Group #34176122 — subgroups

Subgroups — #34176122

Order 20224.

164 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 25 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	#23763064	C₂	1	1	yes	maximal, characteristic
2	#23761565	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₉×#1410	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098940	C₄	1	1	yes	characteristic, complemented
5	#19098939	—	2	1	no	—
6	#19098911	—	2	1	no	—
7	#19098356	—	2	1	no	—
8	#19098284	C₄	1	1	yes	characteristic, complemented
9	#492×C₇₉	C₂²	1	1	yes	characteristic
10	#356×C₇₉	—	2	1	no	—
11	C₂×D₆₃₂	—	4	1	no	—
12	C₂×#11929609	—	4	2	no	—
13	#14579158	—	2	1	no	—
14	#14579081	—	4	2	no	—
15	#14579076	—	4	1	no	—
16	C₄×C₇₉⋊C₈	—	2	1	no	—
17	C₂×SD₁₆×C₇₉	—	4	1	no	—
18	C₄⋊Q₈×C₇₉	D₄	1	1	yes	characteristic
19	#127×C₇₉	D₄	1	1	yes	characteristic
20	#101×C₇₉	—	4	1	no	—
21	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×D₃₁₆	—	4	1	no	—
23	C₂×D₃₁₆	—	4	1	no	—
24	C₂×D₃₁₆	—	4	1	no	—
25	D₆₃₂	—	8	2	no	—
26	#11929609	—	8	4	no	—
27	C₂×C₇₉⋊C₈	—	2	2	no	—
28	C₂²⋊Dic₇₉	—	8	1	no	—
29	C₂×Q₈×C₇₉	—	2	1	no	—
30	C₂×D₄×C₇₉	—	2	1	no	—
31	C₂×D₄×C₇₉	—	4	1	no	—
32	SD₁₆×C₇₉	—	8	2	no	—
33	M₁₆×C₇₉	—	8	1	no	—
34	C₂×C₈×C₇₉	—	2	1	no	—
35	C₄⋊C₄×C₇₉	—	4	1	no	—
36	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂³×C₇₉	—	4	1	no	—
38	C₂²×D₇₉	—	8	1	no	—
39	Q₈×C₇₉	—	8	1	no	—
40	D₄×C₇₉	—	8	1	no	—
41	D₄×C₇₉	—	8	2	no	—
42	C₂×C₄×C₇₉	—	2	1	no	—
43	C₂×C₄×C₇₉	—	4	1	no	—
44	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
45	C₂×Dic₇₉	—	16	1	no	—
46	D₃₁₆	—	8	2	no	—
47	D₃₁₆	—	8	2	no	—
48	D₃₁₆	—	8	2	no	—
49	C₈×C₇₉	—	2	2	no	—
50	C₈×C₇₉	—	16	1	no	—
51	C₇₉⋊C₈	—	4	2	no	—
52	C₂²×C₇₉	—	8	1	no	—
53	C₂²×C₇₉	—	16	1	no	—
54	C₂²×C₇₉	#353	1	1	yes	characteristic
55	C₂×D₇₉	—	16	1	no	—
56	C₂×D₇₉	—	16	1	no	—
57	C₂×D₇₉	—	16	1	no	—
58	C₄×C₇₉	—	2	2	no	—
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	—	8	1	no	—
61	Dic₇₉	—	32	1	no	—
62	#7527	—	79	1	no	maximal, Sylow
63	C₂×C₇₉	—	2	1	no	—
64	C₂×C₇₉	—	16	1	no	—
65	C₂×C₇₉	#1408	1	1	yes	characteristic
66	D₇₉	—	32	1	no	—
67	#1721	—	79	1	no	—
68	#1410	D₇₉	1	1	yes	characteristic, complemented
69	#1409	—	79	1	no	—
70	C₇₉	#7527	1	1	yes	characteristic, complemented, Sylow
71	#493	—	79	1	no	—
72	#493	—	158	1	no	—
73	#492	C₂×D₇₉	1	1	yes	characteristic
74	#486	—	158	1	no	—
75	#356	—	2	1	no	—
76	#353	—	158	1	no	—
77	#322	—	79	1	no	—
78	C₂×SD₁₆	—	4	1	no	—
79	C₂×D₈	—	316	1	no	—
80	C₂×D₈	—	316	2	no	—
81	C₄⋊Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
82	#127	—	158	1	no	—
83	#127	D₃₁₆	1	1	yes	characteristic
84	#102	—	316	2	no	—
85	#101	—	4	1	no	—
86	#99	—	316	1	no	—
87	C₄×C₈	—	158	1	no	—
88	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
89	C₂×Q₈	—	2	1	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	316	1	no	—
93	C₂×D₄	—	316	1	no	—
94	C₂×D₄	—	316	1	no	—
95	SD₁₆	—	8	2	no	—
96	D₈	—	632	2	no	—
97	D₈	—	632	4	no	—
98	M₁₆	—	8	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	158	2	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₂²⋊C₄	—	632	1	no	—
103	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
104	C₂³	—	4	1	no	—
105	C₂³	—	632	1	no	—
106	Q₈	—	8	1	no	—
107	D₄	—	8	1	no	—
108	D₄	—	8	2	no	—
109	D₄	—	632	2	no	—
110	D₄	—	632	2	no	—
111	D₄	—	632	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	1264	1	no	—
115	C₂×C₄	#14579075	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	16	1	no	—
118	C₈	—	316	2	no	—
119	C₂²	—	8	1	no	—
120	C₂²	—	16	1	no	—
121	C₂²	—	1264	1	no	—
122	C₂²	—	1264	1	no	—
123	C₂²	—	1264	1	no	—
124	C₂²	#19098355	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	8	1	no	—
128	C₄	—	2528	1	no	—
129	C₂	—	2	1	no	—
130	C₂	—	16	1	no	—
131	C₂	—	2528	1	no	—
132	C₂	#23761562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763064

C₂

yes

maximal, characteristic

#23761565

C₂

yes

maximal, characteristic, complemented

C₇₉×#1410

C₂

yes

maximal, characteristic, complemented

#19098940

C₄

yes

characteristic, complemented

—

—

—

C₄

yes

characteristic, complemented

#492×C₇₉

C₂²

yes

characteristic

#356×C₇₉

—

C₂×D₆₃₂

—

—

—

—

—

C₄×C₇₉⋊C₈

—

C₂×SD₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

D₄

yes

characteristic

#127×C₇₉

D₄

yes

characteristic

#101×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

D₆₃₂

—

#11929609

—

C₂×C₇₉⋊C₈

—

C₂²⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

SD₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#99

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#353

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

#7527

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1408

yes

characteristic

D₇₉

—

#1721

—

#1410

D₇₉

yes

characteristic, complemented

#1409

—

C₇₉

#7527

yes

characteristic, complemented, Sylow

#493

—

#493

—

158

—

#492

C₂×D₇₉

yes

characteristic

#486

—

158

—

#356

—

#353

—

158

—

#322

—

C₂×SD₁₆

—

C₂×D₈

—

316

—

C₂×D₈

—

316

—

C₄⋊Q₈

C₂²⋊D₇₉

yes

characteristic

#127

—

158

—

#127

D₃₁₆

yes

characteristic

#102

—

316

—

#101

—

#99

—

316

—

C₄×C₈

—

158

—

C₄×C₈

C₄×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

SD₁₆

—

D₈

—

632

—

D₈

—

632

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

158

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

632

—

103

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

104

C₂³

—

105

C₂³

—

632

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

D₄

—

632

—

110

D₄

—

632

—

111

D₄

—

632

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

1264

—

115

C₂×C₄

#14579075

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

316

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

1264

—

122

C₂²

—

1264

—

123

C₂²

—

1264

—

124

C₂²

#19098355

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

2528

—

129

C₂

—

130

C₂

—

131

C₂

—

2528

—

132

C₂

#23761562

yes

characteristic

	subgroup	G/H
center	C₂	#23761562
commutator	C₄×C₈×C₇₉	C₂×C₄
Frattini	#492	C₂×D₇₉
Fitting	C₇₉×#1410	C₂
socle	C₂×C₇₉	#1408
prime radical	#23763064	C₂
cosocle	#492×C₇₉	C₂²
hypercenter	#1410	D₇₉
hypocenter	C₇₉	#7527
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1410

subgroup

G/H

center

C₂

#23761562

commutator

C₄×C₈×C₇₉

C₂×C₄

Frattini

#492

C₂×D₇₉

Fitting

C₇₉×#1410

C₂

C₂×C₇₉

C₂

#492×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1410

N	H ≅ G/N	type
2. #23761565	C₂	semidirect
3. C₇₉×#1410	C₂	semidirect
4. #19098940	C₄	semidirect
8. #19098284	C₄	semidirect
68. #1410	D₇₉	semidirect
70. C₇₉	#7527	semidirect

H ≅ G/N

type

2. #23761565

C₂

semidirect

3. C₇₉×#1410

C₂

semidirect

4. #19098940

C₄

semidirect

8. #19098284

C₄

semidirect

68. #1410

D₇₉

semidirect

70. C₇₉

#7527

semidirect

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

subgroup	index
#23763064	2
#23761565	2
C₇₉×#1410	2
#7527	79

subgroup

index

#23763064

#23761565

C₇₉×#1410

#7527

Subgroups — #34176122

Conjugacy classes of subgroups

Complements

Maximal subgroups