Group #47985244 — subgroups

Subgroups — #47985244

Order 35456.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#31800440	C₂	1	2	yes	maximal, complemented
2	#31800379	C₂	1	1	yes	maximal, characteristic, complemented
3	#31800120	C₂	1	1	yes	maximal, characteristic, complemented
4	#31799816	C₂	1	2	yes	maximal, complemented
5	#364×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#22579902	C₂²	1	1	yes	characteristic
7	C₂×D₂₂₁₆	—	2	1	no	—
8	C₂×D₂₂₁₆	C₂²	1	2	yes	—
9	#22579886	—	2	1	no	—
10	#22579877	—	2	1	no	—
11	#22579825	—	2	2	no	—
12	D₄₄₃₂	—	2	2	no	—
13	#22579816	C₂²	1	1	yes	characteristic
14	#22579808	—	2	1	no	—
15	M₃₂×C₂₇₇	C₂²	1	2	yes	—
16	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
17	#17984563	—	2	1	no	—
18	C₂×D₁₁₀₈	—	2	2	no	—
19	C₂×D₁₁₀₈	—	4	2	no	—
20	Dic₁₁₀₈	D₄	1	1	yes	characteristic, complemented
21	#17984552	—	2	1	no	—
22	D₂₂₁₆	—	2	2	no	—
23	D₂₂₁₆	—	2	2	no	—
24	D₂₂₁₆	—	4	1	no	—
25	D₂₂₁₆	—	4	1	no	—
26	D₂₂₁₆	D₄	1	1	yes	characteristic, complemented
27	#17984547	—	2	1	no	—
28	#17984546	—	4	1	no	—
29	C₈×D₂₇₇	—	4	1	no	—
30	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₇₇	C₂³	1	1	yes	characteristic
32	C₄²×C₂₇₇	D₄	1	1	yes	characteristic
33	C₁₆×C₂₇₇	—	2	2	no	—
34	C₂²×D₂₇₇	—	4	2	no	—
35	Dic₅₅₄	—	2	1	no	—
36	D₁₁₀₈	—	2	1	no	—
37	D₁₁₀₈	—	4	2	no	—
38	D₁₁₀₈	—	4	2	no	—
39	D₁₁₀₈	—	8	4	no	—
40	C₂²⋊D₂₇₇	—	4	1	no	—
41	C₄×D₂₇₇	—	4	1	no	—
42	#14555660	—	4	1	no	—
43	C₂×C₄×C₂₇₇	—	2	1	no	—
44	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
45	C₈×C₂₇₇	D₈	1	1	yes	characteristic
46	C₈×C₂₇₇	D₈	1	1	yes	characteristic
47	C₈×C₂₇₇	C₂×D₄	1	1	yes	characteristic
48	C₈×C₂₇₇	#41	1	1	yes	characteristic
49	C₂²×C₂₇₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₂₇₇	—	4	1	no	—
51	C₂×D₂₇₇	—	8	4	no	—
52	C₂×D₂₇₇	—	8	2	no	—
53	C₄×C₂₇₇	—	2	2	no	—
54	C₄×C₂₇₇	#121	1	1	yes	characteristic
55	C₄×C₂₇₇	C₂×D₈	1	1	yes	characteristic
56	Dic₂₇₇	—	4	1	no	—
57	C₂×C₂₇₇	—	2	1	no	—
58	C₂×C₂₇₇	#459	1	1	yes	characteristic
59	D₂₇₇	—	8	1	no	—
60	D₂₇₇	—	16	2	no	—
61	C₂₇₇	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	277	1	no	maximal, Sylow
63	#509	—	277	2	no	—
64	#493	—	277	1	no	—
65	#443	—	277	1	no	—
66	#364	D₂₇₇	1	1	yes	characteristic, complemented
67	#361	—	277	2	no	—
68	#135	—	277	1	no	—
69	C₂×D₈	—	277	2	no	—
70	C₂×D₈	—	554	1	no	—
71	#131	—	554	1	no	—
72	#127	—	554	1	no	—
73	SD₃₂	—	554	2	no	—
74	D₁₆	—	554	2	no	—
75	M₃₂	C₂×D₂₇₇	1	2	yes	—
76	#108	—	277	1	no	—
77	#104	—	554	1	no	—
78	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
79	#41	—	554	1	no	—
80	C₂×D₄	—	554	2	no	—
81	C₂×D₄	—	1108	2	no	—
82	Q₁₆	—	277	1	no	—
83	SD₁₆	—	554	1	no	—
84	D₈	—	277	1	no	—
85	D₈	—	554	2	no	—
86	D₈	—	554	2	no	—
87	D₈	—	1108	1	no	—
88	D₈	—	1108	1	no	—
89	M₁₆	—	554	1	no	—
90	M₁₆	—	1108	1	no	—
91	C₂×C₈	—	1108	1	no	—
92	C₂×C₈	D₁₁₀₈	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₂₇₇	1	1	yes	characteristic
94	C₄²	D₁₁₀₈	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	1108	2	no	—
97	Q₈	—	554	1	no	—
98	D₄	—	554	1	no	—
99	D₄	—	1108	1	no	—
100	D₄	—	1108	2	no	—
101	D₄	—	1108	2	no	—
102	D₄	—	2216	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	1108	1	no	—
105	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
106	C₈	—	1108	1	no	—
107	C₈	D₂₂₁₆	1	1	yes	characteristic
108	C₈	D₂₂₁₆	1	1	yes	characteristic
109	C₈	D₄×D₂₇₇	1	1	yes	characteristic
110	C₈	#17984564	1	1	yes	characteristic
111	C₂²	—	1108	1	no	—
112	C₂²	—	2216	4	no	—
113	C₂²	—	2216	2	no	—
114	C₂²	#22579908	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	1108	1	no	—
117	C₄	#22579855	1	1	yes	characteristic
118	C₄	C₂×D₂₂₁₆	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	4432	2	no	—
121	C₂	—	2216	1	no	—
122	C₂	#31800198	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800440

C₂

yes

maximal, complemented

#31800379

C₂

yes

maximal, characteristic, complemented

#31800120

C₂

yes

maximal, characteristic, complemented

#31799816

C₂

yes

maximal, complemented

#364×C₂₇₇

C₂

yes

maximal, characteristic, complemented

#22579902

C₂²

yes

characteristic

C₂×D₂₂₁₆

—

C₂×D₂₂₁₆

C₂²

yes

—

#22579886

—

#22579877

—

#22579825

—

D₄₄₃₂

—

#22579816

C₂²

yes

characteristic

#22579808

—

M₃₂×C₂₇₇

C₂²

yes

—

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

#17984563

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

Dic₁₁₀₈

D₄

yes

characteristic, complemented

#17984552

—

D₂₂₁₆

—

D₂₂₁₆

—

D₂₂₁₆

—

D₂₂₁₆

—

D₂₂₁₆

D₄

yes

characteristic, complemented

#17984547

—

#17984546

—

C₈×D₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

characteristic

C₂×C₈×C₂₇₇

C₂³

yes

characteristic

C₄²×C₂₇₇

D₄

yes

characteristic

C₁₆×C₂₇₇

—

C₂²×D₂₇₇

—

Dic₅₅₄

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

C₂²⋊D₂₇₇

—

C₄×D₂₇₇

—

#14555660

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

D₈

yes

characteristic

C₈×C₂₇₇

D₈

yes

characteristic

C₈×C₂₇₇

C₂×D₄

yes

C₈×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#121

yes

characteristic

C₄×C₂₇₇

C₂×D₈

yes

characteristic

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#459

yes

characteristic

D₂₇₇

—

D₂₇₇

—

C₂₇₇

#2213

yes

characteristic, complemented, Sylow

#2213

—

277

maximal, Sylow

#509

—

277

—

#493

—

277

—

#443

—

277

—

#364

D₂₇₇

yes

characteristic, complemented

#361

—

277

—

#135

—

277

—

C₂×D₈

—

277

—

C₂×D₈

—

554

—

#131

—

554

—

#127

—

554

—

SD₃₂

—

554

—

D₁₆

—

554

—

M₃₂

C₂×D₂₇₇

yes

—

#108

—

277

—

#104

—

554

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

#41

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

1108

—

Q₁₆

—

277

—

SD₁₆

—

554

—

D₈

—

277

—

D₈

—

554

—

D₈

—

554

—

D₈

—

1108

—

D₈

—

1108

—

M₁₆

—

554

—

M₁₆

—

1108

—

C₂×C₈

—

1108

—

C₂×C₈

D₁₁₀₈

yes

characteristic

C₂×C₈

C₂²×D₂₇₇

yes

characteristic

C₄²

D₁₁₀₈

yes

characteristic

C₁₆

—

C₂³

—

1108

—

Q₈

—

554

—

D₄

—

554

—

D₄

—

1108

—

100

D₄

—

1108

—

101

D₄

—

1108

—

102

D₄

—

2216

—

103

C₂×C₄

—

104

C₂×C₄

—

1108

—

105

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

106

C₈

—

1108

—

107

C₈

D₂₂₁₆

yes

characteristic

108

C₈

D₂₂₁₆

yes

characteristic

109

C₈

D₄×D₂₇₇

yes

characteristic

110

C₈

#17984564

yes

characteristic

111

C₂²

—

1108

—

112

C₂²

—

2216

—

113

C₂²

—

2216

—

114

C₂²

#22579908

yes

characteristic

115

C₄

—

116

C₄

—

1108

—

117

C₄

#22579855

yes

characteristic

118

C₄

C₂×D₂₂₁₆

yes

characteristic

119

C₂

—

120

C₂

—

4432

—

121

C₂

—

2216

—

122

C₂

#31800198

yes

characteristic

	subgroup	G/H
center	C₂	#31800198
commutator, cosocle	C₂×C₈×C₂₇₇	C₂³
Frattini	C₂×C₈	C₂²×D₂₇₇
Fitting	#364×C₂₇₇	C₂
socle	C₂×C₂₇₇	#459
hypercenter	#364	D₂₇₇
hypocenter	C₂₇₇	#2213
LCS	G ▷ C₂×C₈×C₂₇₇ ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#31800198

commutator, cosocle

C₂×C₈×C₂₇₇

C₂³

Frattini

C₂×C₈

C₂²×D₂₇₇

Fitting

#364×C₂₇₇

C₂

C₂×C₂₇₇

D₂₇₇

C₂₇₇

G ▷ C₂×C₈×C₂₇₇ ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #31800440	C₂	semidirect
2. #31800379	C₂	semidirect
3. #31800120	C₂	semidirect
4. #31799816	C₂	semidirect
5. #364×C₂₇₇	C₂	semidirect
20. Dic₁₁₀₈	D₄	semidirect
26. D₂₂₁₆	D₄	semidirect
61. C₂₇₇	#2213	semidirect
66. #364	D₂₇₇	semidirect

H ≅ G/N

type

1. #31800440

C₂

semidirect

2. #31800379

C₂

semidirect

3. #31800120

C₂

semidirect

4. #31799816

C₂

semidirect

5. #364×C₂₇₇

C₂

semidirect

20. Dic₁₁₀₈

D₄

semidirect

26. D₂₂₁₆

D₄

semidirect

61. C₂₇₇

#2213

semidirect

66. #364

D₂₇₇

semidirect

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

subgroup	index
#31800440	2
#31800379	2
#31800120	2
#31799816	2
#364×C₂₇₇	2
#2213	277

subgroup

index

#364×C₂₇₇

277

Subgroups — #47985244

Conjugacy classes of subgroups

Complements

Maximal subgroups