Subgroups — #31774453

Order 17344.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576159	C₂	1	2	yes	maximal, complemented
2	#22576152	C₂	1	1	yes	maximal, characteristic, complemented
3	#22576138	C₂	1	4	yes	maximal
4	#22576125	C₂	1	4	yes	maximal, complemented
5	#22576118	C₂	1	2	yes	maximal
6	C₄²×D₂₇₁	C₂	1	1	yes	maximal, characteristic
7	#125×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₄₂	C₂²	1	4	yes	—
9	C₂×C₄×D₂₇₁	C₂²	1	2	yes	—
10	C₂×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₇₁	C₂²	1	2	yes	—
12	Dic₂₇₁⋊C₄	C₂²	1	4	yes	—
13	Dic₂₇₁⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₂₇₁)⋊C₄	C₂²	1	4	yes	—
15	C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₇₁	C₂²	1	2	yes	—
17	C₄×Dic₂₇₁	C₂²	1	4	yes	—
18	C₄⋊C₄×C₂₇₁	C₂²	1	4	yes	—
19	C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
20	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
21	C₂²×D₂₇₁	C₂³	1	1	yes	characteristic
22	Dic₅₄₂	—	2	8	no	—
23	C₂×Dic₂₇₁	C₂³	1	4	yes	—
24	C₂×Dic₂₇₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₂₇₁	C₂³	1	2	yes	—
26	C₄×D₂₇₁	—	2	4	no	—
27	C₄×D₂₇₁	Q₈	1	4	yes	complemented
28	C₂×C₄×C₂₇₁	C₂³	1	2	yes	—
29	C₂×C₄×C₂₇₁	C₂³	1	4	yes	—
30	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
31	C₂²×C₂₇₁	C₂⁴	1	1	yes	characteristic
32	C₂×D₂₇₁	—	2	2	no	—
33	C₂×D₂₇₁	C₂×Q₈	1	2	yes	—
34	C₄×C₂₇₁	—	2	2	no	—
35	C₄×C₂₇₁	—	2	4	no	—
36	C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
37	Dic₂₇₁	—	2	4	no	—
38	Dic₂₇₁	C₂×Q₈	1	2	yes	—
39	Dic₂₇₁	#41	1	4	yes	—
40	C₂×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
41	C₂×C₂₇₁	C₂×#41	1	2	yes	—
42	D₂₇₁	—	2	2	no	—
43	C₂₇₁	#533	1	1	yes	characteristic, complemented, Sylow
44	#533	—	271	1	no	maximal, Sylow
45	C₄⋊Q₈	—	271	2	no	—
46	#125	—	271	1	no	—
47	#125	D₂₇₁	1	1	yes	characteristic, complemented
48	C₂²⋊Q₈	—	271	4	no	—
49	C₄×Q₈	—	271	4	no	—
50	#117	—	271	2	no	—
51	C₂×C₄²	—	271	1	no	—
52	C₂×Q₈	—	271	4	no	—
53	C₂²×C₄	—	271	1	no	—
54	C₂²×C₄	—	271	2	no	—
55	C₄⋊C₄	—	271	2	no	—
56	C₄⋊C₄	—	271	4	no	—
57	C₄⋊C₄	—	271	4	no	—
58	C₄⋊C₄	C₂×D₂₇₁	1	2	yes	—
59	C₄⋊C₄	C₂×D₂₇₁	1	4	yes	—
60	C₂²⋊C₄	—	271	4	no	—
61	C₄²	—	271	2	no	—
62	C₄²	—	271	1	no	—
63	C₄²	—	271	4	no	—
64	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
65	C₂³	—	271	1	no	—
66	Q₈	—	542	8	no	—
67	C₂×C₄	—	271	1	no	—
68	C₂×C₄	—	271	2	no	—
69	C₂×C₄	—	271	4	no	—
70	C₂×C₄	—	271	4	no	—
71	C₂×C₄	—	542	4	no	—
72	C₂×C₄	C₂²×D₂₇₁	1	2	yes	—
73	C₂×C₄	C₂²×D₂₇₁	1	4	yes	—
74	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
75	C₂²	—	271	2	no	—
76	C₂²	—	542	2	no	—
77	C₂²	C₂³×D₂₇₁	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	4	no	—
80	C₄	—	271	2	no	—
81	C₄	—	271	4	no	—
82	C₄	—	542	4	no	—
83	C₄	Q₈×D₂₇₁	1	2	yes	—
84	C₂	—	542	2	no	—
85	C₂	C₂×Q₈×D₂₇₁	1	1	yes	characteristic
86	C₂	#41×D₂₇₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₇₁
commutator, socle, cosocle	C₂²×C₂₇₁	C₂⁴
Fitting	#125×C₂₇₁	C₂
prime radical	C₂²×D₂₇₁	C₂³
hypercenter	#125	D₂₇₁
hypocenter	C₂₇₁	#533
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #22576159	C₂	semidirect
2. #22576152	C₂	semidirect
4. #22576125	C₂	semidirect
7. #125×C₂₇₁	C₂	semidirect
27. C₄×D₂₇₁	Q₈	semidirect
43. C₂₇₁	#533	semidirect
47. #125	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22576159	2
#22576152	2
#22576138	2
#22576125	2
#22576118	2
C₄²×D₂₇₁	2
#125×C₂₇₁	2
#533	271