Subgroups — #31886817

Order 18176.

526 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588086	C₂	1	2	yes	maximal
2	C₂×#17985721	C₂	1	4	yes	maximal
3	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇₁⋊C₈	C₂²	1	4	yes	—
5	#17985754	C₄	1	8	yes	—
6	#17985721	C₄	1	16	yes	complemented
7	C₂×C₄×C₈×C₇₁	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
9	C₄³×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₈	C₂×C₄	1	8	yes	—
11	C₄×C₇₁⋊C₈	C₂×C₄	1	16	yes	—
12	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₇₁	—	2	8	no	—
17	C₄×C₈×C₇₁	C₂×C₄	1	8	yes	—
18	C₂×C₇₁⋊C₈	—	2	16	no	—
19	C₂×C₇₁⋊C₈	C₄²	1	16	yes	—
20	C₂²×C₄×C₇₁	C₄²	1	4	yes	—
21	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
22	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₈×C₇₁	—	2	8	no	—
24	C₂×C₈×C₇₁	C₄²	1	8	yes	—
25	C₄²×C₇₁	—	2	8	no	—
26	C₄²×C₇₁	M₁₆	1	8	yes	—
27	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
28	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄×C₇₁	—	2	8	no	—
30	C₂×C₄×C₇₁	#97	1	8	yes	—
31	C₂×C₄×C₇₁	C₂×C₄²	1	4	yes	—
32	C₂×C₄×C₇₁	C₂×C₄²	1	2	yes	—
33	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
34	C₂×C₄×C₇₁	#131	1	8	yes	—
35	C₈×C₇₁	—	2	8	no	—
36	C₇₁⋊C₈	—	2	16	no	—
37	C₂²×C₇₁	C₄³	1	1	yes	characteristic
38	C₂²×C₇₁	C₂×#97	1	2	yes	—
39	C₂²×C₇₁	#405	1	4	yes	—
40	C₄×C₇₁	—	2	8	no	—
41	C₄×C₇₁	C₄×M₁₆	1	4	yes	—
42	C₄×C₇₁	#405	1	8	yes	—
43	#7843	—	71	1	no	maximal, Sylow
44	C₂×C₇₁	#1452	1	4	yes	—
45	C₂×C₇₁	C₄×#97	1	1	yes	characteristic
46	C₂×C₇₁	#1726	1	2	yes	—
47	#1744	—	71	2	no	—
48	C₄²×C₈	D₇₁	1	1	yes	characteristic
49	C₂×#322	—	71	4	no	—
50	C₇₁	#7843	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₄×C₈	—	71	4	no	—
52	C₂×C₄×C₈	Dic₇₁	1	4	yes	—
53	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
54	C₄³	C₂×D₇₁	1	1	yes	characteristic
55	#336	—	71	8	no	—
56	#322	—	71	16	no	—
57	C₂²×C₈	—	71	8	no	—
58	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
59	C₂×C₄²	C₄×D₇₁	1	4	yes	—
60	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
61	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
62	C₄×C₈	—	2	8	no	—
63	C₄×C₈	—	71	16	no	—
64	C₄×C₈	C₄×D₇₁	1	8	yes	—
65	C₂²×C₄	C₄×Dic₇₁	1	4	yes	—
66	C₂²×C₄	C₂×C₄×D₇₁	1	2	yes	—
67	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
68	C₂×C₈	—	2	8	no	—
69	C₂×C₈	—	71	16	no	—
70	C₂×C₈	—	142	16	no	—
71	C₂×C₈	C₄×Dic₇₁	1	8	yes	—
72	C₄²	—	2	8	no	—
73	C₄²	#11765849	1	8	yes	—
74	C₄²	C₂×C₄×D₇₁	1	4	yes	—
75	C₂³	C₂×C₄×Dic₇₁	1	1	yes	characteristic
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	#14556039	1	8	yes	—
78	C₂×C₄	C₄²×D₇₁	1	4	yes	—
79	C₂×C₄	C₂×C₄×Dic₇₁	1	2	yes	—
80	C₂×C₄	C₂×#11765849	1	4	yes	—
81	C₂×C₄	#14556134	1	8	yes	—
82	C₈	—	2	8	no	—
83	C₈	—	142	16	no	—
84	C₂²	C₄²×Dic₇₁	1	1	yes	characteristic
85	C₂²	C₂×#14556039	1	2	yes	—
86	C₂²	#17985954	1	4	yes	—
87	C₄	—	2	8	no	—
88	C₄	C₄×#11765849	1	4	yes	—
89	C₄	#17985953	1	8	yes	—
90	C₂	#22586632	1	4	yes	—
91	C₂	C₄×#14556039	1	1	yes	characteristic
92	C₂	#22588026	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator	C₂²×C₇₁	C₄³
Fitting	C₄²×C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#7843
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #17985721	C₄	semidirect
50. C₇₁	#7843	semidirect

Maximal subgroups

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

subgroup	index
#22588086	2
C₂×#17985721	2
C₄²×C₈×C₇₁	2
#7843	71