Group #47982313 — subgroups

Subgroups — #47982313

Order 35456.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#31800718	C₂	1	1	yes	maximal, characteristic
2	#31800381	C₂	1	1	yes	maximal, characteristic, complemented
3	#31800293	C₂	1	2	yes	maximal, complemented
4	#31799729	C₂	1	2	yes	maximal, complemented
5	#333×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17984553	—	2	2	no	—
7	#22579880	C₂²	1	1	yes	characteristic
8	#22579869	—	2	2	no	—
9	#22579855	—	2	2	no	—
10	C₄×D₁₁₀₈	C₂²	1	2	yes	—
11	#22579838	—	2	2	no	—
12	#22579812	—	2	2	no	—
13	#22579802	—	2	2	no	—
14	C₄×#14555660	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₇₇	C₂²	1	2	yes	—
17	C₂×Dic₅₅₄	—	2	1	no	—
18	C₂×D₁₁₀₈	—	2	2	no	—
19	C₂×D₁₁₀₈	D₄	1	2	yes	—
20	C₂×C₄×D₂₇₇	—	2	2	no	—
21	#17984553	—	4	4	no	—
22	C₂×#14555660	—	2	1	no	—
23	C₄⋊Dic₂₇₇	—	2	2	no	—
24	C₄⋊Dic₂₇₇	D₄	1	2	yes	—
25	#17984540	—	2	4	no	—
26	#17984540	—	2	2	no	—
27	C₄×Dic₂₇₇	—	2	2	no	—
28	C₂×Q₈×C₂₇₇	D₄	1	2	yes	—
29	C₂×C₈×C₂₇₇	—	2	2	no	—
30	C₄⋊C₄×C₂₇₇	—	2	2	no	—
31	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
32	C₂²×D₂₇₇	—	2	2	no	—
33	Dic₅₅₄	—	4	2	no	—
34	D₁₁₀₈	—	2	2	no	—
35	D₁₁₀₈	—	2	2	no	—
36	D₁₁₀₈	—	8	2	no	—
37	C₂×Dic₂₇₇	—	2	2	no	—
38	C₂×Dic₂₇₇	—	4	1	no	—
39	C₄×D₂₇₇	—	4	2	no	—
40	C₄×D₂₇₇	—	4	2	no	—
41	#14555660	—	2	2	no	—
42	Q₈×C₂₇₇	—	4	2	no	—
43	C₂×C₄×C₂₇₇	—	2	2	no	—
44	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
46	C₈×C₂₇₇	—	4	2	no	—
47	C₂²×C₂₇₇	#120	1	1	yes	characteristic
48	C₂×D₂₇₇	—	4	2	no	—
49	C₂×D₂₇₇	—	4	2	no	—
50	C₂×D₂₇₇	—	4	2	no	—
51	C₄×C₂₇₇	—	2	1	no	—
52	C₄×C₂₇₇	—	4	2	no	—
53	C₄×C₂₇₇	#135	1	2	yes	—
54	C₄×C₂₇₇	#137	1	2	yes	—
55	Dic₂₇₇	—	4	2	no	—
56	Dic₂₇₇	—	8	1	no	—
57	C₂×C₂₇₇	#452	1	2	yes	—
58	C₂×C₂₇₇	#453	1	1	yes	characteristic
59	D₂₇₇	—	8	2	no	—
60	C₂₇₇	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	277	1	no	maximal, Sylow
62	#555	—	277	1	no	—
63	#494	—	277	1	no	—
64	#476	—	277	2	no	—
65	#333	D₂₇₇	1	1	yes	characteristic, complemented
66	D₄⋊C₈	—	277	2	no	—
67	C₂×Q₁₆	—	554	2	no	—
68	C₄⋊Q₈	—	277	1	no	—
69	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
70	#124	—	554	2	no	—
71	#121	—	554	2	no	—
72	C₄×D₄	—	277	2	no	—
73	#117	—	554	2	no	—
74	#106	—	554	2	no	—
75	C₄⋊C₈	C₂×D₂₇₇	1	2	yes	—
76	#102	—	554	2	no	—
77	C₄×C₈	—	277	1	no	—
78	C₂×Q₈	—	554	1	no	—
79	C₂×Q₈	D₁₁₀₈	1	2	yes	—
80	C₂×D₄	—	277	2	no	—
81	C₂×D₄	—	554	2	no	—
82	C₂²×C₄	—	554	2	no	—
83	Q₁₆	—	1108	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	554	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	277	2	no	—
88	C₄⋊C₄	—	554	2	no	—
89	C₂²⋊C₄	—	554	4	no	—
90	C₂²⋊C₄	—	554	2	no	—
91	C₄²	—	554	2	no	—
92	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
93	C₂³	—	554	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	1108	2	no	—
96	D₄	—	554	2	no	—
97	D₄	—	554	2	no	—
98	D₄	—	2216	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	554	2	no	—
101	C₂×C₄	—	1108	2	no	—
102	C₂×C₄	—	1108	2	no	—
103	C₂×C₄	—	1108	1	no	—
104	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
105	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
106	C₈	—	4	2	no	—
107	C₈	—	554	2	no	—
108	C₂²	—	1108	2	no	—
109	C₂²	—	1108	2	no	—
110	C₂²	—	1108	2	no	—
111	C₂²	#22579850	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	1108	2	no	—
115	C₄	—	2216	1	no	—
116	C₄	#22579901	1	2	yes	—
117	C₄	#22579912	1	2	yes	—
118	C₂	—	2216	2	no	—
119	C₂	#31800167	1	2	yes	—
120	C₂	#31800172	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800718

C₂

yes

maximal, characteristic

#31800381

C₂

yes

maximal, characteristic, complemented

#31800293

C₂

yes

maximal, complemented

#31799729

C₂

yes

maximal, complemented

#333×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂×#17984553

—

#22579880

C₂²

yes

characteristic

#22579869

—

#22579855

—

C₄×D₁₁₀₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₇₇

C₂²

yes

—

C₂×Dic₅₅₄

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

D₄

yes

—

C₂×C₄×D₂₇₇

—

#17984553

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

D₄

yes

—

#17984540

—

#17984540

—

C₄×Dic₂₇₇

—

C₂×Q₈×C₂₇₇

D₄

yes

—

C₂×C₈×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

C₂²×D₂₇₇

—

Dic₅₅₄

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₄×D₂₇₇

—

C₄×D₂₇₇

—

#14555660

—

Q₈×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

—

C₂²×C₂₇₇

#120

yes

characteristic

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#135

yes

—

C₄×C₂₇₇

#137

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

#452

yes

—

C₂×C₂₇₇

#453

yes

characteristic

D₂₇₇

—

C₂₇₇

#1634

yes

characteristic, complemented, Sylow

#1634

—

277

maximal, Sylow

#555

—

277

—

#494

—

277

—

#476

—

277

—

#333

D₂₇₇

yes

characteristic, complemented

D₄⋊C₈

—

277

—

C₂×Q₁₆

—

554

—

C₄⋊Q₈

—

277

—

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

#124

—

554

—

#121

—

554

—

C₄×D₄

—

277

—

#117

—

554

—

#106

—

554

—

C₄⋊C₈

C₂×D₂₇₇

yes

—

#102

—

554

—

C₄×C₈

—

277

—

C₂×Q₈

—

554

—

C₂×Q₈

D₁₁₀₈

yes

—

C₂×D₄

—

277

—

C₂×D₄

—

554

—

C₂²×C₄

—

554

—

Q₁₆

—

1108

—

C₂×C₈

—

C₂×C₈

—

554

—

C₄⋊C₄

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

554

—

C₂²⋊C₄

—

554

—

C₂²⋊C₄

—

554

—

C₄²

—

554

—

C₄²

C₂²×D₂₇₇

yes

characteristic

C₂³

—

554

—

Q₈

—

Q₈

—

1108

—

D₄

—

554

—

D₄

—

554

—

D₄

—

2216

—

C₂×C₄

—

100

C₂×C₄

—

554

—

101

C₂×C₄

—

1108

—

102

C₂×C₄

—

1108

—

103

C₂×C₄

—

1108

—

104

C₂×C₄

D₄×D₂₇₇

yes

—

105

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

106

C₈

—

107

C₈

—

554

—

108

C₂²

—

1108

—

109

C₂²

—

1108

—

110

C₂²

—

1108

—

111

C₂²

#22579850

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

1108

—

115

C₄

—

2216

—

116

C₄

#22579901

yes

—

117

C₄

#22579912

yes

—

118

C₂

—

2216

—

119

C₂

#31800167

yes

—

120

C₂

#31800172

yes

characteristic

	subgroup	G/H
center	C₂²	#22579850
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#333×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#120
prime radical	#31800718	C₂
hypercenter	#333	D₂₇₇
hypocenter	C₂₇₇	#1634
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#22579850

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#333×C₂₇₇

C₂

C₂²×C₂₇₇

C₂

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #31800381	C₂	semidirect
3. #31800293	C₂	semidirect
4. #31799729	C₂	semidirect
5. #333×C₂₇₇	C₂	semidirect
60. C₂₇₇	#1634	semidirect
65. #333	D₂₇₇	semidirect

H ≅ G/N

type

2. #31800381

C₂

semidirect

3. #31800293

C₂

semidirect

4. #31799729

C₂

semidirect

5. #333×C₂₇₇

C₂

semidirect

60. C₂₇₇

#1634

semidirect

65. #333

D₂₇₇

semidirect

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

subgroup	index
#31800718	2
#31800381	2
#31800293	2
#31799729	2
#333×C₂₇₇	2
#1634	277

subgroup

index

#333×C₂₇₇

277

Subgroups — #47982313

Conjugacy classes of subgroups

Complements

Maximal subgroups