Subgroups — C₄×#410×C₆₇ #30699642

Order 17152.

1666 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#410×C₆₇	C₂	1	2	yes	maximal
3	C₄×#117×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₄×#99×C₆₇	C₂	1	8	yes	maximal, complemented
5	C₆₇×#1740	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	C₄×#41×C₆₇	—	2	2	no	—
8	C₄×#41×C₆₇	—	2	4	no	—
9	C₄×#41×C₆₇	—	2	2	no	—
10	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	complemented
11	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	complemented
12	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
13	C₂×#117×C₆₇	C₄	1	2	yes	complemented
14	C₂×#117×C₆₇	C₂²	1	4	yes	—
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₄	1	2	yes	complemented
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#410×C₆₇	C₄	1	16	yes	direct product factor
19	C₂×#99×C₆₇	C₂²	1	8	yes	—
20	C₄×C₄⋊C₄×C₆₇	—	2	4	no	—
21	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
22	#376×C₆₇	—	2	2	no	—
23	C₄³×C₆₇	—	2	2	no	—
24	#342×C₆₇	C₂²	1	8	yes	—
25	C₂×#41×C₆₇	—	2	2	no	—
26	C₂×#41×C₆₇	—	2	2	no	—
27	C₂×#41×C₆₇	—	2	4	no	—
28	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₆₇	—	2	4	no	—
37	C₄×Q₈×C₆₇	—	2	4	no	—
38	C₄×D₄×C₆₇	—	2	8	no	—
39	C₄×D₄×C₆₇	—	2	8	no	—
40	C₄×D₄×C₆₇	—	2	4	no	—
41	C₄×D₄×C₆₇	—	2	4	no	—
42	#117×C₆₇	—	2	2	no	—
43	#117×C₆₇	—	2	2	no	—
44	#117×C₆₇	—	2	8	no	—
45	#117×C₆₇	—	2	4	no	—
46	#117×C₆₇	—	2	4	no	—
47	#117×C₆₇	C₂×C₄	1	16	yes	complemented
48	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
54	C₂×C₄²×C₆₇	—	2	4	no	—
55	C₂×C₄²×C₆₇	—	2	4	no	—
56	C₂×C₄²×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	8	no	—
59	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	D₄	1	2	yes	—
65	C₂×C₄²×C₆₇	D₄	1	4	yes	—
66	C₂×C₄²×C₆₇	D₄	1	2	yes	—
67	#99×C₆₇	C₂×C₄	1	32	yes	complemented
68	#95×C₆₇	—	2	8	no	—
69	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
71	#41×C₆₇	—	2	16	no	—
72	#41×C₆₇	—	2	16	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×Q₈×C₆₇	—	2	2	no	—
75	C₂×Q₈×C₆₇	C₄²	1	4	yes	complemented
76	C₂×D₄×C₆₇	—	2	4	no	—
77	C₂×D₄×C₆₇	—	2	4	no	—
78	C₂×D₄×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	C₄²	1	4	yes	complemented
81	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	8	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	4	4	no	—
89	C₂²×C₄×C₆₇	—	4	8	no	—
90	C₂²×C₄×C₆₇	C₄²	1	8	yes	complemented
91	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	#41	1	4	yes	—
103	C₄⋊C₄×C₆₇	—	2	8	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	2	16	no	—
106	C₂²⋊C₄×C₆₇	—	2	4	no	—
107	C₂²⋊C₄×C₆₇	—	2	8	no	—
108	C₂²⋊C₄×C₆₇	—	2	8	no	—
109	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	2	4	no	—
113	C₄²×C₆₇	—	2	4	no	—
114	C₄²×C₆₇	—	2	16	no	—
115	C₄²×C₆₇	—	4	16	no	—
116	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
118	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
119	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
120	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
121	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
122	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	4	2	no	—
126	C₂³×C₆₇	—	4	4	no	—
127	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
128	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
129	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
130	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
131	C₂³×C₆₇	C₂×#41	1	2	yes	—
132	Q₈×C₆₇	—	2	8	no	—
133	D₄×C₆₇	—	2	16	no	—
134	D₄×C₆₇	—	2	8	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	8	no	—
138	C₂×C₄×C₆₇	—	2	8	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	4	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	16	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	4	8	no	—
146	C₂×C₄×C₆₇	—	4	16	no	—
147	C₂×C₄×C₆₇	—	4	8	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
150	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
154	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂×C₄×C₆₇	#117	1	2	yes	—
159	C₂×C₄×C₆₇	#117	1	2	yes	—
160	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
161	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
162	C₂²×C₆₇	—	2	4	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	—	2	4	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	4	no	—
168	C₂²×C₆₇	—	4	4	no	—
169	C₂²×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
171	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
173	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
174	C₄×C₆₇	—	2	2	no	—
175	C₄×C₆₇	—	2	4	no	—
176	C₄×C₆₇	—	2	4	no	—
177	C₄×C₆₇	—	4	16	no	—
178	C₄×C₆₇	—	4	4	no	—
179	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
180	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
181	C₄×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
182	C₄×C₆₇	#410	1	4	yes	direct product factor
183	C₄×#410	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂×C₆₇	—	2	4	no	—
185	C₂×C₆₇	—	2	1	no	—
186	C₂×C₆₇	—	2	1	no	—
187	C₂×C₆₇	—	4	4	no	—
188	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₆₇	C₂×#410	1	2	yes	—
190	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
191	C₂×#410	C₂×C₆₇	1	2	yes	—
192	C₄×#117	C₂×C₆₇	1	2	yes	complemented
193	C₄×#99	C₂×C₆₇	1	8	yes	complemented
194	#1740	C₂×C₆₇	1	2	yes	complemented
195	C₆₇	C₄×#410	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
197	C₄×#41	—	2	2	no	—
198	C₄×#41	—	2	2	no	—
199	C₄×#41	—	2	4	no	—
200	C₂×C₄×Q₈	C₄×C₆₇	1	2	yes	complemented
201	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	complemented
202	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
203	C₂×#117	C₄×C₆₇	1	2	yes	complemented
204	C₂×#117	C₂²×C₆₇	1	4	yes	—
205	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
206	C₂²×C₄²	C₄×C₆₇	1	2	yes	complemented
207	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
208	#410	C₄×C₆₇	1	16	yes	direct product factor
209	C₂×#99	C₂²×C₆₇	1	8	yes	—
210	C₄×C₄⋊C₄	—	2	4	no	—
211	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
212	#376	—	2	2	no	—
213	C₄³	—	2	2	no	—
214	#342	C₂²×C₆₇	1	8	yes	—
215	C₂×#41	—	2	2	no	—
216	C₂×#41	—	2	2	no	—
217	C₂×#41	—	2	4	no	—
218	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
219	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
220	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
221	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
222	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
223	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
224	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
225	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
226	C₄×Q₈	—	2	4	no	—
227	C₄×Q₈	—	2	4	no	—
228	C₄×D₄	—	2	8	no	—
229	C₄×D₄	—	2	8	no	—
230	C₄×D₄	—	2	4	no	—
231	C₄×D₄	—	2	4	no	—
232	#117	—	2	2	no	—
233	#117	—	2	2	no	—
234	#117	—	2	8	no	—
235	#117	—	2	4	no	—
236	#117	—	2	4	no	—
237	#117	C₂×C₄×C₆₇	1	16	yes	complemented
238	C₂×C₄⋊C₄	—	2	4	no	—
239	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
240	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
241	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
242	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
243	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
244	C₂×C₄²	—	2	4	no	—
245	C₂×C₄²	—	2	2	no	—
246	C₂×C₄²	—	2	4	no	—
247	C₂×C₄²	—	2	2	no	—
248	C₂×C₄²	—	2	8	no	—
249	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
250	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
251	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
253	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄²	D₄×C₆₇	1	2	yes	—
255	C₂×C₄²	D₄×C₆₇	1	2	yes	—
256	C₂×C₄²	D₄×C₆₇	1	4	yes	—
257	#99	C₂×C₄×C₆₇	1	32	yes	complemented
258	#95	—	2	8	no	—
259	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
260	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
261	#41	—	2	16	no	—
262	#41	—	2	16	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	C₄²×C₆₇	1	4	yes	complemented
266	C₂×D₄	—	2	4	no	—
267	C₂×D₄	—	2	4	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	C₄²×C₆₇	1	4	yes	complemented
271	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	8	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	—	4	4	no	—
279	C₂²×C₄	—	4	8	no	—
280	C₂²×C₄	C₄²×C₆₇	1	8	yes	complemented
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	4	yes	—
284	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
285	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
288	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
289	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
290	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
292	C₂²×C₄	#41×C₆₇	1	4	yes	—
293	C₄⋊C₄	—	2	8	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	16	no	—
296	C₂²⋊C₄	—	2	4	no	—
297	C₂²⋊C₄	—	2	8	no	—
298	C₂²⋊C₄	—	2	8	no	—
299	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	4	no	—
303	C₄²	—	2	4	no	—
304	C₄²	—	2	16	no	—
305	C₄²	—	4	16	no	—
306	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
307	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
308	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
309	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
310	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
311	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
312	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
313	C₂³	—	2	2	no	—
314	C₂³	—	2	2	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	4	4	no	—
317	C₂³	C₂×C₄²×C₆₇	1	4	yes	—
318	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
319	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
320	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
321	C₂³	C₂×#41×C₆₇	1	2	yes	—
322	Q₈	—	2	8	no	—
323	D₄	—	2	16	no	—
324	D₄	—	2	8	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	8	no	—
328	C₂×C₄	—	2	8	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	16	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	4	8	no	—
336	C₂×C₄	—	4	16	no	—
337	C₂×C₄	—	4	8	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
340	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
341	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
342	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
343	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
344	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
346	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
347	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
348	C₂×C₄	#117×C₆₇	1	2	yes	—
349	C₂×C₄	#117×C₆₇	1	2	yes	—
350	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
351	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
352	C₂²	—	2	4	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	4	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	4	no	—
358	C₂²	—	4	4	no	—
359	C₂²	—	4	4	no	—
360	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
361	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
362	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
363	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
364	C₄	—	2	4	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	4	no	—
367	C₄	—	4	16	no	—
368	C₄	—	4	4	no	—
369	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
370	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
371	C₄	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
372	C₄	#410×C₆₇	1	4	yes	direct product factor
373	C₂	—	2	4	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	4	4	no	—
377	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
378	C₂	C₂×#410×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₆₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×#117×C₆₇	C₂	semidirect
4. C₄×#99×C₆₇	C₂	semidirect
5. C₆₇×#1740	C₂	semidirect
10. C₂×C₄×Q₈×C₆₇	C₄	semidirect
11. C₂×C₄×D₄×C₆₇	C₄	semidirect
13. C₂×#117×C₆₇	C₄	semidirect
16. C₂²×C₄²×C₆₇	C₄	semidirect
18. #410×C₆₇	C₄	direct (N × H)
47. #117×C₆₇	C₂×C₄	semidirect
67. #99×C₆₇	C₂×C₄	semidirect
75. C₂×Q₈×C₆₇	C₄²	semidirect
80. C₂×D₄×C₆₇	C₄²	semidirect
90. C₂²×C₄×C₆₇	C₄²	semidirect
182. C₄×C₆₇	#410	direct (N × H)
183. C₄×#410	C₆₇	direct (N × H)
192. C₄×#117	C₂×C₆₇	semidirect
193. C₄×#99	C₂×C₆₇	semidirect
194. #1740	C₂×C₆₇	semidirect
195. C₆₇	C₄×#410	direct (N × H)
200. C₂×C₄×Q₈	C₄×C₆₇	semidirect
201. C₂×C₄×D₄	C₄×C₆₇	semidirect
203. C₂×#117	C₄×C₆₇	semidirect
206. C₂²×C₄²	C₄×C₆₇	semidirect
208. #410	C₄×C₆₇	direct (N × H)
237. #117	C₂×C₄×C₆₇	semidirect
257. #99	C₂×C₄×C₆₇	semidirect
265. C₂×Q₈	C₄²×C₆₇	semidirect
270. C₂×D₄	C₄²×C₆₇	semidirect
280. C₂²×C₄	C₄²×C₆₇	semidirect
372. C₄	#410×C₆₇	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×C₄×#41×C₆₇	2
C₂×#410×C₆₇	2
C₄×#117×C₆₇	2
C₄×#99×C₆₇	2
C₆₇×#1740	2
C₄×#410	67

Subgroups — C4×#410×C67 #30699642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#410×C₆₇ #30699642