Subgroups — #32150879

Order 18176.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597257	C₂	1	2	yes	maximal, complemented
2	#22590226	C₂	1	4	yes	maximal, complemented
3	#22590221	C₂	1	4	yes	maximal, complemented
4	C₂×#17985829	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590068	C₂	1	2	yes	maximal, complemented
6	#22589936	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#508×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#17986766	—	2	2	no	—
9	C₂×#14556068	C₂²	1	1	yes	characteristic
10	#17986121	—	2	4	no	—
11	#17986116	C₂²	1	8	yes	complemented
12	C₂×#14556064	C₂²	1	2	yes	—
13	#17986016	—	2	2	no	—
14	#17985954	C₂²	1	1	yes	characteristic
15	#17985829	C₂²	1	2	yes	complemented
16	#17985829	C₂²	1	2	yes	complemented
17	#17985806	C₂²	1	8	yes	complemented
18	#17985774	C₂²	1	2	yes	complemented
19	C₁₆×Dic₇₁	C₂²	1	2	yes	complemented
20	C₂×#135×C₇₁	C₂²	1	2	yes	—
21	#508×C₇₁	C₄	1	2	yes	complemented
22	#508×C₇₁	C₄	1	2	yes	complemented
23	#508×C₇₁	C₂²	1	2	yes	—
24	#508×C₇₁	C₂²	1	2	yes	—
25	C₂×Q₃₂×C₇₁	C₄	1	1	yes	characteristic, complemented
26	C₂×SD₃₂×C₇₁	C₄	1	2	yes	complemented
27	C₂×D₁₆×C₇₁	C₄	1	1	yes	characteristic, complemented
28	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
29	#14556135	—	2	8	no	—
30	#14556135	—	4	4	no	—
31	C₂×#11765849	—	2	2	no	—
32	C₂×#11765849	—	4	4	no	—
33	C₂×#11765849	D₄	1	1	yes	characteristic
34	#14556087	D₄	1	1	yes	characteristic
35	#14556068	—	2	1	no	—
36	#14556068	—	2	1	no	—
37	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
38	#14556064	—	2	2	no	—
39	#14556064	C₂³	1	4	yes	—
40	#14556055	—	2	8	no	—
41	#14556038	D₄	1	2	yes	—
42	C₈×Dic₇₁	C₂³	1	2	yes	—
43	C₂×#41×C₇₁	—	2	2	no	—
44	#135×C₇₁	—	2	2	no	—
45	#135×C₇₁	—	2	2	no	—
46	#135×C₇₁	C₂×C₄	1	4	yes	—
47	#135×C₇₁	C₂³	1	4	yes	—
48	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₇₁	—	2	2	no	—
50	C₂×D₈×C₇₁	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
52	Q₃₂×C₇₁	C₂×C₄	1	2	yes	—
53	Q₃₂×C₇₁	C₂×C₄	1	2	yes	—
54	SD₃₂×C₇₁	C₂×C₄	1	4	yes	—
55	SD₃₂×C₇₁	C₂×C₄	1	4	yes	—
56	D₁₆×C₇₁	C₂×C₄	1	2	yes	—
57	D₁₆×C₇₁	C₂×C₄	1	2	yes	—
58	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
64	#11765849	—	2	4	no	—
65	#11765849	—	4	2	no	—
66	#11765849	—	4	8	no	—
67	#11765849	D₈	1	2	yes	—
68	#11765849	D₈	1	2	yes	—
69	C₂×C₇₁⋊C₈	—	4	2	no	—
70	C₂×C₇₁⋊C₈	—	4	8	no	—
71	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
72	C₄⋊Dic₇₁	—	2	1	no	—
73	C₂²⋊Dic₇₁	—	2	1	no	—
74	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
75	C₇₁⋊C₁₆	—	2	2	no	—
76	#41×C₇₁	—	2	4	no	—
77	#41×C₇₁	—	2	4	no	—
78	#41×C₇₁	—	4	2	no	—
79	#41×C₇₁	—	4	2	no	—
80	C₂×Q₈×C₇₁	—	2	2	no	—
81	C₂×D₄×C₇₁	—	2	2	no	—
82	C₂×D₄×C₇₁	—	4	2	no	—
83	C₂²×C₄×C₇₁	—	4	2	no	—
84	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
85	Q₁₆×C₇₁	—	2	2	no	—
86	Q₁₆×C₇₁	C₂²×C₄	1	4	yes	—
87	SD₁₆×C₇₁	—	2	4	no	—
88	SD₁₆×C₇₁	—	2	4	no	—
89	D₈×C₇₁	—	2	2	no	—
90	D₈×C₇₁	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
95	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
97	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₇₁	—	4	2	no	—
102	C₂³×C₇₁	#135	1	1	yes	characteristic
103	Q₈×C₇₁	—	2	4	no	—
104	Q₈×C₇₁	—	4	2	no	—
105	D₄×C₇₁	—	2	4	no	—
106	D₄×C₇₁	—	4	4	no	—
107	D₄×C₇₁	—	4	4	no	—
108	D₄×C₇₁	—	4	2	no	—
109	C₂×C₄×C₇₁	—	4	2	no	—
110	C₂×C₄×C₇₁	—	4	4	no	—
111	C₂×C₄×C₇₁	—	4	4	no	—
112	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₇₁	—	2	2	no	—
119	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
120	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
121	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
122	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
123	C₇₁⋊C₈	—	4	4	no	—
124	C₇₁⋊C₈	C₂×D₈	1	4	yes	—
125	C₂²×C₇₁	—	4	1	no	—
126	C₂²×C₇₁	—	4	4	no	—
127	C₂²×C₇₁	—	16	2	no	—
128	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
129	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
130	C₂²×C₇₁	C₂×#135	1	1	yes	characteristic
131	C₄×C₇₁	—	4	4	no	—
132	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
133	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
134	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
135	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
136	Dic₇₁	—	4	2	no	—
137	#33465	—	71	1	no	maximal, Sylow
138	C₂×C₇₁	—	2	1	no	—
139	C₂×C₇₁	—	2	1	no	—
140	C₂×C₇₁	—	2	1	no	—
141	C₂×C₇₁	—	8	4	no	—
142	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
143	C₂×#508	D₇₁	1	1	yes	characteristic
144	#2957	—	71	2	no	—
145	#2179	—	71	4	no	—
146	#2178	—	71	4	no	—
147	C₂×#368	—	71	1	no	—
148	#2143	—	71	2	no	—
149	#2108	—	71	1	no	—
150	C₇₁	#33465	1	1	yes	characteristic, complemented, Sylow
151	C₂×#135	C₂×D₇₁	1	2	yes	—
152	#568	—	142	2	no	—
153	#508	Dic₇₁	1	2	yes	complemented
154	#508	Dic₇₁	1	2	yes	complemented
155	#508	C₂×D₇₁	1	2	yes	—
156	#508	C₂×D₇₁	1	2	yes	—
157	C₂×Q₃₂	Dic₇₁	1	1	yes	characteristic, complemented
158	C₂×SD₃₂	Dic₇₁	1	2	yes	complemented
159	C₂×D₁₆	Dic₇₁	1	1	yes	characteristic, complemented
160	C₂×M₃₂	—	71	1	no	—
161	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
162	#444	—	142	4	no	—
163	#443	—	71	8	no	—
164	C₂×#108	—	71	2	no	—
165	#421	—	142	2	no	—
166	#405	—	71	1	no	—
167	#368	—	71	2	no	—
168	#368	—	71	2	no	—
169	#359	—	71	8	no	—
170	#346	—	71	2	no	—
171	C₄×C₁₆	—	71	2	no	—
172	C₂×#41	—	2	2	no	—
173	#135	—	2	2	no	—
174	#135	—	2	2	no	—
175	#135	C₂×Dic₇₁	1	4	yes	—
176	#135	C₂²×D₇₁	1	4	yes	—
177	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×D₈	C₂×Dic₇₁	1	2	yes	—
180	#131	—	284	4	no	—
181	#131	—	142	8	no	—
182	C₂×M₁₆	—	284	4	no	—
183	C₂×M₁₆	—	71	1	no	—
184	C₂×M₁₆	—	142	2	no	—
185	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
186	#117	—	71	1	no	—
187	Q₃₂	C₂×Dic₇₁	1	2	yes	—
188	Q₃₂	C₂×Dic₇₁	1	2	yes	—
189	SD₃₂	C₂×Dic₇₁	1	4	yes	—
190	SD₃₂	C₂×Dic₇₁	1	4	yes	—
191	D₁₆	C₂×Dic₇₁	1	2	yes	—
192	D₁₆	C₂×Dic₇₁	1	2	yes	—
193	M₃₂	—	142	1	no	—
194	M₃₂	—	142	1	no	—
195	C₂×C₁₆	—	71	2	no	—
196	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
197	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
198	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
199	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
200	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
201	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
202	#108	—	71	4	no	—
203	#108	—	142	2	no	—
204	#104	—	142	8	no	—
205	#97	—	71	2	no	—
206	C₄×C₈	—	71	2	no	—
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	#41	—	4	2	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×Dic₇₁	1	4	yes	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	2	no	—
221	D₈	C₂²×Dic₇₁	1	4	yes	—
222	M₁₆	—	284	2	no	—
223	M₁₆	—	284	8	no	—
224	M₁₆	—	71	2	no	—
225	M₁₆	—	71	2	no	—
226	M₁₆	—	142	4	no	—
227	C₂×C₈	—	284	2	no	—
228	C₂×C₈	—	284	8	no	—
229	C₂×C₈	—	71	2	no	—
230	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
232	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
233	C₂×C₈	#11765864	1	1	yes	characteristic
234	C₂×C₈	#11765864	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
236	C₄⋊C₄	—	142	1	no	—
237	C₂²⋊C₄	—	142	1	no	—
238	C₄²	—	71	2	no	—
239	C₁₆	—	142	2	no	—
240	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
241	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
242	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
243	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	#14556147	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	142	2	no	—
256	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
258	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
259	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
260	C₂×C₄	#14556147	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
262	C₈	—	284	4	no	—
263	C₈	—	71	4	no	—
264	C₈	D₄×Dic₇₁	1	1	yes	characteristic
265	C₈	D₄×Dic₇₁	1	1	yes	characteristic
266	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
267	C₈	C₂×#11765864	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	D₈×Dic₇₁	1	1	yes	characteristic
272	C₂²	D₈×Dic₇₁	1	1	yes	characteristic
273	C₂²	C₂×#14556147	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	284	2	no	—
276	C₄	D₈×Dic₇₁	1	1	yes	characteristic
277	C₄	D₈×Dic₇₁	1	1	yes	characteristic
278	C₄	C₂×D₄×Dic₇₁	1	1	yes	characteristic
279	C₄	C₂×D₈×D₇₁	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×D₈×Dic₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₇₁
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting, prime radical	C₂×#508×C₇₁	C₂
socle	C₂×C₇₁	C₂×C₄×D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×#508	D₇₁
hypocenter	C₇₁	#33465
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #22597257	C₂	semidirect
2. #22590226	C₂	semidirect
3. #22590221	C₂	semidirect
4. C₂×#17985829	C₂	semidirect
5. #22590068	C₂	semidirect
6. #22589936	C₂	semidirect
11. #17986116	C₂²	semidirect
15. #17985829	C₂²	semidirect
16. #17985829	C₂²	semidirect
17. #17985806	C₂²	semidirect
18. #17985774	C₂²	semidirect
19. C₁₆×Dic₇₁	C₂²	semidirect
21. #508×C₇₁	C₄	semidirect
22. #508×C₇₁	C₄	semidirect
25. C₂×Q₃₂×C₇₁	C₄	semidirect
26. C₂×SD₃₂×C₇₁	C₄	semidirect
27. C₂×D₁₆×C₇₁	C₄	semidirect
150. C₇₁	#33465	semidirect
153. #508	Dic₇₁	semidirect
154. #508	Dic₇₁	semidirect
157. C₂×Q₃₂	Dic₇₁	semidirect
158. C₂×SD₃₂	Dic₇₁	semidirect
159. C₂×D₁₆	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22597257	2
#22590226	2
#22590221	2
C₂×#17985829	2
#22590068	2
#22589936	2
C₂×#508×C₇₁	2
#33465	71