Group #47983293 — subgroups

Subgroups — #47983293

Order 35456.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₈×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
2	#31799871	C₂	1	2	yes	maximal
3	#31799765	C₂	1	1	yes	maximal, characteristic
4	#31799764	C₂	1	2	yes	maximal
5	#340×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555660	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
8	C₂×#17984542	—	2	2	no	—
9	C₂×#17984542	—	2	2	no	—
10	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
11	C₈×Dic₂₇₇	C₄	1	4	yes	complemented
12	C₂²×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
14	C₂²×Dic₂₇₇	—	2	2	no	—
15	C₂²×Dic₂₇₇	D₄	1	1	yes	characteristic
16	C₂²×Dic₂₇₇	D₄	1	1	yes	characteristic
17	C₂×#14555660	—	2	2	no	—
18	C₂×#14555660	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₂₇₇	—	2	2	no	—
20	C₄⋊Dic₂₇₇	—	2	2	no	—
21	#17984542	—	4	4	no	—
22	#17984542	—	4	4	no	—
23	C₄×Dic₂₇₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₇₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₇₇	—	2	2	no	—
26	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₇₇	—	2	2	no	—
28	C₂×C₈×C₂₇₇	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₂₇₇	—	2	2	no	—
30	C₄⋊C₄×C₂₇₇	—	2	2	no	—
31	C₂×Dic₂₇₇	—	4	2	no	—
32	C₂×Dic₂₇₇	—	4	2	no	—
33	C₂×Dic₂₇₇	—	4	2	no	—
34	C₂×Dic₂₇₇	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₂₇₇	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₂₇₇	D₈	1	2	yes	—
37	C₂×Dic₂₇₇	SD₁₆	1	4	yes	—
38	C₂×Dic₂₇₇	Q₁₆	1	2	yes	—
39	#14555660	—	2	2	no	—
40	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	—	4	2	no	—
42	C₂×C₄×C₂₇₇	—	4	2	no	—
43	C₂×C₄×C₂₇₇	—	4	2	no	—
44	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₇₇	#41	1	2	yes	—
47	C₂×C₄×C₂₇₇	#41	1	2	yes	—
48	C₈×C₂₇₇	—	2	2	no	—
49	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₇₇	#124	1	2	yes	—
51	C₂²×C₂₇₇	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₂₇₇	C₂×SD₁₆	1	2	yes	—
53	C₂²×C₂₇₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₄×C₂₇₇	—	8	2	no	—
55	C₄×C₂₇₇	#117	1	1	yes	characteristic
56	C₄×C₂₇₇	#117	1	1	yes	characteristic
57	C₄×C₂₇₇	#125	1	2	yes	—
58	Dic₂₇₇	—	8	2	no	—
59	Dic₂₇₇	#106	1	4	yes	—
60	Dic₂₇₇	#107	1	4	yes	—
61	C₂×C₂₇₇	#382	1	1	yes	characteristic
62	C₂×C₂₇₇	C₂×#106	1	1	yes	characteristic
63	C₂×C₂₇₇	C₂×#107	1	1	yes	characteristic
64	C₂×C₂₇₇	#486	1	2	yes	—
65	C₂×C₂₇₇	#487	1	2	yes	—
66	C₂₇₇	#1834	1	1	yes	characteristic, complemented, Sylow
67	#1834	—	277	1	no	maximal, Sylow
68	C₂×C₄×C₈	—	277	1	no	—
69	#384	—	277	2	no	—
70	#340	—	277	1	no	—
71	#340	—	277	2	no	—
72	#340	D₂₇₇	1	1	yes	characteristic
73	C₂²×C₈	—	277	1	no	—
74	C₂²×C₈	C₂×D₂₇₇	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	277	2	no	—
76	C₂×C₄⋊C₄	—	554	2	no	—
77	C₂×C₄⋊C₄	—	554	2	no	—
78	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
79	C₂×C₄²	—	277	1	no	—
80	C₄×C₈	—	277	4	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	277	1	no	—
83	C₂²×C₄	—	277	1	no	—
84	C₂²×C₄	—	554	2	no	—
85	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	277	2	no	—
88	C₂×C₈	—	554	2	no	—
89	C₂×C₈	C₄×D₂₇₇	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	554	2	no	—
93	C₄⋊C₄	—	554	2	no	—
94	C₄⋊C₄	—	1108	4	no	—
95	C₄⋊C₄	—	1108	4	no	—
96	C₄²	—	277	2	no	—
97	C₄²	—	277	2	no	—
98	C₂³	D₄×D₂₇₇	1	1	yes	characteristic
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	277	2	no	—
103	C₂×C₄	—	277	2	no	—
104	C₂×C₄	—	277	2	no	—
105	C₂×C₄	—	277	4	no	—
106	C₂×C₄	—	277	2	no	—
107	C₂×C₄	—	1108	2	no	—
108	C₂×C₄	—	1108	2	no	—
109	C₂×C₄	—	1108	2	no	—
110	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
112	C₂×C₄	#17984562	1	2	yes	—
113	C₂×C₄	#17984563	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	554	2	no	—
116	C₂²	C₄⋊C₄×D₂₇₇	1	1	yes	characteristic
117	C₂²	#22579866	1	2	yes	—
118	C₂²	D₈×D₂₇₇	1	1	yes	characteristic
119	C₂²	SD₁₆×D₂₇₇	1	2	yes	—
120	C₂²	Q₁₆×D₂₇₇	1	1	yes	characteristic
121	C₄	—	8	2	no	—
122	C₄	—	277	4	no	—
123	C₄	—	277	4	no	—
124	C₄	—	2216	2	no	—
125	C₄	#22579836	1	1	yes	characteristic
126	C₄	#22579837	1	1	yes	characteristic
127	C₄	#22579870	1	2	yes	—
128	C₂	#31799866	1	1	yes	characteristic
129	C₂	#106×D₂₇₇	1	1	yes	characteristic
130	C₂	#107×D₂₇₇	1	1	yes	characteristic
131	C₂	#31800347	1	2	yes	—
132	C₂	#31800351	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₂₇₇

C₂

yes

maximal, characteristic

#31799871

C₂

yes

maximal

#31799765

C₂

yes

maximal, characteristic

#31799764

C₂

yes

maximal

#340×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×#14555660

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

C₂×#17984542

—

C₂×#17984542

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

C₈×Dic₂₇₇

C₄

yes

complemented

C₂²×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

D₄

yes

characteristic

C₂²×Dic₂₇₇

D₄

yes

characteristic

C₂×#14555660

—

C₂×#14555660

C₂×C₄

yes

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

#17984542

—

#17984542

—

C₄×Dic₂₇₇

C₂×C₄

yes

—

C₄×Dic₂₇₇

Q₈

yes

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

C₂×C₄

yes

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₄⋊C₄

yes

—

C₂×Dic₂₇₇

C₄⋊C₄

yes

—

C₂×Dic₂₇₇

D₈

yes

—

C₂×Dic₂₇₇

SD₁₆

yes

—

C₂×Dic₂₇₇

Q₁₆

yes

—

#14555660

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²×C₄

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

characteristic

C₂²×C₂₇₇

#124

yes

—

C₂²×C₂₇₇

C₂×D₈

yes

characteristic

C₂²×C₂₇₇

C₂×SD₁₆

yes

—

C₂²×C₂₇₇

C₂×Q₁₆

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

—

Dic₂₇₇

—

Dic₂₇₇

#106

yes

—

Dic₂₇₇

#107

yes

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

—

C₂×C₂₇₇

#487

yes

—

C₂₇₇

#1834

yes

characteristic, complemented, Sylow

#1834

—

277

maximal, Sylow

C₂×C₄×C₈

—

277

—

#384

—

277

—

#340

—

277

—

#340

—

277

—

#340

D₂₇₇

yes

characteristic

C₂²×C₈

—

277

—

C₂²×C₈

C₂×D₂₇₇

yes

characteristic

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

554

—

C₂×C₄⋊C₄

—

554

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

—

C₂×C₄²

—

277

—

C₄×C₈

—

277

—

C₂²×C₄

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

277

—

C₂×C₈

—

554

—

C₂×C₈

C₄×D₂₇₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

1108

—

C₄⋊C₄

—

1108

—

C₄²

—

277

—

C₄²

—

277

—

C₂³

D₄×D₂₇₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

277

—

103

C₂×C₄

—

277

—

104

C₂×C₄

—

277

—

105

C₂×C₄

—

277

—

106

C₂×C₄

—

277

—

107

C₂×C₄

—

1108

—

108

C₂×C₄

—

1108

—

109

C₂×C₄

—

1108

—

110

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

111

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

112

C₂×C₄

#17984562

yes

—

113

C₂×C₄

#17984563

yes

—

114

C₈

—

115

C₈

—

554

—

116

C₂²

C₄⋊C₄×D₂₇₇

yes

characteristic

117

C₂²

#22579866

yes

—

118

C₂²

D₈×D₂₇₇

yes

characteristic

119

C₂²

SD₁₆×D₂₇₇

yes

—

120

C₂²

Q₁₆×D₂₇₇

yes

characteristic

121

C₄

—

122

C₄

—

277

—

123

C₄

—

277

—

124

C₄

—

2216

—

125

C₄

#22579836

yes

characteristic

126

C₄

#22579837

yes

characteristic

127

C₄

#22579870

yes

—

128

C₂

#31799866

yes

characteristic

129

C₂

#106×D₂₇₇

yes

characteristic

130

C₂

#107×D₂₇₇

yes

characteristic

131

C₂

#31800347

yes

—

132

C₂

#31800351

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₇₇
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#340×C₂₇₇	C₂
socle, prime radical	C₂³×C₂₇₇	C₂×D₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#340	D₂₇₇
hypocenter	C₂₇₇	#1834
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₂₇₇

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#340×C₂₇₇

C₂

socle, prime radical

C₂³×C₂₇₇

C₂×D₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
11. C₈×Dic₂₇₇	C₄	semidirect
66. C₂₇₇	#1834	semidirect

H ≅ G/N

type

11. C₈×Dic₂₇₇

C₄

semidirect

66. C₂₇₇

#1834

semidirect

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

subgroup	index
C₂×C₈×Dic₂₇₇	2
#31799871	2
#31799765	2
#31799764	2
#340×C₂₇₇	2
#1834	277

subgroup

index

C₂×C₈×Dic₂₇₇

#340×C₂₇₇

277

Subgroups — #47983293

Conjugacy classes of subgroups

Complements

Maximal subgroups