Subgroups — C₆₇×#2470 #22556901

Order 8576.

532 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#400×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#397×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#396×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#394×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#389×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#389×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#388×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#388×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#387×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#386×C₆₇	C₂	1	1	yes	maximal, characteristic
13	#385×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#385×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
19	#123×C₆₇	—	2	4	no	—
20	#117×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
43	#95×C₆₇	C₂²	1	1	yes	characteristic
44	#95×C₆₇	C₂²	1	1	yes	characteristic
45	#95×C₆₇	C₂²	1	1	yes	characteristic
46	#95×C₆₇	C₂²	1	1	yes	characteristic
47	#95×C₆₇	C₂²	1	1	yes	characteristic
48	#95×C₆₇	C₂²	1	1	yes	characteristic
49	#95×C₆₇	C₂²	1	1	yes	characteristic
50	#95×C₆₇	C₂²	1	1	yes	characteristic
51	#95×C₆₇	C₂²	1	1	yes	characteristic
52	#95×C₆₇	C₂²	1	1	yes	characteristic
53	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
54	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×D₄×C₆₇	—	2	2	no	—
56	C₂×D₄×C₆₇	—	2	2	no	—
57	C₂×D₄×C₆₇	—	4	1	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	4	1	no	—
64	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	4	1	no	—
85	C₄⋊C₄×C₆₇	—	4	1	no	—
86	C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂²⋊C₄×C₆₇	—	2	2	no	—
91	C₂²⋊C₄×C₆₇	—	2	2	no	—
92	C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂²⋊C₄×C₆₇	—	4	1	no	—
95	C₂²⋊C₄×C₆₇	D₄	1	4	yes	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	4	1	no	—
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	—	2	1	no	—
102	C₂³×C₆₇	—	2	1	no	—
103	C₂³×C₆₇	—	2	1	no	—
104	C₂³×C₆₇	—	2	1	no	—
105	C₂³×C₆₇	—	2	1	no	—
106	C₂³×C₆₇	—	2	1	no	—
107	C₂³×C₆₇	—	2	1	no	—
108	C₂³×C₆₇	—	2	1	no	—
109	C₂³×C₆₇	—	2	1	no	—
110	C₂³×C₆₇	—	2	1	no	—
111	C₂³×C₆₇	C₂×D₄	1	2	yes	—
112	C₂³×C₆₇	#41	1	2	yes	—
113	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
114	D₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	2	8	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	4	4	no	—
151	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂×C₄×C₆₇	#41	1	2	yes	—
154	C₂×C₄×C₆₇	#41	1	2	yes	—
155	C₂×C₄×C₆₇	#41	1	2	yes	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	—	4	1	no	—
163	C₂²×C₆₇	—	4	1	no	—
164	C₂²×C₆₇	—	4	1	no	—
165	C₂²×C₆₇	—	4	1	no	—
166	C₂²×C₆₇	—	4	1	no	—
167	C₂²×C₆₇	—	4	1	no	—
168	C₂²×C₆₇	—	4	1	no	—
169	C₂²×C₆₇	—	4	1	no	—
170	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
172	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
173	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
174	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
175	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	2	no	—
180	C₄×C₆₇	—	2	2	no	—
181	C₄×C₆₇	—	2	2	no	—
182	C₄×C₆₇	—	4	1	no	—
183	C₄×C₆₇	—	4	1	no	—
184	C₄×C₆₇	—	4	1	no	—
185	C₄×C₆₇	—	4	1	no	—
186	C₄×C₆₇	—	4	1	no	—
187	C₄×C₆₇	—	4	1	no	—
188	C₄×C₆₇	—	4	1	no	—
189	C₄×C₆₇	—	4	1	no	—
190	C₂×C₆₇	—	4	2	no	—
191	C₂×C₆₇	—	4	2	no	—
192	C₂×C₆₇	#525	1	1	yes	characteristic
193	C₂×C₆₇	#529	1	1	yes	characteristic
194	C₂×C₆₇	#542	1	1	yes	characteristic
195	C₂×C₆₇	#546	1	1	yes	characteristic
196	C₂×C₆₇	#547	1	1	yes	characteristic
197	C₂×C₆₇	#552	1	1	yes	characteristic
198	C₂×C₆₇	#553	1	1	yes	characteristic
199	#2470	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₆₇	#2470	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×#123	C₂×C₆₇	1	1	yes	characteristic
202	C₂×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#400	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#397	C₂×C₆₇	1	1	yes	characteristic, complemented
205	#396	C₂×C₆₇	1	1	yes	characteristic, complemented
206	#394	C₂×C₆₇	1	1	yes	characteristic
207	#389	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#389	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#388	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#388	C₂×C₆₇	1	1	yes	characteristic, complemented
211	#387	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#386	C₂×C₆₇	1	1	yes	characteristic
213	#385	C₂×C₆₇	1	1	yes	characteristic, complemented
214	#385	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
216	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
217	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
218	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
219	#123	—	2	4	no	—
220	#117	—	2	4	no	—
221	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
230	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
243	#95	C₂²×C₆₇	1	1	yes	characteristic
244	#95	C₂²×C₆₇	1	1	yes	characteristic
245	#95	C₂²×C₆₇	1	1	yes	characteristic
246	#95	C₂²×C₆₇	1	1	yes	characteristic
247	#95	C₂²×C₆₇	1	1	yes	characteristic
248	#95	C₂²×C₆₇	1	1	yes	characteristic
249	#95	C₂²×C₆₇	1	1	yes	characteristic
250	#95	C₂²×C₆₇	1	1	yes	characteristic
251	#95	C₂²×C₆₇	1	1	yes	characteristic
252	#95	C₂²×C₆₇	1	1	yes	characteristic
253	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
254	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
267	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
272	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
273	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₂²⋊C₄	—	2	2	no	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	D₄×C₆₇	1	4	yes	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	4	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	2	1	no	—
302	C₂³	—	2	1	no	—
303	C₂³	—	2	1	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	2	1	no	—
309	C₂³	—	2	1	no	—
310	C₂³	—	2	1	no	—
311	C₂³	C₂×D₄×C₆₇	1	2	yes	—
312	C₂³	#41×C₆₇	1	2	yes	—
313	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
314	D₄	—	4	4	no	—
315	C₂×C₄	—	2	8	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
352	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
353	C₂×C₄	#41×C₆₇	1	2	yes	—
354	C₂×C₄	#41×C₆₇	1	2	yes	—
355	C₂×C₄	#41×C₆₇	1	2	yes	—
356	C₂²	—	2	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	2	no	—
362	C₂²	—	4	1	no	—
363	C₂²	—	4	1	no	—
364	C₂²	—	4	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	1	no	—
370	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
371	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
372	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
373	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
374	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
375	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
376	C₂²	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
377	C₄	—	2	2	no	—
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	2	2	no	—
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	4	1	no	—
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	4	1	no	—
390	C₂	—	4	2	no	—
391	C₂	—	4	2	no	—
392	C₂	#525×C₆₇	1	1	yes	characteristic
393	C₂	#529×C₆₇	1	1	yes	characteristic
394	C₂	#542×C₆₇	1	1	yes	characteristic
395	C₂	#546×C₆₇	1	1	yes	characteristic
396	C₂	#547×C₆₇	1	1	yes	characteristic
397	C₂	#552×C₆₇	1	1	yes	characteristic
398	C₂	#553×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#117×C₆₇	C₂	semidirect
3. #400×C₆₇	C₂	semidirect
4. #397×C₆₇	C₂	semidirect
5. #396×C₆₇	C₂	semidirect
7. #389×C₆₇	C₂	semidirect
8. #389×C₆₇	C₂	semidirect
9. #388×C₆₇	C₂	semidirect
10. #388×C₆₇	C₂	semidirect
11. #387×C₆₇	C₂	semidirect
13. #385×C₆₇	C₂	semidirect
14. #385×C₆₇	C₂	semidirect
15. C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
199. #2470	C₆₇	direct (N × H)
200. C₆₇	#2470	direct (N × H)
202. C₂×#117	C₂×C₆₇	semidirect
203. #400	C₂×C₆₇	semidirect
204. #397	C₂×C₆₇	semidirect
205. #396	C₂×C₆₇	semidirect
207. #389	C₂×C₆₇	semidirect
208. #389	C₂×C₆₇	semidirect
209. #388	C₂×C₆₇	semidirect
210. #388	C₂×C₆₇	semidirect
211. #387	C₂×C₆₇	semidirect
213. #385	C₂×C₆₇	semidirect
214. #385	C₂×C₆₇	semidirect
215. C₄×C₂²⋊C₄	C₂×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₆₇	2
C₂×#117×C₆₇	2
#400×C₆₇	2
#397×C₆₇	2
#396×C₆₇	2
#394×C₆₇	2
#389×C₆₇	2
#388×C₆₇	2
#387×C₆₇	2
#386×C₆₇	2
#385×C₆₇	2
C₄×C₂²⋊C₄×C₆₇	2
#2470	67

Subgroups — C67×#2470 #22556901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2470 #22556901