Subgroups — C₂×C₆₇×#3390 #30740615

Order 17152.

1746 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#557×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#3390	C₂	1	16	yes	maximal, direct product factor
3	C₂×#501×C₆₇	C₂	1	2	yes	maximal
4	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#479×C₆₇	C₂	1	2	yes	maximal
6	C₂×#479×C₆₇	C₂	1	2	yes	maximal
7	C₂×#477×C₆₇	C₂	1	2	yes	maximal
8	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#441×C₆₇	C₂	1	2	yes	maximal
10	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
11	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	#557×C₆₇	—	2	8	no	—
13	#557×C₆₇	C₂²	1	16	yes	—
14	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#125×C₆₇	—	2	2	no	—
17	C₂×#125×C₆₇	—	2	2	no	—
18	C₂×#125×C₆₇	C₂²	1	2	yes	—
19	C₂×#125×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	#501×C₆₇	C₂²	1	16	yes	—
26	#499×C₆₇	C₂²	1	8	yes	—
27	#479×C₆₇	C₂²	1	16	yes	—
28	#479×C₆₇	C₂²	1	16	yes	—
29	#477×C₆₇	C₂²	1	16	yes	—
30	#446×C₆₇	C₂²	1	8	yes	—
31	#441×C₆₇	C₂²	1	16	yes	—
32	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	—
33	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#107×C₆₇	C₂²	1	2	yes	—
35	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#106×C₆₇	C₂²	1	2	yes	—
38	C₂×#106×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#103×C₆₇	C₂²	1	2	yes	—
42	C₂×#103×C₆₇	C₂²	1	2	yes	—
43	C₂×#103×C₆₇	C₂²	1	2	yes	—
44	C₂×#97×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₆₇	Q₈	1	8	yes	complemented
50	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₆₇	—	2	16	no	—
55	C₄⋊Q₈×C₆₇	—	2	4	no	—
56	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
57	#125×C₆₇	—	2	16	no	—
58	#125×C₆₇	—	2	16	no	—
59	#125×C₆₇	—	2	4	no	—
60	#125×C₆₇	—	2	4	no	—
61	#125×C₆₇	C₂³	1	8	yes	—
62	#125×C₆₇	C₂³	1	8	yes	—
63	C₄×Q₈×C₆₇	—	2	16	no	—
64	C₄×Q₈×C₆₇	—	2	4	no	—
65	C₄×Q₈×C₆₇	—	2	4	no	—
66	C₄×Q₈×C₆₇	—	2	4	no	—
67	C₄×Q₈×C₆₇	D₄	1	4	yes	—
68	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
69	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
70	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
71	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
85	C₂×C₄²×C₆₇	—	2	2	no	—
86	C₂×C₄²×C₆₇	—	2	2	no	—
87	C₂×C₄²×C₆₇	—	2	2	no	—
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
90	#107×C₆₇	C₂³	1	4	yes	—
91	#107×C₆₇	C₂³	1	8	yes	—
92	#107×C₆₇	C₂³	1	4	yes	—
93	#107×C₆₇	C₂³	1	4	yes	—
94	#106×C₆₇	C₂³	1	8	yes	—
95	#106×C₆₇	C₂³	1	8	yes	—
96	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
97	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
98	#103×C₆₇	C₂³	1	8	yes	—
99	#103×C₆₇	C₂³	1	8	yes	—
100	#103×C₆₇	C₂³	1	8	yes	—
101	#97×C₆₇	C₂³	1	8	yes	—
102	C₄×C₈×C₆₇	C₂³	1	4	yes	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	8	no	—
105	C₂×Q₈×C₆₇	—	2	4	no	—
106	C₂×Q₈×C₆₇	—	4	2	no	—
107	C₂×Q₈×C₆₇	—	4	8	no	—
108	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
109	C₂×Q₈×C₆₇	C₂×Q₈	1	8	yes	—
110	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	Q₁₆×C₆₇	C₂×Q₈	1	16	yes	complemented
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	4	no	—
123	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
124	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
125	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
126	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
127	C₄⋊C₄×C₆₇	—	2	8	no	—
128	C₄⋊C₄×C₆₇	—	2	8	no	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	8	no	—
132	C₄⋊C₄×C₆₇	—	2	8	no	—
133	C₄⋊C₄×C₆₇	—	2	8	no	—
134	C₄⋊C₄×C₆₇	—	2	8	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
139	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
140	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
141	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
142	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₄²×C₆₇	—	2	8	no	—
144	C₄²×C₆₇	—	2	8	no	—
145	C₄²×C₆₇	—	2	8	no	—
146	C₄²×C₆₇	C₂×D₄	1	2	yes	—
147	C₄²×C₆₇	C₂×D₄	1	4	yes	—
148	C₄²×C₆₇	C₂⁴	1	2	yes	—
149	C₄²×C₆₇	C₂⁴	1	4	yes	—
150	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	2	8	no	—
152	Q₈×C₆₇	—	2	4	no	—
153	Q₈×C₆₇	—	4	4	no	—
154	Q₈×C₆₇	—	4	16	no	—
155	Q₈×C₆₇	C₂²×Q₈	1	8	yes	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	4	4	no	—
166	C₂×C₄×C₆₇	—	4	4	no	—
167	C₂×C₄×C₆₇	—	4	4	no	—
168	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
169	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
170	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
171	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
174	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
175	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
176	C₈×C₆₇	—	2	4	no	—
177	C₈×C₆₇	—	2	2	no	—
178	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
179	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
180	C₂²×C₆₇	#575	1	2	yes	—
181	C₂²×C₆₇	#578	1	2	yes	—
182	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
183	C₄×C₆₇	—	2	8	no	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	4	no	—
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	2	4	no	—
188	C₄×C₆₇	—	4	4	no	—
189	C₄×C₆₇	—	4	4	no	—
190	C₄×C₆₇	—	4	4	no	—
191	C₄×C₆₇	D₄×Q₈	1	2	yes	—
192	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
193	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
194	C₂×#3390	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂×C₆₇	#3390	1	4	yes	direct product factor
196	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
199	C₂×#557	C₂×C₆₇	1	2	yes	—
200	#3390	C₂×C₆₇	1	16	yes	direct product factor
201	C₂×#501	C₂×C₆₇	1	2	yes	—
202	C₂×#499	C₂×C₆₇	1	1	yes	characteristic
203	C₂×#479	C₂×C₆₇	1	2	yes	—
204	C₂×#479	C₂×C₆₇	1	2	yes	—
205	C₂×#477	C₂×C₆₇	1	2	yes	—
206	C₂×#446	C₂×C₆₇	1	1	yes	characteristic
207	C₂×#441	C₂×C₆₇	1	2	yes	—
208	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
209	C₆₇	C₂×#3390	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
211	#557	—	2	8	no	—
212	#557	C₂²×C₆₇	1	16	yes	—
213	C₂×C₄⋊Q₈	—	2	2	no	—
214	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
215	C₂×#125	—	2	2	no	—
216	C₂×#125	—	2	2	no	—
217	C₂×#125	C₂²×C₆₇	1	2	yes	—
218	C₂×#125	C₂²×C₆₇	1	2	yes	—
219	C₂×C₄×Q₈	—	2	2	no	—
220	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
221	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
222	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
223	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
224	#501	C₂²×C₆₇	1	16	yes	—
225	#499	C₂²×C₆₇	1	8	yes	—
226	#479	C₂²×C₆₇	1	16	yes	—
227	#479	C₂²×C₆₇	1	16	yes	—
228	#477	C₂²×C₆₇	1	16	yes	—
229	#446	C₂²×C₆₇	1	8	yes	—
230	#441	C₂²×C₆₇	1	16	yes	—
231	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	—
232	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×#107	C₂²×C₆₇	1	2	yes	—
234	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#106	C₂²×C₆₇	1	2	yes	—
237	C₂×#106	C₂²×C₆₇	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
239	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#103	C₂²×C₆₇	1	2	yes	—
241	C₂×#103	C₂²×C₆₇	1	2	yes	—
242	C₂×#103	C₂²×C₆₇	1	2	yes	—
243	C₂×#97	C₂²×C₆₇	1	2	yes	—
244	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
245	C₂²×Q₈	—	2	2	no	—
246	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
247	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
248	C₂×Q₁₆	Q₈×C₆₇	1	8	yes	complemented
249	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
250	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
252	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
253	C₄⋊Q₈	—	2	16	no	—
254	C₄⋊Q₈	—	2	4	no	—
255	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
256	#125	—	2	16	no	—
257	#125	—	2	16	no	—
258	#125	—	2	4	no	—
259	#125	—	2	4	no	—
260	#125	C₂³×C₆₇	1	8	yes	—
261	#125	C₂³×C₆₇	1	8	yes	—
262	C₄×Q₈	—	2	16	no	—
263	C₄×Q₈	—	2	4	no	—
264	C₄×Q₈	—	2	4	no	—
265	C₄×Q₈	—	2	4	no	—
266	C₄×Q₈	D₄×C₆₇	1	4	yes	—
267	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
268	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
269	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
270	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
280	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
289	#107	C₂³×C₆₇	1	4	yes	—
290	#107	C₂³×C₆₇	1	4	yes	—
291	#107	C₂³×C₆₇	1	8	yes	—
292	#107	C₂³×C₆₇	1	4	yes	—
293	#106	C₂³×C₆₇	1	8	yes	—
294	#106	C₂³×C₆₇	1	8	yes	—
295	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
296	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
297	#103	C₂³×C₆₇	1	8	yes	—
298	#103	C₂³×C₆₇	1	8	yes	—
299	#103	C₂³×C₆₇	1	8	yes	—
300	#97	C₂³×C₆₇	1	8	yes	—
301	C₄×C₈	C₂³×C₆₇	1	4	yes	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	8	no	—
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	4	2	no	—
306	C₂×Q₈	—	4	8	no	—
307	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
308	C₂×Q₈	C₂×Q₈×C₆₇	1	8	yes	—
309	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
318	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
319	Q₁₆	C₂×Q₈×C₆₇	1	16	yes	complemented
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	4	no	—
322	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
323	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
324	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
325	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
326	C₄⋊C₄	—	2	8	no	—
327	C₄⋊C₄	—	2	8	no	—
328	C₄⋊C₄	—	2	8	no	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	2	8	no	—
333	C₄⋊C₄	—	2	8	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
338	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
339	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
340	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
341	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
342	C₄²	—	2	8	no	—
343	C₄²	—	2	8	no	—
344	C₄²	—	2	8	no	—
345	C₄²	C₂×D₄×C₆₇	1	2	yes	—
346	C₄²	C₂×D₄×C₆₇	1	4	yes	—
347	C₄²	C₂⁴×C₆₇	1	2	yes	—
348	C₄²	C₂⁴×C₆₇	1	4	yes	—
349	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
350	Q₈	—	2	8	no	—
351	Q₈	—	2	4	no	—
352	Q₈	—	4	4	no	—
353	Q₈	—	4	16	no	—
354	Q₈	C₂²×Q₈×C₆₇	1	8	yes	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
368	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
369	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
370	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
371	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
373	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
374	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
375	C₈	—	2	2	no	—
376	C₈	—	2	4	no	—
377	C₈	C₂²×Q₈×C₆₇	1	4	yes	—
378	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
379	C₂²	#575×C₆₇	1	2	yes	—
380	C₂²	#578×C₆₇	1	2	yes	—
381	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
382	C₄	—	2	4	no	—
383	C₄	—	2	2	no	—
384	C₄	—	2	2	no	—
385	C₄	—	2	4	no	—
386	C₄	—	2	8	no	—
387	C₄	—	4	4	no	—
388	C₄	—	4	4	no	—
389	C₄	—	4	4	no	—
390	C₄	D₄×Q₈×C₆₇	1	2	yes	—
391	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
392	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
393	C₂	C₆₇×#3390	1	4	yes	direct product factor
394	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
395	C₂	C₂×#575×C₆₇	1	1	yes	characteristic
396	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3390	C₂	direct (N × H)
49. C₂×Q₁₆×C₆₇	Q₈	semidirect
120. Q₁₆×C₆₇	C₂×Q₈	semidirect
194. C₂×#3390	C₆₇	direct (N × H)
195. C₂×C₆₇	#3390	direct (N × H)
200. #3390	C₂×C₆₇	direct (N × H)
209. C₆₇	C₂×#3390	direct (N × H)
248. C₂×Q₁₆	Q₈×C₆₇	semidirect
319. Q₁₆	C₂×Q₈×C₆₇	semidirect
393. C₂	C₆₇×#3390	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×#557×C₆₇	2
C₆₇×#3390	2
C₂×#501×C₆₇	2
C₂×#499×C₆₇	2
C₂×#479×C₆₇	2
C₂×#477×C₆₇	2
C₂×#446×C₆₇	2
C₂×#441×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×#3390	67

Subgroups — C2×C67×#3390 #30740615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3390 #30740615