Group #47984148 — subgroups

Subgroups — #47984148

Order 35456.

244 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 93 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	#31800026	C₂	1	1	yes	maximal, characteristic
3	#31799762	C₂	1	4	yes	maximal
4	#376×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#17984547	C₂²	1	2	yes	—
6	#22579880	C₄	1	2	yes	complemented
7	#22579872	C₄	1	2	yes	complemented
8	#22579859	—	2	2	no	—
9	C₄×Dic₅₅₄	—	2	2	no	—
10	#22579839	C₂²	1	2	yes	—
11	C₄⋊#14555660	—	2	2	no	—
12	C₂×C₄²×C₂₇₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
14	#95×C₂₇₇	—	2	2	no	—
15	C₂×Dic₅₅₄	—	2	2	no	—
16	#17984547	—	2	4	no	—
17	C₂×#14555660	—	2	2	no	—
18	C₄⋊Dic₂₇₇	—	2	2	no	—
19	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
20	#17984542	—	2	4	no	—
21	C₂²⋊Dic₂₇₇	—	2	2	no	—
22	C₄×Dic₂₇₇	—	2	2	no	—
23	C₂²×C₄×C₂₇₇	—	2	2	no	—
24	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
25	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
26	C₄²×C₂₇₇	—	2	2	no	—
27	C₄²×C₂₇₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₂₇₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₇₇	D₄	1	2	yes	—
30	C₄²×C₂₇₇	Q₈	1	4	yes	complemented
31	Dic₅₅₄	—	4	4	no	—
32	C₂×Dic₂₇₇	—	2	4	no	—
33	#14555660	—	4	2	no	—
34	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₇₇	—	2	4	no	—
36	C₂×C₄×C₂₇₇	—	4	2	no	—
37	C₂×C₄×C₂₇₇	—	4	4	no	—
38	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₂₇₇	#41	1	2	yes	—
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	—	4	4	no	—
50	C₄×C₂₇₇	C₄×Q₈	1	2	yes	—
51	C₄×C₂₇₇	#124	1	1	yes	characteristic
52	C₄×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
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	—	277	1	no	—
61	#376	D₂₇₇	1	1	yes	characteristic
62	#339	—	277	4	no	—
63	C₂×M₁₆	—	277	2	no	—
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	C₂×D₂₇₇	1	2	yes	—
70	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
71	C₄⋊C₈	—	554	2	no	—
72	#95	—	2	2	no	—
73	C₂×Q₈	—	554	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂²⋊D₂₇₇	1	2	yes	—
76	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
77	M₁₆	—	554	4	no	—
78	C₂×C₈	—	554	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	—	2	2	no	—
84	C₄²	—	554	2	no	—
85	C₄²	C₄×D₂₇₇	1	1	yes	characteristic
86	C₄²	C₄×D₂₇₇	1	1	yes	characteristic
87	C₄²	D₁₁₀₈	1	2	yes	—
88	C₄²	Dic₅₅₄	1	4	yes	complemented
89	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
90	Q₈	—	1108	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	554	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×D₁₁₀₈	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₅₅₄	1	2	yes	—
100	C₂×C₄	#17984563	1	2	yes	—
101	C₈	—	1108	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×#17984540	1	1	yes	characteristic
104	C₂²	#22579859	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	1108	4	no	—
109	C₄	C₄×Dic₅₅₄	1	2	yes	—
110	C₄	#22579868	1	1	yes	characteristic
111	C₄	#22579880	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#31799908	1	1	yes	characteristic
114	C₂	#31800020	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31800559

C₂

yes

maximal, characteristic

#31800026

C₂

yes

maximal, characteristic

#31799762

C₂

yes

maximal

#376×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984547

C₂²

yes

—

#22579880

C₄

yes

complemented

#22579872

C₄

yes

complemented

#22579859

—

C₄×Dic₅₅₄

—

#22579839

C₂²

yes

—

C₄⋊#14555660

—

C₂×C₄²×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

#95×C₂₇₇

—

C₂×Dic₅₅₄

—

#17984547

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

C₂×C₄

yes

—

#17984542

—

C₂²⋊Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄²×C₂₇₇

—

C₄²×C₂₇₇

C₂×C₄

yes

characteristic

C₄²×C₂₇₇

C₂×C₄

yes

characteristic

C₄²×C₂₇₇

D₄

yes

—

C₄²×C₂₇₇

Q₈

yes

complemented

Dic₅₅₄

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

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

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₂²⋊Q₈

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₄×Q₈

yes

—

C₄×C₂₇₇

#124

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

characteristic

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

—

277

—

#376

D₂₇₇

yes

characteristic

#339

—

277

—

C₂×M₁₆

—

277

—

C₄⋊Q₈

—

277

—

#125

—

277

—

C₂²⋊Q₈

—

554

—

C₄×Q₈

—

554

—

#117

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

C₄⋊C₈

—

554

—

#95

—

C₂×Q₈

—

554

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

554

—

C₂×C₈

—

554

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₂²⋊C₄

—

554

—

C₄²

—

C₄²

—

554

—

C₄²

C₄×D₂₇₇

yes

characteristic

C₄²

C₄×D₂₇₇

yes

characteristic

C₄²

D₁₁₀₈

yes

—

C₄²

Dic₅₅₄

yes

complemented

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

Q₈

—

1108

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

554

—

C₂×C₄

#17984540

yes

—

C₂×C₄

#17984540

yes

—

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

C₂×C₄

C₂×Dic₅₅₄

yes

—

100

C₂×C₄

#17984563

yes

—

101

C₈

—

1108

—

102

C₂²

—

103

C₂²

C₂×#17984540

yes

characteristic

104

C₂²

#22579859

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

1108

—

109

C₄

C₄×Dic₅₅₄

yes

—

110

C₄

#22579868

yes

characteristic

111

C₄

#22579880

yes

characteristic

112

C₂

—

113

C₂

#31799908

yes

characteristic

114

C₂

#31800020

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₀₈
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#376×C₂₇₇	C₂
socle	C₂²×C₂₇₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₇₇	C₂×D₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#376	D₂₇₇
hypocenter	C₂₇₇	#1995
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₁₁₀₈

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#376×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 ◁ #376

N	H ≅ G/N	type
6. #22579880	C₄	semidirect
7. #22579872	C₄	semidirect
30. C₄²×C₂₇₇	Q₈	semidirect
57. C₂₇₇	#1995	semidirect
88. C₄²	Dic₅₅₄	semidirect

H ≅ G/N

type

6. #22579880

C₄

semidirect

7. #22579872

C₄

semidirect

30. C₄²×C₂₇₇

Q₈

semidirect

57. C₂₇₇

#1995

semidirect

88. C₄²

Dic₅₅₄

semidirect

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

subgroup	index
#31800559	2
#31800026	2
#31799762	2
#376×C₂₇₇	2
#1995	277

subgroup

index

#376×C₂₇₇

277

Subgroups — #47984148

Conjugacy classes of subgroups

Complements

Maximal subgroups