Subgroups — #25249119

Order 11008.

1422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583279	C₂	1	12	yes	maximal
2	#19142993	C₂	1	2	yes	maximal
3	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₇₂	C₂²	1	16	yes	—
5	C₂×#13063325	C₂²	1	12	yes	—
6	#14583279	—	2	48	no	—
7	#14582769	—	2	8	no	—
8	C₂×C₄×C₈×C₄₃	C₂²	1	6	yes	—
9	C₄³×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₈₆	C₂³	1	8	yes	—
11	C₂×Dic₁₇₂	—	2	96	no	—
12	#13063325	—	2	48	no	—
13	C₂×C₄⋊Dic₄₃	—	2	24	no	—
14	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
15	C₂×C₄²×C₄₃	D₄	1	4	yes	—
16	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
17	C₄×C₈×C₄₃	D₄	1	24	yes	—
18	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
19	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
20	C₂²×Dic₄₃	—	2	8	no	—
21	C₂×Dic₈₆	—	2	48	no	—
22	C₂×C₈×C₄₃	C₂×D₄	1	24	yes	—
23	C₄²×C₄₃	Q₁₆	1	16	yes	complemented
24	C₄²×C₄₃	C₂×D₄	1	12	yes	—
25	C₄⋊Dic₄₃	—	4	48	no	—
26	Dic₁₇₂	—	4	64	no	—
27	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₄×C₄₃	#127	1	12	yes	—
29	C₂×C₄×C₄₃	C₂×Q₁₆	1	24	yes	—
30	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
31	C₂×Dic₄₃	—	4	24	no	—
32	Dic₈₆	—	4	32	no	—
33	C₈×C₄₃	#127	1	16	yes	—
34	#33302	—	43	1	no	maximal, Sylow
35	C₂²×C₄₃	C₂×#127	1	3	yes	—
36	C₂²×C₄₃	C₂²×Q₁₆	1	4	yes	—
37	C₄×C₄₃	#494	1	24	yes	complemented
38	C₄×C₄₃	C₂×#127	1	4	yes	—
39	Dic₄₃	—	8	8	no	—
40	C₂×#494	—	43	12	no	—
41	#2868	—	43	2	no	—
42	C₄²×C₈	D₄₃	1	1	yes	characteristic
43	C₂×C₄₃	#2867	1	1	yes	characteristic
44	C₂×C₄₃	C₂×#494	1	6	yes	—
45	C₂²×Q₁₆	—	43	16	no	—
46	C₂×C₄⋊Q₈	—	43	12	no	—
47	#494	—	86	48	no	—
48	C₂×C₄×C₈	C₂×D₄₃	1	6	yes	—
49	#384	—	86	8	no	—
50	C₄³	C₂×D₄₃	1	1	yes	characteristic
51	C₄₃	#33302	1	1	yes	characteristic, complemented, Sylow
52	C₂²×Q₈	—	43	8	no	—
53	C₂×Q₁₆	—	86	96	no	—
54	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
55	C₄⋊Q₈	—	86	48	no	—
56	C₂×C₄⋊C₄	—	86	24	no	—
57	C₂×C₄²	D₁₇₂	1	4	yes	—
58	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
59	C₄×C₈	D₁₇₂	1	24	yes	—
60	C₂×Q₈	—	86	48	no	—
61	C₂²×C₄	—	86	8	no	—
62	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
63	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
64	Q₁₆	—	172	64	no	—
65	C₂×C₈	C₂×D₁₇₂	1	24	yes	—
66	C₄⋊C₄	—	172	48	no	—
67	C₄²	Dic₁₇₂	1	16	yes	complemented
68	C₄²	C₂×D₁₇₂	1	12	yes	—
69	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
70	Q₈	—	172	32	no	—
71	C₂×C₄	—	172	24	no	—
72	C₂×C₄	#13063322	1	12	yes	—
73	C₂×C₄	C₂×Dic₁₇₂	1	24	yes	—
74	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
75	C₈	#13063322	1	16	yes	—
76	C₂²	C₂×#13063322	1	3	yes	—
77	C₂²	C₂²×Dic₁₇₂	1	4	yes	—
78	C₄	—	344	8	no	—
79	C₄	#14583279	1	24	yes	complemented
80	C₄	C₂×#13063322	1	4	yes	—
81	C₂	#19142990	1	1	yes	characteristic
82	C₂	C₂×#14583279	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, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#33302
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
23. C₄²×C₄₃	Q₁₆	semidirect
37. C₄×C₄₃	#494	semidirect
51. C₄₃	#33302	semidirect
67. C₄²	Dic₁₇₂	semidirect
79. C₄	#14583279	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583279	2
#19142993	2
C₄²×C₈×C₄₃	2
#33302	43