Subgroups — #31774501

Order 17344.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576152	C₂	1	1	yes	maximal, characteristic, complemented
2	#22576141	C₂	1	4	yes	maximal, complemented
3	#22576132	C₂	1	4	yes	maximal, complemented
4	#22576131	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₂₇₁	C₂	1	2	yes	maximal, complemented
6	#22576117	C₂	1	1	yes	maximal, characteristic, complemented
7	#127×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₂₇₁	C₂²	1	4	yes	—
9	C₂²×Dic₂₇₁	C₂²	1	4	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	2	yes	—
14	C₂²⋊Dic₂₇₁	C₂²	1	4	yes	—
15	C₂²⋊Dic₂₇₁	C₂²	1	4	yes	—
16	(C₂×D₂₇₁)⋊C₄	C₂²	1	2	yes	—
17	C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₂₇₁	C₂²	1	2	yes	—
19	C₂×D₄×C₂₇₁	C₂²	1	4	yes	—
20	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
21	C₂²×D₂₇₁	C₂³	1	1	yes	characteristic
22	C₂²⋊D₂₇₁	—	4	4	no	—
23	C₂×Dic₂₇₁	—	4	4	no	—
24	C₂×Dic₂₇₁	C₂³	1	2	yes	—
25	C₂×Dic₂₇₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₂₇₁	C₂³	1	4	yes	—
27	C₄×D₂₇₁	—	2	2	no	—
28	C₂³×C₂₇₁	C₂³	1	4	yes	—
29	D₄×C₂₇₁	—	2	4	no	—
30	D₄×C₂₇₁	—	4	4	no	—
31	C₂×C₄×C₂₇₁	C₂³	1	2	yes	—
32	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
33	C₂²×C₂₇₁	—	2	4	no	—
34	C₂²×C₂₇₁	—	2	4	no	—
35	C₂²×C₂₇₁	—	2	4	no	—
36	C₂²×C₂₇₁	C₂⁴	1	1	yes	characteristic
37	C₂×D₂₇₁	—	2	1	no	—
38	C₂×D₂₇₁	—	2	2	no	—
39	C₄×C₂₇₁	—	2	2	no	—
40	C₄×C₂₇₁	#41	1	2	yes	—
41	Dic₂₇₁	—	2	1	no	—
42	Dic₂₇₁	—	2	2	no	—
43	Dic₂₇₁	—	2	4	no	—
44	C₂×C₂₇₁	—	4	4	no	—
45	C₂×C₂₇₁	C₂×#41	1	1	yes	characteristic
46	C₂×C₂₇₁	H(5,𝔽₂)	1	2	yes	—
47	D₂₇₁	—	4	1	no	—
48	C₂₇₁	#540	1	1	yes	characteristic, complemented, Sylow
49	#540	—	271	1	no	maximal, Sylow
50	#127	D₂₇₁	1	1	yes	characteristic, complemented
51	#125	—	271	1	no	—
52	#123	—	271	4	no	—
53	#121	—	271	2	no	—
54	#121	—	271	4	no	—
55	C₄×D₄	—	271	2	no	—
56	#117	—	271	1	no	—
57	C₂×D₄	—	271	4	no	—
58	C₂×D₄	C₂×D₂₇₁	1	2	yes	—
59	C₂×D₄	C₂×D₂₇₁	1	4	yes	—
60	C₂²×C₄	—	271	4	no	—
61	C₂²×C₄	—	271	1	no	—
62	C₄⋊C₄	—	271	2	no	—
63	C₄⋊C₄	—	271	2	no	—
64	C₄⋊C₄	—	271	4	no	—
65	C₂²⋊C₄	—	271	4	no	—
66	C₂²⋊C₄	—	271	4	no	—
67	C₂²⋊C₄	—	271	2	no	—
68	C₄²	—	271	1	no	—
69	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
70	C₂³	—	271	1	no	—
71	C₂³	C₂²×D₂₇₁	1	4	yes	—
72	D₄	—	2	4	no	—
73	D₄	—	4	4	no	—
74	D₄	—	1084	4	no	—
75	C₂×C₄	—	271	2	no	—
76	C₂×C₄	—	271	1	no	—
77	C₂×C₄	—	271	4	no	—
78	C₂×C₄	—	542	2	no	—
79	C₂×C₄	—	1084	4	no	—
80	C₂×C₄	C₂²×D₂₇₁	1	2	yes	—
81	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
82	C₂²	—	2	4	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	542	1	no	—
86	C₂²	—	542	2	no	—
87	C₂²	C₂³×D₂₇₁	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	542	1	no	—
90	C₄	—	542	2	no	—
91	C₄	—	542	4	no	—
92	C₄	#16864653	1	2	yes	—
93	C₂	—	4	4	no	—
94	C₂	—	1084	1	no	—
95	C₂	C₂×#16864653	1	1	yes	characteristic
96	C₂	#22576205	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₇₁
commutator, socle, cosocle	C₂²×C₂₇₁	C₂⁴
Fitting	#127×C₂₇₁	C₂
hypercenter	#127	D₂₇₁
hypocenter	C₂₇₁	#540
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #22576152	C₂	semidirect
2. #22576141	C₂	semidirect
3. #22576132	C₂	semidirect
4. #22576131	C₂	semidirect
5. D₄×Dic₂₇₁	C₂	semidirect
6. #22576117	C₂	semidirect
7. #127×C₂₇₁	C₂	semidirect
48. C₂₇₁	#540	semidirect
50. #127	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22576152	2
#22576141	2
#22576132	2
#22576131	2
D₄×Dic₂₇₁	2
#22576117	2
#127×C₂₇₁	2
#540	271