Group #34181976 — subgroups

Subgroups — #34181976

Order 20224.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#23762163	C₂	1	2	yes	maximal
2	#23761262	C₂	1	4	yes	maximal
3	C₇₉×#1450	C₂	1	1	yes	maximal, characteristic
4	#19098620	C₂²	1	4	yes	—
5	#19098316	—	2	2	no	—
6	#19098314	—	2	2	no	—
7	#431×C₇₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	#322×C₇₉	C₄	1	2	yes	—
10	C₈²×C₇₉	C₄	1	2	yes	—
11	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
13	#14579072	—	2	4	no	—
14	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
17	C₄⋊C₈×C₇₉	—	2	2	no	—
18	C₂²⋊C₈×C₇₉	—	2	2	no	—
19	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
23	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂×C₇₉⋊C₈	—	4	4	no	—
25	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₇₉	#41	1	2	yes	—
27	C₂×C₈×C₇₉	—	2	2	no	—
28	C₂×C₈×C₇₉	—	2	2	no	—
29	C₂×C₈×C₇₉	—	2	4	no	—
30	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₇₉	#41	1	2	yes	—
37	C₂³×C₇₉	#125	1	1	yes	characteristic
38	C₂×C₄×C₇₉	—	2	1	no	—
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	—	4	2	no	—
41	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	#117	1	2	yes	—
43	C₂×C₄×C₇₉	#124	1	2	yes	—
44	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
45	C₈×C₇₉	—	2	1	no	—
46	C₈×C₇₉	—	2	1	no	—
47	C₈×C₇₉	—	4	4	no	—
48	C₈×C₇₉	#108	1	2	yes	—
49	C₈×C₇₉	#108	1	2	yes	—
50	C₇₉⋊C₈	—	8	4	no	—
51	C₂²×C₇₉	—	2	2	no	—
52	C₂²×C₇₉	—	2	1	no	—
53	C₂²×C₇₉	#382	1	1	yes	characteristic
54	C₄×C₇₉	—	2	2	no	—
55	C₄×C₇₉	—	2	2	no	—
56	C₄×C₇₉	—	2	1	no	—
57	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
58	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
59	#8796	—	79	1	no	maximal, Sylow
60	C₂×C₇₉	—	4	1	no	—
61	C₂×C₇₉	#1833	1	2	yes	—
62	C₂×C₇₉	#1837	1	1	yes	characteristic
63	#1568	—	79	2	no	—
64	#1450	D₇₉	1	1	yes	characteristic
65	#1287	—	79	4	no	—
66	C₇₉	#8796	1	1	yes	characteristic, complemented, Sylow
67	#432	—	79	4	no	—
68	#431	C₂×D₇₉	1	2	yes	—
69	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
70	#335	—	158	2	no	—
71	#334	—	158	2	no	—
72	#322	Dic₇₉	1	2	yes	—
73	C₈²	Dic₇₉	1	2	yes	—
74	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
75	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
76	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	158	4	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	158	4	no	—
81	#97	—	158	4	no	—
82	C₄×C₈	D₃₁₆	1	1	yes	characteristic
83	C₄×C₈	D₃₁₆	1	1	yes	characteristic
84	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
85	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
88	C₂²×C₄	#11929624	1	2	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	316	4	no	—
93	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
95	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
96	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
97	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
98	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
99	C₄²	#11929624	1	2	yes	—
100	C₂³	#14579153	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
105	C₂×C₄	#14579120	1	2	yes	—
106	C₂×C₄	#14579149	1	2	yes	—
107	C₂×C₄	#14579153	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	4	4	no	—
111	C₈	—	632	4	no	—
112	C₈	#14579097	1	2	yes	—
113	C₈	#14579097	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#19098429	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	C₂×#14579097	1	1	yes	characteristic
121	C₄	C₂×#14579097	1	1	yes	characteristic
122	C₂	—	4	1	no	—
123	C₂	#23763524	1	2	yes	—
124	C₂	#23763538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23762163

C₂

yes

maximal

#23761262

C₂

yes

maximal

C₇₉×#1450

C₂

yes

maximal, characteristic

#19098620

C₂²

yes

—

#19098316

—

#19098314

—

#431×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#322×C₇₉

C₄

yes

—

C₈²×C₇₉

C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#125

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#124

yes

—

C₂×C₄×C₇₉

#125

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#108

yes

—

C₈×C₇₉

#108

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#382

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

#1833

yes

—

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

#1287

—

C₇₉

#8796

yes

characteristic, complemented, Sylow

#432

—

#431

C₂×D₇₉

yes

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#335

—

158

—

#334

—

158

—

#322

Dic₇₉

yes

—

C₈²

Dic₇₉

yes

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₂²⋊C₈

—

C₂²⋊C₈

—

158

—

#97

—

158

—

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

C₂×Dic₇₉

yes

—

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

C₂²×C₄

#11929624

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₄²

C₂²×Dic₇₉

yes

characteristic

C₄²

C₂×D₃₁₆

yes

characteristic

C₄²

#11929624

yes

—

100

C₂³

#14579153

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

105

C₂×C₄

#14579120

yes

—

106

C₂×C₄

#14579149

yes

—

107

C₂×C₄

#14579153

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

632

—

112

C₈

#14579097

yes

—

113

C₈

#14579097

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

#19098429

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#14579097

yes

characteristic

121

C₄

C₂×#14579097

yes

characteristic

122

C₂

—

123

C₂

#23763524

yes

—

124

C₂

#23763538

yes

characteristic

	subgroup	G/H
center	C₂²	#19098429
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1450	C₂
socle	C₂²×C₇₉	#382
prime radical	C₂³×C₇₉	#125
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1450	D₇₉
hypocenter	C₇₉	#8796
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#19098429

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1450

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
66. C₇₉	#8796	semidirect

H ≅ G/N

type

66. C₇₉

#8796

semidirect

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

subgroup	index
#23762163	2
#23761262	2
C₇₉×#1450	2
#8796	79

subgroup

index

#23762163

#23761262

C₇₉×#1450

#8796

Subgroups — #34181976

Conjugacy classes of subgroups

Complements

Maximal subgroups