Group #47982790 — subgroups

Subgroups — #47982790

Order 35456.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#31800398	C₂	1	1	yes	maximal, characteristic, complemented
2	#31800376	C₂	1	1	yes	maximal, characteristic
3	#31800375	C₂	1	1	yes	maximal, characteristic
4	#31800352	C₂	1	1	yes	maximal, characteristic, complemented
5	#31800350	C₂	1	1	yes	maximal, characteristic
6	#31799723	C₂	1	1	yes	maximal, characteristic, complemented
7	#500×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#17984552	—	2	2	no	—
9	C₂×#17984550	—	2	2	no	—
10	#22579880	C₂²	1	1	yes	characteristic
11	#22579877	C₂²	1	1	yes	characteristic
12	#22579811	—	2	2	no	—
13	#22579804	—	2	2	no	—
14	#22579800	—	2	2	no	—
15	C₄×#14555660	C₂²	1	1	yes	characteristic
16	C₄×#14555660	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
19	#107×C₂₇₇	—	2	2	no	—
20	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₅₄	—	2	1	no	—
22	C₂×D₁₁₀₈	—	2	1	no	—
23	C₂×D₁₁₀₈	—	2	1	no	—
24	C₂×D₁₁₀₈	—	2	1	no	—
25	#17984552	—	4	4	no	—
26	#17984550	—	4	4	no	—
27	C₂×#14555660	D₄	1	2	yes	—
28	C₂×#14555660	D₄	1	2	yes	—
29	C₄⋊Dic₂₇₇	—	2	1	no	—
30	C₄⋊Dic₂₇₇	—	2	1	no	—
31	C₂×Q₈×C₂₇₇	—	2	1	no	—
32	C₂×Q₈×C₂₇₇	—	2	1	no	—
33	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
34	C₄⋊C₄×C₂₇₇	—	2	1	no	—
35	C₄⋊C₄×C₂₇₇	—	2	1	no	—
36	C₄⋊C₄×C₂₇₇	—	2	1	no	—
37	C₄⋊C₄×C₂₇₇	—	2	1	no	—
38	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
39	C₂²×D₂₇₇	—	4	1	no	—
40	Dic₅₅₄	—	4	2	no	—
41	D₁₁₀₈	—	4	2	no	—
42	D₁₁₀₈	—	4	2	no	—
43	D₁₁₀₈	—	4	2	no	—
44	C₂×Dic₂₇₇	—	4	1	no	—
45	#14555660	—	2	2	no	—
46	#14555660	—	2	2	no	—
47	Q₈×C₂₇₇	—	4	2	no	—
48	Q₈×C₂₇₇	—	4	2	no	—
49	C₂×C₄×C₂₇₇	—	4	1	no	—
50	C₂×C₄×C₂₇₇	—	4	1	no	—
51	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
54	C₈×C₂₇₇	SD₁₆	1	4	yes	complemented
55	C₂²×C₂₇₇	#127	1	1	yes	characteristic
56	C₂×D₂₇₇	—	8	1	no	—
57	C₂×D₂₇₇	—	8	1	no	—
58	C₂×D₂₇₇	—	8	1	no	—
59	C₄×C₂₇₇	—	8	1	no	—
60	C₄×C₂₇₇	—	8	1	no	—
61	C₄×C₂₇₇	C₂×SD₁₆	1	2	yes	—
62	C₄×C₂₇₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₂₇₇	#137	1	2	yes	—
64	Dic₂₇₇	—	8	1	no	—
65	C₂×C₂₇₇	#492	1	1	yes	characteristic
66	C₂×C₂₇₇	#496	1	1	yes	characteristic
67	C₂×C₂₇₇	#497	1	1	yes	characteristic
68	D₂₇₇	—	16	1	no	—
69	C₂₇₇	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	277	1	no	maximal, Sylow
71	#500	D₂₇₇	1	1	yes	characteristic, complemented
72	#498	—	277	1	no	—
73	#492	—	277	1	no	—
74	#492	—	277	1	no	—
75	#487	—	277	1	no	—
76	#486	—	277	1	no	—
77	#322	—	277	1	no	—
78	C₂×SD₁₆	—	554	2	no	—
79	C₂×SD₁₆	—	554	2	no	—
80	C₄⋊Q₈	—	277	1	no	—
81	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
83	#127	—	277	1	no	—
84	#107	—	2	2	no	—
85	#106	—	554	2	no	—
86	#103	—	554	2	no	—
87	#102	—	554	2	no	—
88	C₄×C₈	—	277	1	no	—
89	C₄×C₈	—	277	1	no	—
90	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	554	1	no	—
94	C₂×D₄	—	554	1	no	—
95	C₂×D₄	—	554	1	no	—
96	C₂×D₄	—	554	1	no	—
97	SD₁₆	—	1108	4	no	—
98	SD₁₆	—	1108	4	no	—
99	C₂×C₈	—	277	2	no	—
100	C₂×C₈	—	277	2	no	—
101	C₂×C₈	C₂²⋊D₂₇₇	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	554	1	no	—
107	C₄⋊C₄	—	554	1	no	—
108	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
109	C₂³	—	1108	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	1108	2	no	—
113	D₄	—	1108	2	no	—
114	D₄	—	1108	2	no	—
115	D₄	—	1108	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	1108	1	no	—
119	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
120	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
122	C₈	—	554	2	no	—
123	C₈	—	554	2	no	—
124	C₈	#17984550	1	4	yes	complemented
125	C₂²	—	2216	1	no	—
126	C₂²	—	2216	1	no	—
127	C₂²	—	2216	1	no	—
128	C₂²	#22579876	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	2216	1	no	—
132	C₄	C₂×#17984550	1	2	yes	—
133	C₄	#22579904	1	2	yes	—
134	C₄	#22579912	1	2	yes	—
135	C₂	—	4432	1	no	—
136	C₂	#31800376	1	1	yes	characteristic
137	C₂	#31800390	1	1	yes	characteristic
138	C₂	#31800395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800398

C₂

yes

maximal, characteristic, complemented

#31800376

C₂

yes

maximal, characteristic

#31800375

C₂

yes

maximal, characteristic

#31800352

C₂

yes

maximal, characteristic, complemented

#31800350

C₂

yes

maximal, characteristic

#31799723

C₂

yes

maximal, characteristic, complemented

#500×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂×#17984552

—

C₂×#17984550

—

#22579880

C₂²

yes

characteristic

#22579877

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#14555660

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

#107×C₂₇₇

—

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

#17984552

—

#17984550

—

C₂×#14555660

D₄

yes

—

C₂×#14555660

D₄

yes

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

C₂²×D₂₇₇

—

Dic₅₅₄

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

C₂×Dic₂₇₇

—

#14555660

—

#14555660

—

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

SD₁₆

yes

complemented

C₂²×C₂₇₇

#127

yes

characteristic

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×SD₁₆

yes

—

C₄×C₂₇₇

AGL(1,ℤ/8)

yes

—

C₄×C₂₇₇

#137

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

D₂₇₇

—

C₂₇₇

#1714

yes

characteristic, complemented, Sylow

#1714

—

277

maximal, Sylow

#500

D₂₇₇

yes

characteristic, complemented

#498

—

277

—

#492

—

277

—

#492

—

277

—

#487

—

277

—

#486

—

277

—

#322

—

277

—

C₂×SD₁₆

—

554

—

C₂×SD₁₆

—

554

—

C₄⋊Q₈

—

277

—

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

#127

—

277

—

#107

—

#106

—

554

—

#103

—

554

—

#102

—

554

—

C₄×C₈

—

277

—

C₄×C₈

—

277

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

SD₁₆

—

1108

—

SD₁₆

—

1108

—

C₂×C₈

—

277

—

100

C₂×C₈

—

277

—

101

C₂×C₈

C₂²⋊D₂₇₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

554

—

107

C₄⋊C₄

—

554

—

108

C₄²

C₂²×D₂₇₇

yes

characteristic

109

C₂³

—

1108

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

1108

—

113

D₄

—

1108

—

114

D₄

—

1108

—

115

D₄

—

1108

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

1108

—

119

C₂×C₄

D₄×D₂₇₇

yes

characteristic

120

C₂×C₄

D₄×D₂₇₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

122

C₈

—

554

—

123

C₈

—

554

—

124

C₈

#17984550

yes

complemented

125

C₂²

—

2216

—

126

C₂²

—

2216

—

127

C₂²

—

2216

—

128

C₂²

#22579876

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

2216

—

132

C₄

C₂×#17984550

yes

—

133

C₄

#22579904

yes

—

134

C₄

#22579912

yes

—

135

C₂

—

4432

—

136

C₂

#31800376

yes

characteristic

137

C₂

#31800390

yes

characteristic

138

C₂

#31800395

yes

characteristic

	subgroup	G/H
center	C₂²	#22579876
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#500×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#127
prime radical	#22579877	C₂²
hypercenter	#500	D₂₇₇
hypocenter	C₂₇₇	#1714
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#22579876

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#500×C₂₇₇

C₂

C₂²×C₂₇₇

C₂²

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #31800398	C₂	semidirect
4. #31800352	C₂	semidirect
6. #31799723	C₂	semidirect
7. #500×C₂₇₇	C₂	semidirect
54. C₈×C₂₇₇	SD₁₆	semidirect
69. C₂₇₇	#1714	semidirect
71. #500	D₂₇₇	semidirect
124. C₈	#17984550	semidirect

H ≅ G/N

type

1. #31800398

C₂

semidirect

4. #31800352

C₂

semidirect

6. #31799723

C₂

semidirect

7. #500×C₂₇₇

C₂

semidirect

54. C₈×C₂₇₇

SD₁₆

semidirect

69. C₂₇₇

#1714

semidirect

71. #500

D₂₇₇

semidirect

124. C₈

#17984550

semidirect

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

subgroup	index
#31800398	2
#31800376	2
#31800375	2
#31800352	2
#31800350	2
#31799723	2
#500×C₂₇₇	2
#1714	277

subgroup

index

#500×C₂₇₇

277

Subgroups — #47982790

Conjugacy classes of subgroups

Complements

Maximal subgroups