Group #47983285 — subgroups

Subgroups — #47983285

Order 35456.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#22579809	C₂	1	2	yes	maximal
2	#31799848	C₂	1	1	yes	maximal, characteristic
3	#31799757	C₂	1	1	yes	maximal, characteristic
4	#31799756	C₂	1	2	yes	maximal
5	#336×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555660	C₂²	1	2	yes	—
7	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
9	#22579809	—	2	4	no	—
10	#22579787	C₄	1	2	yes	—
11	#22579787	C₄	1	2	yes	—
12	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₂₇₇	C₂×C₄	1	2	yes	—
17	C₂×#14555660	—	2	2	no	—
18	C₂×#14555660	—	2	2	no	—
19	C₂×#14555660	—	2	2	no	—
20	C₄×Dic₂₇₇	—	2	2	no	—
21	C₄×Dic₂₇₇	D₄	1	2	yes	—
22	C₄×Dic₂₇₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₇₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₇₇	—	2	2	no	—
26	C₂×C₈×C₂₇₇	—	2	2	no	—
27	C₂×C₈×C₂₇₇	—	2	2	no	—
28	C₄²×C₂₇₇	—	2	2	no	—
29	C₂×Dic₂₇₇	—	2	2	no	—
30	C₂×Dic₂₇₇	—	2	2	no	—
31	C₂×Dic₂₇₇	—	2	1	no	—
32	C₂×Dic₂₇₇	—	2	1	no	—
33	C₂×Dic₂₇₇	—	2	2	no	—
34	C₂×Dic₂₇₇	—	2	2	no	—
35	C₂×Dic₂₇₇	C₄⋊C₄	1	4	yes	—
36	#14555660	—	4	2	no	—
37	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₇₇	—	2	2	no	—
39	C₂×C₄×C₂₇₇	—	2	2	no	—
40	C₂×C₄×C₂₇₇	—	2	2	no	—
41	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	#41	1	2	yes	—
44	C₂×C₄×C₂₇₇	#41	1	2	yes	—
45	C₈×C₂₇₇	—	4	2	no	—
46	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₇₇	#117	1	1	yes	characteristic
48	C₂²×C₂₇₇	#117	1	1	yes	characteristic
49	C₂²×C₂₇₇	#131	1	2	yes	—
50	C₂²×C₂₇₇	#131	1	2	yes	—
51	C₄×C₂₇₇	—	4	2	no	—
52	C₄×C₂₇₇	#124	1	2	yes	—
53	C₄×C₂₇₇	#125	1	2	yes	—
54	Dic₂₇₇	—	2	4	no	—
55	Dic₂₇₇	—	4	1	no	—
56	Dic₂₇₇	—	4	1	no	—
57	C₂×C₂₇₇	#382	1	1	yes	characteristic
58	C₂×C₂₇₇	#424	1	2	yes	—
59	C₂×C₂₇₇	#433	1	1	yes	characteristic
60	C₂×C₂₇₇	#433	1	2	yes	—
61	C₂×C₂₇₇	#433	1	1	yes	characteristic
62	C₂₇₇	#1832	1	1	yes	characteristic, complemented, Sylow
63	#1832	—	277	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	277	2	no	—
65	#376	—	277	1	no	—
66	#336	—	277	2	no	—
67	#336	—	277	1	no	—
68	#336	D₂₇₇	1	1	yes	characteristic
69	C₂²×C₈	—	277	2	no	—
70	C₂²×C₈	C₂×D₂₇₇	1	2	yes	—
71	C₂×C₄²	—	277	1	no	—
72	C₂×C₄²	—	277	1	no	—
73	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
74	C₄⋊C₈	—	554	4	no	—
75	#95	—	277	2	no	—
76	#95	—	277	2	no	—
77	C₂²×C₄	—	277	2	no	—
78	C₂²×C₄	—	277	1	no	—
79	C₂²×C₄	—	277	1	no	—
80	C₂²×C₄	C₄×D₂₇₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	554	2	no	—
86	C₂×C₈	—	554	2	no	—
87	C₂×C₈	—	554	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	277	2	no	—
90	C₄²	—	277	2	no	—
91	C₄²	—	554	2	no	—
92	C₂³	C₂×C₄×D₂₇₇	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	277	4	no	—
97	C₂×C₄	—	554	2	no	—
98	C₂×C₄	—	554	2	no	—
99	C₂×C₄	—	554	1	no	—
100	C₂×C₄	—	554	1	no	—
101	C₂×C₄	—	554	2	no	—
102	C₂×C₄	—	554	2	no	—
103	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
105	C₂×C₄	#17984562	1	2	yes	—
106	C₂×C₄	#17984563	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	1108	2	no	—
109	C₂²	C₄⋊C₄×D₂₇₇	1	1	yes	characteristic
110	C₂²	#22579836	1	1	yes	characteristic
111	C₂²	#22579837	1	1	yes	characteristic
112	C₂²	#22579886	1	2	yes	—
113	C₂²	#22579888	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	554	4	no	—
116	C₄	—	1108	1	no	—
117	C₄	—	1108	1	no	—
118	C₄	#22579866	1	2	yes	—
119	C₄	#22579870	1	2	yes	—
120	C₂	#31799866	1	1	yes	characteristic
121	C₂	#31800027	1	2	yes	—
122	C₂	#31800060	1	2	yes	—
123	C₂	#31800061	1	1	yes	characteristic
124	C₂	#31800064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579809

C₂

yes

maximal

#31799848

C₂

yes

maximal, characteristic

#31799757

C₂

yes

maximal, characteristic

#31799756

C₂

yes

maximal

#336×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×#14555660

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

#22579809

—

#22579787

C₄

yes

—

#22579787

C₄

yes

—

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₂²×Dic₂₇₇

C₂×C₄

yes

characteristic

C₂²×Dic₂₇₇

C₂×C₄

yes

characteristic

C₂²×Dic₂₇₇

C₂×C₄

yes

—

C₂×#14555660

—

C₂×#14555660

—

C₂×#14555660

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

D₄

yes

—

C₄×Dic₂₇₇

Q₈

yes

—

C₂²×C₄×C₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄²×C₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₄⋊C₄

yes

—

#14555660

—

C₂³×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

—

C₂²×C₂₇₇

#131

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#124

yes

—

C₄×C₂₇₇

#125

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

#382

yes

characteristic

C₂×C₂₇₇

#424

yes

—

C₂×C₂₇₇

#433

yes

characteristic

C₂×C₂₇₇

#433

yes

—

C₂×C₂₇₇

#433

yes

characteristic

C₂₇₇

#1832

yes

characteristic, complemented, Sylow

#1832

—

277

maximal, Sylow

C₂×C₄⋊C₈

—

277

—

#376

—

277

—

#336

—

277

—

#336

—

277

—

#336

D₂₇₇

yes

characteristic

C₂²×C₈

—

277

—

C₂²×C₈

C₂×D₂₇₇

yes

—

C₂×C₄²

—

277

—

C₂×C₄²

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

C₄⋊C₈

—

554

—

#95

—

277

—

#95

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

C₄×D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₄²

—

C₄²

—

277

—

C₄²

—

277

—

C₄²

—

554

—

C₂³

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

277

—

C₂×C₄

—

554

—

C₂×C₄

—

554

—

C₂×C₄

—

554

—

100

C₂×C₄

—

554

—

101

C₂×C₄

—

554

—

102

C₂×C₄

—

554

—

103

C₂×C₄

D₄×D₂₇₇

yes

characteristic

104

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

105

C₂×C₄

#17984562

yes

—

106

C₂×C₄

#17984563

yes

—

107

C₈

—

108

C₈

—

1108

—

109

C₂²

C₄⋊C₄×D₂₇₇

yes

characteristic

110

C₂²

#22579836

yes

characteristic

111

C₂²

#22579837

yes

characteristic

112

C₂²

#22579886

yes

—

113

C₂²

#22579888

yes

—

114

C₄

—

115

C₄

—

554

—

116

C₄

—

1108

—

117

C₄

—

1108

—

118

C₄

#22579866

yes

—

119

C₄

#22579870

yes

—

120

C₂

#31799866

yes

characteristic

121

C₂

#31800027

yes

—

122

C₂

#31800060

yes

—

123

C₂

#31800061

yes

characteristic

124

C₂

#31800064

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₇₇
commutator, socle, prime radical	C₂³×C₂₇₇	C₂²×C₄
Fitting	#336×C₂₇₇	C₂
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#336	D₂₇₇
hypocenter	C₂₇₇	#1832
LCS	G ▷ C₂³×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₇₇

commutator, socle, prime radical

C₂³×C₂₇₇

C₂²×C₄

Fitting

#336×C₂₇₇

C₂

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂³×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₂₇₇	#1832	semidirect

H ≅ G/N

type

62. C₂₇₇

#1832

semidirect

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

subgroup	index
C₂×#22579809	2
#31799848	2
#31799757	2
#31799756	2
#336×C₂₇₇	2
#1832	277

subgroup

index

#336×C₂₇₇

277

Subgroups — #47983285

Conjugacy classes of subgroups

Complements

Maximal subgroups