Subgroups — C₆₇×#34388 #30713639

Order 17152.

2438 conjugacy classes of subgroups.

The subgroup lattice has 450 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×#2381	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2380	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2364	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2353	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2339	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2337	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2325	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2318	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#383×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	2	yes	—
13	#532×C₆₇	C₄	1	16	yes	complemented
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	2	yes	—
17	C₂×#121×C₆₇	C₂²	1	2	yes	—
18	C₂×#121×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#391×C₆₇	C₂²	1	8	yes	—
24	#388×C₆₇	C₂²	1	8	yes	—
25	#388×C₆₇	C₂²	1	8	yes	—
26	#387×C₆₇	C₂²	1	8	yes	—
27	#387×C₆₇	C₂²	1	8	yes	—
28	#386×C₆₇	C₂²	1	8	yes	—
29	#386×C₆₇	C₂²	1	8	yes	—
30	#386×C₆₇	C₂²	1	8	yes	—
31	#386×C₆₇	C₂²	1	8	yes	—
32	#385×C₆₇	C₂²	1	8	yes	—
33	#383×C₆₇	C₂²	1	2	yes	—
34	#383×C₆₇	C₂²	1	2	yes	—
35	#383×C₆₇	C₂²	1	2	yes	—
36	#383×C₆₇	C₂²	1	2	yes	—
37	#383×C₆₇	C₂²	1	2	yes	—
38	#382×C₆₇	C₂²	1	2	yes	—
39	#381×C₆₇	C₂²	1	2	yes	—
40	#381×C₆₇	C₂²	1	4	yes	—
41	#380×C₆₇	C₂²	1	2	yes	—
42	#380×C₆₇	C₂²	1	4	yes	—
43	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
44	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
45	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
46	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
47	#376×C₆₇	C₂²	1	2	yes	—
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂×#95×C₆₇	C₂²	1	2	yes	—
50	C₂×#95×C₆₇	C₂²	1	2	yes	—
51	C₂×#95×C₆₇	C₂²	1	4	yes	—
52	C₂×#41×C₆₇	—	4	8	no	—
53	C₂×#41×C₆₇	C₂×C₄	1	16	yes	complemented
54	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
56	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
58	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
65	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
66	#127×C₆₇	C₂×C₄	1	8	yes	complemented
67	#124×C₆₇	C₂×C₄	1	16	yes	complemented
68	#121×C₆₇	C₂×C₄	1	16	yes	complemented
69	#121×C₆₇	C₂×C₄	1	16	yes	complemented
70	C₄×D₄×C₆₇	—	4	16	no	—
71	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
72	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
83	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
84	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
85	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
86	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	#95×C₆₇	C₂³	1	4	yes	—
91	#95×C₆₇	C₂³	1	4	yes	—
92	#95×C₆₇	C₂³	1	4	yes	—
93	#95×C₆₇	C₂³	1	4	yes	—
94	#95×C₆₇	C₂³	1	4	yes	—
95	#95×C₆₇	C₂³	1	4	yes	—
96	#95×C₆₇	C₂³	1	4	yes	—
97	#95×C₆₇	C₂³	1	8	yes	—
98	#95×C₆₇	C₂³	1	4	yes	—
99	#95×C₆₇	C₂³	1	8	yes	—
100	#95×C₆₇	C₂³	1	4	yes	—
101	#95×C₆₇	C₂³	1	4	yes	—
102	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
103	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
104	#41×C₆₇	—	4	32	no	—
105	C₂×Q₈×C₆₇	—	4	4	no	—
106	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
107	C₂×D₄×C₆₇	—	2	16	no	—
108	C₂×D₄×C₆₇	—	4	2	no	—
109	C₂×D₄×C₆₇	—	4	4	no	—
110	C₂×D₄×C₆₇	—	4	2	no	—
111	C₂×D₄×C₆₇	—	4	4	no	—
112	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
113	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
114	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
115	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	8	no	—
118	C₂²×C₄×C₆₇	—	4	2	no	—
119	C₂²×C₄×C₆₇	—	4	2	no	—
120	C₂²×C₄×C₆₇	—	4	8	no	—
121	C₂²×C₄×C₆₇	—	4	8	no	—
122	C₂²×C₄×C₆₇	—	4	8	no	—
123	C₂²×C₄×C₆₇	—	4	8	no	—
124	C₂²×C₄×C₆₇	—	4	4	no	—
125	C₂²×C₄×C₆₇	—	4	4	no	—
126	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
131	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
139	C₄⋊C₄×C₆₇	—	4	4	no	—
140	C₄⋊C₄×C₆₇	—	4	8	no	—
141	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
142	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
143	C₂²⋊C₄×C₆₇	—	2	8	no	—
144	C₂²⋊C₄×C₆₇	—	4	8	no	—
145	C₂²⋊C₄×C₆₇	—	4	8	no	—
146	C₂²⋊C₄×C₆₇	—	4	8	no	—
147	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
148	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
149	C₄²×C₆₇	—	4	4	no	—
150	C₄²×C₆₇	—	4	8	no	—
151	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
152	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
153	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
154	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	2	8	no	—
158	C₂³×C₆₇	—	2	4	no	—
159	C₂³×C₆₇	—	2	8	no	—
160	C₂³×C₆₇	—	2	4	no	—
161	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
162	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
163	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
164	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
165	Q₈×C₆₇	—	4	8	no	—
166	D₄×C₆₇	—	2	32	no	—
167	D₄×C₆₇	—	4	8	no	—
168	D₄×C₆₇	—	4	8	no	—
169	D₄×C₆₇	—	4	8	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	8	no	—
179	C₂×C₄×C₆₇	—	2	8	no	—
180	C₂×C₄×C₆₇	—	2	8	no	—
181	C₂×C₄×C₆₇	—	2	8	no	—
182	C₂×C₄×C₆₇	—	2	8	no	—
183	C₂×C₄×C₆₇	—	2	8	no	—
184	C₂×C₄×C₆₇	—	4	4	no	—
185	C₂×C₄×C₆₇	—	4	4	no	—
186	C₂×C₄×C₆₇	—	4	16	no	—
187	C₂×C₄×C₆₇	—	8	8	no	—
188	C₂×C₄×C₆₇	—	8	16	no	—
189	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
192	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
193	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
196	C₂²×C₆₇	—	2	2	no	—
197	C₂²×C₆₇	—	2	2	no	—
198	C₂²×C₆₇	—	2	8	no	—
199	C₂²×C₆₇	—	2	8	no	—
200	C₂²×C₆₇	—	2	8	no	—
201	C₂²×C₆₇	—	2	8	no	—
202	C₂²×C₆₇	—	4	8	no	—
203	C₂²×C₆₇	—	4	8	no	—
204	C₂²×C₆₇	#518	1	4	yes	—
205	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
206	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
207	C₂²×C₆₇	#585	1	2	yes	—
208	C₂²×C₆₇	#585	1	2	yes	—
209	C₄×C₆₇	—	2	2	no	—
210	C₄×C₆₇	—	2	2	no	—
211	C₄×C₆₇	—	2	8	no	—
212	C₄×C₆₇	—	2	2	no	—
213	C₄×C₆₇	—	2	2	no	—
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	4	8	no	—
217	C₄×C₆₇	—	4	8	no	—
218	#34388	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂×C₆₇	—	2	4	no	—
220	C₂×C₆₇	—	4	8	no	—
221	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
222	C₂×C₆₇	#3428	1	1	yes	characteristic
223	C₂×C₆₇	#3428	1	1	yes	characteristic
224	C₂×C₆₇	#3560	1	4	yes	—
225	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
226	#2381	C₂×C₆₇	1	2	yes	complemented
227	#2380	C₂×C₆₇	1	2	yes	complemented
228	#2364	C₂×C₆₇	1	4	yes	complemented
229	#2353	C₂×C₆₇	1	4	yes	complemented
230	#2339	C₂×C₆₇	1	4	yes	complemented
231	#2337	C₂×C₆₇	1	4	yes	complemented
232	#2325	C₂×C₆₇	1	4	yes	complemented
233	#2318	C₂×C₆₇	1	4	yes	complemented
234	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
235	C₂×#383	C₂×C₆₇	1	1	yes	characteristic, complemented
236	C₆₇	#34388	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²×#41	C₂²×C₆₇	1	2	yes	—
238	#532	C₄×C₆₇	1	16	yes	complemented
239	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×#124	C₂²×C₆₇	1	2	yes	—
242	C₂×#121	C₂²×C₆₇	1	2	yes	—
243	C₂×#121	C₂²×C₆₇	1	2	yes	—
244	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
245	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
246	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
247	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
248	#391	C₂²×C₆₇	1	8	yes	—
249	#388	C₂²×C₆₇	1	8	yes	—
250	#388	C₂²×C₆₇	1	8	yes	—
251	#387	C₂²×C₆₇	1	8	yes	—
252	#387	C₂²×C₆₇	1	8	yes	—
253	#386	C₂²×C₆₇	1	8	yes	—
254	#386	C₂²×C₆₇	1	8	yes	—
255	#386	C₂²×C₆₇	1	8	yes	—
256	#386	C₂²×C₆₇	1	8	yes	—
257	#385	C₂²×C₆₇	1	8	yes	—
258	#383	C₂²×C₆₇	1	2	yes	—
259	#383	C₂²×C₆₇	1	2	yes	—
260	#383	C₂²×C₆₇	1	2	yes	—
261	#383	C₂²×C₆₇	1	2	yes	—
262	#383	C₂²×C₆₇	1	2	yes	—
263	#382	C₂²×C₆₇	1	2	yes	—
264	#381	C₂²×C₆₇	1	4	yes	—
265	#381	C₂²×C₆₇	1	2	yes	—
266	#380	C₂²×C₆₇	1	4	yes	—
267	#380	C₂²×C₆₇	1	2	yes	—
268	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
269	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
270	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
271	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
272	#376	C₂²×C₆₇	1	2	yes	—
273	C₂×#95	C₂²×C₆₇	1	4	yes	—
274	C₂×#95	C₂²×C₆₇	1	2	yes	—
275	C₂×#95	C₂²×C₆₇	1	2	yes	—
276	C₂×#95	C₂²×C₆₇	1	2	yes	—
277	C₂×#41	—	4	8	no	—
278	C₂×#41	C₂×C₄×C₆₇	1	16	yes	complemented
279	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
280	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
281	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
282	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
283	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
284	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
286	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
287	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
288	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
290	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
291	#127	C₂×C₄×C₆₇	1	8	yes	complemented
292	#124	C₂×C₄×C₆₇	1	16	yes	complemented
293	#121	C₂×C₄×C₆₇	1	16	yes	complemented
294	#121	C₂×C₄×C₆₇	1	16	yes	complemented
295	C₄×D₄	—	4	16	no	—
296	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
297	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
302	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
303	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
304	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
305	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
306	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
307	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
308	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
309	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
310	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
311	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
315	#95	C₂³×C₆₇	1	4	yes	—
316	#95	C₂³×C₆₇	1	4	yes	—
317	#95	C₂³×C₆₇	1	4	yes	—
318	#95	C₂³×C₆₇	1	4	yes	—
319	#95	C₂³×C₆₇	1	4	yes	—
320	#95	C₂³×C₆₇	1	4	yes	—
321	#95	C₂³×C₆₇	1	4	yes	—
322	#95	C₂³×C₆₇	1	8	yes	—
323	#95	C₂³×C₆₇	1	4	yes	—
324	#95	C₂³×C₆₇	1	8	yes	—
325	#95	C₂³×C₆₇	1	4	yes	—
326	#95	C₂³×C₆₇	1	4	yes	—
327	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
328	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
329	#41	—	4	32	no	—
330	C₂×Q₈	—	4	4	no	—
331	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
332	C₂×D₄	—	2	16	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	—	4	2	no	—
335	C₂×D₄	—	4	4	no	—
336	C₂×D₄	—	4	2	no	—
337	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
338	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
339	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
340	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	8	no	—
343	C₂²×C₄	—	4	8	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	—	4	2	no	—
346	C₂²×C₄	—	4	8	no	—
347	C₂²×C₄	—	4	8	no	—
348	C₂²×C₄	—	4	8	no	—
349	C₂²×C₄	—	4	4	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
352	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
353	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
354	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
355	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
356	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
357	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
363	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
364	C₄⋊C₄	—	4	8	no	—
365	C₄⋊C₄	—	4	4	no	—
366	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
367	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
368	C₂²⋊C₄	—	2	8	no	—
369	C₂²⋊C₄	—	4	8	no	—
370	C₂²⋊C₄	—	4	8	no	—
371	C₂²⋊C₄	—	4	8	no	—
372	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
373	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
374	C₄²	—	4	8	no	—
375	C₄²	—	4	4	no	—
376	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
377	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
378	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
379	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	8	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	2	8	no	—
385	C₂³	—	2	4	no	—
386	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
387	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
388	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
389	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
390	Q₈	—	4	8	no	—
391	D₄	—	2	32	no	—
392	D₄	—	4	8	no	—
393	D₄	—	4	8	no	—
394	D₄	—	4	8	no	—
395	C₂×C₄	—	2	8	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	2	8	no	—
406	C₂×C₄	—	2	8	no	—
407	C₂×C₄	—	2	8	no	—
408	C₂×C₄	—	2	8	no	—
409	C₂×C₄	—	4	16	no	—
410	C₂×C₄	—	4	4	no	—
411	C₂×C₄	—	4	4	no	—
412	C₂×C₄	—	8	8	no	—
413	C₂×C₄	—	8	16	no	—
414	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
415	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
416	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
417	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
418	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
419	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
420	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
421	C₂²	—	2	8	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	8	no	—
425	C₂²	—	2	8	no	—
426	C₂²	—	2	8	no	—
427	C₂²	—	4	8	no	—
428	C₂²	—	4	8	no	—
429	C₂²	#518×C₆₇	1	4	yes	—
430	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
431	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
432	C₂²	#585×C₆₇	1	2	yes	—
433	C₂²	#585×C₆₇	1	2	yes	—
434	C₄	—	2	2	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	8	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	4	8	no	—
442	C₄	—	4	8	no	—
443	C₂	—	2	4	no	—
444	C₂	—	4	8	no	—
445	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
446	C₂	C₆₇×#3428	1	1	yes	characteristic
447	C₂	C₆₇×#3428	1	1	yes	characteristic
448	C₂	C₆₇×#3560	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×#2381	C₂	semidirect
3. C₆₇×#2380	C₂	semidirect
4. C₆₇×#2364	C₂	semidirect
5. C₆₇×#2353	C₂	semidirect
6. C₆₇×#2339	C₂	semidirect
7. C₆₇×#2337	C₂	semidirect
8. C₆₇×#2325	C₂	semidirect
9. C₆₇×#2318	C₂	semidirect
10. C₆₇×#2283	C₂	semidirect
11. C₂×#383×C₆₇	C₂	semidirect
13. #532×C₆₇	C₄	semidirect
53. C₂×#41×C₆₇	C₂×C₄	semidirect
65. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
66. #127×C₆₇	C₂×C₄	semidirect
67. #124×C₆₇	C₂×C₄	semidirect
68. #121×C₆₇	C₂×C₄	semidirect
69. #121×C₆₇	C₂×C₄	semidirect
71. C₄×D₄×C₆₇	C₂×C₄	semidirect
72. C₄×D₄×C₆₇	C₂×C₄	semidirect
83. C₂×C₄²×C₆₇	C₂×C₄	semidirect
218. #34388	C₆₇	direct (N × H)
226. #2381	C₂×C₆₇	semidirect
227. #2380	C₂×C₆₇	semidirect
228. #2364	C₂×C₆₇	semidirect
229. #2353	C₂×C₆₇	semidirect
230. #2339	C₂×C₆₇	semidirect
231. #2337	C₂×C₆₇	semidirect
232. #2325	C₂×C₆₇	semidirect
233. #2318	C₂×C₆₇	semidirect
234. #2283	C₂×C₆₇	semidirect
235. C₂×#383	C₂×C₆₇	semidirect
236. C₆₇	#34388	direct (N × H)
238. #532	C₄×C₆₇	semidirect
278. C₂×#41	C₂×C₄×C₆₇	semidirect
290. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
291. #127	C₂×C₄×C₆₇	semidirect
292. #124	C₂×C₄×C₆₇	semidirect
293. #121	C₂×C₄×C₆₇	semidirect
294. #121	C₂×C₄×C₆₇	semidirect
296. C₄×D₄	C₂×C₄×C₆₇	semidirect
297. C₄×D₄	C₂×C₄×C₆₇	semidirect
308. 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×#532×C₆₇	2
C₆₇×#2381	2
C₆₇×#2380	2
C₆₇×#2364	2
C₆₇×#2353	2
C₆₇×#2339	2
C₆₇×#2337	2
C₆₇×#2325	2
C₆₇×#2318	2
C₆₇×#2283	2
C₂×#383×C₆₇	2
#34388	67

Subgroups — C67×#34388 #30713639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34388 #30713639