Group #47984106 — subgroups

Subgroups — #47984106

Order 35456.

280 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄×Dic₅₅₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579791	C₂	1	1	yes	maximal, characteristic
3	#31799756	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₅₄	C₄	1	1	yes	characteristic
6	C₂²×#14555660	C₂²	1	2	yes	—
7	C₄×Dic₅₅₄	—	2	2	no	—
8	C₄×Dic₅₅₄	—	2	2	no	—
9	C₂×C₄⋊Dic₂₇₇	C₄	1	1	yes	characteristic
10	C₂×#17984542	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
12	#22579791	—	2	2	no	—
13	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₂₇₇	—	2	2	no	—
16	C₂×Dic₅₅₄	—	2	4	no	—
17	C₂×Dic₅₅₄	—	2	2	no	—
18	C₂²×Dic₂₇₇	C₂×C₄	1	2	yes	—
19	C₂²×Dic₂₇₇	C₂×C₄	1	2	yes	—
20	C₂×#14555660	—	2	2	no	—
21	C₂×#14555660	—	2	2	no	—
22	C₂×#14555660	Q₈	1	4	yes	complemented
23	C₄⋊Dic₂₇₇	—	2	2	no	—
24	#17984542	—	2	4	no	—
25	C₄×Dic₂₇₇	—	2	2	no	—
26	C₄×Dic₂₇₇	—	2	2	no	—
27	C₂²×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₇₇	—	2	2	no	—
31	C₂×C₈×C₂₇₇	—	2	4	no	—
32	C₄²×C₂₇₇	D₄	1	2	yes	—
33	C₄²×C₂₇₇	D₄	1	2	yes	—
34	Dic₅₅₄	—	2	8	no	—
35	C₂×Dic₂₇₇	—	2	2	no	—
36	C₂×Dic₂₇₇	—	2	2	no	—
37	C₂×Dic₂₇₇	—	2	2	no	—
38	C₂×Dic₂₇₇	—	2	4	no	—
39	C₂×Dic₂₇₇	M₁₆	1	4	yes	—
40	#14555660	—	2	4	no	—
41	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₇₇	—	2	2	no	—
43	C₂×C₄×C₂₇₇	—	2	2	no	—
44	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₇₇	#41	1	2	yes	—
50	C₈×C₂₇₇	—	4	2	no	—
51	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₇₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₂₇₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₂₇₇	#131	1	2	yes	—
55	C₄×C₂₇₇	—	2	2	no	—
56	C₄×C₂₇₇	—	2	2	no	—
57	C₄×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₂₇₇	#124	1	1	yes	characteristic
59	C₄×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₂₇₇	—	2	4	no	—
61	Dic₂₇₇	—	4	2	no	—
62	C₂×C₂₇₇	#391	1	1	yes	characteristic
63	C₂×C₂₇₇	#407	1	1	yes	characteristic
64	C₂×C₂₇₇	#408	1	1	yes	characteristic
65	C₂×C₂₇₇	#446	1	4	yes	—
66	C₂₇₇	#1985	1	1	yes	characteristic, complemented, Sylow
67	#1985	—	277	1	no	maximal, Sylow
68	C₂×C₄×Q₈	—	277	1	no	—
69	C₂×C₄⋊C₈	D₂₇₇	1	1	yes	characteristic
70	C₂×#97	—	277	1	no	—
71	#336	—	277	4	no	—
72	C₂²×Q₈	—	277	1	no	—
73	C₂²×C₈	—	277	2	no	—
74	C₂²×C₈	C₂×D₂₇₇	1	2	yes	—
75	C₄×Q₈	—	554	2	no	—
76	C₄×Q₈	—	554	2	no	—
77	C₂×C₄⋊C₄	—	277	2	no	—
78	C₂×C₄⋊C₄	—	277	1	no	—
79	C₂×C₄²	—	277	2	no	—
80	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#97	—	554	2	no	—
83	C₂×Q₈	—	554	4	no	—
84	C₂×Q₈	—	554	2	no	—
85	C₂²×C₄	—	277	2	no	—
86	C₂²×C₄	—	277	2	no	—
87	C₂²×C₄	C₄×D₂₇₇	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₂₇₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	277	4	no	—
93	C₂×C₈	—	554	2	no	—
94	C₂×C₈	—	554	2	no	—
95	C₄⋊C₄	—	554	4	no	—
96	C₄⋊C₄	—	554	2	no	—
97	C₄²	—	554	2	no	—
98	C₄²	—	554	2	no	—
99	C₄²	C₂²⋊D₂₇₇	1	2	yes	—
100	C₄²	D₁₁₀₈	1	2	yes	—
101	C₂³	C₂×C₄×D₂₇₇	1	1	yes	characteristic
102	Q₈	—	554	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	277	4	no	—
106	C₂×C₄	—	554	2	no	—
107	C₂×C₄	—	554	2	no	—
108	C₂×C₄	—	554	2	no	—
109	C₂×C₄	—	554	4	no	—
110	C₂×C₄	#17984540	1	2	yes	—
111	C₂×C₄	#17984540	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
113	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
115	C₂×C₄	#17984562	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	554	4	no	—
118	C₂²	C₂×#17984540	1	1	yes	characteristic
119	C₂²	#22579845	1	2	yes	—
120	C₂²	M₁₆×D₂₇₇	1	2	yes	—
121	C₂²	#22579886	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	554	4	no	—
125	C₄	—	1108	2	no	—
126	C₄	#22579860	1	2	yes	—
127	C₄	#22579867	1	1	yes	characteristic
128	C₄	#22579878	1	1	yes	characteristic
129	C₂	#31799907	1	1	yes	characteristic
130	C₂	#31799965	1	1	yes	characteristic
131	C₂	#31799968	1	1	yes	characteristic
132	C₂	#31800131	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₅₅₄

C₂

yes

maximal, characteristic

C₂×#22579791

C₂

yes

maximal, characteristic

#31799756

C₂

yes

maximal

C₂×C₄⋊C₈×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₅₄

C₄

yes

characteristic

C₂²×#14555660

C₂²

yes

—

C₄×Dic₅₅₄

—

C₄×Dic₅₅₄

—

C₂×C₄⋊Dic₂₇₇

C₄

yes

characteristic

C₂×#17984542

C₄

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

—

#22579791

—

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₇₇

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂²×Dic₂₇₇

C₂×C₄

yes

—

C₂²×Dic₂₇₇

C₂×C₄

yes

—

C₂×#14555660

—

C₂×#14555660

—

C₂×#14555660

Q₈

yes

complemented

C₄⋊Dic₂₇₇

—

#17984542

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₂²×C₄×C₂₇₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₇₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄²×C₂₇₇

D₄

yes

—

C₄²×C₂₇₇

D₄

yes

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

M₁₆

yes

—

#14555660

—

C₂³×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

—

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₄×Q₈

yes

—

C₂²×C₂₇₇

C₂×M₁₆

yes

—

C₂²×C₂₇₇

#131

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

—

C₄×C₂₇₇

#124

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

characteristic

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

—

C₂₇₇

#1985

yes

characteristic, complemented, Sylow

#1985

—

277

maximal, Sylow

C₂×C₄×Q₈

—

277

—

C₂×C₄⋊C₈

D₂₇₇

yes

characteristic

C₂×#97

—

277

—

#336

—

277

—

C₂²×Q₈

—

277

—

C₂²×C₈

—

277

—

C₂²×C₈

C₂×D₂₇₇

yes

—

C₄×Q₈

—

554

—

C₄×Q₈

—

554

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄²

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

C₄⋊C₈

—

#97

—

554

—

C₂×Q₈

—

554

—

C₂×Q₈

—

554

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

C₄×D₂₇₇

yes

characteristic

C₂²×C₄

C₄×D₂₇₇

yes

characteristic

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

277

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄²

—

554

—

C₄²

—

554

—

C₄²

C₂²⋊D₂₇₇

yes

—

100

C₄²

D₁₁₀₈

yes

—

101

C₂³

C₂×C₄×D₂₇₇

yes

characteristic

102

Q₈

—

554

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

277

—

106

C₂×C₄

—

554

—

107

C₂×C₄

—

554

—

108

C₂×C₄

—

554

—

109

C₂×C₄

—

554

—

110

C₂×C₄

#17984540

yes

—

111

C₂×C₄

#17984540

yes

—

112

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

113

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

114

C₂×C₄

Q₈×D₂₇₇

yes

—

115

C₂×C₄

#17984562

yes

—

116

C₈

—

117

C₈

—

554

—

118

C₂²

C₂×#17984540

yes

characteristic

119

C₂²

#22579845

yes

—

120

C₂²

M₁₆×D₂₇₇

yes

—

121

C₂²

#22579886

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

554

—

125

C₄

—

1108

—

126

C₄

#22579860

yes

—

127

C₄

#22579867

yes

characteristic

128

C₄

#22579878

yes

characteristic

129

C₂

#31799907

yes

characteristic

130

C₂

#31799965

yes

characteristic

131

C₂

#31799968

yes

characteristic

132

C₂

#31800131

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₇₇
commutator, socle, prime radical	C₂³×C₂₇₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₂₇₇	C₂
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₇₇
hypocenter	C₂₇₇	#1985
LCS	G ▷ C₂³×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₇₇

commutator, socle, prime radical

C₂³×C₂₇₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₂₇₇

C₂

cosocle

C₂²×C₄×C₂₇₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₇₇

C₂₇₇

G ▷ C₂³×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₂×#14555660	Q₈	semidirect
66. C₂₇₇	#1985	semidirect

H ≅ G/N

type

22. C₂×#14555660

Q₈

semidirect

66. C₂₇₇

#1985

semidirect

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

subgroup	index
C₂×C₄×Dic₅₅₄	2
C₂×#22579791	2
#31799756	2
C₂×C₄⋊C₈×C₂₇₇	2
#1985	277

subgroup

index

C₂×C₄×Dic₅₅₄

C₂×#22579791

#31799756

C₂×C₄⋊C₈×C₂₇₇

#1985

277

Subgroups — #47984106

Conjugacy classes of subgroups

Complements

Maximal subgroups