	1	2	yes	—
75	C₄³×C₆₇	C₂²	1	2	yes	—
76	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	#126×C₆₇	C₂×C₄	1	16	yes	complemented
85	#125×C₆₇	C₂×C₄	1	8	yes	complemented
86	#124×C₆₇	C₂×C₄	1	8	yes	—
87	#123×C₆₇	C₂×C₄	1	16	yes	complemented
88	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
89	#121×C₆₇	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
91	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
93	#117×C₆₇	—	4	2	no	—
94	#117×C₆₇	—	4	4	no	—
95	#117×C₆₇	—	4	2	no	—
96	#117×C₆₇	C₂×C₄	1	16	yes	complemented
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₄²×C₆₇	—	4	4	no	—
124	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
125	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
128	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
130	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
137	#95×C₆₇	C₂³	1	4	yes	—
138	#95×C₆₇	C₂³	1	4	yes	—
139	#95×C₆₇	C₂³	1	4	yes	—
140	#95×C₆₇	C₂³	1	4	yes	—
141	#95×C₆₇	C₂³	1	4	yes	—
142	#95×C₆₇	C₂³	1	4	yes	—
143	#95×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	4	yes	—
146	#95×C₆₇	C₂³	1	2	yes	—
147	#95×C₆₇	C₂³	1	2	yes	—
148	#95×C₆₇	C₂³	1	4	yes	—
149	#95×C₆₇	C₂³	1	4	yes	—
150	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
151	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
152	C₂×Q₈×C₆₇	—	4	2	no	—
153	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
154	C₂×D₄×C₆₇	—	4	4	no	—
155	C₂×D₄×C₆₇	—	4	2	no	—
156	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
157	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	4	4	no	—
163	C₂²×C₄×C₆₇	—	4	2	no	—
164	C₂²×C₄×C₆₇	—	4	2	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	—	4	4	no	—
167	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
169	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
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	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₄⋊C₄×C₆₇	—	4	4	no	—
185	C₄⋊C₄×C₆₇	—	4	4	no	—
186	C₄⋊C₄×C₆₇	—	4	2	no	—
187	C₄⋊C₄×C₆₇	—	4	4	no	—
188	C₄⋊C₄×C₆₇	—	4	4	no	—
189	C₄⋊C₄×C₆₇	—	4	2	no	—
190	C₄⋊C₄×C₆₇	—	4	4	no	—
191	C₄⋊C₄×C₆₇	—	4	2	no	—
192	C₄⋊C₄×C₆₇	—	4	2	no	—
193	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
194	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
195	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
196	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
197	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
198	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
199	C₂²⋊C₄×C₆₇	—	4	4	no	—
200	C₂²⋊C₄×C₆₇	—	4	4	no	—
201	C₂²⋊C₄×C₆₇	—	4	4	no	—
202	C₂²⋊C₄×C₆₇	—	4	2	no	—
203	C₂²⋊C₄×C₆₇	—	4	4	no	—
204	C₂²⋊C₄×C₆₇	—	4	2	no	—
205	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
206	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
207	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
208	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
209	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
210	C₄²×C₆₇	—	2	4	no	—
211	C₄²×C₆₇	—	2	4	no	—
212	C₄²×C₆₇	—	2	4	no	—
213	C₄²×C₆₇	—	2	4	no	—
214	C₄²×C₆₇	—	4	4	no	—
215	C₄²×C₆₇	—	4	4	no	—
216	C₄²×C₆₇	—	4	4	no	—
217	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
218	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
219	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
220	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
221	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
222	C₂³×C₆₇	—	2	1	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	1	no	—
225	C₂³×C₆₇	—	2	2	no	—
226	C₂³×C₆₇	—	2	2	no	—
227	C₂³×C₆₇	—	2	4	no	—
228	C₂³×C₆₇	—	2	4	no	—
229	C₂³×C₆₇	—	2	2	no	—
230	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
231	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
232	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
233	Q₈×C₆₇	—	4	4	no	—
234	D₄×C₆₇	—	4	4	no	—
235	D₄×C₆₇	—	4	8	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	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	16	no	—
247	C₂×C₄×C₆₇	—	2	8	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	2	4	no	—
260	C₂×C₄×C₆₇	—	4	4	no	—
261	C₂×C₄×C₆₇	—	4	4	no	—
262	C₂×C₄×C₆₇	—	4	16	no	—
263	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
264	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
265	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
269	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
270	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
271	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
272	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
273	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
274	C₂²×C₆₇	—	2	2	no	—
275	C₂²×C₆₇	—	2	4	no	—
276	C₂²×C₆₇	—	2	2	no	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	—	2	4	no	—
279	C₂²×C₆₇	—	2	4	no	—
280	C₂²×C₆₇	—	4	4	no	—
281	C₂²×C₆₇	—	4	2	no	—
282	C₂²×C₆₇	—	4	2	no	—
283	C₂²×C₆₇	—	4	4	no	—
284	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
285	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
287	C₂²×C₆₇	#585	1	2	yes	—
288	C₂²×C₆₇	#585	1	2	yes	—
289	C₄×C₆₇	—	2	4	no	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	2	2	no	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	2	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	2	4	no	—
296	C₄×C₆₇	—	2	4	no	—
297	C₄×C₆₇	—	4	4	no	—
298	C₄×C₆₇	—	4	4	no	—
299	C₄×C₆₇	—	4	4	no	—
300	C₄×C₆₇	—	4	4	no	—
301	C₄×C₆₇	#520	1	4	yes	—
302	#34350	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
303	C₂×C₆₇	—	4	2	no	—
304	C₂×C₆₇	—	4	4	no	—
305	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
306	C₂×C₆₇	#3428	1	1	yes	characteristic
307	C₂×C₆₇	#3428	1	1	yes	characteristic
308	C₂×C₆₇	#3529	1	2	yes	—
309	C₂×C₆₇	#3531	1	2	yes	—
310	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
311	#2383	C₂×C₆₇	1	4	yes	complemented
312	#2382	C₂×C₆₇	1	2	yes	complemented
313	#2379	C₂×C₆₇	1	2	yes	complemented
314	#2378	C₂×C₆₇	1	2	yes	complemented
315	#2377	C₂×C₆₇	1	2	yes	—
316	#2376	C₂×C₆₇	1	4	yes	complemented
317	#2298	C₂×C₆₇	1	2	yes	complemented
318	#2296	C₂×C₆₇	1	2	yes	complemented
319	#2295	C₂×C₆₇	1	2	yes	complemented
320	#2293	C₂×C₆₇	1	2	yes	complemented
321	#2290	C₂×C₆₇	1	2	yes	complemented
322	#2290	C₂×C₆₇	1	2	yes	complemented
323	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
324	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
325	C₆₇	#34350	1	1	yes	characteristic, direct product factor, Sylow
326	#529	C₄×C₆₇	1	16	yes	complemented
327	C₂×#126	C₂²×C₆₇	1	2	yes	—
328	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×#123	C₂²×C₆₇	1	2	yes	—
331	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
334	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
336	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×#117	C₂²×C₆₇	1	2	yes	—
339	C₂×#117	C₂²×C₆₇	1	2	yes	—
340	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
341	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
342	#391	C₂²×C₆₇	1	4	yes	—
343	#390	C₂²×C₆₇	1	4	yes	—
344	#389	C₂²×C₆₇	1	4	yes	—
345	#389	C₂²×C₆₇	1	4	yes	—
346	#389	C₂²×C₆₇	1	4	yes	—
347	#389	C₂²×C₆₇	1	4	yes	—
348	#388	C₂²×C₆₇	1	4	yes	—
349	#388	C₂²×C₆₇	1	4	yes	—
350	#388	C₂²×C₆₇	1	4	yes	—
351	#388	C₂²×C₆₇	1	4	yes	—
352	#388	C₂²×C₆₇	1	4	yes	—
353	#388	C₂²×C₆₇	1	4	yes	—
354	#387	C₂²×C₆₇	1	4	yes	—
355	#387	C₂²×C₆₇	1	4	yes	—
356	#387	C₂²×C₆₇	1	4	yes	—
357	#387	C₂²×C₆₇	1	4	yes	—
358	#387	C₂²×C₆₇	1	4	yes	—
359	#387	C₂²×C₆₇	1	4	yes	—
360	#386	C₂²×C₆₇	1	4	yes	—
361	#386	C₂²×C₆₇	1	4	yes	—
362	#385	C₂²×C₆₇	1	4	yes	—
363	#385	C₂²×C₆₇	1	4	yes	—
364	#385	C₂²×C₆₇	1	4	yes	—
365	#385	C₂²×C₆₇	1	4	yes	—
366	#384	C₂²×C₆₇	1	2	yes	—
367	#383	C₂²×C₆₇	1	2	yes	—
368	#382	C₂²×C₆₇	1	2	yes	—
369	#382	C₂²×C₆₇	1	2	yes	—
370	#381	C₂²×C₆₇	1	2	yes	—
371	#380	C₂²×C₆₇	1	2	yes	—
372	#380	C₂²×C₆₇	1	2	yes	—
373	#380	C₂²×C₆₇	1	2	yes	—
374	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
375	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
376	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
377	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
378	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
379	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
380	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
381	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
382	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
383	#376	C₂²×C₆₇	1	2	yes	—
384	#376	C₂²×C₆₇	1	2	yes	—
385	C₄³	C₂²×C₆₇	1	2	yes	—
386	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
387	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
388	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
390	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
391	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
392	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
393	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
394	#126	C₂×C₄×C₆₇	1	16	yes	complemented
395	#125	C₂×C₄×C₆₇	1	8	yes	complemented
396	#124	C₂×C₄×C₆₇	1	8	yes	—
397	#123	C₂×C₄×C₆₇	1	16	yes	complemented
398	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
399	#121	C₂×C₄×C₆₇	1	8	yes	—
400	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
401	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
402	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
403	#117	—	4	2	no	—
404	#117	—	4	4	no	—
405	#117	—	4	2	no	—
406	#117	C₂×C₄×C₆₇	1	16	yes	complemented
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄²	—	4	4	no	—
434	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
435	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
436	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
437	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
439	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
440	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
441	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
442	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
444	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
447	#95	C₂³×C₆₇	1	4	yes	—
448	#95	C₂³×C₆₇	1	4	yes	—
449	#95	C₂³×C₆₇	1	4	yes	—
450	#95	C₂³×C₆₇	1	4	yes	—
451	#95	C₂³×C₆₇	1	4	yes	—
452	#95	C₂³×C₆₇	1	4	yes	—
453	#95	C₂³×C₆₇	1	4	yes	—
454	#95	C₂³×C₆₇	1	4	yes	—
455	#95	C₂³×C₆₇	1	4	yes	—
456	#95	C₂³×C₆₇	1	2	yes	—
457	#95	C₂³×C₆₇	1	2	yes	—
458	#95	C₂³×C₆₇	1	4	yes	—
459	#95	C₂³×C₆₇	1	4	yes	—
460	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
461	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
462	C₂×Q₈	—	4	2	no	—
463	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	—	4	2	no	—
466	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
467	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
478	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
479	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
480	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
487	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
492	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
493	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
494	C₄⋊C₄	—	4	4	no	—
495	C₄⋊C₄	—	4	2	no	—
496	C₄⋊C₄	—	4	2	no	—
497	C₄⋊C₄	—	4	4	no	—
498	C₄⋊C₄	—	4	4	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	—	4	4	no	—
501	C₄⋊C₄	—	4	4	no	—
502	C₄⋊C₄	—	4	2	no	—
503	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
504	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
505	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
506	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
507	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
508	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
509	C₂²⋊C₄	—	4	2	no	—
510	C₂²⋊C₄	—	4	4	no	—
511	C₂²⋊C₄	—	4	2	no	—
512	C₂²⋊C₄	—	4	4	no	—
513	C₂²⋊C₄	—	4	4	no	—
514	C₂²⋊C₄	—	4	4	no	—
515	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
516	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
517	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
518	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
519	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
520	C₄²	—	2	4	no	—
521	C₄²	—	2	4	no	—
522	C₄²	—	2	4	no	—
523	C₄²	—	2	4	no	—
524	C₄²	—	4	4	no	—
525	C₄²	—	4	4	no	—
526	C₄²	—	4	4	no	—
527	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
528	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
529	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
530	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
531	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
532	C₂³	—	2	1	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	2	no	—
540	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
541	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
542	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
543	Q₈	—	4	4	no	—
544	D₄	—	4	4	no	—
545	D₄	—	4	8	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	16	no	—
557	C₂×C₄	—	2	8	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	16	no	—
573	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
574	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
575	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
576	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
577	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
578	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
579	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
580	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
581	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
582	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
583	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	4	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	2	4	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	4	no	—
594	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
595	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
596	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
597	C₂²	#585×C₆₇	1	2	yes	—
598	C₂²	#585×C₆₇	1	2	yes	—
599	C₄	—	2	4	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	2	no	—
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	4	no	—
606	C₄	—	2	4	no	—
607	C₄	—	4	4	no	—
608	C₄	—	4	4	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	4	no	—
611	C₄	#520×C₆₇	1	4	yes	—
612	C₂	—	4	2	no	—
613	C₂	—	4	4	no	—
614	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
615	C₂	C₆₇×#3428	1	1	yes	characteristic
616	C₂	C₆₇×#3428	1	1	yes	characteristic
617	C₂	C₆₇×#3529	1	2	yes	—
618	C₂	C₆₇×#3531	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2383	C₂	semidirect
3. C₆₇×#2382	C₂	semidirect
4. C₆₇×#2379	C₂	semidirect
5. C₆₇×#2378	C₂	semidirect
7. C₆₇×#2376	C₂	semidirect
8. C₆₇×#2298	C₂	semidirect
9. C₆₇×#2296	C₂	semidirect
10. C₆₇×#2295	C₂	semidirect
11. C₆₇×#2293	C₂	semidirect
12. C₆₇×#2290	C₂	semidirect
13. C₆₇×#2290	C₂	semidirect
14. C₆₇×#2285	C₂	semidirect
15. C₆₇×#2283	C₂	semidirect
16. #529×C₆₇	C₄	semidirect
84. #126×C₆₇	C₂×C₄	semidirect
85. #125×C₆₇	C₂×C₄	semidirect
87. #123×C₆₇	C₂×C₄	semidirect
88. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
90. C₄×Q₈×C₆₇	C₂×C₄	semidirect
92. C₄×D₄×C₆₇	C₂×C₄	semidirect
96. #117×C₆₇	C₂×C₄	semidirect
124. C₂×C₄²×C₆₇	C₂×C₄	semidirect
302. #34350	C₆₇	direct (N × H)
311. #2383	C₂×C₆₇	semidirect
312. #2382	C₂×C₆₇	semidirect
313. #2379	C₂×C₆₇	semidirect
314. #2378	C₂×C₆₇	semidirect
316. #2376	C₂×C₆₇	semidirect
317. #2298	C₂×C₆₇	semidirect
318. #2296	C₂×C₆₇	semidirect
319. #2295	C₂×C₆₇	semidirect
320. #2293	C₂×C₆₇	semidirect
321. #2290	C₂×C₆₇	semidirect
322. #2290	C₂×C₆₇	semidirect
323. #2285	C₂×C₆₇	semidirect
324. #2283	C₂×C₆₇	semidirect
325. C₆₇	#34350	direct (N × H)
326. #529	C₄×C₆₇	semidirect
394. #126	C₂×C₄×C₆₇	semidirect
395. #125	C₂×C₄×C₆₇	semidirect
397. #123	C₂×C₄×C₆₇	semidirect
398. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
400. C₄×Q₈	C₂×C₄×C₆₇	semidirect
402. C₄×D₄	C₂×C₄×C₆₇	semidirect
406. #117	C₂×C₄×C₆₇	semidirect
434. C₂×C₄²	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₆₇×#2383	2
C₆₇×#2382	2
C₆₇×#2379	2
C₆₇×#2378	2
C₆₇×#2377	2
C₆₇×#2376	2
C₆₇×#2298	2
C₆₇×#2296	2
C₆₇×#2295	2
C₆₇×#2293	2
C₆₇×#2290	2
C₆₇×#2285	2
C₆₇×#2283	2
#34350	67

Subgroups — C67×#34350 #30713601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34350 #30713601