Group #47985887 — subgroups

Subgroups — #47985887

Order 35456.

516 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₂₇₇	C₂	1	2	yes	maximal
2	#31799909	C₂	1	2	yes	maximal
3	#31799888	C₂	1	8	yes	maximal
4	#31799868	C₂	1	1	yes	maximal, characteristic
5	#31799848	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
7	Q₈×Dic₂₇₇	—	2	8	no	—
8	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₇₇	C₂²	1	4	yes	—
10	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₇₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
13	#22579787	C₂²	1	4	yes	—
14	#22579787	C₂²	1	4	yes	—
15	#22579787	C₂²	1	8	yes	—
16	C₂²×Q₈×C₂₇₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₂₇₇	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
21	C₂²×Dic₂₇₇	C₂³	1	4	yes	—
22	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
23	C₄⋊Dic₂₇₇	—	2	8	no	—
24	C₄⋊Dic₂₇₇	—	2	4	no	—
25	C₄×Dic₂₇₇	—	2	8	no	—
26	C₄×Dic₂₇₇	—	2	2	no	—
27	C₄×Dic₂₇₇	D₄	1	4	yes	—
28	C₂×Q₈×C₂₇₇	—	2	4	no	—
29	C₂×Q₈×C₂₇₇	—	2	4	no	—
30	C₂×Q₈×C₂₇₇	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₇₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₂₇₇	C₂×C₄	1	16	yes	—
35	C₄²×C₂₇₇	C₂×C₄	1	4	yes	—
36	C₂×Dic₂₇₇	—	2	4	no	—
37	C₂×Dic₂₇₇	—	2	4	no	—
38	C₂×Dic₂₇₇	—	2	2	no	—
39	C₂×Dic₂₇₇	—	2	8	no	—
40	C₂×Dic₂₇₇	—	2	4	no	—
41	C₂×Dic₂₇₇	C₂×D₄	1	4	yes	—
42	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
43	Q₈×C₂₇₇	—	2	16	no	—
44	C₂×C₄×C₂₇₇	—	2	4	no	—
45	C₂×C₄×C₂₇₇	—	2	4	no	—
46	C₂×C₄×C₂₇₇	—	2	4	no	—
47	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₂₇₇	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₇₇	#41	1	2	yes	—
52	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
55	C₂²×C₂₇₇	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₂₇₇	—	2	8	no	—
57	C₄×C₂₇₇	—	2	4	no	—
58	C₄×C₂₇₇	C₄×D₄	1	4	yes	—
59	Dic₂₇₇	—	2	4	no	—
60	Dic₂₇₇	—	4	4	no	—
61	Dic₂₇₇	—	4	2	no	—
62	C₂×C₂₇₇	C₂×C₄×D₄	1	1	yes	characteristic
63	C₂×C₂₇₇	#519	1	2	yes	—
64	C₂×C₂₇₇	#536	1	2	yes	—
65	C₂×C₂₇₇	#541	1	2	yes	—
66	C₂₇₇	#2327	1	1	yes	characteristic, complemented, Sylow
67	#2327	—	277	1	no	maximal, Sylow
68	C₂×C₄⋊Q₈	D₂₇₇	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	277	2	no	—
70	#391	—	277	2	no	—
71	#387	—	277	8	no	—
72	#382	—	277	1	no	—
73	#376	—	277	1	no	—
74	C₂²×Q₈	C₂×D₂₇₇	1	2	yes	—
75	C₄⋊Q₈	Dic₂₇₇	1	8	yes	complemented
76	C₄×Q₈	—	554	8	no	—
77	C₂×C₄⋊C₄	—	277	2	no	—
78	C₂×C₄⋊C₄	—	277	4	no	—
79	C₂×C₄⋊C₄	C₂×D₂₇₇	1	4	yes	—
80	C₂×C₄²	—	277	1	no	—
81	C₂×C₄²	—	277	4	no	—
82	C₂×C₄²	—	277	1	no	—
83	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
84	#95	—	277	4	no	—
85	#95	—	277	4	no	—
86	#95	—	277	8	no	—
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×Dic₂₇₇	1	8	yes	—
90	C₂²×C₄	—	277	2	no	—
91	C₂²×C₄	—	277	2	no	—
92	C₂²×C₄	—	277	4	no	—
93	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
94	C₂²×C₄	C₂²×D₂₇₇	1	4	yes	—
95	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
96	C₄⋊C₄	—	554	8	no	—
97	C₄⋊C₄	—	554	4	no	—
98	C₄⋊C₄	C₂×Dic₂₇₇	1	16	yes	—
99	C₄²	—	277	4	no	—
100	C₄²	—	554	8	no	—
101	C₄²	—	554	2	no	—
102	C₄²	C₂×Dic₂₇₇	1	4	yes	—
103	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	277	4	no	—
109	C₂×C₄	—	554	4	no	—
110	C₂×C₄	—	554	4	no	—
111	C₂×C₄	—	554	2	no	—
112	C₂×C₄	—	554	8	no	—
113	C₂×C₄	—	554	4	no	—
114	C₂×C₄	C₂²×Dic₂₇₇	1	4	yes	—
115	C₂×C₄	C₂²×Dic₂₇₇	1	8	yes	—
116	C₂×C₄	C₂²×Dic₂₇₇	1	2	yes	—
117	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
118	C₂×C₄	#17984562	1	2	yes	—
119	C₂²	C₂³×Dic₂₇₇	1	1	yes	characteristic
120	C₂²	C₂×D₄×D₂₇₇	1	1	yes	characteristic
121	C₂²	C₂×#17984562	1	1	yes	characteristic
122	C₂²	#22579928	1	4	yes	—
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	554	4	no	—
126	C₄	—	1108	4	no	—
127	C₄	—	1108	2	no	—
128	C₄	D₄×Dic₂₇₇	1	4	yes	—
129	C₂	C₂×D₄×Dic₂₇₇	1	1	yes	characteristic
130	C₂	#31800480	1	2	yes	—
131	C₂	#31800577	1	2	yes	—
132	C₂	#31800613	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₂₇₇

C₂

yes

maximal

#31799909

C₂

yes

maximal

#31799888

C₂

yes

maximal

#31799868

C₂

yes

maximal, characteristic

#31799848

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₂₇₇

C₂

yes

maximal, characteristic

Q₈×Dic₂₇₇

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

C₂×C₄×Dic₂₇₇

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

#22579787

C₂²

yes

—

#22579787

C₂²

yes

—

#22579787

C₂²

yes

—

C₂²×Q₈×C₂₇₇

C₂²

yes

—

C₄⋊Q₈×C₂₇₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₂²×Dic₂₇₇

C₂³

yes

—

C₂²×Dic₂₇₇

C₂³

yes

—

C₂²×Dic₂₇₇

C₂³

yes

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

D₄

yes

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

C₂×C₄

yes

—

C₄²×C₂₇₇

C₂×C₄

yes

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₂×D₄

yes

—

C₂³×C₂₇₇

C₂⁴

yes

characteristic

Q₈×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

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

#41

yes

—

C₂²×C₂₇₇

C₂³×C₄

yes

characteristic

C₂²×C₂₇₇

C₂²×D₄

yes

characteristic

C₂²×C₂₇₇

C₂×#41

yes

characteristic

C₂²×C₂₇₇

2₋¹⁺⁴

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₄×D₄

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

C₂×C₄×D₄

yes

characteristic

C₂×C₂₇₇

#519

yes

—

C₂×C₂₇₇

#536

yes

—

C₂×C₂₇₇

#541

yes

—

C₂₇₇

#2327

yes

characteristic, complemented, Sylow

#2327

—

277

maximal, Sylow

C₂×C₄⋊Q₈

D₂₇₇

yes

characteristic

C₂×C₄×Q₈

—

277

—

#391

—

277

—

#387

—

277

—

#382

—

277

—

#376

—

277

—

C₂²×Q₈

C₂×D₂₇₇

yes

—

C₄⋊Q₈

Dic₂₇₇

yes

complemented

C₄×Q₈

—

554

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

—

C₂×C₄²

—

277

—

C₂×C₄²

—

277

—

C₂×C₄²

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

#95

—

277

—

#95

—

277

—

#95

—

277

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₇₇

yes

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

C₂²×D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

C₂×Dic₂₇₇

yes

—

C₄²

—

277

—

100

C₄²

—

554

—

101

C₄²

—

554

—

102

C₄²

C₂×Dic₂₇₇

yes

—

103

C₂³

C₂³×D₂₇₇

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

277

—

109

C₂×C₄

—

554

—

110

C₂×C₄

—

554

—

111

C₂×C₄

—

554

—

112

C₂×C₄

—

554

—

113

C₂×C₄

—

554

—

114

C₂×C₄

C₂²×Dic₂₇₇

yes

—

115

C₂×C₄

C₂²×Dic₂₇₇

yes

—

116

C₂×C₄

C₂²×Dic₂₇₇

yes

—

117

C₂×C₄

D₄×D₂₇₇

yes

—

118

C₂×C₄

#17984562

yes

—

119

C₂²

C₂³×Dic₂₇₇

yes

characteristic

120

C₂²

C₂×D₄×D₂₇₇

yes

characteristic

121

C₂²

C₂×#17984562

yes

characteristic

122

C₂²

#22579928

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

554

—

126

C₄

—

1108

—

127

C₄

—

1108

—

128

C₄

D₄×Dic₂₇₇

yes

—

129

C₂

C₂×D₄×Dic₂₇₇

yes

characteristic

130

C₂

#31800480

yes

—

131

C₂

#31800577

yes

—

132

C₂

#31800613

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₂₇₇	C₂
socle, prime radical, cosocle	C₂³×C₂₇₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₂₇₇
hypocenter	C₂₇₇	#2327
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₇₇

commutator

C₂²×C₂₇₇

C₂³×C₄

Fitting

C₂×C₄⋊Q₈×C₂₇₇

C₂

socle, prime radical, cosocle

C₂³×C₂₇₇

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₂₇₇

C₂₇₇

G ▷ C₂²×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
17. C₄⋊Q₈×C₂₇₇	C₄	semidirect
66. C₂₇₇	#2327	semidirect
75. C₄⋊Q₈	Dic₂₇₇	semidirect

H ≅ G/N

type

17. C₄⋊Q₈×C₂₇₇

C₄

semidirect

66. C₂₇₇

#2327

semidirect

75. C₄⋊Q₈

Dic₂₇₇

semidirect

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

subgroup	index
C₂×Q₈×Dic₂₇₇	2
#31799909	2
#31799888	2
#31799868	2
#31799848	2
C₂×C₄⋊Q₈×C₂₇₇	2
#2327	277

subgroup

index

C₂×Q₈×Dic₂₇₇

C₂×C₄⋊Q₈×C₂₇₇

277

Subgroups — #47985887

Conjugacy classes of subgroups

Complements

Maximal subgroups