Group #47982243 — subgroups

Subgroups — #47982243

Order 35456.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#31800732	C₂	1	1	yes	maximal, characteristic, complemented
2	#31800199	C₂	1	2	yes	maximal
3	#31799751	C₂	1	1	yes	maximal, characteristic, complemented
4	#31799734	C₂	1	2	yes	maximal, complemented
5	#492×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984551	—	2	2	no	—
7	#22579879	—	2	2	no	—
8	#22579878	—	2	2	no	—
9	C₄×Dic₅₅₄	C₂²	1	2	yes	—
10	#22579845	—	2	2	no	—
11	C₄⋊#14555660	C₂²	1	2	yes	—
12	#22579801	—	2	2	no	—
13	C₂×SD₁₆×C₂₇₇	—	2	2	no	—
14	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	#127×C₂₇₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₄	—	2	2	no	—
18	C₂×Dic₅₅₄	D₄	1	2	yes	—
19	#17984551	—	4	4	no	—
20	C₂×#14555660	—	2	2	no	—
21	C₄⋊Dic₂₇₇	D₄	1	2	yes	—
22	#17984542	—	2	4	no	—
23	#17984542	—	2	2	no	—
24	C₄×Dic₂₇₇	—	2	2	no	—
25	C₄×Dic₂₇₇	—	2	2	no	—
26	C₂×Q₈×C₂₇₇	D₄	1	2	yes	—
27	C₂×D₄×C₂₇₇	—	2	1	no	—
28	C₂×D₄×C₂₇₇	—	2	2	no	—
29	SD₁₆×C₂₇₇	—	4	4	no	—
30	C₂×C₈×C₂₇₇	—	2	1	no	—
31	C₄⋊C₄×C₂₇₇	—	2	2	no	—
32	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
33	Dic₅₅₄	—	2	2	no	—
34	Dic₅₅₄	—	8	2	no	—
35	Dic₅₅₄	SD₁₆	1	4	yes	complemented
36	C₂×Dic₂₇₇	—	2	2	no	—
37	C₂×Dic₂₇₇	—	2	2	no	—
38	#14555660	—	4	2	no	—
39	C₂³×C₂₇₇	—	4	1	no	—
40	Q₈×C₂₇₇	—	4	2	no	—
41	D₄×C₂₇₇	—	4	2	no	—
42	D₄×C₂₇₇	—	4	2	no	—
43	D₄×C₂₇₇	—	4	2	no	—
44	C₂×C₄×C₂₇₇	—	2	2	no	—
45	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₇₇	—	2	2	no	—
48	C₂²×C₂₇₇	—	8	2	no	—
49	C₂²×C₂₇₇	—	8	1	no	—
50	C₂²×C₂₇₇	#120	1	1	yes	characteristic
51	C₄×C₂₇₇	—	2	1	no	—
52	C₄×C₂₇₇	—	4	2	no	—
53	C₄×C₂₇₇	C₂×SD₁₆	1	2	yes	—
54	C₄×C₂₇₇	#137	1	2	yes	—
55	Dic₂₇₇	—	2	4	no	—
56	Dic₂₇₇	—	4	2	no	—
57	C₂×C₂₇₇	—	16	1	no	—
58	C₂×C₂₇₇	#448	1	2	yes	—
59	C₂×C₂₇₇	#453	1	1	yes	characteristic
60	C₂₇₇	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	277	1	no	maximal, Sylow
62	Q₈²	—	277	1	no	—
63	#492	D₂₇₇	1	1	yes	characteristic
64	#460	—	277	2	no	—
65	#333	—	277	1	no	—
66	Q₈⋊C₈	—	277	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	554	2	no	—
69	C₄⋊Q₈	—	554	2	no	—
70	C₄⋊Q₈	—	554	2	no	—
71	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
72	#127	C₂×D₂₇₇	1	1	yes	characteristic
73	C₄×Q₈	—	277	2	no	—
74	C₄×Q₈	—	554	2	no	—
75	C₄⋊C₈	—	277	2	no	—
76	#102	—	554	2	no	—
77	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
78	C₂×Q₈	—	277	2	no	—
79	C₂×Q₈	—	554	2	no	—
80	C₂×Q₈	C₂²⋊D₂₇₇	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	1108	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	554	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	277	2	no	—
89	C₄⋊C₄	—	554	4	no	—
90	C₄⋊C₄	—	554	2	no	—
91	C₄²	—	554	2	no	—
92	C₄²	—	554	2	no	—
93	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	277	4	no	—
97	Q₈	—	554	2	no	—
98	Q₈	—	2216	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	554	2	no	—
104	C₂×C₄	—	554	2	no	—
105	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	1108	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#22579848	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	554	4	no	—
115	C₄	—	1108	2	no	—
116	C₄	SD₁₆×D₂₇₇	1	2	yes	—
117	C₄	#22579909	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#31800139	1	2	yes	—
120	C₂	#31800169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800732

C₂

yes

maximal, characteristic, complemented

#31800199

C₂

yes

maximal

#31799751

C₂

yes

maximal, characteristic, complemented

#31799734

C₂

yes

maximal, complemented

#492×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984551

—

#22579879

—

#22579878

—

C₄×Dic₅₅₄

C₂²

yes

—

#22579845

—

C₄⋊#14555660

C₂²

yes

—

#22579801

—

C₂×SD₁₆×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

#127×C₂₇₇

C₂²

yes

characteristic

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

D₄

yes

—

#17984551

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

D₄

yes

—

#17984542

—

#17984542

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₂×Q₈×C₂₇₇

D₄

yes

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

—

SD₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

Dic₅₅₄

—

Dic₅₅₄

—

Dic₅₅₄

SD₁₆

yes

complemented

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

—

Q₈×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#120

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×SD₁₆

yes

—

C₄×C₂₇₇

#137

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#448

yes

—

C₂×C₂₇₇

#453

yes

characteristic

C₂₇₇

#1623

yes

characteristic, complemented, Sylow

#1623

—

277

maximal, Sylow

Q₈²

—

277

—

#492

D₂₇₇

yes

characteristic

#460

—

277

—

#333

—

277

—

Q₈⋊C₈

—

277

—

C₂×SD₁₆

—

C₂×SD₁₆

—

554

—

C₄⋊Q₈

—

554

—

C₄⋊Q₈

—

554

—

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

#127

C₂×D₂₇₇

yes

characteristic

C₄×Q₈

—

277

—

C₄×Q₈

—

554

—

C₄⋊C₈

—

277

—

#102

—

554

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

—

277

—

C₂×Q₈

—

554

—

C₂×Q₈

C₂²⋊D₂₇₇

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

1108

—

C₂×C₈

—

C₂×C₈

—

554

—

C₄⋊C₄

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄²

—

554

—

C₄²

—

554

—

C₄²

C₂²×D₂₇₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

277

—

Q₈

—

554

—

Q₈

—

2216

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

554

—

104

C₂×C₄

—

554

—

105

C₂×C₄

D₄×D₂₇₇

yes

—

106

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

107

C₈

—

108

C₈

—

1108

—

109

C₂²

—

110

C₂²

—

111

C₂²

#22579848

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

554

—

115

C₄

—

1108

—

116

C₄

SD₁₆×D₂₇₇

yes

—

117

C₄

#22579909

yes

—

118

C₂

—

119

C₂

#31800139

yes

—

120

C₂

#31800169

yes

characteristic

	subgroup	G/H
center	C₂²	#22579848
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#492×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#120
prime radical	#127×C₂₇₇	C₂²
hypercenter	#492	D₂₇₇
hypocenter	C₂₇₇	#1623
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22579848

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#492×C₂₇₇

C₂

socle

C₂²×C₂₇₇

#120

prime radical

#127×C₂₇₇

C₂²

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #31800732	C₂	semidirect
3. #31799751	C₂	semidirect
4. #31799734	C₂	semidirect
35. Dic₅₅₄	SD₁₆	semidirect
60. C₂₇₇	#1623	semidirect

H ≅ G/N

type

1. #31800732

C₂

semidirect

3. #31799751

C₂

semidirect

4. #31799734

C₂

semidirect

35. Dic₅₅₄

SD₁₆

semidirect

60. C₂₇₇

#1623

semidirect

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

subgroup	index
#31800732	2
#31800199	2
#31799751	2
#31799734	2
#492×C₂₇₇	2
#1623	277

subgroup

index

#492×C₂₇₇

277

Subgroups — #47982243

Conjugacy classes of subgroups

Complements

Maximal subgroups