Group #47982842 — subgroups

Subgroups — #47982842

Order 35456.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#31800405	C₂	1	1	yes	maximal, characteristic
2	#31800405	C₂	1	1	yes	maximal, characteristic
3	#31800398	C₂	1	1	yes	maximal, characteristic
4	#31800354	C₂	1	1	yes	maximal, characteristic
5	#31800353	C₂	1	1	yes	maximal, characteristic
6	#31799724	C₂	1	1	yes	maximal, characteristic
7	#494×C₂₇₇	C₂	1	1	yes	maximal, characteristic
8	#22579880	C₂²	1	1	yes	characteristic
9	#22579880	C₂²	1	1	yes	characteristic
10	#22579813	—	2	2	no	—
11	#22579813	—	2	2	no	—
12	#22579811	—	2	2	no	—
13	#22579805	—	2	2	no	—
14	#22579803	—	2	2	no	—
15	C₄×#14555660	C₂²	1	1	yes	characteristic
16	C₄×#14555660	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₂₇₇	—	2	2	no	—
18	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₅₄	—	2	1	no	—
22	C₂×Dic₅₅₄	—	2	1	no	—
23	C₂×#14555660	D₄	1	2	yes	—
24	C₂×#14555660	D₄	1	2	yes	—
25	C₄⋊Dic₂₇₇	—	2	1	no	—
26	C₄⋊Dic₂₇₇	—	2	1	no	—
27	C₄⋊Dic₂₇₇	—	2	1	no	—
28	C₄⋊Dic₂₇₇	—	2	1	no	—
29	C₂×Q₈×C₂₇₇	—	2	1	no	—
30	C₂×Q₈×C₂₇₇	—	2	1	no	—
31	Q₁₆×C₂₇₇	—	4	4	no	—
32	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₂₇₇	—	2	1	no	—
34	C₄⋊C₄×C₂₇₇	—	2	1	no	—
35	C₄⋊C₄×C₂₇₇	—	2	1	no	—
36	C₄⋊C₄×C₂₇₇	—	2	1	no	—
37	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
38	Dic₅₅₄	—	4	2	no	—
39	Dic₅₅₄	—	4	2	no	—
40	C₂×Dic₂₇₇	—	4	1	no	—
41	C₂×Dic₂₇₇	—	4	1	no	—
42	#14555660	—	2	2	no	—
43	#14555660	Q₁₆	1	4	yes	complemented
44	Q₈×C₂₇₇	—	4	2	no	—
45	Q₈×C₂₇₇	—	4	2	no	—
46	C₂×C₄×C₂₇₇	—	4	1	no	—
47	C₂×C₄×C₂₇₇	—	4	1	no	—
48	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₂₇₇	—	2	2	no	—
52	C₂²×C₂₇₇	#127	1	1	yes	characteristic
53	C₄×C₂₇₇	—	8	1	no	—
54	C₄×C₂₇₇	—	8	1	no	—
55	C₄×C₂₇₇	C₂×Q₁₆	1	2	yes	—
56	C₄×C₂₇₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₂₇₇	#137	1	2	yes	—
58	Dic₂₇₇	—	8	1	no	—
59	Dic₂₇₇	—	8	1	no	—
60	C₂×C₂₇₇	#494	1	1	yes	characteristic
61	C₂×C₂₇₇	#496	1	1	yes	characteristic
62	C₂×C₂₇₇	#497	1	1	yes	characteristic
63	C₂₇₇	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	277	1	no	maximal, Sylow
65	#500	—	277	1	no	—
66	#500	—	277	1	no	—
67	#498	—	277	1	no	—
68	#494	D₂₇₇	1	1	yes	characteristic
69	#487	—	277	1	no	—
70	#487	—	277	1	no	—
71	#322	—	277	1	no	—
72	C₂×Q₁₆	—	2	2	no	—
73	C₄⋊Q₈	—	277	1	no	—
74	C₄⋊Q₈	—	277	1	no	—
75	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
77	#107	—	554	2	no	—
78	#107	—	554	2	no	—
79	#106	—	554	2	no	—
80	#103	—	554	2	no	—
81	#103	—	554	2	no	—
82	C₄×C₈	—	277	1	no	—
83	C₄×C₈	—	277	1	no	—
84	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	554	1	no	—
88	C₂×Q₈	—	554	1	no	—
89	Q₁₆	—	4	4	no	—
90	C₂×C₈	—	277	2	no	—
91	C₂×C₈	—	277	2	no	—
92	C₂×C₈	C₂²⋊D₂₇₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	554	1	no	—
98	C₄⋊C₄	—	554	1	no	—
99	C₄⋊C₄	—	554	1	no	—
100	C₄⋊C₄	—	554	1	no	—
101	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	1108	2	no	—
105	Q₈	—	1108	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	1108	1	no	—
109	C₂×C₄	—	1108	1	no	—
110	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	277	4	no	—
115	C₈	—	554	2	no	—
116	C₂²	#22579876	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	2216	1	no	—
120	C₄	—	2216	1	no	—
121	C₄	Q₁₆×D₂₇₇	1	2	yes	—
122	C₄	#22579906	1	2	yes	—
123	C₄	#22579912	1	2	yes	—
124	C₂	#31800380	1	1	yes	characteristic
125	C₂	#31800389	1	1	yes	characteristic
126	C₂	#31800395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800405

C₂

yes

maximal, characteristic

#31800405

C₂

yes

maximal, characteristic

#31800398

C₂

yes

maximal, characteristic

#31800354

C₂

yes

maximal, characteristic

#31800353

C₂

yes

maximal, characteristic

#31799724

C₂

yes

maximal, characteristic

#494×C₂₇₇

C₂

yes

maximal, characteristic

#22579880

C₂²

yes

characteristic

#22579880

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#14555660

C₂²

yes

characteristic

C₂×Q₁₆×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×#14555660

D₄

yes

—

C₂×#14555660

D₄

yes

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

Q₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

Dic₅₅₄

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

#14555660

Q₁₆

yes

complemented

Q₈×C₂₇₇

—

Q₈×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

—

C₂²×C₂₇₇

#127

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×Q₁₆

yes

—

C₄×C₂₇₇

AGL(1,ℤ/8)

yes

—

C₄×C₂₇₇

#137

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, complemented, Sylow

#1723

—

277

maximal, Sylow

#500

—

277

—

#500

—

277

—

#498

—

277

—

#494

D₂₇₇

yes

characteristic

#487

—

277

—

#487

—

277

—

#322

—

277

—

C₂×Q₁₆

—

C₄⋊Q₈

—

277

—

C₄⋊Q₈

—

277

—

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

#107

—

554

—

#107

—

554

—

#106

—

554

—

#103

—

554

—

#103

—

554

—

C₄×C₈

—

277

—

C₄×C₈

—

277

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

554

—

C₂×Q₈

—

554

—

Q₁₆

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

C₂²⋊D₂₇₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

100

C₄⋊C₄

—

554

—

101

C₄²

C₂²×D₂₇₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

1108

—

105

Q₈

—

1108

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

1108

—

109

C₂×C₄

—

1108

—

110

C₂×C₄

D₄×D₂₇₇

yes

characteristic

111

C₂×C₄

D₄×D₂₇₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

113

C₈

—

114

C₈

—

277

—

115

C₈

—

554

—

116

C₂²

#22579876

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

2216

—

120

C₄

—

2216

—

121

C₄

Q₁₆×D₂₇₇

yes

—

122

C₄

#22579906

yes

—

123

C₄

#22579912

yes

—

124

C₂

#31800380

yes

characteristic

125

C₂

#31800389

yes

characteristic

126

C₂

#31800395

yes

characteristic

	subgroup	G/H
center	C₂²	#22579876
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#494×C₂₇₇	C₂
socle, prime radical	C₂²×C₂₇₇	#127
hypercenter	#494	D₂₇₇
hypocenter	C₂₇₇	#1723
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#22579876

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#494×C₂₇₇

C₂

C₂²×C₂₇₇

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
43. #14555660	Q₁₆	semidirect
63. C₂₇₇	#1723	semidirect

H ≅ G/N

type

43. #14555660

Q₁₆

semidirect

63. C₂₇₇

#1723

semidirect

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

subgroup	index
#31800405	2
#31800398	2
#31800354	2
#31800353	2
#31799724	2
#494×C₂₇₇	2
#1723	277

subgroup

index

#494×C₂₇₇

277

Subgroups — #47982842

Conjugacy classes of subgroups

Complements

Maximal subgroups