Group #47983968 — subgroups

Subgroups — #47983968

Order 35456.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₂×#22579880	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579803	C₂	1	4	yes	maximal
3	C₂×C₄×#14555660	C₂	1	1	yes	maximal, characteristic
4	#384×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₅₄	C₂²	1	2	yes	—
6	C₂²×#14555660	C₂²	1	2	yes	—
7	#22579880	—	2	2	no	—
8	#22579880	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₇₇	—	2	2	no	—
10	#22579803	—	2	8	no	—
11	C₄×#14555660	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₇₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₅₄	—	2	2	no	—
16	C₂×Dic₅₅₄	—	2	4	no	—
17	C₂×Dic₅₅₄	—	2	4	no	—
18	C₂×Dic₅₅₄	C₂×C₄	1	4	yes	—
19	C₂²×Dic₂₇₇	—	2	2	no	—
20	C₂×#14555660	—	2	2	no	—
21	C₂×#14555660	—	2	2	no	—
22	C₂×#14555660	D₄	1	4	yes	—
23	C₄⋊Dic₂₇₇	—	2	8	no	—
24	C₂²×C₄×C₂₇₇	—	2	2	no	—
25	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
26	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₇₇	—	2	4	no	—
28	C₄⋊C₄×C₂₇₇	—	4	4	no	—
29	C₄²×C₂₇₇	C₂×C₄	1	2	yes	—
30	C₄²×C₂₇₇	D₄	1	2	yes	—
31	Dic₅₅₄	—	2	8	no	—
32	Dic₅₅₄	—	4	4	no	—
33	C₂×Dic₂₇₇	—	2	4	no	—
34	C₂×Dic₂₇₇	—	4	2	no	—
35	C₂×Dic₂₇₇	—	4	2	no	—
36	#14555660	—	2	4	no	—
37	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₇₇	—	4	2	no	—
39	C₂×C₄×C₂₇₇	—	4	4	no	—
40	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₇₇	SD₁₆	1	4	yes	—
42	C₂×C₄×C₂₇₇	Q₁₆	1	4	yes	—
43	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₇₇	#41	1	2	yes	—
47	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₇₇	#124	1	1	yes	characteristic
49	C₂²×C₂₇₇	#127	1	1	yes	characteristic
50	C₂²×C₂₇₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₂₇₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₂₇₇	—	8	2	no	—
53	C₄×C₂₇₇	#103	1	8	yes	complemented
54	C₄×C₂₇₇	C₄×D₄	1	2	yes	—
55	C₄×C₂₇₇	#121	1	2	yes	—
56	Dic₂₇₇	—	4	4	no	—
57	C₂×C₂₇₇	#390	1	1	yes	characteristic
58	C₂×C₂₇₇	C₂×#103	1	2	yes	—
59	C₂×C₂₇₇	#487	1	2	yes	—
60	C₂×C₂₇₇	#492	1	1	yes	characteristic
61	C₂×C₂₇₇	#494	1	1	yes	characteristic
62	C₂₇₇	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	277	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	277	1	no	—
65	C₂×#103	—	277	4	no	—
66	C₂×C₄×C₈	—	277	1	no	—
67	#384	D₂₇₇	1	1	yes	characteristic
68	C₂²×Q₈	—	277	2	no	—
69	C₂²×C₈	—	277	2	no	—
70	C₄⋊Q₈	—	277	4	no	—
71	C₄⋊Q₈	—	554	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	554	2	no	—
74	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
75	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
76	#103	—	554	8	no	—
77	C₄×C₈	—	554	2	no	—
78	C₂×Q₈	—	277	4	no	—
79	C₂×Q₈	—	554	2	no	—
80	C₂×Q₈	—	554	4	no	—
81	C₂×Q₈	—	554	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	554	2	no	—
84	C₂²×C₄	C₂²⋊D₂₇₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
86	C₂×C₈	—	277	4	no	—
87	C₂×C₈	—	554	2	no	—
88	C₂×C₈	—	554	2	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	554	8	no	—
92	C₄²	C₄×D₂₇₇	1	2	yes	—
93	C₄²	D₁₁₀₈	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
95	Q₈	—	554	8	no	—
96	Q₈	—	1108	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	554	4	no	—
100	C₂×C₄	—	1108	2	no	—
101	C₂×C₄	—	1108	2	no	—
102	C₂×C₄	#17984540	1	4	yes	—
103	C₂×C₄	#17984551	1	4	yes	—
104	C₂×C₄	#17984553	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
106	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
107	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
108	C₂×C₄	#17984564	1	2	yes	—
109	C₈	—	554	4	no	—
110	C₂²	C₂×#17984540	1	1	yes	characteristic
111	C₂²	#22579869	1	1	yes	characteristic
112	C₂²	#22579876	1	1	yes	characteristic
113	C₂²	C₂×#17984551	1	2	yes	—
114	C₂²	C₂×#17984553	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	1108	4	no	—
117	C₄	#22579803	1	8	yes	complemented
118	C₄	#22579842	1	2	yes	—
119	C₄	#22579855	1	2	yes	—
120	C₂	#31799903	1	1	yes	characteristic
121	C₂	C₂×#22579803	1	2	yes	—
122	C₂	#31800354	1	2	yes	—
123	C₂	#31800374	1	1	yes	characteristic
124	C₂	#31800381	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579880

C₂

yes

maximal, characteristic

C₂×#22579803

C₂

yes

maximal

C₂×C₄×#14555660

C₂

yes

maximal, characteristic

#384×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₅₄

C₂²

yes

—

C₂²×#14555660

C₂²

yes

—

#22579880

—

#22579880

C₄

yes

complemented

C₂×C₄⋊Dic₂₇₇

—

#22579803

—

C₄×#14555660

—

C₂×C₄⋊C₄×C₂₇₇

—

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

C₂×C₄

yes

—

C₂²×Dic₂₇₇

—

C₂×#14555660

—

C₂×#14555660

—

C₂×#14555660

D₄

yes

—

C₄⋊Dic₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂×C₄

yes

—

C₄²×C₂₇₇

D₄

yes

—

Dic₅₅₄

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

SD₁₆

yes

—

C₂×C₄×C₂₇₇

Q₁₆

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

C₂×SD₁₆

yes

—

C₂²×C₂₇₇

C₂×Q₁₆

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#103

yes

complemented

C₄×C₂₇₇

C₄×D₄

yes

—

C₄×C₂₇₇

#121

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

#390

yes

characteristic

C₂×C₂₇₇

C₂×#103

yes

—

C₂×C₂₇₇

#487

yes

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, complemented, Sylow

#1956

—

277

maximal, Sylow

C₂×C₄⋊Q₈

—

277

—

C₂×#103

—

277

—

C₂×C₄×C₈

—

277

—

#384

D₂₇₇

yes

characteristic

C₂²×Q₈

—

277

—

C₂²×C₈

—

277

—

C₄⋊Q₈

—

277

—

C₄⋊Q₈

—

554

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

554

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

#103

—

554

—

C₄×C₈

—

554

—

C₂×Q₈

—

277

—

C₂×Q₈

—

554

—

C₂×Q₈

—

554

—

C₂×Q₈

—

554

—

C₂²×C₄

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²⋊D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

277

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₄²

C₄×D₂₇₇

yes

—

C₄²

D₁₁₀₈

yes

—

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

Q₈

—

554

—

Q₈

—

1108

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

554

—

100

C₂×C₄

—

1108

—

101

C₂×C₄

—

1108

—

102

C₂×C₄

#17984540

yes

—

103

C₂×C₄

#17984551

yes

—

104

C₂×C₄

#17984553

yes

—

105

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

106

C₂×C₄

D₄×D₂₇₇

yes

—

107

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

108

C₂×C₄

#17984564

yes

—

109

C₈

—

554

—

110

C₂²

C₂×#17984540

yes

characteristic

111

C₂²

#22579869

yes

characteristic

112

C₂²

#22579876

yes

characteristic

113

C₂²

C₂×#17984551

yes

—

114

C₂²

C₂×#17984553

yes

—

115

C₄

—

116

C₄

—

1108

—

117

C₄

#22579803

yes

complemented

118

C₄

#22579842

yes

—

119

C₄

#22579855

yes

—

120

C₂

#31799903

yes

characteristic

121

C₂

C₂×#22579803

yes

—

122

C₂

#31800354

yes

—

123

C₂

#31800374

yes

characteristic

124

C₂

#31800381

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₇₇
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#384×C₂₇₇	C₂
socle, prime radical	C₂³×C₂₇₇	C₂×D₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#384	D₂₇₇
hypocenter	C₂₇₇	#1956
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₇₇

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#384×C₂₇₇

C₂

socle, prime radical

C₂³×C₂₇₇

C₂×D₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #22579880	C₄	semidirect
53. C₄×C₂₇₇	#103	semidirect
62. C₂₇₇	#1956	semidirect
117. C₄	#22579803	semidirect

H ≅ G/N

type

8. #22579880

C₄

semidirect

53. C₄×C₂₇₇

#103

semidirect

62. C₂₇₇

#1956

semidirect

117. C₄

#22579803

semidirect

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

subgroup	index
C₂×#22579880	2
C₂×#22579803	2
C₂×C₄×#14555660	2
#384×C₂₇₇	2
#1956	277

subgroup

index

#384×C₂₇₇

277

Subgroups — #47983968

Conjugacy classes of subgroups

Complements

Maximal subgroups