Subgroups — #25249116

Order 11008.

2046 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583276	C₂	1	12	yes	maximal, complemented
2	#19142990	C₂	1	2	yes	maximal, complemented
3	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₃₄₄	C₂²	1	16	yes	—
5	C₂×#13063322	—	2	8	no	—
6	C₂×#13063322	C₂²	1	12	yes	—
7	#14583276	—	2	48	no	—
8	C₂×C₄×C₈×C₄₃	C₂²	1	6	yes	—
9	C₄³×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×D₁₇₂	—	2	24	no	—
11	C₂²×D₁₇₂	C₂³	1	8	yes	—
12	C₂×D₃₄₄	—	2	96	no	—
13	#13063322	—	2	48	no	—
14	#13063322	—	4	32	no	—
15	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
16	C₂×C₄²×C₄₃	D₄	1	4	yes	—
17	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
18	C₄×C₈×C₄₃	D₄	1	24	yes	complemented
19	C₂³×D₄₃	—	2	8	no	—
20	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
21	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
22	C₂×D₁₇₂	—	2	48	no	—
23	C₂×D₁₇₂	—	4	48	no	—
24	C₂×D₁₇₂	—	4	96	no	—
25	C₂×C₈×C₄₃	C₂×D₄	1	24	yes	—
26	C₄²×C₄₃	D₈	1	16	yes	complemented
27	C₄²×C₄₃	C₂×D₄	1	12	yes	—
28	D₃₄₄	—	4	64	no	—
29	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
30	C₂²×D₄₃	—	4	24	no	—
31	C₂²×D₄₃	—	4	32	no	—
32	C₂×C₄×C₄₃	#127	1	12	yes	—
33	C₂×C₄×C₄₃	C₂×D₈	1	24	yes	—
34	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
35	D₁₇₂	—	4	32	no	—
36	D₁₇₂	—	8	96	no	—
37	C₈×C₄₃	#127	1	16	yes	complemented
38	#33301	—	43	1	no	maximal, Sylow
39	C₂²×C₄₃	C₂×#127	1	3	yes	—
40	C₂²×C₄₃	C₂²×D₈	1	4	yes	—
41	C₂×D₄₃	—	8	8	no	—
42	C₂×D₄₃	—	8	48	no	—
43	C₄×C₄₃	#493	1	24	yes	complemented
44	C₄×C₄₃	C₂×#127	1	4	yes	—
45	C₂×#493	—	43	12	no	—
46	#2867	—	43	2	no	—
47	C₄²×C₈	D₄₃	1	1	yes	characteristic, complemented
48	C₂×C₄₃	#2867	1	1	yes	characteristic
49	C₂×C₄₃	C₂×#493	1	6	yes	—
50	D₄₃	—	16	8	no	—
51	C₂²×D₈	—	43	16	no	—
52	C₂×#127	—	43	12	no	—
53	C₂×#127	—	86	8	no	—
54	#493	—	86	48	no	—
55	C₂×C₄×C₈	C₂×D₄₃	1	6	yes	—
56	C₄³	C₂×D₄₃	1	1	yes	characteristic
57	C₄₃	#33301	1	1	yes	characteristic, complemented, Sylow
58	C₂²×D₄	—	43	8	no	—
59	C₂²×D₄	—	86	24	no	—
60	C₂×D₈	—	86	96	no	—
61	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
62	#127	—	86	48	no	—
63	#127	—	172	32	no	—
64	C₂×C₄²	D₁₇₂	1	4	yes	—
65	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
66	C₄×C₈	D₁₇₂	1	24	yes	complemented
67	C₂⁴	—	86	8	no	—
68	C₂×D₄	—	86	48	no	—
69	C₂×D₄	—	172	48	no	—
70	C₂×D₄	—	172	96	no	—
71	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
72	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
73	D₈	—	172	64	no	—
74	C₂×C₈	C₂×D₁₇₂	1	24	yes	—
75	C₄²	D₃₄₄	1	16	yes	complemented
76	C₄²	C₂×D₁₇₂	1	12	yes	—
77	C₂³	—	172	32	no	—
78	C₂³	—	172	24	no	—
79	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
80	D₄	—	344	96	no	—
81	D₄	—	172	32	no	—
82	C₂×C₄	#13063322	1	12	yes	—
83	C₂×C₄	C₂×D₃₄₄	1	24	yes	—
84	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
85	C₈	#13063322	1	16	yes	complemented
86	C₂²	—	344	48	no	—
87	C₂²	—	344	8	no	—
88	C₂²	C₂×#13063322	1	3	yes	—
89	C₂²	C₂²×D₃₄₄	1	4	yes	—
90	C₄	#14583276	1	24	yes	complemented
91	C₄	C₂×#13063322	1	4	yes	—
92	C₂	—	688	8	no	—
93	C₂	#19142990	1	1	yes	characteristic
94	C₂	C₂×#14583276	1	6	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#33301
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14583276	C₂	semidirect
2. #19142990	C₂	semidirect
3. C₄²×C₈×C₄₃	C₂	semidirect
18. C₄×C₈×C₄₃	D₄	semidirect
26. C₄²×C₄₃	D₈	semidirect
37. C₈×C₄₃	#127	semidirect
43. C₄×C₄₃	#493	semidirect
47. C₄²×C₈	D₄₃	semidirect
57. C₄₃	#33301	semidirect
66. C₄×C₈	D₁₇₂	semidirect
75. C₄²	D₃₄₄	semidirect
85. C₈	#13063322	semidirect
90. C₄	#14583276	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583276	2
#19142990	2
C₄²×C₈×C₄₃	2
#33301	43