Group #34175692 — subgroups

Subgroups — #34175692

Order 20224.

148 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 39 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	#23763574	C₂	1	1	yes	maximal, characteristic
2	#23761351	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1327	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579096	—	2	1	no	—
5	C₂×#14579096	—	2	1	no	—
6	#19098587	C₄	1	1	yes	characteristic
7	C₂×#14579072	C₄	1	1	yes	characteristic
8	#19098345	—	2	1	no	—
9	#423×C₇₉	C₂²	1	1	yes	characteristic
10	#349×C₇₉	—	2	1	no	—
11	C₂×#11929608	—	2	1	no	—
12	C₂²×C₇₉⋊C₈	—	2	1	no	—
13	#14579101	—	4	1	no	—
14	#14579096	—	4	2	no	—
15	#14579096	—	4	2	no	—
16	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
17	#14579072	—	2	1	no	—
18	#14579072	C₈	1	2	yes	—
19	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₇₉	—	4	1	no	—
23	C₄⋊C₈×C₇₉	—	4	1	no	—
24	#11929608	—	4	1	no	—
25	#11929608	—	4	1	no	—
26	C₂×C₇₉⋊C₈	—	2	2	no	—
27	C₂×C₇₉⋊C₈	—	2	2	no	—
28	C₂×C₇₉⋊C₈	—	2	2	no	—
29	C₂×C₇₉⋊C₈	—	4	1	no	—
30	C₇₉⋊C₁₆	—	8	1	no	—
31	C₂²×C₄×C₇₉	—	2	1	no	—
32	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
33	M₁₆×C₇₉	—	4	1	no	—
34	M₁₆×C₇₉	—	4	1	no	—
35	C₂×C₈×C₇₉	—	2	1	no	—
36	C₂×C₈×C₇₉	—	2	1	no	—
37	C₄²×C₇₉	—	2	1	no	—
38	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
40	C₁₆×C₇₉	—	8	1	no	—
41	C₂³×C₇₉	#101	1	1	yes	characteristic
42	C₂×C₄×C₇₉	—	2	1	no	—
43	C₂×C₄×C₇₉	—	2	1	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	8	1	no	—
47	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
49	C₈×C₇₉	—	4	1	no	—
50	C₈×C₇₉	—	4	1	no	—
51	C₇₉⋊C₈	—	2	4	no	—
52	C₇₉⋊C₈	—	8	1	no	—
53	C₂²×C₇₉	—	2	1	no	—
54	C₂²×C₇₉	—	4	1	no	—
55	C₂²×C₇₉	#324	1	1	yes	characteristic
56	C₄×C₇₉	—	2	1	no	—
57	C₄×C₇₉	—	4	1	no	—
58	C₄×C₇₉	—	4	1	no	—
59	C₄×C₇₉	#353	1	1	yes	characteristic
60	C₄×C₇₉	#356	1	1	yes	characteristic
61	#7374	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	—	2	1	no	—
63	C₂×C₇₉	—	4	1	no	—
64	C₂×C₇₉	#1325	1	1	yes	characteristic
65	#1849	—	79	1	no	—
66	#1327	—	79	1	no	—
67	#1327	D₇₉	1	1	yes	characteristic
68	C₇₉	#7374	1	1	yes	characteristic, complemented, Sylow
69	C₂×#108	—	158	1	no	—
70	C₂×#108	—	158	1	no	—
71	#423	—	79	1	no	—
72	#423	C₂×D₇₉	1	1	yes	characteristic
73	C₂×#97	—	79	1	no	—
74	#349	—	2	1	no	—
75	#349	—	158	1	no	—
76	C₂×M₁₆	—	158	1	no	—
77	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
78	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
79	C₂²×C₈	—	158	1	no	—
80	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
81	M₃₂	—	4	1	no	—
82	M₃₂	—	316	1	no	—
83	#108	—	316	2	no	—
84	#108	—	316	2	no	—
85	C₄⋊C₈	—	4	1	no	—
86	C₄⋊C₈	—	316	1	no	—
87	#97	—	79	2	no	—
88	#97	—	158	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
91	M₁₆	—	4	1	no	—
92	M₁₆	—	4	1	no	—
93	M₁₆	—	316	1	no	—
94	M₁₆	—	316	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	316	1	no	—
98	C₂×C₈	—	158	2	no	—
99	C₂×C₈	—	158	2	no	—
100	C₂×C₈	—	158	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₈×D₇₉	1	1	yes	characteristic
103	C₄²	#11929607	1	1	yes	characteristic
104	C₁₆	—	8	1	no	—
105	C₁₆	—	632	1	no	—
106	C₂³	#14579079	1	1	yes	characteristic
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
113	C₂×C₄	#14579075	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	632	1	no	—
117	C₈	—	158	4	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	4	1	no	—
120	C₂²	#19098288	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	4	1	no	—
123	C₄	—	4	1	no	—
124	C₄	#19098355	1	1	yes	characteristic
125	C₄	#19098361	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	1	no	—
128	C₂	#23761345	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763574

C₂

yes

maximal, characteristic

#23761351

C₂

yes

maximal, characteristic

C₇₉×#1327

C₂

yes

maximal, characteristic

C₂×#14579096

—

C₂×#14579096

—

#19098587

C₄

yes

characteristic

C₂×#14579072

C₄

yes

characteristic

#19098345

—

#423×C₇₉

C₂²

yes

characteristic

#349×C₇₉

—

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

#14579101

—

#14579096

—

#14579096

—

C₄⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

C₈

yes

—

C₂×M₁₆×C₇₉

D₄

yes

characteristic

C₂×M₁₆×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

C₄⋊C₈×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₄²×C₇₉

M₁₆

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

#101

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

#99

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#324

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

C₇₉

#7374

yes

characteristic, complemented, Sylow

C₂×#108

—

158

—

C₂×#108

—

158

—

#423

—

#423

C₂×D₇₉

yes

characteristic

C₂×#97

—

#349

—

#349

—

158

—

C₂×M₁₆

—

158

—

C₂×M₁₆

D₃₁₆

yes

characteristic

C₂×M₁₆

C₂²⋊D₇₉

yes

characteristic

C₂²×C₈

—

158

—

C₂×C₄²

C₄×D₇₉

yes

characteristic

M₃₂

—

M₃₂

—

316

—

#108

—

316

—

#108

—

316

—

C₄⋊C₈

—

C₄⋊C₈

—

316

—

#97

—

#97

—

158

—

C₂²×C₄

—

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

316

—

M₁₆

—

316

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

100

C₂×C₈

—

158

—

101

C₄²

—

102

C₄²

C₈×D₇₉

yes

characteristic

103

C₄²

#11929607

yes

characteristic

104

C₁₆

—

105

C₁₆

—

632

—

106

C₂³

#14579079

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

characteristic

113

C₂×C₄

#14579075

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

632

—

117

C₈

—

158

—

118

C₂²

—

119

C₂²

—

120

C₂²

#19098288

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#19098355

yes

characteristic

125

C₄

#19098361

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#23761345

yes

characteristic

	subgroup	G/H
center	C₄	#19098355
commutator	C₄²×C₇₉	C₂×C₈
Frattini	#423	C₂×D₇₉
Fitting	C₇₉×#1327	C₂
socle	C₂×C₇₉	#1325
prime radical	C₂³×C₇₉	#101
cosocle	#423×C₇₉	C₂²
hypercenter	#1327	D₇₉
hypocenter	C₇₉	#7374
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1327

subgroup

G/H

center

C₄

#19098355

commutator

C₄²×C₇₉

C₂×C₈

Frattini

#423

C₂×D₇₉

Fitting

C₇₉×#1327

C₂

C₂×C₇₉

C₂³×C₇₉

#423×C₇₉

C₂²

D₇₉

C₇₉

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

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1327

N	H ≅ G/N	type
68. C₇₉	#7374	semidirect

H ≅ G/N

type

68. C₇₉

#7374

semidirect

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

subgroup	index
#23763574	2
#23761351	2
C₇₉×#1327	2
#7374	79

subgroup

index

#23763574

#23761351

C₇₉×#1327

#7374

Subgroups — #34175692

Conjugacy classes of subgroups

Complements

Maximal subgroups