Group #47985336 — subgroups

Subgroups — #47985336

Order 35456.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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	#31800399	C₂	1	1	yes	maximal, characteristic, complemented
2	#31800119	C₂	1	1	yes	maximal, characteristic
3	#31799825	C₂	1	1	yes	maximal, characteristic, complemented
4	#31799824	C₂	1	1	yes	maximal, characteristic, complemented
5	#31799822	C₂	1	1	yes	maximal, characteristic, complemented
6	#31799812	C₂	1	1	yes	maximal, characteristic
7	#360×C₂₇₇	C₂	1	1	yes	maximal, characteristic
8	#22579887	—	2	1	no	—
9	#22579879	—	2	1	no	—
10	#22579820	C₂²	1	1	yes	characteristic
11	#22579816	C₂²	1	1	yes	characteristic
12	#22579812	C₂²	1	1	yes	characteristic
13	#22579811	—	2	1	no	—
14	#22579807	—	2	1	no	—
15	C₈×Dic₂₇₇	C₂²	1	1	yes	characteristic
16	#135×C₂₇₇	C₂²	1	1	yes	characteristic
17	M₃₂×C₂₇₇	C₂²	1	1	yes	characteristic
18	#106×C₂₇₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₅₄	—	4	1	no	—
20	#17984547	—	2	1	no	—
21	#17984547	—	4	1	no	—
22	C₂×#14555660	—	4	1	no	—
23	C₂×#14555660	D₄	1	1	yes	characteristic
24	C₄⋊Dic₂₇₇	—	2	1	no	—
25	#17984542	—	4	1	no	—
26	C₄×Dic₂₇₇	D₄	1	1	yes	characteristic
27	#17984538	—	2	1	no	—
28	#41×C₂₇₇	—	2	1	no	—
29	Q₁₆×C₂₇₇	Q₈	1	1	yes	characteristic, complemented
30	SD₁₆×C₂₇₇	—	2	1	no	—
31	D₈×C₂₇₇	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₈×C₂₇₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₇₇	—	2	1	no	—
34	C₁₆×C₂₇₇	—	2	1	no	—
35	Dic₅₅₄	—	8	2	no	—
36	C₂×Dic₂₇₇	—	2	1	no	—
37	C₂×Dic₂₇₇	—	4	1	no	—
38	#14555660	—	4	1	no	—
39	#14555660	SD₁₆	1	1	yes	characteristic
40	#14555660	SD₁₆	1	1	yes	characteristic
41	Q₈×C₂₇₇	—	2	1	no	—
42	D₄×C₂₇₇	—	2	1	no	—
43	D₄×C₂₇₇	—	4	1	no	—
44	C₂×C₄×C₂₇₇	—	4	1	no	—
45	C₂×C₄×C₂₇₇	—	4	1	no	—
46	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
48	C₈×C₂₇₇	#41	1	1	yes	characteristic
49	C₂²×C₂₇₇	—	4	1	no	—
50	C₂²×C₂₇₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₂₇₇	—	4	1	no	—
52	C₄×C₂₇₇	—	8	1	no	—
53	C₄×C₂₇₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₂₇₇	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₂₇₇	—	2	2	no	—
56	Dic₂₇₇	—	8	1	no	—
57	C₂×C₂₇₇	—	2	1	no	—
58	C₂×C₂₇₇	—	8	1	no	—
59	C₂×C₂₇₇	#476	1	1	yes	characteristic
60	C₂₇₇	#2230	1	1	yes	characteristic, complemented, Sylow
61	#2230	—	277	1	no	maximal, Sylow
62	#498	—	277	1	no	—
63	#443	—	277	1	no	—
64	#365	—	277	1	no	—
65	#365	—	277	1	no	—
66	#364	—	277	1	no	—
67	#360	—	277	1	no	—
68	#360	D₂₇₇	1	1	yes	characteristic
69	#135	C₂×D₂₇₇	1	1	yes	characteristic
70	#131	—	554	1	no	—
71	C₄⋊Q₈	—	554	1	no	—
72	M₃₂	—	277	1	no	—
73	M₃₂	C₂×D₂₇₇	1	1	yes	characteristic
74	#108	—	277	1	no	—
75	#106	—	277	1	no	—
76	#106	—	554	1	no	—
77	#106	C₂×D₂₇₇	1	1	yes	characteristic
78	#104	—	554	1	no	—
79	C₄×C₈	—	277	1	no	—
80	#41	—	2	1	no	—
81	C₂×Q₈	—	1108	1	no	—
82	Q₁₆	Dic₅₅₄	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₅₅₄	1	1	yes	characteristic, complemented
85	M₁₆	—	554	1	no	—
86	M₁₆	—	1108	1	no	—
87	C₂×C₈	—	277	1	no	—
88	C₂×C₈	—	1108	1	no	—
89	C₂×C₈	C₂²×D₂₇₇	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	554	1	no	—
92	C₄⋊C₄	—	1108	1	no	—
93	C₄²	—	277	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	554	1	no	—
96	Q₈	—	2	1	no	—
97	Q₈	—	2216	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	554	1	no	—
103	C₂×C₄	—	1108	1	no	—
104	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
105	C₈	—	277	1	no	—
106	C₈	—	277	1	no	—
107	C₈	—	1108	1	no	—
108	C₈	C₂×Dic₅₅₄	1	1	yes	characteristic
109	C₈	#17984562	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#22579904	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	554	2	no	—
115	C₄	—	2216	1	no	—
116	C₄	#22579857	1	1	yes	characteristic
117	C₄	SD₁₆×D₂₇₇	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#31800290	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800399

C₂

yes

maximal, characteristic, complemented

#31800119

C₂

yes

maximal, characteristic

#31799825

C₂

yes

maximal, characteristic, complemented

#31799824

C₂

yes

maximal, characteristic, complemented

#31799822

C₂

yes

maximal, characteristic, complemented

#31799812

C₂

yes

maximal, characteristic

#360×C₂₇₇

C₂

yes

maximal, characteristic

#22579887

—

#22579879

—

#22579820

C₂²

yes

characteristic

#22579816

C₂²

yes

characteristic

#22579812

C₂²

yes

characteristic

#22579811

—

#22579807

—

C₈×Dic₂₇₇

C₂²

yes

characteristic

#135×C₂₇₇

C₂²

yes

characteristic

M₃₂×C₂₇₇

C₂²

yes

characteristic

#106×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

—

—

—

D₄

yes

characteristic

C₄⋊Dic₂₇₇

—

#17984542

—

C₄×Dic₂₇₇

D₄

yes

characteristic

#17984538

—

#41×C₂₇₇

—

Q₁₆×C₂₇₇

Q₈

yes

characteristic, complemented

SD₁₆×C₂₇₇

—

D₈×C₂₇₇

Q₈

yes

characteristic, complemented

C₂×C₈×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₁₆×C₂₇₇

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

#14555660

SD₁₆

yes

characteristic

#14555660

SD₁₆

yes

characteristic

Q₈×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

C₂×Q₈

yes

characteristic

C₈×C₂₇₇

#41

yes

characteristic

C₂²×C₂₇₇

—

C₂²×C₂₇₇

AGL(1,ℤ/8)

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

characteristic

C₄×C₂₇₇

C₂×SD₁₆

yes

characteristic

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#476

yes

characteristic

C₂₇₇

#2230

yes

characteristic, complemented, Sylow

#2230

—

277

maximal, Sylow

#498

—

277

—

#443

—

277

—

#365

—

277

—

#365

—

277

—

#364

—

277

—

#360

—

277

—

#360

D₂₇₇

yes

characteristic

#135

C₂×D₂₇₇

yes

characteristic

#131

—

554

—

C₄⋊Q₈

—

554

—

M₃₂

—

277

—

M₃₂

C₂×D₂₇₇

yes

characteristic

#108

—

277

—

#106

—

277

—

#106

—

554

—

#106

C₂×D₂₇₇

yes

characteristic

#104

—

554

—

C₄×C₈

—

277

—

#41

—

C₂×Q₈

—

1108

—

Q₁₆

Dic₅₅₄

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₅₅₄

yes

characteristic, complemented

M₁₆

—

554

—

M₁₆

—

1108

—

C₂×C₈

—

277

—

C₂×C₈

—

1108

—

C₂×C₈

C₂²×D₂₇₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

1108

—

C₄²

—

277

—

C₁₆

—

C₁₆

—

554

—

Q₈

—

Q₈

—

2216

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

554

—

103

C₂×C₄

—

1108

—

104

C₂×C₄

D₄×D₂₇₇

yes

characteristic

105

C₈

—

277

—

106

C₈

—

277

—

107

C₈

—

1108

—

108

C₈

C₂×Dic₅₅₄

yes

characteristic

109

C₈

#17984562

yes

characteristic

110

C₂²

—

111

C₂²

#22579904

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

554

—

115

C₄

—

2216

—

116

C₄

#22579857

yes

characteristic

117

C₄

SD₁₆×D₂₇₇

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#31800290

yes

characteristic

	subgroup	G/H
center	C₂	#31800290
commutator, cosocle	C₂×C₈×C₂₇₇	C₂³
Frattini	C₂×C₈	C₂²×D₂₇₇
Fitting	#360×C₂₇₇	C₂
socle	C₂×C₂₇₇	#476
prime radical	#135×C₂₇₇	C₂²
hypercenter	#360	D₂₇₇
hypocenter	C₂₇₇	#2230
LCS	G ▷ C₂×C₈×C₂₇₇ ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#31800290

commutator, cosocle

C₂×C₈×C₂₇₇

C₂³

Frattini

C₂×C₈

C₂²×D₂₇₇

Fitting

#360×C₂₇₇

C₂

socle

C₂×C₂₇₇

#476

prime radical

#135×C₂₇₇

C₂²

D₂₇₇

C₂₇₇

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #31800399	C₂	semidirect
3. #31799825	C₂	semidirect
4. #31799824	C₂	semidirect
5. #31799822	C₂	semidirect
29. Q₁₆×C₂₇₇	Q₈	semidirect
31. D₈×C₂₇₇	Q₈	semidirect
60. C₂₇₇	#2230	semidirect
82. Q₁₆	Dic₅₅₄	semidirect
84. D₈	Dic₅₅₄	semidirect

H ≅ G/N

type

1. #31800399

C₂

semidirect

3. #31799825

C₂

semidirect

4. #31799824

C₂

semidirect

5. #31799822

C₂

semidirect

29. Q₁₆×C₂₇₇

Q₈

semidirect

31. D₈×C₂₇₇

Q₈

semidirect

60. C₂₇₇

#2230

semidirect

82. Q₁₆

Dic₅₅₄

semidirect

84. D₈

Dic₅₅₄

semidirect

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

subgroup	index
#31800399	2
#31800119	2
#31799825	2
#31799824	2
#31799822	2
#31799812	2
#360×C₂₇₇	2
#2230	277

subgroup

index

#360×C₂₇₇

277

Subgroups — #47985336

Conjugacy classes of subgroups

Complements

Maximal subgroups