Subgroups — #26102895

Order 12032.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241306	C₂	1	1	yes	maximal, characteristic
2	#20241305	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1312	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
5	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
7	C₄×C₁₆×C₄₇	C₄	1	2	yes	—
8	C₈²×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄₇⋊C₁₆	—	2	2	no	—
10	C₂×C₄₇⋊C₁₆	C₈	1	2	yes	—
11	C₂×C₄₇⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₁₆×C₄₇	—	2	2	no	—
13	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
18	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
19	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
20	C₄²×C₄₇	C₄²	1	1	yes	characteristic
21	C₁₆×C₄₇	—	4	2	no	—
22	C₄₇⋊C₁₆	—	2	2	no	—
23	C₄₇⋊C₁₆	—	4	2	no	—
24	C₄₇⋊C₁₆	C₁₆	1	4	yes	complemented
25	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
27	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
28	C₈×C₄₇	—	2	2	no	—
29	C₈×C₄₇	—	2	2	no	—
30	C₈×C₄₇	C₂×C₁₆	1	2	yes	—
31	C₈×C₄₇	M₃₂	1	2	yes	—
32	#7053	—	47	1	no	maximal, Sylow
33	C₂²×C₄₇	#322	1	1	yes	characteristic
34	C₄×C₄₇	C₄×C₁₆	1	1	yes	characteristic
35	C₄×C₄₇	#346	1	1	yes	characteristic
36	C₄×C₄₇	#347	1	2	yes	—
37	C₄×C₄₇	#347	1	2	yes	—
38	#1312	—	47	1	no	—
39	#1312	D₄₇	1	1	yes	characteristic
40	#1311	—	47	1	no	—
41	C₂×C₄₇	#1312	1	1	yes	characteristic
42	C₂×C₄₇	#1313	1	1	yes	characteristic
43	C₂×C₄₇	#1313	1	1	yes	characteristic
44	C₄×C₁₆	—	47	1	no	—
45	C₄×C₁₆	—	47	1	no	—
46	C₄×C₁₆	—	47	2	no	—
47	C₄×C₁₆	Dic₄₇	1	2	yes	—
48	C₈²	C₂×D₄₇	1	1	yes	characteristic
49	C₄₇	#7053	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	47	2	no	—
52	C₂×C₁₆	—	47	2	no	—
53	C₂×C₁₆	—	94	2	no	—
54	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
55	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
56	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
57	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
58	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
59	C₂×C₈	#10625324	1	2	yes	—
60	C₂×C₈	#10625329	1	2	yes	—
61	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
62	C₁₆	—	4	2	no	—
63	C₁₆	—	47	4	no	—
64	C₁₆	—	94	2	no	—
65	C₁₆	—	188	2	no	—
66	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
67	C₂×C₄	#13093934	1	1	yes	characteristic
68	C₂×C₄	#13093935	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	—	2	2	no	—
71	C₈	C₁₆×D₄₇	1	2	yes	—
72	C₈	#13093963	1	2	yes	—
73	C₂²	#15688845	1	1	yes	characteristic
74	C₄	C₁₆×Dic₄₇	1	1	yes	characteristic
75	C₄	#15688898	1	1	yes	characteristic
76	C₄	#15688900	1	2	yes	—
77	C₄	#15688901	1	2	yes	—
78	C₂	#20241307	1	1	yes	characteristic
79	C₂	#20241309	1	1	yes	characteristic
80	C₂	#20241310	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄×Dic₄₇
commutator	C₄×C₄₇	C₄×C₁₆
Frattini	C₈²	C₂×D₄₇
Fitting	C₄₇×#1312	C₂
socle, prime radical	C₂²×C₄₇	#322
cosocle	C₈²×C₄₇	C₂²
hypercenter	#1312	D₄₇
hypocenter	C₄₇	#7053
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #1312

Complements

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

N	H ≅ G/N	type
24. C₄₇⋊C₁₆	C₁₆	semidirect
49. C₄₇	#7053	semidirect

Maximal subgroups

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

subgroup	index
#20241306	2
#20241305	2
C₄₇×#1312	2
#7053	47