Subgroups — C₆₇×#21936 #30701187

Order 17152.

1030 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#408×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2297	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1851	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1850	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1798	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1797	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	#408×C₆₇	—	2	8	no	—
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
19	#391×C₆₇	C₄	1	1	yes	characteristic
20	#391×C₆₇	C₄	1	1	yes	characteristic
21	#390×C₆₇	C₄	1	1	yes	characteristic
22	#390×C₆₇	C₄	1	1	yes	characteristic
23	#386×C₆₇	C₄	1	4	yes	—
24	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
25	#336×C₆₇	C₂²	1	4	yes	—
26	#336×C₆₇	C₂²	1	4	yes	—
27	C₂×#41×C₆₇	—	4	1	no	—
28	C₂×#41×C₆₇	—	4	1	no	—
29	C₂×#41×C₆₇	—	4	1	no	—
30	C₂×#41×C₆₇	—	4	1	no	—
31	C₂×#41×C₆₇	D₄	1	4	yes	—
32	C₂×#41×C₆₇	Q₈	1	4	yes	—
33	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
39	C₂×M₁₆×C₆₇	—	2	4	no	—
40	C₂×M₁₆×C₆₇	—	4	2	no	—
41	C₂×M₁₆×C₆₇	—	4	2	no	—
42	C₂²×C₈×C₆₇	—	2	4	no	—
43	C₂²×C₈×C₆₇	—	4	2	no	—
44	C₂²×C₈×C₆₇	—	4	2	no	—
45	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
47	#117×C₆₇	—	4	1	no	—
48	#117×C₆₇	—	4	1	no	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₆₇	—	4	1	no	—
55	C₂×C₄²×C₆₇	—	4	1	no	—
56	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
60	C₄⋊C₈×C₆₇	—	4	2	no	—
61	C₄⋊C₈×C₆₇	—	4	2	no	—
62	C₄⋊C₈×C₆₇	—	4	2	no	—
63	C₄⋊C₈×C₆₇	—	4	2	no	—
64	C₂²⋊C₈×C₆₇	—	2	8	no	—
65	C₂²⋊C₈×C₆₇	—	2	8	no	—
66	#95×C₆₇	C₂×C₄	1	4	yes	—
67	#95×C₆₇	C₂×C₄	1	4	yes	—
68	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	#41×C₆₇	—	4	4	no	—
71	#41×C₆₇	—	4	4	no	—
72	#41×C₆₇	—	4	4	no	—
73	#41×C₆₇	—	4	4	no	—
74	C₂×Q₈×C₆₇	—	4	1	no	—
75	C₂×Q₈×C₆₇	—	4	1	no	—
76	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
77	C₂×D₄×C₆₇	—	4	1	no	—
78	C₂×D₄×C₆₇	—	4	1	no	—
79	C₂×D₄×C₆₇	—	4	2	no	—
80	C₂×D₄×C₆₇	—	4	2	no	—
81	C₂×D₄×C₆₇	C₄⋊C₄	1	8	yes	—
82	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
83	C₂²×C₄×C₆₇	—	2	1	no	—
84	C₂²×C₄×C₆₇	—	2	1	no	—
85	C₂²×C₄×C₆₇	—	2	1	no	—
86	C₂²×C₄×C₆₇	—	2	1	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	4	no	—
92	C₂²×C₄×C₆₇	—	4	2	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₆₇	—	4	8	no	—
106	C₂×C₈×C₆₇	—	2	4	no	—
107	C₂×C₈×C₆₇	—	2	4	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	2	4	no	—
110	C₂×C₈×C₆₇	—	2	4	no	—
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	4	8	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₂²⋊C₄×C₆₇	—	2	8	no	—
116	C₂²⋊C₄×C₆₇	—	4	4	no	—
117	C₂²⋊C₄×C₆₇	—	4	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	4	1	no	—
123	C₄²×C₆₇	—	4	1	no	—
124	C₂³×C₆₇	—	2	1	no	—
125	C₂³×C₆₇	—	2	1	no	—
126	C₂³×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	4	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	2	4	no	—
131	C₂³×C₆₇	—	2	2	no	—
132	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
133	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	Q₈×C₆₇	—	4	4	no	—
136	D₄×C₆₇	—	4	4	no	—
137	D₄×C₆₇	—	4	8	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	4	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	4	8	no	—
160	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
162	C₂×C₄×C₆₇	#131	1	4	yes	—
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
169	C₈×C₆₇	—	4	4	no	—
170	C₈×C₆₇	—	4	4	no	—
171	C₂²×C₆₇	—	2	4	no	—
172	C₂²×C₆₇	—	2	2	no	—
173	C₂²×C₆₇	—	2	8	no	—
174	C₂²×C₆₇	—	2	4	no	—
175	C₂²×C₆₇	—	4	2	no	—
176	C₂²×C₆₇	—	4	4	no	—
177	C₂²×C₆₇	—	4	1	no	—
178	C₂²×C₆₇	—	4	2	no	—
179	C₂²×C₆₇	—	4	1	no	—
180	C₂²×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
182	C₂²×C₆₇	#518	1	1	yes	characteristic
183	C₂²×C₆₇	#518	1	1	yes	characteristic
184	C₂²×C₆₇	C₂×#131	1	2	yes	—
185	C₂²×C₆₇	#568	1	2	yes	—
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	2	4	no	—
188	C₄×C₆₇	—	2	4	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	—	4	2	no	—
192	C₄×C₆₇	#534	1	1	yes	characteristic
193	C₄×C₆₇	#534	1	1	yes	characteristic
194	C₄×C₆₇	#543	1	1	yes	characteristic
195	C₄×C₆₇	#543	1	1	yes	characteristic
196	#21936	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂×C₆₇	—	4	2	no	—
198	C₂×C₆₇	—	4	4	no	—
199	C₂×C₆₇	#2315	1	1	yes	characteristic
200	C₂×C₆₇	#2906	1	2	yes	—
201	C₂×C₆₇	#2976	1	2	yes	—
202	C₂×C₆₇	#2979	1	1	yes	characteristic
203	C₂×C₆₇	#2980	1	1	yes	characteristic
204	C₂×#408	C₂×C₆₇	1	2	yes	—
205	#2297	C₂×C₆₇	1	1	yes	characteristic
206	#1851	C₂×C₆₇	1	4	yes	complemented
207	#1850	C₂×C₆₇	1	4	yes	complemented
208	#1798	C₂×C₆₇	1	2	yes	complemented
209	#1797	C₂×C₆₇	1	2	yes	complemented
210	C₆₇	#21936	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
212	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
213	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
214	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
215	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
217	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
218	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
219	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
220	#408	—	2	8	no	—
221	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
222	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
223	#391	C₄×C₆₇	1	1	yes	characteristic
224	#391	C₄×C₆₇	1	1	yes	characteristic
225	#390	C₄×C₆₇	1	1	yes	characteristic
226	#390	C₄×C₆₇	1	1	yes	characteristic
227	#386	C₄×C₆₇	1	4	yes	—
228	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
229	#336	C₂²×C₆₇	1	4	yes	—
230	#336	C₂²×C₆₇	1	4	yes	—
231	C₂×#41	—	4	1	no	—
232	C₂×#41	—	4	1	no	—
233	C₂×#41	—	4	1	no	—
234	C₂×#41	—	4	1	no	—
235	C₂×#41	D₄×C₆₇	1	4	yes	—
236	C₂×#41	Q₈×C₆₇	1	4	yes	—
237	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
238	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
239	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
240	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
241	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
242	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
243	C₂×M₁₆	—	2	4	no	—
244	C₂×M₁₆	—	4	2	no	—
245	C₂×M₁₆	—	4	2	no	—
246	C₂²×C₈	—	2	4	no	—
247	C₂²×C₈	—	4	2	no	—
248	C₂²×C₈	—	4	2	no	—
249	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
250	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
251	#117	—	4	1	no	—
252	#117	—	4	1	no	—
253	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
255	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
256	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
257	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
258	C₂×C₄²	—	4	1	no	—
259	C₂×C₄²	—	4	1	no	—
260	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
261	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
263	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
264	C₄⋊C₈	—	4	2	no	—
265	C₄⋊C₈	—	4	2	no	—
266	C₄⋊C₈	—	4	2	no	—
267	C₄⋊C₈	—	4	2	no	—
268	C₂²⋊C₈	—	2	8	no	—
269	C₂²⋊C₈	—	2	8	no	—
270	#95	C₂×C₄×C₆₇	1	4	yes	—
271	#95	C₂×C₄×C₆₇	1	4	yes	—
272	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
273	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
274	#41	—	4	4	no	—
275	#41	—	4	4	no	—
276	#41	—	4	4	no	—
277	#41	—	4	4	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	—	4	1	no	—
280	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
281	C₂×D₄	—	4	1	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	4	2	no	—
285	C₂×D₄	C₄⋊C₄×C₆₇	1	8	yes	—
286	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
299	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
300	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
301	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
302	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
303	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
306	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
308	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
309	M₁₆	—	4	8	no	—
310	C₂×C₈	—	2	4	no	—
311	C₂×C₈	—	2	4	no	—
312	C₂×C₈	—	2	4	no	—
313	C₂×C₈	—	2	4	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	2	4	no	—
316	C₂×C₈	—	4	8	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₂²⋊C₄	—	2	8	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	—	4	4	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	4	1	no	—
327	C₄²	—	4	1	no	—
328	C₂³	—	2	1	no	—
329	C₂³	—	2	1	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	2	2	no	—
332	C₂³	—	2	4	no	—
333	C₂³	—	2	2	no	—
334	C₂³	—	2	4	no	—
335	C₂³	—	2	2	no	—
336	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
337	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
338	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
339	Q₈	—	4	4	no	—
340	D₄	—	4	4	no	—
341	D₄	—	4	8	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	4	8	no	—
364	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
365	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
366	C₂×C₄	#131×C₆₇	1	4	yes	—
367	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
369	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
370	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
371	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
372	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
373	C₈	—	4	4	no	—
374	C₈	—	4	4	no	—
375	C₂²	—	2	4	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	8	no	—
378	C₂²	—	2	4	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	4	no	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	2	no	—
385	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
386	C₂²	#518×C₆₇	1	1	yes	characteristic
387	C₂²	#518×C₆₇	1	1	yes	characteristic
388	C₂²	C₂×#131×C₆₇	1	2	yes	—
389	C₂²	#568×C₆₇	1	2	yes	—
390	C₄	—	2	2	no	—
391	C₄	—	2	4	no	—
392	C₄	—	2	4	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	#534×C₆₇	1	1	yes	characteristic
397	C₄	#534×C₆₇	1	1	yes	characteristic
398	C₄	#543×C₆₇	1	1	yes	characteristic
399	C₄	#543×C₆₇	1	1	yes	characteristic
400	C₂	—	4	2	no	—
401	C₂	—	4	4	no	—
402	C₂	C₆₇×#2315	1	1	yes	characteristic
403	C₂	C₆₇×#2906	1	2	yes	—
404	C₂	C₆₇×#2976	1	2	yes	—
405	C₂	C₆₇×#2979	1	1	yes	characteristic
406	C₂	C₆₇×#2980	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1851	C₂	semidirect
4. C₆₇×#1850	C₂	semidirect
5. C₆₇×#1798	C₂	semidirect
6. C₆₇×#1797	C₂	semidirect
196. #21936	C₆₇	direct (N × H)
206. #1851	C₂×C₆₇	semidirect
207. #1850	C₂×C₆₇	semidirect
208. #1798	C₂×C₆₇	semidirect
209. #1797	C₂×C₆₇	semidirect
210. C₆₇	#21936	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#408×C₆₇	2
C₆₇×#2297	2
C₆₇×#1851	2
C₆₇×#1850	2
C₆₇×#1798	2
C₆₇×#1797	2
#21936	67

Subgroups — C67×#21936 #30701187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21936 #30701187