Group #47986347 — subgroups

Subgroups — #47986347

Order 35456.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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	#31799906	C₂	1	4	yes	maximal
3	#31799887	C₂	1	4	yes	maximal
4	#31799885	C₂	1	4	yes	maximal
5	#31799870	C₂	1	1	yes	maximal, characteristic
6	#383×C₂₇₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₅₄	C₂²	1	2	yes	—
8	#22579880	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
11	C₂×#17984542	C₂²	1	4	yes	—
12	C₂×C₄×Dic₂₇₇	C₂²	1	4	yes	—
13	#22579787	C₂²	1	2	yes	—
14	#22579787	C₂²	1	2	yes	—
15	#22579787	C₂²	1	2	yes	—
16	#22579787	C₂²	1	8	yes	—
17	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
18	#95×C₂₇₇	C₂²	1	2	yes	—
19	#95×C₂₇₇	C₂²	1	2	yes	—
20	#95×C₂₇₇	C₂²	1	2	yes	—
21	C₂×Dic₅₅₄	—	4	2	no	—
22	C₂×Dic₅₅₄	—	4	2	no	—
23	C₂×Dic₅₅₄	C₂×C₄	1	8	yes	—
24	C₂²×Dic₂₇₇	C₂³	1	4	yes	—
25	C₂²×Dic₂₇₇	C₂³	1	4	yes	—
26	C₄⋊Dic₂₇₇	C₂×C₄	1	8	yes	—
27	C₄⋊Dic₂₇₇	C₂×C₄	1	8	yes	—
28	#17984542	—	4	4	no	—
29	C₄×Dic₂₇₇	—	4	4	no	—
30	C₂²×C₄×C₂₇₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
34	C₄²×C₂₇₇	C₂×C₄	1	4	yes	—
35	Dic₅₅₄	—	4	8	no	—
36	C₂×Dic₂₇₇	—	2	4	no	—
37	C₂×Dic₂₇₇	—	2	4	no	—
38	C₂×Dic₂₇₇	—	2	4	no	—
39	C₂×Dic₂₇₇	—	2	4	no	—
40	C₂×Dic₂₇₇	—	2	4	no	—
41	C₂×Dic₂₇₇	C₂²×C₄	1	8	yes	—
42	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	—	2	2	no	—
44	C₂×C₄×C₂₇₇	—	2	2	no	—
45	C₂×C₄×C₂₇₇	—	2	2	no	—
46	C₂×C₄×C₂₇₇	—	2	4	no	—
47	C₂×C₄×C₂₇₇	—	2	4	no	—
48	C₂×C₄×C₂₇₇	—	2	4	no	—
49	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₂₇₇	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₂₇₇	2₋¹⁺⁴	1	2	yes	—
55	C₂²×C₂₇₇	2₋¹⁺⁴	1	2	yes	—
56	C₄×C₂₇₇	—	2	2	no	—
57	C₄×C₂₇₇	—	2	2	no	—
58	C₄×C₂₇₇	—	2	2	no	—
59	C₄×C₂₇₇	—	4	4	no	—
60	Dic₂₇₇	—	2	8	no	—
61	Dic₂₇₇	—	4	4	no	—
62	C₂×C₂₇₇	#518	1	1	yes	characteristic
63	C₂×C₂₇₇	#519	1	1	yes	characteristic
64	C₂×C₂₇₇	#519	1	1	yes	characteristic
65	C₂×C₂₇₇	#563	1	4	yes	—
66	C₂₇₇	#2381	1	1	yes	characteristic, complemented, Sylow
67	#2381	—	277	1	no	maximal, Sylow
68	C₂×C₄⋊Q₈	—	277	1	no	—
69	#391	—	277	4	no	—
70	#387	—	277	4	no	—
71	#387	—	277	4	no	—
72	#383	—	277	1	no	—
73	#383	D₂₇₇	1	1	yes	characteristic
74	C₂²×Q₈	—	277	2	no	—
75	C₄⋊Q₈	—	277	8	no	—
76	C₂×C₄⋊C₄	—	277	2	no	—
77	C₂×C₄⋊C₄	—	277	2	no	—
78	C₂×C₄⋊C₄	—	277	4	no	—
79	C₂×C₄²	—	277	4	no	—
80	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
81	#95	—	277	2	no	—
82	#95	—	277	2	no	—
83	#95	—	277	2	no	—
84	#95	—	277	8	no	—
85	#95	C₂×D₂₇₇	1	2	yes	—
86	#95	C₂×D₂₇₇	1	2	yes	—
87	#95	C₂×D₂₇₇	1	2	yes	—
88	C₂×Q₈	—	277	8	no	—
89	C₂×Q₈	—	1108	2	no	—
90	C₂×Q₈	—	1108	2	no	—
91	C₂²×C₄	—	277	4	no	—
92	C₂²×C₄	—	277	4	no	—
93	C₂²×C₄	C₂²×D₂₇₇	1	4	yes	—
94	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
97	C₄⋊C₄	—	277	8	no	—
98	C₄⋊C₄	—	277	8	no	—
99	C₄⋊C₄	—	1108	4	no	—
100	C₄²	—	1108	4	no	—
101	C₄²	C₄×D₂₇₇	1	4	yes	—
102	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
103	Q₈	—	1108	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	277	8	no	—
111	C₂×C₄	—	554	4	no	—
112	C₂×C₄	—	554	4	no	—
113	C₂×C₄	—	554	4	no	—
114	C₂×C₄	—	554	4	no	—
115	C₂×C₄	—	554	4	no	—
116	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
118	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
119	C₂²	C₂²×C₄×D₂₇₇	1	1	yes	characteristic
120	C₂²	#22579926	1	2	yes	—
121	C₂²	#22579927	1	2	yes	—
122	C₂²	#22579927	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	554	8	no	—
128	C₄	—	1108	4	no	—
129	C₂	#31800476	1	1	yes	characteristic
130	C₂	#31800479	1	1	yes	characteristic
131	C₂	#31800479	1	1	yes	characteristic
132	C₂	#31800754	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579880

C₂

yes

maximal, characteristic

#31799906

C₂

yes

maximal

#31799887

C₂

yes

maximal

#31799885

C₂

yes

maximal

#31799870

C₂

yes

maximal, characteristic

#383×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₅₄

C₂²

yes

—

#22579880

C₄

yes

complemented

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

C₂×#17984542

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

—

#22579787

C₂²

yes

—

#22579787

C₂²

yes

—

#22579787

C₂²

yes

—

#22579787

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

#95×C₂₇₇

C₂²

yes

—

#95×C₂₇₇

C₂²

yes

—

#95×C₂₇₇

C₂²

yes

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

C₂×C₄

yes

—

C₂²×Dic₂₇₇

C₂³

yes

—

C₂²×Dic₂₇₇

C₂³

yes

—

C₄⋊Dic₂₇₇

C₂×C₄

yes

—

C₄⋊Dic₂₇₇

C₂×C₄

yes

—

#17984542

—

C₄×Dic₂₇₇

—

C₂²×C₄×C₂₇₇

C₂³

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄²×C₂₇₇

C₂×C₄

yes

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₂²×C₄

yes

—

C₂³×C₂₇₇

C₂⁴

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

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

—

C₂²×C₂₇₇

C₂³×C₄

yes

characteristic

C₂²×C₂₇₇

H(5,𝔽₂)

yes

—

C₂²×C₂₇₇

2₋¹⁺⁴

yes

—

C₂²×C₂₇₇

2₋¹⁺⁴

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

—

C₂₇₇

#2381

yes

characteristic, complemented, Sylow

#2381

—

277

maximal, Sylow

C₂×C₄⋊Q₈

—

277

—

#391

—

277

—

#387

—

277

—

#387

—

277

—

#383

—

277

—

#383

D₂₇₇

yes

characteristic

C₂²×Q₈

—

277

—

C₄⋊Q₈

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄²

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

#95

—

277

—

#95

—

277

—

#95

—

277

—

#95

—

277

—

#95

C₂×D₂₇₇

yes

—

#95

C₂×D₂₇₇

yes

—

#95

C₂×D₂₇₇

yes

—

C₂×Q₈

—

277

—

C₂×Q₈

—

1108

—

C₂×Q₈

—

1108

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

C₂²×D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₄⋊C₄

—

277

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

1108

—

100

C₄²

—

1108

—

101

C₄²

C₄×D₂₇₇

yes

—

102

C₂³

C₂³×D₂₇₇

yes

characteristic

103

Q₈

—

1108

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

277

—

111

C₂×C₄

—

554

—

112

C₂×C₄

—

554

—

113

C₂×C₄

—

554

—

114

C₂×C₄

—

554

—

115

C₂×C₄

—

554

—

116

C₂×C₄

C₂×C₄×D₂₇₇

yes

—

117

C₂×C₄

C₂×C₄×D₂₇₇

yes

—

118

C₂×C₄

C₂×C₄×D₂₇₇

yes

—

119

C₂²

C₂²×C₄×D₂₇₇

yes

characteristic

120

C₂²

#22579926

yes

—

121

C₂²

#22579927

yes

—

122

C₂²

#22579927

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

554

—

128

C₄

—

1108

—

129

C₂

#31800476

yes

characteristic

130

C₂

#31800479

yes

characteristic

131

C₂

#31800479

yes

characteristic

132

C₂

#31800754

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Fitting	#383×C₂₇₇	C₂
socle, prime radical, cosocle	C₂³×C₂₇₇	C₂⁴
hypercenter	#383	D₂₇₇
hypocenter	C₂₇₇	#2381
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₇₇

commutator

C₂²×C₂₇₇

C₂³×C₄

Fitting

#383×C₂₇₇

C₂

socle, prime radical, cosocle

C₂³×C₂₇₇

C₂⁴

D₂₇₇

C₂₇₇

G ▷ C₂²×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
8. #22579880	C₄	semidirect
66. C₂₇₇	#2381	semidirect

H ≅ G/N

type

8. #22579880

C₄

semidirect

66. C₂₇₇

#2381

semidirect

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

subgroup	index
C₂×#22579880	2
#31799906	2
#31799887	2
#31799885	2
#31799870	2
#383×C₂₇₇	2
#2381	277

subgroup

index

#383×C₂₇₇

277

Subgroups — #47986347

Conjugacy classes of subgroups

Complements

Maximal subgroups