Group #47984721 — subgroups

Subgroups — #47984721

Order 35456.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#31800051	C₂	1	1	yes	maximal, characteristic, complemented
2	#31800029	C₂	1	2	yes	maximal, complemented
3	#31800020	C₂	1	2	yes	maximal
4	#31799970	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17984563	C₂²	1	1	yes	characteristic
7	C₂×#17984547	C₂²	1	2	yes	—
8	C₂²×#14555660	C₂²	1	1	yes	characteristic
9	#22579816	—	2	2	no	—
10	C₄⋊#14555660	—	2	4	no	—
11	#22579808	—	2	4	no	—
12	#22579792	—	2	2	no	—
13	C₂×M₁₆×C₂₇₇	C₂²	1	1	yes	characteristic
14	#117×C₂₇₇	C₂²	1	2	yes	—
15	#17984563	—	2	2	no	—
16	#17984563	—	2	2	no	—
17	C₂×Dic₅₅₄	Q₈	1	1	yes	characteristic
18	C₂×D₁₁₀₈	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₇₇	—	2	1	no	—
20	C₂×C₄×D₂₇₇	—	2	1	no	—
21	#17984547	—	2	2	no	—
22	#17984547	—	2	2	no	—
23	C₂×#14555660	—	2	1	no	—
24	C₂×#14555660	—	2	2	no	—
25	C₂×#14555660	—	2	2	no	—
26	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₇₇	—	2	2	no	—
28	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₇₇	—	2	2	no	—
30	C₂²⋊C₄×C₂₇₇	—	2	2	no	—
31	C₄²×C₂₇₇	D₄	1	4	yes	—
32	C₂²×D₂₇₇	—	2	1	no	—
33	Dic₅₅₄	—	2	2	no	—
34	D₁₁₀₈	—	2	2	no	—
35	C₂²⋊D₂₇₇	—	4	2	no	—
36	C₂×Dic₂₇₇	—	2	1	no	—
37	C₄×D₂₇₇	—	4	1	no	—
38	C₄×D₂₇₇	—	4	1	no	—
39	#14555660	—	4	1	no	—
40	#14555660	—	4	2	no	—
41	C₂³×C₂₇₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₂₇₇	—	2	4	no	—
43	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₇₇	#41	1	2	yes	—
47	C₈×C₂₇₇	—	2	2	no	—
48	C₂²×C₂₇₇	—	4	1	no	—
49	C₂²×C₂₇₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₇₇	#123	1	2	yes	—
51	C₂×D₂₇₇	—	4	2	no	—
52	C₂×D₂₇₇	—	4	1	no	—
53	C₄×C₂₇₇	—	4	4	no	—
54	C₄×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₂₇₇	#123	1	1	yes	characteristic
56	C₄×C₂₇₇	#127	1	1	yes	characteristic
57	Dic₂₇₇	—	4	1	no	—
58	C₂×C₂₇₇	—	2	2	no	—
59	C₂×C₂₇₇	—	2	1	no	—
60	C₂×C₂₇₇	#399	1	1	yes	characteristic
61	D₂₇₇	—	8	1	no	—
62	C₂₇₇	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	277	1	no	maximal, Sylow
64	#430	—	277	1	no	—
65	#424	—	277	2	no	—
66	#421	—	277	2	no	—
67	#408	—	277	1	no	—
68	H(3,ℤ/4)	D₂₇₇	1	1	yes	characteristic, complemented
69	C₂×#41	—	277	1	no	—
70	C₂×M₁₆	—	277	2	no	—
71	C₂×M₁₆	C₂×D₂₇₇	1	1	yes	characteristic
72	C₂²×C₈	—	277	1	no	—
73	#117	C₂×D₂₇₇	1	2	yes	—
74	#108	—	554	2	no	—
75	C₄⋊C₈	—	554	4	no	—
76	#104	—	554	4	no	—
77	C₂²⋊C₈	—	554	2	no	—
78	#41	—	554	2	no	—
79	#41	—	554	2	no	—
80	C₂×Q₈	—	277	1	no	—
81	C₂×D₄	—	277	1	no	—
82	C₂×D₄	—	554	1	no	—
83	C₂²×C₄	—	554	1	no	—
84	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	554	2	no	—
87	M₁₆	—	554	2	no	—
88	C₂×C₈	—	554	1	no	—
89	C₂×C₈	—	554	2	no	—
90	C₂×C₈	—	554	2	no	—
91	C₂×C₈	D₁₁₀₈	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₁₁₀₈	1	4	yes	—
95	C₂³	—	554	1	no	—
96	C₂³	#17984562	1	1	yes	characteristic
97	Q₈	—	554	2	no	—
98	D₄	—	554	2	no	—
99	D₄	—	1108	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	554	1	no	—
102	C₂×C₄	—	1108	1	no	—
103	C₂×C₄	—	1108	1	no	—
104	C₂×C₄	C₂×D₁₁₀₈	1	2	yes	—
105	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
107	C₂×C₄	#17984562	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	1108	1	no	—
110	C₈	—	1108	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	1108	2	no	—
113	C₂²	—	1108	1	no	—
114	C₂²	#22579860	1	1	yes	characteristic
115	C₂²	#22579864	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	1108	1	no	—
118	C₄	#22579860	1	2	yes	—
119	C₄	#22579864	1	1	yes	characteristic
120	C₄	#22579877	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	2216	1	no	—
124	C₂	#31799939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800051

C₂

yes

maximal, characteristic, complemented

#31800029

C₂

yes

maximal, complemented

#31800020

C₂

yes

maximal

#31799970

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂×#17984563

C₂²

yes

characteristic

C₂×#17984547

C₂²

yes

—

C₂²×#14555660

C₂²

yes

—

—

—

—

C₂×M₁₆×C₂₇₇

C₂²

yes

characteristic

#117×C₂₇₇

C₂²

yes

—

#17984563

—

#17984563

—

C₂×Dic₅₅₄

Q₈

yes

characteristic

C₂×D₁₁₀₈

Q₈

yes

characteristic

C₂×C₂²⋊D₂₇₇

—

C₂×C₄×D₂₇₇

—

—

—

—

—

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₄⋊C₄×C₂₇₇

—

C₂²⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

D₄

yes

—

C₂²×D₂₇₇

—

Dic₅₅₄

—

D₁₁₀₈

—

C₂²⋊D₂₇₇

—

C₂×Dic₂₇₇

—

C₄×D₂₇₇

—

C₄×D₂₇₇

—

#14555660

—

#14555660

—

C₂³×C₂₇₇

#41

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₇₇

#123

yes

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

—

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#399

yes

characteristic

D₂₇₇

—

C₂₇₇

#2098

yes

characteristic, complemented, Sylow

#2098

—

277

maximal, Sylow

#430

—

277

—

#424

—

277

—

#421

—

277

—

#408

—

277

—

H(3,ℤ/4)

D₂₇₇

yes

characteristic, complemented

C₂×#41

—

277

—

C₂×M₁₆

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

characteristic

C₂²×C₈

—

277

—

#117

C₂×D₂₇₇

yes

—

#108

—

554

—

C₄⋊C₈

—

554

—

#104

—

554

—

C₂²⋊C₈

—

554

—

#41

—

554

—

#41

—

554

—

C₂×Q₈

—

277

—

C₂×D₄

—

277

—

C₂×D₄

—

554

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

M₁₆

—

554

—

M₁₆

—

554

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₂×C₈

D₁₁₀₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₁₁₀₈

yes

—

C₂³

—

554

—

C₂³

#17984562

yes

characteristic

Q₈

—

554

—

D₄

—

554

—

D₄

—

1108

—

100

C₂×C₄

—

101

C₂×C₄

—

554

—

102

C₂×C₄

—

1108

—

103

C₂×C₄

—

1108

—

104

C₂×C₄

C₂×D₁₁₀₈

yes

—

105

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

106

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

107

C₂×C₄

#17984562

yes

—

108

C₈

—

109

C₈

—

1108

—

110

C₈

—

1108

—

111

C₂²

—

112

C₂²

—

1108

—

113

C₂²

—

1108

—

114

C₂²

#22579860

yes

characteristic

115

C₂²

#22579864

yes

—

116

C₄

—

117

C₄

—

1108

—

118

C₄

#22579860

yes

—

119

C₄

#22579864

yes

characteristic

120

C₄

#22579877

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

2216

—

124

C₂

#31799939

yes

characteristic

	subgroup	G/H
center	C₄	#22579877
commutator, cosocle	C₂²×C₄×C₂₇₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	H(3,ℤ/4)×C₂₇₇	C₂
socle	C₂×C₂₇₇	#399
prime radical	C₂×#17984563	C₂²
hypercenter	H(3,ℤ/4)	D₂₇₇
hypocenter	C₂₇₇	#2098
LCS	G ▷ C₂²×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#22579877

commutator, cosocle

C₂²×C₄×C₂₇₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

H(3,ℤ/4)×C₂₇₇

C₂

C₂×C₂₇₇

C₂²

D₂₇₇

C₂₇₇

G ▷ C₂²×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #31800051	C₂	semidirect
2. #31800029	C₂	semidirect
5. H(3,ℤ/4)×C₂₇₇	C₂	semidirect
62. C₂₇₇	#2098	semidirect
68. H(3,ℤ/4)	D₂₇₇	semidirect

H ≅ G/N

type

1. #31800051

C₂

semidirect

2. #31800029

C₂

semidirect

5. H(3,ℤ/4)×C₂₇₇

C₂

semidirect

62. C₂₇₇

#2098

semidirect

68. H(3,ℤ/4)

D₂₇₇

semidirect

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

subgroup	index
#31800051	2
#31800029	2
#31800020	2
#31799970	2
H(3,ℤ/4)×C₂₇₇	2
#2098	277

subgroup

index

H(3,ℤ/4)×C₂₇₇

277

Subgroups — #47984721

Conjugacy classes of subgroups

Complements

Maximal subgroups