Subgroups — #26103015

Order 12032.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241452	C₂	1	1	yes	maximal, characteristic
2	#20241305	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1367	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
5	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
7	C₄₇×C₄⋊C₁₆	—	2	1	no	—
8	C₈²×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄₇⋊C₁₆	—	2	1	no	—
10	C₂×C₄₇⋊C₁₆	—	4	1	no	—
11	C₂×C₄₇⋊C₁₆	C₈	1	1	yes	characteristic
12	C₂×C₄₇⋊C₁₆	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₄₇	—	4	1	no	—
14	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
16	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₄₇	—	2	1	no	—
18	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
19	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
20	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
21	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
22	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
23	C₁₆×C₄₇	—	8	1	no	—
24	C₄₇⋊C₁₆	—	2	2	no	—
25	C₄₇⋊C₁₆	—	2	1	no	—
26	C₄₇⋊C₁₆	—	8	1	no	—
27	C₄₇⋊C₁₆	C₁₆	1	2	yes	complemented
28	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
29	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
30	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
31	C₈×C₄₇	—	2	1	no	—
32	C₈×C₄₇	—	2	1	no	—
33	C₈×C₄₇	—	2	1	no	—
34	C₈×C₄₇	—	4	1	no	—
35	C₈×C₄₇	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₄₇	M₃₂	1	1	yes	characteristic
37	#7099	—	47	1	no	maximal, Sylow
38	C₂²×C₄₇	#334	1	1	yes	characteristic
39	C₄×C₄₇	—	2	1	no	—
40	C₄×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
41	C₄×C₄₇	#364	1	1	yes	characteristic
42	C₄×C₄₇	#365	1	1	yes	characteristic
43	C₄×C₄₇	#365	1	1	yes	characteristic
44	#1367	—	47	1	no	—
45	#1367	D₄₇	1	1	yes	characteristic
46	#1311	—	47	1	no	—
47	C₂×C₄₇	#1367	1	1	yes	characteristic
48	C₂×C₄₇	#1368	1	1	yes	characteristic
49	C₂×C₄₇	#1369	1	1	yes	characteristic
50	C₄⋊C₁₆	—	2	1	no	—
51	C₄⋊C₁₆	—	94	1	no	—
52	C₄×C₁₆	—	47	1	no	—
53	C₄×C₁₆	—	47	1	no	—
54	C₈²	C₂×D₄₇	1	1	yes	characteristic
55	C₄₇	#7099	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₁₆	—	4	1	no	—
57	C₂×C₁₆	—	47	1	no	—
58	C₂×C₁₆	—	47	1	no	—
59	C₂×C₁₆	—	94	1	no	—
60	C₂×C₁₆	—	188	1	no	—
61	C₄×C₈	Dic₉₄	1	1	yes	characteristic
62	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
63	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
66	C₂×C₈	#10625324	1	1	yes	characteristic
67	C₂×C₈	#10625335	1	1	yes	characteristic
68	C₂×C₈	#10625336	1	1	yes	characteristic
69	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
70	C₁₆	—	8	1	no	—
71	C₁₆	—	47	2	no	—
72	C₁₆	—	94	2	no	—
73	C₁₆	—	94	1	no	—
74	C₁₆	—	376	1	no	—
75	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
76	C₂×C₄	#13093955	1	1	yes	characteristic
77	C₂×C₄	#13093959	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	2	1	no	—
80	C₈	—	2	1	no	—
81	C₈	—	4	1	no	—
82	C₈	C₁₆×D₄₇	1	1	yes	characteristic
83	C₈	#13093963	1	1	yes	characteristic
84	C₂²	#15688873	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	Dic₄₇⋊C₁₆	1	1	yes	characteristic
87	C₄	#15688943	1	1	yes	characteristic
88	C₄	#15688945	1	1	yes	characteristic
89	C₄	#15688947	1	1	yes	characteristic
90	C₂	#20241451	1	1	yes	characteristic
91	C₂	#20241453	1	1	yes	characteristic
92	C₂	#20241455	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093955
commutator	C₈×C₄₇	C₂×C₁₆
Frattini	C₈²	C₂×D₄₇
Fitting	C₄₇×#1367	C₂
socle, prime radical	C₂²×C₄₇	#334
cosocle	C₈²×C₄₇	C₂²
hypercenter	#1367	D₄₇
hypocenter	C₄₇	#7099
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

Complements

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

N	H ≅ G/N	type
27. C₄₇⋊C₁₆	C₁₆	semidirect
55. C₄₇	#7099	semidirect

Maximal subgroups

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

subgroup	index
#20241452	2
#20241305	2
C₄₇×#1367	2
#7099	47