Subgroups — C₂×#22569144 #31720197

Order 17152.

1982 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569144	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16863527	C₂	1	4	yes	maximal, complemented
3	C₂×Q₈×C₆₇⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×#16863423	C₂	1	2	yes	maximal
5	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863352	C₂	1	2	yes	maximal
7	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11761612	C₂²	1	2	yes	—
10	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	#16863527	C₂²	1	32	yes	complemented
13	Q₈×C₆₇⋊C₈	C₂²	1	32	yes	complemented
14	#16863423	C₂²	1	16	yes	—
15	#16863420	C₂²	1	8	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
18	#16863352	C₂²	1	16	yes	—
19	C₂×#14523954	C₂²	1	4	yes	—
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₆₇⋊C₈	C₂²	1	8	yes	—
23	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
24	#533×C₆₇	C₄	1	8	yes	—
25	#533×C₆₇	C₂²	1	8	yes	—
26	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
27	C₂×#125×C₆₇	C₄	1	2	yes	—
28	C₂×C₂²⋊Q₈×C₆₇	C₄	1	4	yes	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
30	C₂×#117×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#11761612	—	2	8	no	—
33	C₂×#11761612	C₂³	1	8	yes	—
34	C₂²×C₆₇⋊C₈	—	2	4	no	—
35	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	complemented
36	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
38	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
39	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
42	C₄⋊(C₆₇⋊C₈)	C₂³	1	16	yes	—
43	C₄⋊(C₆₇⋊C₈)	C₂³	1	32	yes	—
44	#14523954	C₂³	1	16	yes	—
45	C₄×C₆₇⋊C₈	C₂³	1	8	yes	—
46	C₄×C₆₇⋊C₈	C₂³	1	16	yes	—
47	C₄×C₆₇⋊C₈	C₂³	1	8	yes	—
48	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₆₇	C₂×C₄	1	16	yes	—
52	#125×C₆₇	C₂×C₄	1	16	yes	—
53	C₂²⋊Q₈×C₆₇	C₂×C₄	1	32	yes	—
54	C₄×Q₈×C₆₇	C₂×C₄	1	16	yes	—
55	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
56	#117×C₆₇	C₂×C₄	1	8	yes	—
57	#117×C₆₇	C₂³	1	8	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
61	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
68	#11761612	—	2	16	no	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	—	2	8	no	—
71	C₂×C₆₇⋊C₈	—	2	8	no	—
72	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
73	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
74	C₂×C₆₇⋊C₈	C₂×Q₈	1	16	yes	complemented
75	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
76	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
77	C₂×C₆₇⋊C₈	C₂⁴	1	2	yes	—
78	C₂×C₆₇⋊C₈	C₂⁴	1	2	yes	—
79	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Q₈×C₆₇	—	2	16	no	—
81	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
82	C₂²×C₄×C₆₇	—	2	8	no	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
90	C₂²×C₄×C₆₇	#41	1	4	yes	—
91	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
94	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
95	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
96	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
97	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
98	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
99	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
100	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
101	C₄²×C₆₇	C₂⁴	1	4	yes	—
102	C₄²×C₆₇	C₂⁴	1	8	yes	—
103	C₂³×C₆₇	—	2	4	no	—
104	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
105	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
106	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
107	Q₈×C₆₇	—	2	32	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	16	no	—
110	C₂×C₄×C₆₇	—	2	16	no	—
111	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
112	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₆₇	#131	1	8	yes	—
114	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	16	yes	—
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
121	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
123	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
124	C₆₇⋊C₈	—	2	4	no	—
125	C₆₇⋊C₈	—	2	8	no	—
126	C₆₇⋊C₈	C₂²×Q₈	1	4	yes	—
127	C₆₇⋊C₈	C₂²×Q₈	1	4	yes	—
128	C₂²×C₆₇	—	2	8	no	—
129	C₂²×C₆₇	—	2	4	no	—
130	C₂²×C₆₇	C₂×C₄×Q₈	1	4	yes	—
131	C₂²×C₆₇	C₂×C₄×Q₈	1	2	yes	—
132	C₂²×C₆₇	C₂×#131	1	4	yes	—
133	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	—	2	16	no	—
136	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
137	C₄×C₆₇	C₂×C₄×Q₈	1	4	yes	—
138	C₄×C₆₇	C₂×#131	1	8	yes	—
139	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
140	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
141	C₂×#2960	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	4	no	—
143	C₂×C₆₇	#2960	1	4	yes	complemented
144	C₂×C₆₇	C₂²×C₄×Q₈	1	1	yes	characteristic
145	C₂×C₆₇	C₂²×#131	1	2	yes	—
146	C₂×#533	D₆₇	1	1	yes	characteristic
147	#2960	—	67	16	no	—
148	C₂×#446	—	67	4	no	—
149	C₂×C₈×Q₈	—	67	4	no	—
150	C₂×#424	—	67	2	no	—
151	C₂×#423	—	67	1	no	—
152	C₂×#405	—	67	2	no	—
153	C₂²×C₄×C₈	—	67	1	no	—
154	C₆₇	C₂×#2960	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	67	2	no	—
156	C₂³×C₈	—	67	1	no	—
157	C₂³×C₈	—	67	1	no	—
158	#533	Dic₆₇	1	8	yes	—
159	#533	C₂×D₆₇	1	8	yes	—
160	C₂×C₄⋊Q₈	Dic₆₇	1	2	yes	—
161	C₂×#125	Dic₆₇	1	2	yes	—
162	C₂×C₂²⋊Q₈	Dic₆₇	1	4	yes	—
163	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
164	C₂×#117	C₂×D₆₇	1	2	yes	—
165	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
166	#446	—	67	32	no	—
167	C₈×Q₈	—	67	32	no	—
168	#424	—	67	16	no	—
169	#423	—	67	8	no	—
170	C₂×C₄⋊C₈	—	67	4	no	—
171	C₂×C₄⋊C₈	—	67	8	no	—
172	#405	—	67	16	no	—
173	C₂×#97	—	67	4	no	—
174	C₂×C₄×C₈	—	67	2	no	—
175	C₂×C₄×C₈	—	67	2	no	—
176	C₂×C₄×C₈	—	67	8	no	—
177	C₂×C₄×C₈	—	67	4	no	—
178	C₂²×Q₈	C₂×Dic₆₇	1	4	yes	—
179	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
180	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
181	C₂×M₁₆	—	67	8	no	—
182	C₂×M₁₆	—	134	8	no	—
183	C₂²×C₈	—	67	8	no	—
184	C₂²×C₈	—	67	4	no	—
185	C₂²×C₈	—	67	1	no	—
186	C₂²×C₈	—	67	1	no	—
187	C₂²×C₈	—	67	4	no	—
188	C₂²×C₈	—	67	2	no	—
189	C₂²×C₈	—	67	4	no	—
190	C₂²×C₈	—	134	4	no	—
191	C₄⋊Q₈	C₂×Dic₆₇	1	16	yes	—
192	#125	C₂×Dic₆₇	1	16	yes	—
193	C₂²⋊Q₈	C₂×Dic₆₇	1	32	yes	—
194	C₄×Q₈	C₂×Dic₆₇	1	16	yes	—
195	C₄×Q₈	C₂²×D₆₇	1	16	yes	—
196	#117	C₂×Dic₆₇	1	8	yes	—
197	#117	C₂²×D₆₇	1	8	yes	—
198	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
199	C₂×C₄⋊C₄	C₂×Dic₆₇	1	8	yes	—
200	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
201	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
202	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
205	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
206	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
207	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
208	C₄⋊C₈	—	67	16	no	—
209	C₄⋊C₈	—	67	32	no	—
210	#97	—	67	16	no	—
211	C₄×C₈	—	67	8	no	—
212	C₄×C₈	—	67	8	no	—
213	C₄×C₈	—	67	16	no	—
214	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
215	C₂×Q₈	—	2	16	no	—
216	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
220	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
221	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
224	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
225	C₂²×C₄	#11761629	1	4	yes	—
226	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
227	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
228	M₁₆	—	134	16	no	—
229	C₂×C₈	—	67	4	no	—
230	C₂×C₈	—	67	4	no	—
231	C₂×C₈	—	67	16	no	—
232	C₂×C₈	—	67	2	no	—
233	C₂×C₈	—	67	2	no	—
234	C₂×C₈	—	67	4	no	—
235	C₂×C₈	—	67	8	no	—
236	C₂×C₈	—	134	4	no	—
237	C₂×C₈	—	134	8	no	—
238	C₂×C₈	—	134	8	no	—
239	C₄⋊C₄	C₂²×Dic₆₇	1	16	yes	—
240	C₄⋊C₄	C₂²×Dic₆₇	1	32	yes	—
241	C₄⋊C₄	C₂²×Dic₆₇	1	16	yes	—
242	C₂²⋊C₄	C₂²×Dic₆₇	1	16	yes	—
243	C₄²	C₂²×Dic₆₇	1	4	yes	—
244	C₄²	C₂²×Dic₆₇	1	4	yes	—
245	C₄²	C₂²×Dic₆₇	1	4	yes	—
246	C₄²	C₂²×Dic₆₇	1	8	yes	—
247	C₄²	C₂³×D₆₇	1	4	yes	—
248	C₄²	C₂³×D₆₇	1	8	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	Q₈×Dic₆₇	1	4	yes	—
251	C₂³	C₂³×Dic₆₇	1	2	yes	—
252	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
253	Q₈	—	2	32	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	Q₈×Dic₆₇	1	8	yes	—
258	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
259	C₂×C₄	#14524050	1	8	yes	—
260	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
261	C₂×C₄	C₂³×Dic₆₇	1	16	yes	—
262	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
264	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
265	C₂×C₄	C₂×Q₈×D₆₇	1	4	yes	—
266	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
267	C₂×C₄	C₂×#11761629	1	2	yes	—
268	C₂×C₄	C₂×#11761629	1	4	yes	—
269	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
270	C₈	—	67	4	no	—
271	C₈	—	67	4	no	—
272	C₈	—	134	4	no	—
273	C₈	—	134	8	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	8	no	—
276	C₂²	C₂×Q₈×Dic₆₇	1	4	yes	—
277	C₂²	C₂×Q₈×Dic₆₇	1	2	yes	—
278	C₂²	C₂×#14524050	1	4	yes	—
279	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	16	no	—
282	C₄	C₂×Q₈×Dic₆₇	1	2	yes	—
283	C₄	C₂×Q₈×Dic₆₇	1	4	yes	—
284	C₄	C₂×#14524050	1	8	yes	—
285	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
286	C₄	C₂²×#11761629	1	1	yes	characteristic
287	C₂	—	2	4	no	—
288	C₂	#22569144	1	4	yes	direct product factor
289	C₂	C₂²×Q₈×Dic₆₇	1	1	yes	characteristic
290	C₂	C₂²×#14524050	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#533×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂×#533	D₆₇
hypocenter	C₆₇	C₂×#2960
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #22569144	C₂	direct (N × H)
2. C₂×#16863527	C₂	semidirect
3. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
12. #16863527	C₂²	semidirect
13. Q₈×C₆₇⋊C₈	C₂²	semidirect
35. C₂²×C₆₇⋊C₈	Q₈	semidirect
74. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
143. C₂×C₆₇	#2960	semidirect
154. C₆₇	C₂×#2960	semidirect
288. C₂	#22569144	direct (N × H)

Maximal subgroups

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

subgroup	index
#22569144	2
C₂×#16863527	2
C₂×Q₈×C₆₇⋊C₈	2
C₂×#16863423	2
C₂×#16863420	2
C₂×#16863352	2
C₂²×C₄×C₆₇⋊C₈	2
C₂×#533×C₆₇	2
C₂×#2960	67

Subgroups — C2×#22569144 #31720197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569144 #31720197