Subgroups — #26103610

Order 12032.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊C₃₂	C₄	1	2	yes	—
5	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₄₇	C₄	1	2	yes	—
8	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄₇⋊C₁₆	C₈	1	2	yes	—
10	C₄×C₄₇⋊C₈	C₈	1	1	yes	characteristic
11	C₄×C₄₇⋊C₈	C₈	1	1	yes	characteristic
12	C₄₇⋊C₃₂	—	2	2	no	—
13	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₄₇	—	2	2	no	—
17	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
18	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
19	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
20	C₂×C₄₇⋊C₈	C₁₆	1	2	yes	—
21	C₂×C₄₇⋊C₈	C₁₆	1	2	yes	—
22	C₁₆×C₄₇	M₁₆	1	2	yes	—
23	C₁₆×C₄₇	M₁₆	1	2	yes	—
24	C₄₇⋊C₁₆	—	2	2	no	—
25	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
27	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₄₇	#97	1	1	yes	characteristic
29	C₈×C₄₇	#97	1	1	yes	characteristic
30	C₈×C₄₇	M₃₂	1	2	yes	—
31	C₄₇⋊C₈	—	2	2	no	—
32	C₄₇⋊C₈	C₃₂	1	4	yes	complemented
33	#7331	—	47	1	no	maximal, Sylow
34	C₂²×C₄₇	C₄×C₁₆	1	1	yes	characteristic
35	C₄×C₄₇	#322	1	1	yes	characteristic
36	C₄×C₄₇	#346	1	1	yes	characteristic
37	C₄×C₄₇	C₂×C₃₂	1	2	yes	—
38	C₄×C₄₇	M₆₄	1	2	yes	—
39	C₄×C₃₂	—	47	1	no	—
40	C₄×C₃₂	D₄₇	1	1	yes	characteristic
41	C₈×C₁₆	—	47	1	no	—
42	C₂×C₄₇	#1312	1	1	yes	characteristic
43	C₂×C₄₇	C₄×C₃₂	1	1	yes	characteristic
44	C₂×C₄₇	#1397	1	1	yes	characteristic
45	C₂×C₃₂	—	47	2	no	—
46	C₂×C₃₂	Dic₄₇	1	2	yes	—
47	C₄×C₁₆	—	47	1	no	—
48	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
49	C₈²	—	47	1	no	—
50	C₄₇	#7331	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₁₆	—	47	2	no	—
52	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
53	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
54	C₄×C₈	—	47	1	no	—
55	C₄×C₈	—	47	1	no	—
56	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
57	C₃₂	—	2	2	no	—
58	C₃₂	—	94	2	no	—
59	C₂×C₈	—	47	2	no	—
60	C₂×C₈	—	47	2	no	—
61	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
62	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
63	C₄²	C₈×D₄₇	1	1	yes	characteristic
64	C₁₆	—	94	2	no	—
65	C₁₆	#10625324	1	2	yes	—
66	C₁₆	#10625329	1	2	yes	—
67	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
68	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
69	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
70	C₈	—	47	4	no	—
71	C₈	—	94	2	no	—
72	C₈	#13093934	1	1	yes	characteristic
73	C₈	#13093935	1	1	yes	characteristic
74	C₈	#13093963	1	2	yes	—
75	C₂²	C₁₆×Dic₄₇	1	1	yes	characteristic
76	C₄	#15688845	1	1	yes	characteristic
77	C₄	#15688898	1	1	yes	characteristic
78	C₄	C₃₂×D₄₇	1	2	yes	—
79	C₄	#15688956	1	2	yes	—
80	C₂	#20241307	1	1	yes	characteristic
81	C₂	C₃₂×Dic₄₇	1	1	yes	characteristic
82	C₂	#20241521	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₁₆	C₂×D₄₇
commutator	C₂×C₄₇	C₄×C₃₂
Fitting	C₄×C₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₄×C₁₆
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	C₄×C₃₂	D₄₇
hypocenter	C₄₇	#7331
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
32. C₄₇⋊C₈	C₃₂	semidirect
50. C₄₇	#7331	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄₇⋊C₃₂	2
C₁₆×C₄₇⋊C₈	2
C₄×C₃₂×C₄₇	2
#7331	47