Subgroups — #31913852

Order 18176.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586418	C₂	1	6	yes	maximal
2	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₇₁	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
6	C₄³×C₇₁	C₄	1	1	yes	characteristic
7	C₂×C₇₁⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₇₁	C₈	1	3	yes	—
11	C₂×C₄²×C₇₁	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₇₁	—	2	6	no	—
14	C₄×C₈×C₇₁	—	2	6	no	—
15	C₇₁⋊C₁₆	—	4	4	no	—
16	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
18	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₈×C₇₁	—	2	3	no	—
20	C₂×C₈×C₇₁	—	2	3	no	—
21	C₂×C₈×C₇₁	M₁₆	1	6	yes	—
22	C₂×C₈×C₇₁	#41	1	6	yes	—
23	C₄²×C₇₁	—	2	6	no	—
24	C₄²×C₇₁	—	2	3	no	—
25	C₄²×C₇₁	—	2	3	no	—
26	C₄²×C₇₁	C₁₆	1	4	yes	complemented
27	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₇₁	—	2	3	no	—
29	C₂×C₄×C₇₁	—	2	3	no	—
30	C₂×C₄×C₇₁	—	2	3	no	—
31	C₂×C₄×C₇₁	—	2	3	no	—
32	C₂×C₄×C₇₁	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₇₁	M₃₂	1	6	yes	—
34	C₂×C₄×C₇₁	#117	1	3	yes	—
35	C₂×C₄×C₇₁	C₂×M₁₆	1	3	yes	—
36	C₈×C₇₁	—	2	6	no	—
37	C₈×C₇₁	#126	1	4	yes	—
38	C₂²×C₇₁	#431	1	3	yes	—
39	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
40	C₂²×C₇₁	C₂×M₃₂	1	3	yes	—
41	C₄×C₇₁	—	2	6	no	—
42	C₄×C₇₁	—	2	6	no	—
43	C₄×C₇₁	#383	1	1	yes	characteristic
44	C₄×C₇₁	#432	1	3	yes	—
45	#12894	—	71	1	no	maximal, Sylow
46	C₂×C₇₁	#1839	1	1	yes	characteristic
47	C₂×C₇₁	#2162	1	3	yes	—
48	C₂×C₇₁	#2163	1	3	yes	—
49	C₄²×C₈	D₇₁	1	1	yes	characteristic
50	#1374	—	71	6	no	—
51	C₇₁	#12894	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	71	4	no	—
53	C₂×C₄×C₈	Dic₇₁	1	3	yes	—
54	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
55	C₄³	Dic₇₁	1	1	yes	characteristic
56	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
57	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
58	C₂×C₄²	C₇₁⋊C₈	1	3	yes	—
59	C₂×C₄²	C₇₁⋊C₈	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₇₁	1	3	yes	—
61	C₂×C₁₆	—	142	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
65	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#11765849	1	6	yes	—
70	C₂×C₈	#11765865	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	C₇₁⋊C₁₆	1	4	yes	complemented
75	C₁₆	—	284	4	no	—
76	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×C₇₁⋊C₁₆	1	6	yes	—
82	C₂×C₄	#14556068	1	6	yes	—
83	C₂×C₄	#14556087	1	3	yes	—
84	C₂×C₄	C₂×#11765849	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#14556122	1	4	yes	—
87	C₂²	#17986052	1	3	yes	—
88	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
89	C₂²	C₂×#14556068	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#17985867	1	1	yes	characteristic
93	C₄	#17986056	1	3	yes	—
94	C₂	#22588483	1	1	yes	characteristic
95	C₂	#22590139	1	3	yes	—
96	C₂	#22590143	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₄²×C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#12894
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
26. C₄²×C₇₁	C₁₆	semidirect
51. C₇₁	#12894	semidirect
74. C₄²	C₇₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22586418	2
C₄²×C₈×C₇₁	2
#12894	71