Subgroups — C₂×C₆₇×#1503 #30696783

Order 17152.

1194 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1503	C₂	1	8	yes	maximal, direct product factor
5	C₂×#328×C₆₇	C₂	1	4	yes	maximal
6	#533×C₆₇	—	2	2	no	—
7	#533×C₆₇	—	2	2	no	—
8	#533×C₆₇	C₄	1	4	yes	complemented
9	#533×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
11	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
12	C₂×#125×C₆₇	—	2	1	no	—
13	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
16	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	complemented
17	C₂×#117×C₆₇	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₂²×C₄²×C₆₇	—	2	1	no	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#431×C₆₇	—	2	2	no	—
22	#431×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	—	2	1	no	—
26	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
28	C₂×#95×C₆₇	C₂²	1	2	yes	—
29	#328×C₆₇	C₂²	1	16	yes	—
30	C₂²×Q₈×C₆₇	—	2	1	no	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	—	2	2	no	—
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₄⋊Q₈×C₆₇	—	2	2	no	—
38	C₄⋊Q₈×C₆₇	—	2	2	no	—
39	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
40	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
41	#125×C₆₇	—	2	4	no	—
42	#125×C₆₇	—	2	4	no	—
43	C₂²⋊Q₈×C₆₇	—	2	4	no	—
44	C₂²⋊Q₈×C₆₇	—	2	4	no	—
45	C₂²⋊Q₈×C₆₇	—	2	2	no	—
46	C₂²⋊Q₈×C₆₇	—	2	2	no	—
47	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
48	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
49	C₄×Q₈×C₆₇	—	2	4	no	—
50	C₄×Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	—	2	4	no	—
52	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
53	#117×C₆₇	—	2	4	no	—
54	#117×C₆₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	—	2	4	no	—
66	C₂×C₄²×C₆₇	—	4	4	no	—
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	D₄	1	2	yes	—
71	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	D₄	1	2	yes	—
73	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
75	C₄⋊C₈×C₆₇	—	2	4	no	—
76	C₂²⋊C₈×C₆₇	—	2	2	no	—
77	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
78	C₄×C₈×C₆₇	—	2	4	no	—
79	#95×C₆₇	—	2	2	no	—
80	#95×C₆₇	C₂³	1	4	yes	—
81	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×Q₈×C₆₇	—	4	4	no	—
85	C₂×Q₈×C₆₇	—	4	4	no	—
86	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	8	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	4	no	—
94	C₂²×C₄×C₆₇	—	4	4	no	—
95	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
102	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	4	4	no	—
113	C₄⋊C₄×C₆₇	—	2	16	no	—
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	2	8	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
122	C₂²⋊C₄×C₆₇	—	2	4	no	—
123	C₂²⋊C₄×C₆₇	—	2	4	no	—
124	C₄²×C₆₇	—	2	8	no	—
125	C₄²×C₆₇	—	4	8	no	—
126	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
127	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
128	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
129	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
130	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
131	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
132	C₄²×C₆₇	C₂×D₄	1	2	yes	—
133	C₄²×C₆₇	C₂×D₄	1	2	yes	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
139	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
140	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	Q₈×C₆₇	—	4	4	no	—
142	Q₈×C₆₇	—	4	4	no	—
143	Q₈×C₆₇	—	4	8	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	8	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	1	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	—	4	4	no	—
157	C₂×C₄×C₆₇	—	4	16	no	—
158	C₂×C₄×C₆₇	—	4	4	no	—
159	C₂×C₄×C₆₇	—	4	4	no	—
160	C₂×C₄×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	#103	1	8	yes	—
162	C₂×C₄×C₆₇	#103	1	4	yes	—
163	C₂×C₄×C₆₇	#103	1	4	yes	—
164	C₂×C₄×C₆₇	#104	1	4	yes	—
165	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
171	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
172	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
173	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₈×C₆₇	—	4	4	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	4	no	—
181	C₂²×C₆₇	—	2	4	no	—
182	C₂²×C₆₇	#410	1	2	yes	—
183	C₂²×C₆₇	C₂×#103	1	2	yes	—
184	C₂²×C₆₇	C₂×#103	1	2	yes	—
185	C₂²×C₆₇	C₂×#104	1	2	yes	—
186	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
188	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	2	2	no	—
194	C₄×C₆₇	—	4	8	no	—
195	C₄×C₆₇	—	4	4	no	—
196	C₄×C₆₇	—	4	4	no	—
197	C₄×C₆₇	C₂×#103	1	4	yes	—
198	C₄×C₆₇	C₂×#103	1	4	yes	—
199	C₄×C₆₇	C₂×#104	1	4	yes	—
200	C₂×#1503	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	—	2	1	no	—
203	C₂×C₆₇	—	2	2	no	—
204	C₂×C₆₇	#1503	1	4	yes	direct product factor
205	C₂×C₆₇	C₂×#410	1	1	yes	characteristic
206	C₂×C₆₇	C₂²×#103	1	1	yes	characteristic
207	C₂×C₆₇	C₂²×#104	1	1	yes	characteristic
208	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
209	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
210	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
211	#1503	C₂×C₆₇	1	8	yes	direct product factor
212	C₂×#328	C₂×C₆₇	1	4	yes	—
213	C₆₇	C₂×#1503	1	1	yes	characteristic, direct product factor, Sylow
214	#533	—	2	2	no	—
215	#533	—	2	2	no	—
216	#533	C₄×C₆₇	1	4	yes	complemented
217	#533	C₂²×C₆₇	1	4	yes	—
218	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic, complemented
219	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic, complemented
220	C₂×#125	—	2	1	no	—
221	C₂×C₂²⋊Q₈	—	2	1	no	—
222	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
223	C₂×C₄×Q₈	—	2	1	no	—
224	C₂×C₄×Q₈	C₄×C₆₇	1	2	yes	complemented
225	C₂×#117	—	2	1	no	—
226	C₂²×C₄⋊C₄	—	2	1	no	—
227	C₂²×C₄²	—	2	1	no	—
228	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
229	#431	—	2	2	no	—
230	#431	C₂²×C₆₇	1	4	yes	—
231	C₂×C₄⋊C₈	—	2	1	no	—
232	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
233	C₂×C₄×C₈	—	2	1	no	—
234	C₄×C₄⋊C₄	—	2	2	no	—
235	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
236	C₂×#95	C₂²×C₆₇	1	2	yes	—
237	#328	C₂²×C₆₇	1	16	yes	—
238	C₂²×Q₈	—	2	1	no	—
239	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
240	C₂³×C₄	—	2	2	no	—
241	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
242	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
243	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
244	C₂²×C₈	—	2	2	no	—
245	C₄⋊Q₈	—	2	2	no	—
246	C₄⋊Q₈	—	2	2	no	—
247	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
248	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
249	#125	—	2	4	no	—
250	#125	—	2	4	no	—
251	C₂²⋊Q₈	—	2	4	no	—
252	C₂²⋊Q₈	—	2	4	no	—
253	C₂²⋊Q₈	—	2	2	no	—
254	C₂²⋊Q₈	—	2	2	no	—
255	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
256	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
257	C₄×Q₈	—	2	4	no	—
258	C₄×Q₈	—	2	4	no	—
259	C₄×Q₈	—	2	4	no	—
260	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
261	#117	—	2	4	no	—
262	#117	—	2	4	no	—
263	C₂×C₄⋊C₄	—	2	8	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	4	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	—	2	4	no	—
274	C₂×C₄²	—	4	4	no	—
275	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
276	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
277	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
278	C₂×C₄²	D₄×C₆₇	1	2	yes	—
279	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
280	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
281	C₂×C₄²	D₄×C₆₇	1	2	yes	—
282	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
283	C₄⋊C₈	—	2	4	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
286	C₄×C₈	—	2	4	no	—
287	#95	—	2	2	no	—
288	#95	C₂³×C₆₇	1	4	yes	—
289	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	4	no	—
292	C₂×Q₈	—	4	4	no	—
293	C₂×Q₈	—	4	4	no	—
294	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	8	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	4	2	no	—
301	C₂²×C₄	—	4	4	no	—
302	C₂²×C₄	—	4	4	no	—
303	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
304	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
310	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
311	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
318	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	4	4	no	—
321	C₄⋊C₄	—	2	16	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	4	no	—
326	C₄⋊C₄	—	2	8	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
330	C₂²⋊C₄	—	2	4	no	—
331	C₂²⋊C₄	—	2	4	no	—
332	C₄²	—	2	8	no	—
333	C₄²	—	4	8	no	—
334	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
335	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
336	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
337	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
338	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
339	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
340	C₄²	C₂×D₄×C₆₇	1	2	yes	—
341	C₄²	C₂×D₄×C₆₇	1	2	yes	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
345	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
346	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
347	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
348	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
349	Q₈	—	4	4	no	—
350	Q₈	—	4	4	no	—
351	Q₈	—	4	8	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	8	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	16	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	4	no	—
369	C₂×C₄	#103×C₆₇	1	8	yes	—
370	C₂×C₄	#103×C₆₇	1	4	yes	—
371	C₂×C₄	#103×C₆₇	1	4	yes	—
372	C₂×C₄	#104×C₆₇	1	4	yes	—
373	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
376	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
377	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
378	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
379	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
380	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
381	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
383	C₈	—	4	4	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	2	4	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	#410×C₆₇	1	2	yes	—
391	C₂²	C₂×#103×C₆₇	1	2	yes	—
392	C₂²	C₂×#103×C₆₇	1	2	yes	—
393	C₂²	C₂×#104×C₆₇	1	2	yes	—
394	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
395	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
396	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	2	2	no	—
402	C₄	—	4	8	no	—
403	C₄	—	4	4	no	—
404	C₄	—	4	4	no	—
405	C₄	C₂×#103×C₆₇	1	4	yes	—
406	C₄	C₂×#103×C₆₇	1	4	yes	—
407	C₄	C₂×#104×C₆₇	1	4	yes	—
408	C₂	—	2	1	no	—
409	C₂	—	2	1	no	—
410	C₂	—	2	2	no	—
411	C₂	C₆₇×#1503	1	4	yes	direct product factor
412	C₂	C₂×#410×C₆₇	1	1	yes	characteristic
413	C₂	C₂²×#103×C₆₇	1	1	yes	characteristic
414	C₂	C₂²×#104×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1503	C₂	direct (N × H)
8. #533×C₆₇	C₄	semidirect
10. C₂×C₄⋊Q₈×C₆₇	C₄	semidirect
11. C₂×C₄⋊Q₈×C₆₇	C₄	semidirect
16. C₂×C₄×Q₈×C₆₇	C₄	semidirect
39. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
40. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
52. C₄×Q₈×C₆₇	C₂×C₄	semidirect
200. C₂×#1503	C₆₇	direct (N × H)
204. C₂×C₆₇	#1503	direct (N × H)
211. #1503	C₂×C₆₇	direct (N × H)
213. C₆₇	C₂×#1503	direct (N × H)
216. #533	C₄×C₆₇	semidirect
218. C₂×C₄⋊Q₈	C₄×C₆₇	semidirect
219. C₂×C₄⋊Q₈	C₄×C₆₇	semidirect
224. C₂×C₄×Q₈	C₄×C₆₇	semidirect
247. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
248. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
260. C₄×Q₈	C₂×C₄×C₆₇	semidirect
411. C₂	C₆₇×#1503	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×#533×C₆₇	2
C₂×#431×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1503	2
C₂×#328×C₆₇	2
C₂×#1503	67

Subgroups — C2×C67×#1503 #30696783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1503 #30696783