Group #47984147 — subgroups

Subgroups — #47984147

Order 35456.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31800559	C₂	1	1	yes	maximal, characteristic
2	#31799848	C₂	1	1	yes	maximal, characteristic
3	#31799763	C₂	1	4	yes	maximal
4	#423×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	#22579880	C₄	1	2	yes	complemented
6	#22579872	C₄	1	2	yes	complemented
7	#22579859	—	2	2	no	—
8	C₄×Dic₅₅₄	—	2	2	no	—
9	#22579839	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
11	#22579787	—	2	2	no	—
12	C₂×M₁₆×C₂₇₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₇₇	—	2	2	no	—
15	C₂×Dic₅₅₄	—	2	2	no	—
16	C₂²×Dic₂₇₇	—	2	2	no	—
17	C₄⋊Dic₂₇₇	—	2	2	no	—
18	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
19	#17984542	—	2	4	no	—
20	C₂²⋊Dic₂₇₇	—	2	2	no	—
21	C₄×Dic₂₇₇	—	2	2	no	—
22	C₄×Dic₂₇₇	—	2	2	no	—
23	C₄×Dic₂₇₇	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
25	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₇₇	—	2	4	no	—
27	C₂×C₈×C₂₇₇	—	2	2	no	—
28	C₄²×C₂₇₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₇₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₇₇	D₄	1	2	yes	—
31	Dic₅₅₄	—	4	4	no	—
32	C₂×Dic₂₇₇	—	2	4	no	—
33	C₂×Dic₂₇₇	—	2	4	no	—
34	C₂×Dic₂₇₇	—	4	4	no	—
35	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₇₇	—	4	2	no	—
37	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₂₇₇	#41	1	2	yes	—
43	C₈×C₂₇₇	—	4	2	no	—
44	C₂²×C₂₇₇	—	2	2	no	—
45	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₂₇₇	—	2	2	no	—
48	C₄×C₂₇₇	—	2	2	no	—
49	C₄×C₂₇₇	C₄×Q₈	1	2	yes	—
50	C₄×C₂₇₇	#124	1	1	yes	characteristic
51	C₄×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
52	Dic₂₇₇	—	4	4	no	—
53	Dic₂₇₇	—	4	4	no	—
54	C₂×C₂₇₇	—	2	2	no	—
55	C₂×C₂₇₇	#391	1	1	yes	characteristic
56	C₂×C₂₇₇	#421	1	2	yes	—
57	C₂₇₇	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	277	1	no	maximal, Sylow
59	#533	—	277	1	no	—
60	#423	D₂₇₇	1	1	yes	characteristic
61	#376	—	277	1	no	—
62	#339	—	277	4	no	—
63	C₂×M₁₆	C₂×D₂₇₇	1	2	yes	—
64	C₄⋊Q₈	—	277	2	no	—
65	#125	—	277	2	no	—
66	C₂²⋊Q₈	—	554	2	no	—
67	C₄×Q₈	—	554	2	no	—
68	#117	—	277	2	no	—
69	C₂×C₄²	—	277	2	no	—
70	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	#95	—	554	2	no	—
73	C₂×Q₈	—	554	2	no	—
74	C₂²×C₄	—	554	2	no	—
75	C₂²×C₄	D₁₁₀₈	1	2	yes	—
76	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	277	4	no	—
80	C₄⋊C₄	—	554	4	no	—
81	C₄⋊C₄	—	554	2	no	—
82	C₂²⋊C₄	—	554	2	no	—
83	C₄²	—	277	4	no	—
84	C₄²	—	554	2	no	—
85	C₄²	—	554	2	no	—
86	C₄²	C₄×D₂₇₇	1	1	yes	characteristic
87	C₄²	C₄×D₂₇₇	1	1	yes	characteristic
88	C₄²	C₂²⋊D₂₇₇	1	2	yes	—
89	C₂³	C₂×D₁₁₀₈	1	1	yes	characteristic
90	Q₈	—	1108	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	554	4	no	—
93	C₂×C₄	—	554	4	no	—
94	C₂×C₄	—	1108	4	no	—
95	C₂×C₄	#17984540	1	2	yes	—
96	C₂×C₄	#17984540	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
100	C₂×C₄	#17984562	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×#17984540	1	1	yes	characteristic
104	C₂²	#22579860	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	1108	4	no	—
108	C₄	—	1108	4	no	—
109	C₄	#22579845	1	2	yes	—
110	C₄	#22579867	1	1	yes	characteristic
111	C₄	#22579878	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#31799907	1	1	yes	characteristic
114	C₂	#31800018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31800559

C₂

yes

maximal, characteristic

#31799848

C₂

yes

maximal, characteristic

#31799763

C₂

yes

maximal

#423×C₂₇₇

C₂

yes

maximal, characteristic

#22579880

C₄

yes

complemented

#22579872

C₄

yes

complemented

#22579859

—

C₄×Dic₅₅₄

—

#22579839

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

—

#22579787

—

C₂×M₁₆×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₇₇

—

C₂×Dic₅₅₄

—

C₂²×Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

C₂×C₄

yes

—

#17984542

—

C₂²⋊Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

Q₈

yes

complemented

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄²×C₂₇₇

C₂×C₄

yes

characteristic

C₄²×C₂₇₇

C₂×C₄

yes

characteristic

C₄²×C₂₇₇

D₄

yes

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

—

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₂²⋊Q₈

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₄×Q₈

yes

—

C₄×C₂₇₇

#124

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

characteristic

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#391

yes

characteristic

C₂×C₂₇₇

#421

yes

—

C₂₇₇

#1995

yes

characteristic, complemented, Sylow

#1995

—

277

maximal, Sylow

#533

—

277

—

#423

D₂₇₇

yes

characteristic

#376

—

277

—

#339

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

—

C₄⋊Q₈

—

277

—

#125

—

277

—

C₂²⋊Q₈

—

554

—

C₄×Q₈

—

554

—

#117

—

277

—

C₂×C₄²

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

C₄⋊C₈

—

#95

—

554

—

C₂×Q₈

—

554

—

C₂²×C₄

—

554

—

C₂²×C₄

D₁₁₀₈

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₂²⋊C₄

—

554

—

C₄²

—

277

—

C₄²

—

554

—

C₄²

—

554

—

C₄²

C₄×D₂₇₇

yes

characteristic

C₄²

C₄×D₂₇₇

yes

characteristic

C₄²

C₂²⋊D₂₇₇

yes

—

C₂³

C₂×D₁₁₀₈

yes

characteristic

Q₈

—

1108

—

C₂×C₄

—

C₂×C₄

—

554

—

C₂×C₄

—

554

—

C₂×C₄

—

1108

—

C₂×C₄

#17984540

yes

—

C₂×C₄

#17984540

yes

—

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₂×C₄

Q₈×D₂₇₇

yes

—

100

C₂×C₄

#17984562

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂×#17984540

yes

characteristic

104

C₂²

#22579860

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

1108

—

108

C₄

—

1108

—

109

C₄

#22579845

yes

—

110

C₄

#22579867

yes

characteristic

111

C₄

#22579878

yes

characteristic

112

C₂

—

113

C₂

#31799907

yes

characteristic

114

C₂

#31800018

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₇₇
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#423×C₂₇₇	C₂
socle	C₂²×C₂₇₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₇₇	C₂×D₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#423	D₂₇₇
hypocenter	C₂₇₇	#1995
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₇₇

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#423×C₂₇₇

C₂

socle

C₂²×C₂₇₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₇₇

C₂×D₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #22579880	C₄	semidirect
6. #22579872	C₄	semidirect
23. C₄×Dic₂₇₇	Q₈	semidirect
57. C₂₇₇	#1995	semidirect

H ≅ G/N

type

5. #22579880

C₄

semidirect

6. #22579872

C₄

semidirect

23. C₄×Dic₂₇₇

Q₈

semidirect

57. C₂₇₇

#1995

semidirect

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

subgroup	index
#31800559	2
#31799848	2
#31799763	2
#423×C₂₇₇	2
#1995	277

subgroup

index

#423×C₂₇₇

277

Subgroups — #47984147

Conjugacy classes of subgroups

Complements

Maximal subgroups