Subgroups — #31887891

Order 18176.

354 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588026	C₂	1	1	yes	maximal, characteristic
2	#22586623	C₂	1	1	yes	maximal, characteristic
3	#22586170	C₂	1	2	yes	maximal
4	#22586168	C₂	1	2	yes	maximal
5	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×#14556039	C₂²	1	1	yes	characteristic
8	C₂×#14556038	C₄	1	2	yes	—
9	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
10	#17985845	C₂²	1	1	yes	characteristic
11	#17985720	C₄	1	2	yes	complemented
12	C₈×C₇₁⋊C₈	C₄	1	2	yes	complemented
13	#431×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#322×C₇₁	C₄	1	2	yes	complemented
16	C₈²×C₇₁	C₄	1	2	yes	complemented
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
19	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
21	#14556039	—	2	2	no	—
22	#14556038	—	2	4	no	—
23	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
24	C₈×Dic₇₁	C₈	1	8	yes	complemented
25	C₇₁⋊#95	—	2	2	no	—
26	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₇₁	—	2	2	no	—
30	C₂²⋊C₈×C₇₁	—	2	2	no	—
31	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₇₁	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	2	no	—
38	C₂×C₇₁⋊C₈	—	2	2	no	—
39	C₂×C₇₁⋊C₈	—	2	4	no	—
40	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
41	C₄×Dic₇₁	—	2	2	no	—
42	C₄×Dic₇₁	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	#41	1	2	yes	—
45	C₂×C₈×C₇₁	—	2	4	no	—
46	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
52	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₇₁	#41	1	2	yes	—
55	C₂³×C₇₁	#117	1	1	yes	characteristic
56	C₂×C₄×C₇₁	—	4	2	no	—
57	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#117	1	2	yes	—
59	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#117	1	2	yes	—
61	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₇₁	—	2	4	no	—
64	C₂×Dic₇₁	—	4	4	no	—
65	C₈×C₇₁	—	4	4	no	—
66	C₈×C₇₁	C₄×C₈	1	2	yes	—
67	C₈×C₇₁	C₄×C₈	1	2	yes	—
68	C₈×C₇₁	#97	1	2	yes	—
69	C₈×C₇₁	#97	1	2	yes	—
70	C₇₁⋊C₈	—	2	4	no	—
71	C₇₁⋊C₈	—	4	4	no	—
72	C₂²×C₇₁	—	2	2	no	—
73	C₂²×C₇₁	#376	1	1	yes	characteristic
74	C₂²×C₇₁	#431	1	2	yes	—
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	—	2	2	no	—
77	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
78	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
79	C₄×C₇₁	#431	1	2	yes	—
80	Dic₇₁	—	4	4	no	—
81	#8137	—	71	1	no	maximal, Sylow
82	C₂×C₇₁	—	2	2	no	—
83	C₂×C₇₁	#1744	1	1	yes	characteristic
84	C₂×C₇₁	#1747	1	2	yes	—
85	#1726	—	71	1	no	—
86	#1450	—	71	1	no	—
87	#1450	D₇₁	1	1	yes	characteristic
88	#1275	—	71	2	no	—
89	#1274	—	71	2	no	—
90	C₇₁	#8137	1	1	yes	characteristic, complemented, Sylow
91	#431	—	71	2	no	—
92	#431	C₂×D₇₁	1	2	yes	—
93	C₂×#97	—	71	2	no	—
94	C₂×#97	—	71	1	no	—
95	C₂×C₄×C₈	—	71	2	no	—
96	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
97	#376	—	71	1	no	—
98	#322	—	71	2	no	—
99	#322	Dic₇₁	1	2	yes	complemented
100	C₈²	—	71	2	no	—
101	C₈²	Dic₇₁	1	2	yes	complemented
102	C₂²×C₈	—	71	2	no	—
103	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
105	C₂×C₄²	—	71	2	no	—
106	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	142	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₂²⋊C₈	—	142	2	no	—
111	#97	—	142	4	no	—
112	#97	—	142	2	no	—
113	C₄×C₈	—	71	8	no	—
114	C₄×C₈	—	71	4	no	—
115	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
116	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₇₁	1	4	yes	—
118	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
120	#95	—	142	2	no	—
121	C₂²×C₄	—	142	2	no	—
122	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
123	C₂²×C₄	#11765865	1	2	yes	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	71	4	no	—
126	C₂×C₈	—	142	4	no	—
127	C₂×C₈	—	142	2	no	—
128	C₂×C₈	—	142	2	no	—
129	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
130	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
132	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
133	C₂×C₈	C₈×D₇₁	1	4	yes	—
134	C₂×C₈	#11765848	1	4	yes	—
135	C₄²	—	71	4	no	—
136	C₄²	—	142	2	no	—
137	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
138	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
139	C₄²	#11765865	1	2	yes	—
140	C₂³	#14556085	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	284	4	no	—
143	C₂×C₄	—	142	4	no	—
144	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
145	C₂×C₄	#14556085	1	2	yes	—
146	C₂×C₄	#14556085	1	1	yes	characteristic
147	C₂×C₄	#14556087	1	2	yes	—
148	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
149	C₂×C₄	C₂×#11765848	1	2	yes	—
150	C₈	—	4	4	no	—
151	C₈	—	284	4	no	—
152	C₈	—	142	4	no	—
153	C₈	C₈×Dic₇₁	1	2	yes	—
154	C₈	C₈×Dic₇₁	1	2	yes	—
155	C₈	#14556038	1	2	yes	—
156	C₈	#14556038	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	#17985845	1	1	yes	characteristic
159	C₂²	#17986050	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	284	4	no	—
163	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
164	C₄	C₂×#14556038	1	1	yes	characteristic
165	C₄	#17986050	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#22588085	1	1	yes	characteristic
168	C₂	#22588095	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1450	C₂
socle	C₂²×C₇₁	#376
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1450	D₇₁
hypocenter	C₇₁	#8137
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
11. #17985720	C₄	semidirect
12. C₈×C₇₁⋊C₈	C₄	semidirect
15. #322×C₇₁	C₄	semidirect
16. C₈²×C₇₁	C₄	semidirect
24. C₈×Dic₇₁	C₈	semidirect
90. C₇₁	#8137	semidirect
99. #322	Dic₇₁	semidirect
101. C₈²	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22588026	2
#22586623	2
#22586170	2
#22586168	2
C₇₁×#1450	2
#8137	71