Subgroups — #42119541

Order 27584.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20293801	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₇₂₄	C₂	1	3	yes	maximal, complemented
4	#28352151	C₂	1	3	yes	maximal, complemented
5	#28352146	C₂	1	3	yes	maximal, complemented
6	C₄⋊C₄×D₄₃₁	C₂	1	3	yes	maximal, complemented
7	C₄×Q₈×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#20293801	C₄	1	8	yes	complemented
9	C₂×D₁₇₂₄	C₂²	1	3	yes	—
10	C₂×C₄×D₄₃₁	C₂²	1	6	yes	—
11	C₂×C₄×D₄₃₁	C₂²	1	3	yes	—
12	C₄⋊Dic₄₃₁	C₂²	1	3	yes	—
13	Dic₄₃₁⋊C₄	C₂²	1	3	yes	—
14	Dic₄₃₁⋊C₄	C₂²	1	1	yes	characteristic
15	(C₂×D₄₃₁)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₄₃₁)⋊C₄	C₂²	1	3	yes	—
17	C₄×Dic₄₃₁	C₂²	1	3	yes	—
18	C₂×Q₈×C₄₃₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₃₁	C₂²	1	3	yes	—
20	C₄²×C₄₃₁	C₂²	1	3	yes	—
21	C₂²×D₄₃₁	C₂³	1	3	yes	—
22	D₁₇₂₄	C₂×C₄	1	12	yes	complemented
23	C₂×Dic₄₃₁	C₂³	1	1	yes	characteristic
24	C₂×Dic₄₃₁	C₂³	1	3	yes	—
25	C₂×Dic₄₃₁	C₂³	1	1	yes	characteristic
26	C₄×D₄₃₁	—	4	6	no	—
27	C₄×D₄₃₁	C₂×C₄	1	12	yes	complemented
28	Q₈×C₄₃₁	C₂×C₄	1	4	yes	complemented
29	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₄₃₁	C₂³	1	3	yes	—
31	C₂×C₄×C₄₃₁	C₂³	1	3	yes	—
32	C₂²×C₄₃₁	C₂⁴	1	1	yes	characteristic
33	C₂×D₄₃₁	—	2	3	no	—
34	C₂×D₄₃₁	—	2	3	no	—
35	C₂×D₄₃₁	C₂²×C₄	1	6	yes	—
36	C₄×C₄₃₁	—	2	3	no	—
37	C₄×C₄₃₁	—	2	1	no	—
38	C₄×C₄₃₁	C₂²×C₄	1	6	yes	—
39	Dic₄₃₁	—	2	3	no	—
40	Dic₄₃₁	—	2	1	no	—
41	Dic₄₃₁	C₂²×C₄	1	2	yes	—
42	C₂×C₄₃₁	C₂³×C₄	1	1	yes	characteristic
43	C₂×C₄₃₁	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂×C₄₃₁	2₋¹⁺⁴	1	1	yes	characteristic
45	D₄₃₁	—	2	6	no	—
46	C₄₃₁	#520	1	1	yes	characteristic, complemented, Sylow
47	#520	—	431	1	no	maximal, Sylow
48	C₂×#41	—	431	1	no	—
49	C₄×Q₈	—	431	1	no	—
50	C₄×Q₈	D₄₃₁	1	1	yes	characteristic, complemented
51	C₄×D₄	—	431	3	no	—
52	C₄×D₄	—	431	3	no	—
53	#117	—	431	3	no	—
54	C₂×C₄⋊C₄	—	431	3	no	—
55	#41	—	431	8	no	—
56	C₂×Q₈	C₂×D₄₃₁	1	1	yes	characteristic
57	C₂×D₄	—	431	3	no	—
58	C₂²×C₄	—	431	3	no	—
59	C₂²×C₄	—	431	6	no	—
60	C₄⋊C₄	—	431	3	no	—
61	C₄⋊C₄	—	431	3	no	—
62	C₄⋊C₄	—	431	1	no	—
63	C₄⋊C₄	C₂×D₄₃₁	1	3	yes	—
64	C₂²⋊C₄	—	431	3	no	—
65	C₂²⋊C₄	—	431	3	no	—
66	C₄²	—	431	3	no	—
67	C₄²	C₂×D₄₃₁	1	3	yes	—
68	C₂³	—	431	3	no	—
69	Q₈	C₄×D₄₃₁	1	4	yes	complemented
70	D₄	—	431	12	no	—
71	C₂×C₄	—	431	1	no	—
72	C₂×C₄	—	431	3	no	—
73	C₂×C₄	—	431	1	no	—
74	C₂×C₄	—	431	12	no	—
75	C₂×C₄	—	1724	6	no	—
76	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₄₃₁	1	3	yes	—
78	C₂×C₄	C₂²×D₄₃₁	1	3	yes	—
79	C₂²	—	862	3	no	—
80	C₂²	—	862	3	no	—
81	C₂²	—	431	6	no	—
82	C₂²	C₂³×D₄₃₁	1	1	yes	characteristic
83	C₄	—	2	3	no	—
84	C₄	—	2	1	no	—
85	C₄	—	862	3	no	—
86	C₄	—	862	1	no	—
87	C₄	—	431	2	no	—
88	C₄	C₂×C₄×D₄₃₁	1	6	yes	—
89	C₂	—	862	6	no	—
90	C₂	C₂²×C₄×D₄₃₁	1	1	yes	characteristic
91	C₂	#28352235	1	1	yes	characteristic
92	C₂	#28352237	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃₁
commutator	C₂×C₄₃₁	C₂³×C₄
Fitting	C₄×Q₈×C₄₃₁	C₂
socle, cosocle	C₂²×C₄₃₁	C₂⁴
prime radical	C₂×#20293801	C₂
hypercenter	C₄×Q₈	D₄₃₁
hypocenter	C₄₃₁	#520
LCS	G ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₄₃₁	C₂	semidirect
3. C₄×D₁₇₂₄	C₂	semidirect
4. #28352151	C₂	semidirect
5. #28352146	C₂	semidirect
6. C₄⋊C₄×D₄₃₁	C₂	semidirect
7. C₄×Q₈×C₄₃₁	C₂	semidirect
8. #20293801	C₄	semidirect
22. D₁₇₂₄	C₂×C₄	semidirect
27. C₄×D₄₃₁	C₂×C₄	semidirect
28. Q₈×C₄₃₁	C₂×C₄	semidirect
46. C₄₃₁	#520	semidirect
50. C₄×Q₈	D₄₃₁	semidirect
69. Q₈	C₄×D₄₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20293801	2
Q₈×Dic₄₃₁	2
C₄×D₁₇₂₄	2
#28352151	2
#28352146	2
C₄⋊C₄×D₄₃₁	2
C₄×Q₈×C₄₃₁	2
#520	431