Subgroups — #32150814

Order 18176.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597192	C₂	1	2	yes	maximal, complemented
2	#22590216	C₂	1	4	yes	maximal, complemented
3	D₁₆×Dic₇₁	C₂	1	4	yes	maximal, complemented
4	#22590124	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590056	C₂	1	2	yes	maximal, complemented
6	#22589936	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#17986474	—	2	2	no	—
9	C₂×#14556068	C₂²	1	1	yes	characteristic
10	#17986111	—	2	4	no	—
11	D₈×Dic₇₁	C₂²	1	8	yes	complemented
12	#17986037	C₂²	1	2	yes	—
13	#17986004	—	2	2	no	—
14	#17985954	C₂²	1	1	yes	characteristic
15	#17985827	C₂²	1	2	yes	complemented
16	#17985825	C₂²	1	2	yes	complemented
17	#17985800	C₂²	1	8	yes	complemented
18	#17985774	C₂²	1	2	yes	complemented
19	C₁₆×Dic₇₁	C₂²	1	2	yes	complemented
20	C₂²×D₈×C₇₁	C₂²	1	2	yes	—
21	C₂×D₁₆×C₇₁	C₄	1	8	yes	complemented
22	C₂×D₁₆×C₇₁	C₂²	1	2	yes	—
23	C₂×D₁₆×C₇₁	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#11765849	D₄	1	1	yes	characteristic
26	D₄×Dic₇₁	—	2	8	no	—
27	D₄×Dic₇₁	—	4	4	no	—
28	#14556087	—	2	2	no	—
29	#14556087	D₄	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₇₁	—	4	4	no	—
31	#14556068	—	2	2	no	—
32	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
33	#14556062	C₂³	1	4	yes	—
34	#14556060	—	2	2	no	—
35	#14556049	—	2	8	no	—
36	#14556038	D₄	1	2	yes	—
37	C₈×Dic₇₁	C₂³	1	2	yes	—
38	C₂²×D₄×C₇₁	—	2	2	no	—
39	C₂×D₈×C₇₁	—	2	2	no	—
40	C₂×D₈×C₇₁	—	2	4	no	—
41	C₂×D₈×C₇₁	C₂×C₄	1	8	yes	complemented
42	C₂×D₈×C₇₁	C₂³	1	4	yes	—
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	D₁₆×C₇₁	C₂×C₄	1	8	yes	complemented
45	D₁₆×C₇₁	C₂×C₄	1	8	yes	complemented
46	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₇₁	—	4	8	no	—
50	#11765849	—	2	2	no	—
51	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
52	C₄⋊Dic₇₁	—	2	4	no	—
53	C₄⋊Dic₇₁	D₈	1	2	yes	—
54	C₂²⋊Dic₇₁	—	4	2	no	—
55	C₂²⋊Dic₇₁	—	4	8	no	—
56	C₂²⋊Dic₇₁	D₈	1	2	yes	—
57	C₄×Dic₇₁	—	4	2	no	—
58	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
59	C₇₁⋊C₁₆	—	2	2	no	—
60	C₂⁴×C₇₁	—	4	2	no	—
61	C₂×D₄×C₇₁	—	2	8	no	—
62	C₂×D₄×C₇₁	—	2	4	no	—
63	C₂×D₄×C₇₁	—	4	2	no	—
64	C₂×D₄×C₇₁	—	4	4	no	—
65	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	D₈×C₇₁	—	2	8	no	—
67	D₈×C₇₁	—	2	4	no	—
68	D₈×C₇₁	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	#41	1	2	yes	—
72	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
73	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
74	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
75	C₂³×C₇₁	—	4	4	no	—
76	C₂³×C₇₁	—	4	8	no	—
77	C₂³×C₇₁	—	16	4	no	—
78	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
79	D₄×C₇₁	—	2	8	no	—
80	D₄×C₇₁	—	4	4	no	—
81	D₄×C₇₁	—	4	8	no	—
82	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#135	1	2	yes	—
85	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₇₁	—	4	4	no	—
87	C₂×Dic₇₁	—	16	8	no	—
88	C₂×Dic₇₁	C₂×D₈	1	4	yes	—
89	C₈×C₇₁	C₄×D₄	1	2	yes	—
90	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
92	C₇₁⋊C₈	—	2	2	no	—
93	C₂²×C₇₁	—	2	2	no	—
94	C₂²×C₇₁	—	4	8	no	—
95	C₂²×C₇₁	—	8	8	no	—
96	C₂²×C₇₁	—	16	8	no	—
97	C₂²×C₇₁	C₄×D₈	1	2	yes	—
98	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
99	C₄×C₇₁	C₄×D₈	1	2	yes	—
100	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
101	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
102	Dic₇₁	—	2	4	no	—
103	Dic₇₁	—	8	4	no	—
104	#33456	—	71	1	no	maximal, Sylow
105	C₂×C₇₁	—	2	2	no	—
106	C₂×C₇₁	—	8	8	no	—
107	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
108	C₂×C₇₁	#3415	1	2	yes	—
109	C₂²×D₁₆	D₇₁	1	1	yes	characteristic
110	#2948	—	71	2	no	—
111	#2177	—	71	4	no	—
112	C₄×D₁₆	—	71	4	no	—
113	#2158	—	71	1	no	—
114	#2140	—	71	2	no	—
115	#2108	—	71	1	no	—
116	C₇₁	#33456	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₈	C₂×D₇₁	1	2	yes	—
118	#518	—	142	2	no	—
119	C₂×D₁₆	Dic₇₁	1	8	yes	complemented
120	C₂×D₁₆	C₂×D₇₁	1	2	yes	—
121	C₂×D₁₆	C₂×D₇₁	1	2	yes	—
122	C₂×M₃₂	—	71	1	no	—
123	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
124	#442	—	142	4	no	—
125	C₄×D₈	—	71	8	no	—
126	#427	—	71	2	no	—
127	#418	—	142	2	no	—
128	#405	—	71	1	no	—
129	#367	—	71	2	no	—
130	#366	—	71	2	no	—
131	#357	—	71	8	no	—
132	#346	—	71	2	no	—
133	C₄×C₁₆	—	71	2	no	—
134	C₂²×D₄	—	2	2	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	C₂×Dic₇₁	1	8	yes	complemented
138	C₂×D₈	C₂²×D₇₁	1	4	yes	—
139	C₂×M₁₆	—	71	1	no	—
140	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
141	C₄×D₄	—	284	4	no	—
142	C₄×D₄	—	142	8	no	—
143	#117	—	71	1	no	—
144	#117	—	142	2	no	—
145	C₂×C₂²⋊C₄	—	284	4	no	—
146	D₁₆	C₂×Dic₇₁	1	8	yes	complemented
147	D₁₆	C₂×Dic₇₁	1	8	yes	complemented
148	M₃₂	—	142	2	no	—
149	C₂×C₁₆	—	71	2	no	—
150	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
153	#107	—	71	4	no	—
154	#106	—	142	2	no	—
155	#102	—	142	8	no	—
156	#97	—	71	2	no	—
157	C₄×C₈	—	71	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	8	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	284	8	no	—
164	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
165	D₈	—	2	4	no	—
166	D₈	—	2	8	no	—
167	D₈	C₂²×Dic₇₁	1	8	yes	—
168	M₁₆	—	142	2	no	—
169	C₂×C₈	—	71	2	no	—
170	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
171	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
172	C₂×C₈	#11765864	1	2	yes	—
173	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
174	C₄⋊C₄	—	71	2	no	—
175	C₄⋊C₄	—	142	4	no	—
176	C₂²⋊C₄	—	284	2	no	—
177	C₂²⋊C₄	—	284	8	no	—
178	C₂²⋊C₄	—	71	2	no	—
179	C₄²	—	284	2	no	—
180	C₄²	—	71	2	no	—
181	C₁₆	—	142	2	no	—
182	C₁₆	C₂²×Dic₇₁	1	2	yes	—
183	C₁₆	C₂²×Dic₇₁	1	2	yes	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	16	4	no	—
187	C₂³	D₈×D₇₁	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	D₄	—	4	8	no	—
190	D₄	—	4	4	no	—
191	C₂×C₄	—	284	4	no	—
192	C₂×C₄	—	71	4	no	—
193	C₂×C₄	—	1136	8	no	—
194	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
195	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
196	C₂×C₄	#14556147	1	2	yes	—
197	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
198	C₈	—	142	2	no	—
199	C₈	D₄×Dic₇₁	1	2	yes	—
200	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
201	C₈	C₂×#11765864	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	D₈×Dic₇₁	1	2	yes	—
207	C₂²	C₂×D₈×D₇₁	1	1	yes	characteristic
208	C₄	—	568	4	no	—
209	C₄	—	142	4	no	—
210	C₄	D₈×Dic₇₁	1	2	yes	—
211	C₄	C₂×D₄×Dic₇₁	1	1	yes	characteristic
212	C₄	C₂×#14556147	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×D₈×Dic₇₁	1	1	yes	characteristic
216	C₂	#22601477	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₁
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting, prime radical	C₂²×D₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂²×D₁₆	D₇₁
hypocenter	C₇₁	#33456
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. #22597192	C₂	semidirect
2. #22590216	C₂	semidirect
3. D₁₆×Dic₇₁	C₂	semidirect
4. #22590124	C₂	semidirect
5. #22590056	C₂	semidirect
6. #22589936	C₂	semidirect
11. D₈×Dic₇₁	C₂²	semidirect
15. #17985827	C₂²	semidirect
16. #17985825	C₂²	semidirect
17. #17985800	C₂²	semidirect
18. #17985774	C₂²	semidirect
19. C₁₆×Dic₇₁	C₂²	semidirect
21. C₂×D₁₆×C₇₁	C₄	semidirect
41. C₂×D₈×C₇₁	C₂×C₄	semidirect
44. D₁₆×C₇₁	C₂×C₄	semidirect
45. D₁₆×C₇₁	C₂×C₄	semidirect
46. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
116. C₇₁	#33456	semidirect
119. C₂×D₁₆	Dic₇₁	semidirect
137. C₂×D₈	C₂×Dic₇₁	semidirect
146. D₁₆	C₂×Dic₇₁	semidirect
147. D₁₆	C₂×Dic₇₁	semidirect
150. C₂×C₁₆	C₂×Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22597192	2
#22590216	2
D₁₆×Dic₇₁	2
#22590124	2
#22590056	2
#22589936	2
C₂²×D₁₆×C₇₁	2
#33456	71