Group #34197305 — subgroups

Subgroups — #34197305

Order 20224.

260 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 131 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	#23765197	C₂	1	2	yes	maximal
2	#23761682	C₂	1	1	yes	maximal, characteristic
3	#23761319	C₂	1	2	yes	maximal, complemented
4	#23761317	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098614	C₄	1	2	yes	—
7	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
8	Dic₇₉⋊C₁₆	C₄	1	4	yes	—
9	(C₂×D₇₉)⋊C₁₆	C₄	1	4	yes	—
10	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
11	#19098285	C₄	1	2	yes	—
12	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
14	C₈²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
16	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
18	Dic₇₉⋊C₈	C₈	1	4	yes	—
19	(C₂×D₇₉)⋊C₈	C₈	1	4	yes	—
20	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₇₉	C₄²	1	1	yes	characteristic
27	C₂×C₄×D₇₉	C₂×C₈	1	2	yes	—
28	C₂×C₇₉⋊C₈	C₄²	1	2	yes	—
29	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
30	C₈×D₇₉	—	2	4	no	—
31	C₄×Dic₇₉	C₄²	1	1	yes	characteristic
32	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
33	C₇₉⋊C₁₆	—	2	4	no	—
34	C₂×C₈×C₇₉	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
36	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
37	C₁₆×C₇₉	—	2	4	no	—
38	C₂²×D₇₉	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
41	C₂×Dic₇₉	C₄×C₈	1	2	yes	—
42	C₂×Dic₇₉	C₄×C₈	1	1	yes	characteristic
43	C₄×D₇₉	—	2	2	no	—
44	C₄×D₇₉	—	2	4	no	—
45	C₈×C₇₉	—	2	4	no	—
46	C₈×C₇₉	#131	1	4	yes	—
47	C₇₉⋊C₈	—	2	2	no	—
48	C₇₉⋊C₈	—	2	4	no	—
49	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
50	C₂×D₇₉	—	2	2	no	—
51	C₂×D₇₉	—	2	1	no	—
52	C₄×C₇₉	#405	1	2	yes	—
53	C₄×C₇₉	#504	1	4	yes	—
54	Dic₇₉	—	2	2	no	—
55	Dic₇₉	—	2	1	no	—
56	#11412	—	79	1	no	maximal, Sylow
57	C₂×C₇₉	#1450	1	1	yes	characteristic
58	C₂×C₇₉	#2108	1	2	yes	—
59	D₇₉	—	4	1	no	—
60	#2162	—	79	2	no	—
61	#1450	—	79	1	no	—
62	#1312	—	79	2	no	—
63	#1311	—	79	1	no	—
64	C₈×C₁₆	D₇₉	1	1	yes	characteristic, complemented
65	C₇₉	#11412	1	1	yes	characteristic, complemented, Sylow
66	#431	—	79	2	no	—
67	C₂×C₄×C₈	—	79	1	no	—
68	C₄⋊C₁₆	—	79	4	no	—
69	C₂²⋊C₁₆	—	79	4	no	—
70	C₄×C₁₆	—	79	2	no	—
71	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
72	#322	—	79	2	no	—
73	C₈²	—	79	1	no	—
74	C₈²	C₂×D₇₉	1	1	yes	characteristic
75	C₂²×C₈	—	79	2	no	—
76	C₂×C₄²	—	79	1	no	—
77	C₂×C₁₆	—	79	4	no	—
78	C₂×C₁₆	C₄×D₇₉	1	4	yes	—
79	C₄⋊C₈	—	79	4	no	—
80	C₂²⋊C₈	—	79	4	no	—
81	C₄×C₈	—	79	2	no	—
82	C₄×C₈	—	79	2	no	—
83	C₄×C₈	—	79	1	no	—
84	C₄×C₈	C₄×D₇₉	1	2	yes	—
85	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
86	C₂²×C₄	—	79	1	no	—
87	C₂²×C₄	—	79	2	no	—
88	C₂×C₈	—	79	2	no	—
89	C₂×C₈	—	79	4	no	—
90	C₂×C₈	—	158	4	no	—
91	C₂×C₈	C₈×D₇₉	1	4	yes	—
92	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
93	C₄²	—	79	1	no	—
94	C₄²	—	79	2	no	—
95	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₆	—	158	4	no	—
98	C₂³	—	79	1	no	—
99	C₂×C₄	—	79	2	no	—
100	C₂×C₄	—	79	1	no	—
101	C₂×C₄	—	158	4	no	—
102	C₂×C₄	—	158	2	no	—
103	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	158	4	no	—
107	C₈	—	158	2	no	—
108	C₈	#14579167	1	4	yes	—
109	C₂²	—	158	2	no	—
110	C₂²	—	158	1	no	—
111	C₂²	C₄×C₈×D₇₉	1	1	yes	characteristic
112	C₄	—	158	2	no	—
113	C₄	—	158	1	no	—
114	C₄	#19098517	1	2	yes	—
115	C₄	#19098979	1	4	yes	—
116	C₂	—	316	1	no	—
117	C₂	#23761682	1	1	yes	characteristic
118	C₂	#23764995	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765197

C₂

yes

maximal

#23761682

C₂

yes

maximal, characteristic

#23761319

C₂

yes

maximal, complemented

#23761317

C₂

yes

maximal, characteristic, complemented

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

#19098614

C₄

yes

—

C₄×C₈×D₇₉

C₂²

yes

characteristic

Dic₇₉⋊C₁₆

C₄

yes

—

(C₂×D₇₉)⋊C₁₆

C₄

yes

—

C₄×C₇₉⋊C₁₆

C₂²

yes

—

#19098285

C₄

yes

—

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₂×C₄

yes

—

C₄²×D₇₉

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

Dic₇₉⋊C₈

C₈

yes

—

(C₂×D₇₉)⋊C₈

C₈

yes

—

C₄×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₉⋊C₈

C₂×C₄

yes

—

C₈×Dic₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄×D₇₉

C₄²

yes

characteristic

C₂×C₄×D₇₉

C₂×C₈

yes

—

C₂×C₇₉⋊C₈

C₄²

yes

—

C₂×C₇₉⋊C₈

C₂×C₈

yes

—

C₈×D₇₉

—

C₄×Dic₇₉

C₄²

yes

characteristic

C₄×Dic₇₉

C₂×C₈

yes

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

—

C₂²×D₇₉

C₄×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₈

yes

—

C₂×Dic₇₉

C₄×C₈

yes

—

C₂×Dic₇₉

C₄×C₈

yes

characteristic

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#131

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×C₄×C₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

#405

yes

—

C₄×C₇₉

#504

yes

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

—

—

—

—

C₈×C₁₆

D₇₉

yes

characteristic, complemented

C₇₉

#11412

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

—

#322

—

C₈²

—

C₈²

C₂×D₇₉

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₉

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

C₈×D₇₉

yes

—

C₂×C₈

C₂×C₄×D₇₉

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₇₉

yes

characteristic

C₁₆

—

C₁₆

—

158

—

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

158

—

102

C₂×C₄

—

158

—

103

C₂×C₄

C₄²×D₇₉

yes

characteristic

104

C₂×C₄

C₂×C₈×D₇₉

yes

—

105

C₈

—

106

C₈

—

158

—

107

C₈

—

158

—

108

C₈

#14579167

yes

—

109

C₂²

—

158

—

110

C₂²

—

158

—

111

C₂²

C₄×C₈×D₇₉

yes

characteristic

112

C₄

—

158

—

113

C₄

—

158

—

114

C₄

#19098517

yes

—

115

C₄

#19098979

yes

—

116

C₂

—

316

—

117

C₂

#23761682

yes

characteristic

118

C₂

#23764995

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₇₉
commutator, socle	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₈×C₁₆×C₇₉	C₂
prime radical	C₂²×D₇₉	C₄×C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	#11412
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄×C₈ ◁ C₈×C₁₆

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₇₉

commutator, socle

C₂²×C₇₉

C₂×C₄×C₈

Fitting

C₈×C₁₆×C₇₉

C₂

prime radical

C₂²×D₇₉

C₄×C₈

cosocle

C₄×C₈×C₇₉

C₂³

hypercenter

C₈×C₁₆

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₄×C₈ ◁ C₈×C₁₆

N	H ≅ G/N	type
3. #23761319	C₂	semidirect
4. #23761317	C₂	semidirect
5. C₈×C₁₆×C₇₉	C₂	semidirect
64. C₈×C₁₆	D₇₉	semidirect
65. C₇₉	#11412	semidirect

H ≅ G/N

type

3. #23761319

C₂

semidirect

4. #23761317

C₂

semidirect

5. C₈×C₁₆×C₇₉

C₂

semidirect

64. C₈×C₁₆

D₇₉

semidirect

65. C₇₉

#11412

semidirect

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

subgroup	index
#23765197	2
#23761682	2
#23761319	2
#23761317	2
C₈×C₁₆×C₇₉	2
#11412	79

subgroup

index

C₈×C₁₆×C₇₉

Subgroups — #34197305

Conjugacy classes of subgroups

Complements

Maximal subgroups