Subgroups — C₆₇×#34222 #30713473

Order 17152.

2650 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2372	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2364	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2353	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2339	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2331	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2328	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2319	C₂	1	2	yes	maximal, complemented
9	C₄×#121×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2275	C₂	1	2	yes	maximal, complemented
12	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	2	yes	—
14	#532×C₆₇	C₄	1	16	yes	complemented
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	2	yes	—
18	C₂×#121×C₆₇	C₂²	1	2	yes	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	#391×C₆₇	C₂²	1	8	yes	—
27	#390×C₆₇	C₂²	1	4	yes	—
28	#389×C₆₇	C₂²	1	4	yes	—
29	#388×C₆₇	C₂²	1	8	yes	—
30	#388×C₆₇	C₂²	1	8	yes	—
31	#387×C₆₇	C₂²	1	8	yes	—
32	#386×C₆₇	C₂²	1	8	yes	—
33	#386×C₆₇	C₂²	1	8	yes	—
34	#386×C₆₇	C₂²	1	8	yes	—
35	#383×C₆₇	C₂²	1	2	yes	—
36	#382×C₆₇	C₂²	1	2	yes	—
37	#381×C₆₇	C₂²	1	2	yes	—
38	#381×C₆₇	C₂²	1	4	yes	—
39	#380×C₆₇	C₂²	1	2	yes	—
40	#380×C₆₇	C₂²	1	4	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
42	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
43	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
44	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
45	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
46	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
47	#376×C₆₇	C₂²	1	1	yes	characteristic
48	#376×C₆₇	C₂²	1	1	yes	characteristic
49	#376×C₆₇	C₂²	1	2	yes	—
50	#376×C₆₇	C₂²	1	2	yes	—
51	#376×C₆₇	C₂²	1	2	yes	—
52	#376×C₆₇	C₂²	1	2	yes	—
53	C₂×#95×C₆₇	C₂²	1	2	yes	—
54	C₂×#95×C₆₇	C₂²	1	2	yes	—
55	C₂×#95×C₆₇	C₂²	1	4	yes	—
56	C₂×#41×C₆₇	—	4	8	no	—
57	C₂×#41×C₆₇	C₂×C₄	1	16	yes	complemented
58	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
60	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
61	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
69	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
70	#127×C₆₇	C₂×C₄	1	8	yes	complemented
71	#124×C₆₇	C₂×C₄	1	16	yes	complemented
72	#121×C₆₇	C₂×C₄	1	16	yes	complemented
73	#121×C₆₇	C₂×C₄	1	16	yes	complemented
74	C₄×D₄×C₆₇	—	2	16	no	—
75	C₄×D₄×C₆₇	—	4	16	no	—
76	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
77	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄²×C₆₇	—	2	8	no	—
89	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
90	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
91	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
93	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
99	#95×C₆₇	C₂³	1	2	yes	—
100	#95×C₆₇	C₂³	1	2	yes	—
101	#95×C₆₇	C₂³	1	2	yes	—
102	#95×C₆₇	C₂³	1	4	yes	—
103	#95×C₆₇	C₂³	1	4	yes	—
104	#95×C₆₇	C₂³	1	2	yes	—
105	#95×C₆₇	C₂³	1	4	yes	—
106	#95×C₆₇	C₂³	1	8	yes	—
107	#95×C₆₇	C₂³	1	4	yes	—
108	#95×C₆₇	C₂³	1	4	yes	—
109	#95×C₆₇	C₂³	1	8	yes	—
110	#95×C₆₇	C₂³	1	4	yes	—
111	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
112	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
113	#41×C₆₇	—	4	32	no	—
114	C₂×Q₈×C₆₇	—	4	4	no	—
115	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
116	C₂×D₄×C₆₇	—	2	16	no	—
117	C₂×D₄×C₆₇	—	2	8	no	—
118	C₂×D₄×C₆₇	—	4	4	no	—
119	C₂×D₄×C₆₇	—	4	4	no	—
120	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
121	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
122	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
123	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	8	no	—
127	C₂²×C₄×C₆₇	—	2	8	no	—
128	C₂²×C₄×C₆₇	—	2	16	no	—
129	C₂²×C₄×C₆₇	—	4	2	no	—
130	C₂²×C₄×C₆₇	—	4	4	no	—
131	C₂²×C₄×C₆₇	—	4	8	no	—
132	C₂²×C₄×C₆₇	—	4	8	no	—
133	C₂²×C₄×C₆₇	—	4	8	no	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₄⋊C₄×C₆₇	—	2	8	no	—
149	C₄⋊C₄×C₆₇	—	4	8	no	—
150	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
152	C₂²⋊C₄×C₆₇	—	2	8	no	—
153	C₂²⋊C₄×C₆₇	—	2	16	no	—
154	C₂²⋊C₄×C₆₇	—	4	8	no	—
155	C₂²⋊C₄×C₆₇	—	4	8	no	—
156	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
157	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
158	C₄²×C₆₇	—	2	8	no	—
159	C₄²×C₆₇	—	2	4	no	—
160	C₄²×C₆₇	—	4	4	no	—
161	C₄²×C₆₇	—	4	8	no	—
162	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
163	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
164	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
165	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
166	C₄²×C₆₇	#41	1	8	yes	complemented
167	C₂³×C₆₇	—	2	2	no	—
168	C₂³×C₆₇	—	2	2	no	—
169	C₂³×C₆₇	—	2	8	no	—
170	C₂³×C₆₇	—	2	8	no	—
171	C₂³×C₆₇	—	2	4	no	—
172	C₂³×C₆₇	—	2	4	no	—
173	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
174	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
175	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
176	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
177	Q₈×C₆₇	—	4	8	no	—
178	D₄×C₆₇	—	2	32	no	—
179	D₄×C₆₇	—	2	16	no	—
180	D₄×C₆₇	—	4	8	no	—
181	D₄×C₆₇	—	4	8	no	—
182	C₂×C₄×C₆₇	—	2	8	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	8	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	8	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	8	no	—
194	C₂×C₄×C₆₇	—	2	8	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	8	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	4	4	no	—
199	C₂×C₄×C₆₇	—	4	16	no	—
200	C₂×C₄×C₆₇	—	4	16	no	—
201	C₂×C₄×C₆₇	—	8	16	no	—
202	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
203	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
204	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
205	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
206	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
207	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
208	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
209	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
210	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
211	C₂²×C₆₇	—	2	2	no	—
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	2	8	no	—
214	C₂²×C₆₇	—	2	8	no	—
215	C₂²×C₆₇	—	2	8	no	—
216	C₂²×C₆₇	—	2	8	no	—
217	C₂²×C₆₇	—	4	8	no	—
218	C₂²×C₆₇	—	4	8	no	—
219	C₂²×C₆₇	#518	1	4	yes	—
220	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
221	C₂²×C₆₇	C₂²×#41	1	2	yes	—
222	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
223	C₂²×C₆₇	#585	1	2	yes	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	8	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	2	8	no	—
231	C₄×C₆₇	—	4	4	no	—
232	C₄×C₆₇	—	4	8	no	—
233	C₄×C₆₇	C₄×#41	1	4	yes	—
234	#34222	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂×C₆₇	—	2	4	no	—
236	C₂×C₆₇	—	4	8	no	—
237	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
238	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
239	C₂×C₆₇	#3428	1	1	yes	characteristic
240	C₂×C₆₇	#3494	1	4	yes	—
241	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
242	#2372	C₂×C₆₇	1	4	yes	complemented
243	#2364	C₂×C₆₇	1	4	yes	complemented
244	#2353	C₂×C₆₇	1	4	yes	complemented
245	#2339	C₂×C₆₇	1	4	yes	complemented
246	#2331	C₂×C₆₇	1	2	yes	complemented
247	#2328	C₂×C₆₇	1	2	yes	complemented
248	#2319	C₂×C₆₇	1	2	yes	complemented
249	C₄×#121	C₂×C₆₇	1	4	yes	complemented
250	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
251	#2275	C₂×C₆₇	1	2	yes	complemented
252	C₂×#376	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₆₇	#34222	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²×#41	C₂²×C₆₇	1	2	yes	—
255	#532	C₄×C₆₇	1	16	yes	complemented
256	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×#124	C₂²×C₆₇	1	2	yes	—
259	C₂×#121	C₂²×C₆₇	1	2	yes	—
260	C₂×#121	C₂²×C₆₇	1	2	yes	—
261	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
262	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
263	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
264	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
265	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
266	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
267	#391	C₂²×C₆₇	1	8	yes	—
268	#390	C₂²×C₆₇	1	4	yes	—
269	#389	C₂²×C₆₇	1	4	yes	—
270	#388	C₂²×C₆₇	1	8	yes	—
271	#388	C₂²×C₆₇	1	8	yes	—
272	#387	C₂²×C₆₇	1	8	yes	—
273	#386	C₂²×C₆₇	1	8	yes	—
274	#386	C₂²×C₆₇	1	8	yes	—
275	#386	C₂²×C₆₇	1	8	yes	—
276	#383	C₂²×C₆₇	1	2	yes	—
277	#382	C₂²×C₆₇	1	2	yes	—
278	#381	C₂²×C₆₇	1	4	yes	—
279	#381	C₂²×C₆₇	1	2	yes	—
280	#380	C₂²×C₆₇	1	4	yes	—
281	#380	C₂²×C₆₇	1	2	yes	—
282	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
283	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
284	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
285	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
286	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
287	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
288	#376	C₂²×C₆₇	1	1	yes	characteristic
289	#376	C₂²×C₆₇	1	2	yes	—
290	#376	C₂²×C₆₇	1	2	yes	—
291	#376	C₂²×C₆₇	1	2	yes	—
292	#376	C₂²×C₆₇	1	2	yes	—
293	#376	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×#95	C₂²×C₆₇	1	2	yes	—
295	C₂×#95	C₂²×C₆₇	1	4	yes	—
296	C₂×#95	C₂²×C₆₇	1	2	yes	—
297	C₂×#41	—	4	8	no	—
298	C₂×#41	C₂×C₄×C₆₇	1	16	yes	complemented
299	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
300	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
301	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
302	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
303	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
304	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
305	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
306	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
308	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
309	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
311	#127	C₂×C₄×C₆₇	1	8	yes	complemented
312	#124	C₂×C₄×C₆₇	1	16	yes	complemented
313	#121	C₂×C₄×C₆₇	1	16	yes	complemented
314	#121	C₂×C₄×C₆₇	1	16	yes	complemented
315	C₄×D₄	—	2	16	no	—
316	C₄×D₄	—	4	16	no	—
317	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
318	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
319	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
320	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
321	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
322	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
329	C₂×C₄²	—	2	8	no	—
330	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
331	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
332	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
333	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
334	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
335	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
336	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
337	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
338	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
339	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
340	#95	C₂³×C₆₇	1	2	yes	—
341	#95	C₂³×C₆₇	1	2	yes	—
342	#95	C₂³×C₆₇	1	2	yes	—
343	#95	C₂³×C₆₇	1	4	yes	—
344	#95	C₂³×C₆₇	1	4	yes	—
345	#95	C₂³×C₆₇	1	2	yes	—
346	#95	C₂³×C₆₇	1	4	yes	—
347	#95	C₂³×C₆₇	1	8	yes	—
348	#95	C₂³×C₆₇	1	4	yes	—
349	#95	C₂³×C₆₇	1	4	yes	—
350	#95	C₂³×C₆₇	1	8	yes	—
351	#95	C₂³×C₆₇	1	4	yes	—
352	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
353	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
354	#41	—	4	32	no	—
355	C₂×Q₈	—	4	4	no	—
356	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
357	C₂×D₄	—	2	16	no	—
358	C₂×D₄	—	2	8	no	—
359	C₂×D₄	—	4	4	no	—
360	C₂×D₄	—	4	4	no	—
361	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
362	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
363	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
364	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
365	C₂²×C₄	—	2	4	no	—
366	C₂²×C₄	—	2	4	no	—
367	C₂²×C₄	—	2	8	no	—
368	C₂²×C₄	—	2	8	no	—
369	C₂²×C₄	—	2	16	no	—
370	C₂²×C₄	—	4	8	no	—
371	C₂²×C₄	—	4	2	no	—
372	C₂²×C₄	—	4	4	no	—
373	C₂²×C₄	—	4	8	no	—
374	C₂²×C₄	—	4	8	no	—
375	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
376	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
377	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
378	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
379	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
380	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
381	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
382	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
383	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
386	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
387	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
388	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
389	C₄⋊C₄	—	2	8	no	—
390	C₄⋊C₄	—	4	8	no	—
391	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
392	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	2	16	no	—
395	C₂²⋊C₄	—	4	8	no	—
396	C₂²⋊C₄	—	4	8	no	—
397	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
398	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
399	C₄²	—	2	8	no	—
400	C₄²	—	2	4	no	—
401	C₄²	—	4	8	no	—
402	C₄²	—	4	4	no	—
403	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
404	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
405	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
406	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
407	C₄²	#41×C₆₇	1	8	yes	complemented
408	C₂³	—	2	2	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	2	8	no	—
411	C₂³	—	2	8	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	4	no	—
414	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
415	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
416	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
417	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
418	Q₈	—	4	8	no	—
419	D₄	—	2	32	no	—
420	D₄	—	2	16	no	—
421	D₄	—	4	8	no	—
422	D₄	—	4	8	no	—
423	C₂×C₄	—	2	8	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	8	no	—
435	C₂×C₄	—	2	8	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	8	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	4	16	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	16	no	—
442	C₂×C₄	—	8	16	no	—
443	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
444	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
445	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
446	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
447	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
448	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
449	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
450	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
451	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	2	8	no	—
455	C₂²	—	2	8	no	—
456	C₂²	—	2	8	no	—
457	C₂²	—	2	8	no	—
458	C₂²	—	4	8	no	—
459	C₂²	—	4	8	no	—
460	C₂²	#518×C₆₇	1	4	yes	—
461	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
462	C₂²	C₂²×#41×C₆₇	1	2	yes	—
463	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
464	C₂²	#585×C₆₇	1	2	yes	—
465	C₄	—	2	2	no	—
466	C₄	—	2	2	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	8	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	8	no	—
472	C₄	—	4	4	no	—
473	C₄	—	4	8	no	—
474	C₄	C₄×#41×C₆₇	1	4	yes	—
475	C₂	—	2	4	no	—
476	C₂	—	4	8	no	—
477	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
478	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
479	C₂	C₆₇×#3428	1	1	yes	characteristic
480	C₂	C₆₇×#3494	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2372	C₂	semidirect
3. C₆₇×#2364	C₂	semidirect
4. C₆₇×#2353	C₂	semidirect
5. C₆₇×#2339	C₂	semidirect
6. C₆₇×#2331	C₂	semidirect
7. C₆₇×#2328	C₂	semidirect
8. C₆₇×#2319	C₂	semidirect
9. C₄×#121×C₆₇	C₂	semidirect
10. C₆₇×#2283	C₂	semidirect
11. C₆₇×#2275	C₂	semidirect
12. C₂×#376×C₆₇	C₂	semidirect
14. #532×C₆₇	C₄	semidirect
57. C₂×#41×C₆₇	C₂×C₄	semidirect
69. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
70. #127×C₆₇	C₂×C₄	semidirect
71. #124×C₆₇	C₂×C₄	semidirect
72. #121×C₆₇	C₂×C₄	semidirect
73. #121×C₆₇	C₂×C₄	semidirect
76. C₄×D₄×C₆₇	C₂×C₄	semidirect
77. C₄×D₄×C₆₇	C₂×C₄	semidirect
89. C₂×C₄²×C₆₇	C₂×C₄	semidirect
166. C₄²×C₆₇	#41	semidirect
234. #34222	C₆₇	direct (N × H)
242. #2372	C₂×C₆₇	semidirect
243. #2364	C₂×C₆₇	semidirect
244. #2353	C₂×C₆₇	semidirect
245. #2339	C₂×C₆₇	semidirect
246. #2331	C₂×C₆₇	semidirect
247. #2328	C₂×C₆₇	semidirect
248. #2319	C₂×C₆₇	semidirect
249. C₄×#121	C₂×C₆₇	semidirect
250. #2283	C₂×C₆₇	semidirect
251. #2275	C₂×C₆₇	semidirect
252. C₂×#376	C₂×C₆₇	semidirect
253. C₆₇	#34222	direct (N × H)
255. #532	C₄×C₆₇	semidirect
298. C₂×#41	C₂×C₄×C₆₇	semidirect
310. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
311. #127	C₂×C₄×C₆₇	semidirect
312. #124	C₂×C₄×C₆₇	semidirect
313. #121	C₂×C₄×C₆₇	semidirect
314. #121	C₂×C₄×C₆₇	semidirect
317. C₄×D₄	C₂×C₄×C₆₇	semidirect
318. C₄×D₄	C₂×C₄×C₆₇	semidirect
330. C₂×C₄²	C₂×C₄×C₆₇	semidirect
407. C₄²	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₆₇×#2372	2
C₆₇×#2364	2
C₆₇×#2353	2
C₆₇×#2339	2
C₆₇×#2331	2
C₆₇×#2328	2
C₆₇×#2319	2
C₄×#121×C₆₇	2
C₆₇×#2283	2
C₆₇×#2275	2
C₂×#376×C₆₇	2
#34222	67

Subgroups — C67×#34222 #30713473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34222 #30713473