Group #23761679 — subgroups

Subgroups — #23761679

Order 10112.

224 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 121 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	#19098614	C₂	1	1	yes	maximal, characteristic
2	C₄×#11929607	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×D₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#322×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929607	C₄	1	2	yes	—
9	C₂×C₈×D₇₉	C₄	1	2	yes	—
10	C₄²×D₇₉	C₂²	1	1	yes	characteristic
11	Dic₇₉⋊C₈	C₄	1	2	yes	—
12	(C₂×D₇₉)⋊C₈	C₄	1	2	yes	—
13	#14579071	C₄	1	2	yes	—
14	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₈×Dic₇₉	C₄	1	2	yes	—
18	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
24	#11929607	C₈	1	8	yes	complemented
25	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₈×D₇₉	—	2	4	no	—
29	C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₄²×C₇₉	C₂³	1	1	yes	characteristic
36	C₂²×D₇₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₇₉	C₄²	1	1	yes	characteristic
41	C₂×Dic₇₉	C₄²	1	1	yes	characteristic
42	C₂×Dic₇₉	C₄²	1	1	yes	characteristic
43	C₄×D₇₉	—	2	2	no	—
44	C₄×D₇₉	—	2	2	no	—
45	C₄×D₇₉	C₂×C₈	1	4	yes	—
46	C₈×C₇₉	—	2	2	no	—
47	C₈×C₇₉	—	2	2	no	—
48	C₈×C₇₉	C₂×C₈	1	4	yes	complemented
49	C₇₉⋊C₈	—	2	2	no	—
50	C₇₉⋊C₈	C₂×C₈	1	4	yes	complemented
51	C₇₉⋊C₈	M₁₆	1	4	yes	complemented
52	C₂²×C₇₉	C₂×C₄²	1	1	yes	characteristic
53	C₂×D₇₉	—	2	1	no	—
54	C₂×D₇₉	—	2	1	no	—
55	C₂×D₇₉	C₄×C₈	1	2	yes	—
56	C₄×C₇₉	C₂²×C₈	1	2	yes	—
57	C₄×C₇₉	C₂×M₁₆	1	2	yes	—
58	C₄×C₇₉	#131	1	2	yes	—
59	Dic₇₉	—	2	1	no	—
60	Dic₇₉	—	2	1	no	—
61	Dic₇₉	C₄×C₈	1	2	yes	—
62	C₂×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂×C₇₉	C₄×M₁₆	1	1	yes	characteristic
64	C₂×C₇₉	#405	1	1	yes	characteristic
65	D₇₉	—	2	2	no	—
66	C₈×M₁₆	—	79	1	no	maximal, Sylow
67	C₇₉	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
68	#431	—	79	1	no	—
69	C₄×M₁₆	—	79	1	no	—
70	C₂×C₄×C₈	—	79	1	no	—
71	#322	—	79	1	no	—
72	#322	D₇₉	1	1	yes	characteristic, complemented
73	C₈²	—	79	1	no	—
74	C₈²	—	79	1	no	—
75	C₂×M₁₆	—	79	2	no	—
76	C₂²×C₈	—	79	2	no	—
77	C₂×C₄²	—	79	1	no	—
78	C₄⋊C₈	—	79	2	no	—
79	C₂²⋊C₈	—	79	2	no	—
80	#97	—	79	2	no	—
81	C₄×C₈	—	79	2	no	—
82	C₄×C₈	—	79	1	no	—
83	C₄×C₈	—	79	1	no	—
84	C₄×C₈	—	79	1	no	—
85	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
86	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
87	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
88	C₂²×C₄	—	79	1	no	—
89	C₂²×C₄	—	79	1	no	—
90	C₂²×C₄	—	79	1	no	—
91	M₁₆	—	79	8	no	—
92	C₂×C₈	—	79	2	no	—
93	C₂×C₈	—	79	2	no	—
94	C₂×C₈	—	79	2	no	—
95	C₂×C₈	—	158	4	no	—
96	C₂×C₈	C₄×D₇₉	1	2	yes	—
97	C₂×C₈	C₄×D₇₉	1	2	yes	—
98	C₂×C₈	C₄×D₇₉	1	2	yes	—
99	C₄²	—	79	1	no	—
100	C₄²	—	79	1	no	—
101	C₄²	—	79	1	no	—
102	C₄²	C₂²×D₇₉	1	1	yes	characteristic
103	C₂³	—	79	1	no	—
104	C₂×C₄	—	79	1	no	—
105	C₂×C₄	—	79	4	no	—
106	C₂×C₄	—	79	1	no	—
107	C₂×C₄	—	79	1	no	—
108	C₂×C₄	—	158	2	no	—
109	C₂×C₄	—	158	2	no	—
110	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	—	79	4	no	—
116	C₈	—	79	4	no	—
117	C₈	—	158	2	no	—
118	C₈	C₈×D₇₉	1	4	yes	complemented
119	C₂²	—	79	2	no	—
120	C₂²	—	158	1	no	—
121	C₂²	—	158	1	no	—
122	C₂²	C₄²×D₇₉	1	1	yes	characteristic
123	C₄	—	79	2	no	—
124	C₄	—	158	1	no	—
125	C₄	—	158	1	no	—
126	C₄	C₂×C₈×D₇₉	1	2	yes	—
127	C₄	M₁₆×D₇₉	1	2	yes	—
128	C₄	#14579169	1	2	yes	—
129	C₂	—	158	2	no	—
130	C₂	C₄×C₈×D₇₉	1	1	yes	characteristic
131	C₂	#19098515	1	1	yes	characteristic
132	C₂	#19098520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098614

C₂

yes

maximal, characteristic

C₄×#11929607

C₂

yes

maximal, characteristic

C₄×C₈×D₇₉

C₂

yes

maximal, characteristic

#19098285

C₂

yes

maximal, characteristic, complemented

C₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#322×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929607

C₄

yes

—

C₂×C₈×D₇₉

C₄

yes

—

C₄²×D₇₉

C₂²

yes

characteristic

Dic₇₉⋊C₈

C₄

yes

—

(C₂×D₇₉)⋊C₈

C₄

yes

—

#14579071

C₄

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₇₉

C₄

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄×D₇₉

C₂×C₄

yes

characteristic

C₂×C₄×D₇₉

C₂×C₄

yes

characteristic

C₂×C₄×D₇₉

C₂×C₄

yes

characteristic

#11929607

C₈

yes

complemented

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₈×D₇₉

—

C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

C₄²

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×Dic₇₉

C₄²

yes

characteristic

C₂×Dic₇₉

C₄²

yes

characteristic

C₂×Dic₇₉

C₄²

yes

characteristic

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

C₂×C₈

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×C₈

yes

complemented

C₇₉⋊C₈

—

C₇₉⋊C₈

C₂×C₈

yes

complemented

C₇₉⋊C₈

M₁₆

yes

complemented

C₂²×C₇₉

C₂×C₄²

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

C₄×C₈

yes

—

C₄×C₇₉

C₂²×C₈

yes

—

C₄×C₇₉

C₂×M₁₆

yes

—

C₄×C₇₉

#131

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

C₄×C₈

yes

—

C₂×C₇₉

C₂×C₄×C₈

yes

characteristic

C₂×C₇₉

C₄×M₁₆

yes

characteristic

C₂×C₇₉

#405

yes

characteristic

D₇₉

—

C₈×M₁₆

—

maximal, Sylow

C₇₉

C₈×M₁₆

yes

characteristic, complemented, Sylow

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

D₇₉

yes

characteristic, complemented

C₈²

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₄×C₈

C₂×D₇₉

yes

characteristic

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

C₄×D₇₉

yes

—

C₂×C₈

C₄×D₇₉

yes

—

C₂×C₈

C₄×D₇₉

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×D₇₉

yes

characteristic

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

158

—

109

C₂×C₄

—

158

—

110

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

111

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

112

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

158

—

118

C₈

C₈×D₇₉

yes

complemented

119

C₂²

—

120

C₂²

—

158

—

121

C₂²

—

158

—

122

C₂²

C₄²×D₇₉

yes

characteristic

123

C₄

—

124

C₄

—

158

—

125

C₄

—

158

—

126

C₄

C₂×C₈×D₇₉

yes

—

127

C₄

M₁₆×D₇₉

yes

—

128

C₄

#14579169

yes

—

129

C₂

—

158

—

130

C₂

C₄×C₈×D₇₉

yes

characteristic

131

C₂

#19098515

yes

characteristic

132

C₂

#19098520

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₇₉
commutator	C₂×C₇₉	C₂×C₄×C₈
Fitting	#322×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄²
prime radical	C₂²×D₇₉	C₄²
cosocle	C₄²×C₇₉	C₂³
hypercenter	#322	D₇₉
hypocenter	C₇₉	C₈×M₁₆
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₇₉

commutator

C₂×C₇₉

C₂×C₄×C₈

Fitting

#322×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₄²

prime radical

C₂²×D₇₉

C₄²

cosocle

C₄²×C₇₉

C₂³

hypercenter

#322

D₇₉

hypocenter

C₇₉

C₈×M₁₆

LCS

G ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #19098285	C₂	semidirect
5. C₈×C₇₉⋊C₈	C₂	semidirect
6. C₈×C₇₉⋊C₈	C₂	semidirect
7. #322×C₇₉	C₂	semidirect
24. #11929607	C₈	semidirect
48. C₈×C₇₉	C₂×C₈	semidirect
50. C₇₉⋊C₈	C₂×C₈	semidirect
51. C₇₉⋊C₈	M₁₆	semidirect
67. C₇₉	C₈×M₁₆	semidirect
72. #322	D₇₉	semidirect
118. C₈	C₈×D₇₉	semidirect

H ≅ G/N

type

4. #19098285

C₂

semidirect

5. C₈×C₇₉⋊C₈

C₂

semidirect

6. C₈×C₇₉⋊C₈

C₂

semidirect

7. #322×C₇₉

C₂

semidirect

24. #11929607

C₈

semidirect

48. C₈×C₇₉

C₂×C₈

semidirect

50. C₇₉⋊C₈

C₂×C₈

semidirect

51. C₇₉⋊C₈

M₁₆

semidirect

67. C₇₉

C₈×M₁₆

semidirect

72. #322

D₇₉

semidirect

118. C₈

C₈×D₇₉

semidirect

Subgroups — #23761679

Conjugacy classes of subgroups

Complements