Subgroups — C₆₇×#21902 #30701153

Order 17152.

1122 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1923	C₂	1	8	yes	maximal, complemented
3	C₆₇×#1797	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1779	C₂	1	2	yes	maximal
5	C₄×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
12	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	complemented
16	C₄×C₂²⋊C₄×C₆₇	C₄	1	4	yes	—
17	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
18	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
20	#336×C₆₇	C₂²	1	4	yes	complemented
21	#336×C₆₇	C₂²	1	4	yes	complemented
22	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	—	2	4	no	—
24	C₂³×C₄×C₆₇	—	4	2	no	—
25	C₂³×C₄×C₆₇	—	4	2	no	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
35	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₆₇	C₈	1	8	yes	complemented
37	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₆₇	—	4	2	no	—
40	C₂×C₄²×C₆₇	—	4	2	no	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₄⋊C₈×C₆₇	—	4	4	no	—
46	C₂²⋊C₈×C₆₇	—	2	8	no	—
47	C₂²⋊C₈×C₆₇	—	4	4	no	—
48	C₂²⋊C₈×C₆₇	—	4	4	no	—
49	C₄×C₈×C₆₇	—	4	4	no	—
50	#95×C₆₇	C₂×C₄	1	4	yes	—
51	#95×C₆₇	C₂×C₄	1	4	yes	—
52	C₂⁴×C₆₇	—	2	2	no	—
53	C₂⁴×C₆₇	—	2	2	no	—
54	C₂⁴×C₆₇	—	4	1	no	—
55	C₂⁴×C₆₇	—	4	1	no	—
56	C₂⁴×C₆₇	—	4	1	no	—
57	C₂⁴×C₆₇	—	4	1	no	—
58	C₂⁴×C₆₇	C₂×C₈	1	4	yes	—
59	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	4	1	no	—
70	C₂²×C₄×C₆₇	—	4	1	no	—
71	C₂²×C₄×C₆₇	—	4	1	no	—
72	C₂²×C₄×C₆₇	—	4	1	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	4	1	no	—
76	C₂²×C₄×C₆₇	—	4	1	no	—
77	C₂²×C₄×C₆₇	—	4	1	no	—
78	C₂²×C₄×C₆₇	—	4	1	no	—
79	C₂²×C₄×C₆₇	—	4	1	no	—
80	C₂²×C₄×C₆₇	—	4	1	no	—
81	C₂²×C₄×C₆₇	—	4	16	no	—
82	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
83	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
84	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₆₇	#41	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₆₇	—	2	8	no	—
94	C₂×C₈×C₆₇	—	2	8	no	—
95	C₂×C₈×C₆₇	—	2	8	no	—
96	C₂²⋊C₄×C₆₇	—	2	8	no	—
97	C₂²⋊C₄×C₆₇	C₂×C₈	1	16	yes	—
98	C₄²×C₆₇	—	2	4	no	—
99	C₄²×C₆₇	—	2	4	no	—
100	C₄²×C₆₇	—	4	2	no	—
101	C₄²×C₆₇	—	4	2	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	2	no	—
107	C₂³×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	—	2	2	no	—
109	C₂³×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	—	2	2	no	—
111	C₂³×C₆₇	—	2	2	no	—
112	C₂³×C₆₇	—	2	8	no	—
113	C₂³×C₆₇	—	4	4	no	—
114	C₂³×C₆₇	—	4	4	no	—
115	C₂³×C₆₇	—	4	4	no	—
116	C₂³×C₆₇	—	4	4	no	—
117	C₂³×C₆₇	—	4	4	no	—
118	C₂³×C₆₇	—	4	4	no	—
119	C₂³×C₆₇	#117	1	2	yes	—
120	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
121	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
122	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
123	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	8	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	—	4	4	no	—
142	C₂×C₄×C₆₇	—	4	8	no	—
143	C₂×C₄×C₆₇	—	4	4	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	4	8	no	—
146	C₂×C₄×C₆₇	#117	1	2	yes	—
147	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
148	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
149	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
151	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
152	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
153	C₈×C₆₇	—	4	8	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	—	2	2	no	—
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	4	no	—
161	C₂²×C₆₇	—	2	4	no	—
162	C₂²×C₆₇	—	2	4	no	—
163	C₂²×C₆₇	—	4	16	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	#431	1	4	yes	—
169	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
170	C₂²×C₆₇	#518	1	1	yes	characteristic
171	C₂²×C₆₇	#518	1	1	yes	characteristic
172	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
173	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
174	C₂²×C₆₇	#568	1	1	yes	characteristic
175	C₂²×C₆₇	#568	1	1	yes	characteristic
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	2	4	no	—
178	C₄×C₆₇	—	2	4	no	—
179	C₄×C₆₇	—	2	4	no	—
180	C₄×C₆₇	—	4	2	no	—
181	C₄×C₆₇	—	4	2	no	—
182	C₄×C₆₇	#538	1	4	yes	—
183	#21902	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂×C₆₇	—	2	4	no	—
185	C₂×C₆₇	—	4	4	no	—
186	C₂×C₆₇	#2312	1	1	yes	characteristic
187	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
188	C₂×C₆₇	#2921	1	1	yes	characteristic
189	C₂×C₆₇	#2972	1	1	yes	characteristic
190	C₂×C₆₇	#2972	1	1	yes	characteristic
191	C₂×C₆₇	#2973	1	1	yes	characteristic
192	C₂×C₆₇	#2975	1	1	yes	characteristic
193	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
194	#1923	C₂×C₆₇	1	8	yes	complemented
195	#1797	C₂×C₆₇	1	2	yes	complemented
196	#1779	C₂×C₆₇	1	2	yes	—
197	C₄×C₂²⋊C₈	C₂×C₆₇	1	2	yes	complemented
198	C₆₇	#21902	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
200	C₂²×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
201	C₂²×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
202	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
203	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
205	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
206	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
207	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
208	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	complemented
209	C₄×C₂²⋊C₄	C₄×C₆₇	1	4	yes	—
210	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
211	C₂×#95	C₄×C₆₇	1	1	yes	characteristic
212	C₂×#95	C₄×C₆₇	1	1	yes	characteristic
213	#336	C₂²×C₆₇	1	4	yes	complemented
214	#336	C₂²×C₆₇	1	4	yes	complemented
215	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
216	C₂³×C₄	—	2	4	no	—
217	C₂³×C₄	—	4	2	no	—
218	C₂³×C₄	—	4	2	no	—
219	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
220	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
221	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
222	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
223	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
224	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
225	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
226	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
228	C₂×C₂²⋊C₄	—	2	4	no	—
229	C₂×C₂²⋊C₄	C₈×C₆₇	1	8	yes	complemented
230	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
231	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
232	C₂×C₄²	—	4	2	no	—
233	C₂×C₄²	—	4	2	no	—
234	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
235	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
236	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
237	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
238	C₄⋊C₈	—	4	4	no	—
239	C₂²⋊C₈	—	2	8	no	—
240	C₂²⋊C₈	—	4	4	no	—
241	C₂²⋊C₈	—	4	4	no	—
242	C₄×C₈	—	4	4	no	—
243	#95	C₂×C₄×C₆₇	1	4	yes	—
244	#95	C₂×C₄×C₆₇	1	4	yes	—
245	C₂⁴	—	2	2	no	—
246	C₂⁴	—	2	2	no	—
247	C₂⁴	—	4	1	no	—
248	C₂⁴	—	4	1	no	—
249	C₂⁴	—	4	1	no	—
250	C₂⁴	—	4	1	no	—
251	C₂⁴	C₂×C₈×C₆₇	1	4	yes	—
252	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
253	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	4	16	no	—
275	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
276	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
277	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
278	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
279	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
280	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
282	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
283	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
284	C₂²×C₄	#41×C₆₇	1	4	yes	—
285	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
286	C₂×C₈	—	2	8	no	—
287	C₂×C₈	—	2	8	no	—
288	C₂×C₈	—	2	8	no	—
289	C₂²⋊C₄	—	2	8	no	—
290	C₂²⋊C₄	C₂×C₈×C₆₇	1	16	yes	—
291	C₄²	—	2	4	no	—
292	C₄²	—	2	4	no	—
293	C₄²	—	4	2	no	—
294	C₄²	—	4	2	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	8	no	—
306	C₂³	—	4	4	no	—
307	C₂³	—	4	4	no	—
308	C₂³	—	4	4	no	—
309	C₂³	—	4	4	no	—
310	C₂³	—	4	4	no	—
311	C₂³	—	4	4	no	—
312	C₂³	#117×C₆₇	1	2	yes	—
313	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
314	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
315	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
316	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	4	8	no	—
336	C₂×C₄	—	4	4	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	8	no	—
339	C₂×C₄	#117×C₆₇	1	2	yes	—
340	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
341	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
342	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
343	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
344	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
345	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
346	C₈	—	4	8	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	4	no	—
354	C₂²	—	2	4	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	4	16	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	#431×C₆₇	1	4	yes	—
362	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
363	C₂²	#518×C₆₇	1	1	yes	characteristic
364	C₂²	#518×C₆₇	1	1	yes	characteristic
365	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
366	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
367	C₂²	#568×C₆₇	1	1	yes	characteristic
368	C₂²	#568×C₆₇	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	2	4	no	—
371	C₄	—	2	4	no	—
372	C₄	—	2	4	no	—
373	C₄	—	4	2	no	—
374	C₄	—	4	2	no	—
375	C₄	#538×C₆₇	1	4	yes	—
376	C₂	—	2	4	no	—
377	C₂	—	4	4	no	—
378	C₂	C₆₇×#2312	1	1	yes	characteristic
379	C₂	C₂×#431×C₆₇	1	1	yes	characteristic
380	C₂	C₆₇×#2921	1	1	yes	characteristic
381	C₂	C₆₇×#2972	1	1	yes	characteristic
382	C₂	C₆₇×#2972	1	1	yes	characteristic
383	C₂	C₆₇×#2973	1	1	yes	characteristic
384	C₂	C₆₇×#2975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
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×#1923	C₂	semidirect
3. C₆₇×#1797	C₂	semidirect
5. C₄×C₂²⋊C₈×C₆₇	C₂	semidirect
11. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
15. C₂×C₄×C₈×C₆₇	C₂²	semidirect
20. #336×C₆₇	C₂²	semidirect
21. #336×C₆₇	C₂²	semidirect
36. C₂×C₂²⋊C₄×C₆₇	C₈	semidirect
183. #21902	C₆₇	direct (N × H)
194. #1923	C₂×C₆₇	semidirect
195. #1797	C₂×C₆₇	semidirect
197. C₄×C₂²⋊C₈	C₂×C₆₇	semidirect
198. C₆₇	#21902	direct (N × H)
204. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
208. C₂×C₄×C₈	C₂²×C₆₇	semidirect
213. #336	C₂²×C₆₇	semidirect
214. #336	C₂²×C₆₇	semidirect
229. C₂×C₂²⋊C₄	C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1923	2
C₆₇×#1797	2
C₆₇×#1779	2
C₄×C₂²⋊C₈×C₆₇	2
#21902	67

Subgroups — C67×#21902 #30701153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21902 #30701153