Group #34202997 — subgroups

Subgroups — #34202997

Order 20224.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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	#23765656	C₂	1	2	yes	maximal, complemented
2	#23765650	C₂	1	2	yes	maximal, complemented
3	#23763015	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763006	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579106	—	2	4	no	—
7	C₂×D₁₂₆₄	—	2	4	no	—
8	#19098943	C₂²	1	1	yes	characteristic
9	#19098940	—	2	2	no	—
10	#19098940	C₂²	1	1	yes	characteristic
11	#19098940	C₂²	1	2	yes	—
12	#19098910	—	2	2	no	—
13	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
14	C₈²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₆	—	2	2	no	—
16	C₂×D₆₃₂	—	2	2	no	—
17	C₂×D₆₃₂	—	2	2	no	—
18	C₂×D₆₃₂	—	2	2	no	—
19	C₂×D₆₃₂	—	4	4	no	—
20	#14579161	—	2	1	no	—
21	#14579158	—	2	1	no	—
22	#14579158	—	2	2	no	—
23	#14579106	—	4	8	no	—
24	D₁₂₆₄	—	4	8	no	—
25	#14579083	—	4	4	no	—
26	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
27	C₄×C₈×C₇₉	D₄	1	2	yes	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₅₈	—	4	1	no	—
30	C₂×D₃₁₆	—	4	1	no	—
31	C₂×D₃₁₆	—	4	2	no	—
32	C₂×D₃₁₆	—	4	2	no	—
33	C₂×D₃₁₆	—	4	4	no	—
34	Dic₃₁₆	—	4	4	no	—
35	D₆₃₂	—	4	4	no	—
36	D₆₃₂	—	4	4	no	—
37	D₆₃₂	—	4	4	no	—
38	D₆₃₂	—	8	8	no	—
39	C₄⋊Dic₇₉	—	4	2	no	—
40	C₂×C₈×C₇₉	D₈	1	4	yes	—
41	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
42	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₉	D₈	1	8	yes	complemented
44	C₂²×D₇₉	—	8	1	no	—
45	C₂²×D₇₉	—	8	2	no	—
46	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
48	C₂×Dic₇₉	—	8	1	no	—
49	D₃₁₆	—	8	2	no	—
50	D₃₁₆	—	8	2	no	—
51	D₃₁₆	—	8	2	no	—
52	D₃₁₆	—	8	8	no	—
53	D₃₁₆	—	8	4	no	—
54	Dic₁₅₈	—	8	2	no	—
55	C₈×C₇₉	—	2	4	no	—
56	C₈×C₇₉	C₂×D₈	1	4	yes	—
57	C₂²×C₇₉	#493	1	1	yes	characteristic
58	C₂×D₇₉	—	16	1	no	—
59	C₂×D₇₉	—	16	2	no	—
60	C₂×D₇₉	—	16	4	no	—
61	C₂×D₇₉	—	16	2	no	—
62	C₄×C₇₉	#493	1	2	yes	—
63	C₄×C₇₉	#509	1	4	yes	—
64	Dic₇₉	—	16	1	no	—
65	#12519	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	#1712	1	1	yes	characteristic
67	C₂×C₇₉	#2250	1	2	yes	—
68	D₇₉	—	32	2	no	—
69	D₇₉	—	32	1	no	—
70	#2248	—	79	2	no	—
71	#2246	—	79	2	no	—
72	#1712	—	79	1	no	—
73	#1709	—	79	1	no	—
74	#1311	D₇₉	1	1	yes	characteristic, complemented
75	C₇₉	#12519	1	1	yes	characteristic, complemented, Sylow
76	C₂×SD₃₂	—	158	4	no	—
77	C₂×D₁₆	—	158	4	no	—
78	#494	—	79	1	no	—
79	#493	—	79	1	no	—
80	#493	—	79	2	no	—
81	#493	—	158	2	no	—
82	#486	—	158	2	no	—
83	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
84	C₈²	C₂×D₇₉	1	1	yes	characteristic
85	C₂×Q₁₆	—	158	2	no	—
86	C₂×D₈	—	316	4	no	—
87	C₂×D₈	—	158	2	no	—
88	C₂×D₈	—	158	2	no	—
89	C₂×D₈	—	158	2	no	—
90	C₄⋊Q₈	—	158	1	no	—
91	#127	—	158	1	no	—
92	#127	—	158	2	no	—
93	SD₃₂	—	316	8	no	—
94	D₁₆	—	316	8	no	—
95	C₂×C₁₆	D₃₁₆	1	4	yes	—
96	#102	—	316	4	no	—
97	C₄×C₈	D₃₁₆	1	2	yes	—
98	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
99	C₂×Q₈	—	316	1	no	—
100	C₂×D₄	—	316	1	no	—
101	C₂×D₄	—	316	2	no	—
102	C₂×D₄	—	316	2	no	—
103	C₂×D₄	—	316	4	no	—
104	Q₁₆	—	316	4	no	—
105	D₈	—	316	4	no	—
106	D₈	—	316	4	no	—
107	D₈	—	316	4	no	—
108	D₈	—	632	8	no	—
109	C₂×C₈	D₆₃₂	1	4	yes	—
110	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
111	C₄⋊C₄	—	316	2	no	—
112	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
113	C₁₆	D₆₃₂	1	8	yes	complemented
114	C₂³	—	632	1	no	—
115	C₂³	—	632	2	no	—
116	Q₈	—	632	2	no	—
117	D₄	—	632	2	no	—
118	D₄	—	632	2	no	—
119	D₄	—	632	2	no	—
120	D₄	—	632	8	no	—
121	D₄	—	632	4	no	—
122	C₂×C₄	—	632	1	no	—
123	C₂×C₄	#14579158	1	1	yes	characteristic
124	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	C₂×D₆₃₂	1	4	yes	—
127	C₂²	—	1264	1	no	—
128	C₂²	—	1264	2	no	—
129	C₂²	—	1264	4	no	—
130	C₂²	—	1264	2	no	—
131	C₂²	#19098940	1	1	yes	characteristic
132	C₄	—	1264	1	no	—
133	C₄	#19098940	1	2	yes	—
134	C₄	#19099001	1	4	yes	—
135	C₂	—	2528	2	no	—
136	C₂	—	2528	1	no	—
137	C₂	#23763015	1	1	yes	characteristic
138	C₂	#23765666	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765656

C₂

yes

maximal, complemented

#23765650

C₂

yes

maximal, complemented

#23763015

C₂

yes

maximal, characteristic, complemented

#23763006

C₂

yes

maximal, characteristic, complemented

C₇₉×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#14579106

—

C₂×D₁₂₆₄

—

#19098943

C₂²

yes

characteristic

#19098940

—

#19098940

C₂²

yes

characteristic

#19098940

C₂²

yes

—

#19098910

—

C₄×C₁₆×C₇₉

C₂²

yes

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

—

—

—

—

D₁₂₆₄

—

#14579083

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

Dic₃₁₆

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

D₈

yes

complemented

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₈

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

#493

yes

—

C₄×C₇₉

#509

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

D₇₉

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₇₉

#12519

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

158

—

C₂×D₁₆

—

158

—

—

—

—

—

158

—

#486

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

158

—

C₂×D₈

—

316

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

158

—

#127

—

158

—

#127

—

158

—

SD₃₂

—

316

—

D₁₆

—

316

—

C₂×C₁₆

D₃₁₆

yes

—

#102

—

316

—

C₄×C₈

D₃₁₆

yes

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

316

—

100

C₂×D₄

—

316

—

101

C₂×D₄

—

316

—

102

C₂×D₄

—

316

—

103

C₂×D₄

—

316

—

104

Q₁₆

—

316

—

105

D₈

—

316

—

106

D₈

—

316

—

107

D₈

—

316

—

108

D₈

—

632

—

109

C₂×C₈

D₆₃₂

yes

—

110

C₂×C₈

C₂×D₃₁₆

yes

—

111

C₄⋊C₄

—

316

—

112

C₄²

C₂×D₃₁₆

yes

characteristic

113

C₁₆

D₆₃₂

yes

complemented

114

C₂³

—

632

—

115

C₂³

—

632

—

116

Q₈

—

632

—

117

D₄

—

632

—

118

D₄

—

632

—

119

D₄

—

632

—

120

D₄

—

632

—

121

D₄

—

632

—

122

C₂×C₄

—

632

—

123

C₂×C₄

#14579158

yes

characteristic

124

C₂×C₄

C₂×D₆₃₂

yes

—

125

C₈

—

126

C₈

C₂×D₆₃₂

yes

—

127

C₂²

—

1264

—

128

C₂²

—

1264

—

129

C₂²

—

1264

—

130

C₂²

—

1264

—

131

C₂²

#19098940

yes

characteristic

132

C₄

—

1264

—

133

C₄

#19098940

yes

—

134

C₄

#19099001

yes

—

135

C₂

—

2528

—

136

C₂

—

2528

—

137

C₂

#23763015

yes

characteristic

138

C₂

#23765666

yes

—

	subgroup	G/H
center	C₂²	#19098940
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1311	C₂
socle	C₂²×C₇₉	#493
hypercenter	#1311	D₇₉
hypocenter	C₇₉	#12519
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#19098940

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1311

C₂

C₂²×C₇₉

D₇₉

C₇₉

G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #23765656	C₂	semidirect
2. #23765650	C₂	semidirect
3. #23763015	C₂	semidirect
4. #23763006	C₂	semidirect
5. C₇₉×#1311	C₂	semidirect
43. C₁₆×C₇₉	D₈	semidirect
74. #1311	D₇₉	semidirect
75. C₇₉	#12519	semidirect
113. C₁₆	D₆₃₂	semidirect

H ≅ G/N

type

1. #23765656

C₂

semidirect

2. #23765650

C₂

semidirect

3. #23763015

C₂

semidirect

4. #23763006

C₂

semidirect

5. C₇₉×#1311

C₂

semidirect

43. C₁₆×C₇₉

D₈

semidirect

74. #1311

D₇₉

semidirect

75. C₇₉

#12519

semidirect

113. C₁₆

D₆₃₂

semidirect

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

subgroup	index
#23765656	2
#23765650	2
#23763015	2
#23763006	2
C₇₉×#1311	2
#12519	79

subgroup

index

C₇₉×#1311

Subgroups — #34202997

Conjugacy classes of subgroups

Complements

Maximal subgroups