Group #23772002 — subgroups

Subgroups — C₂²×#14579096 #23772002

Order 10112.

480 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 269 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	C₂²×#11929608	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579096	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11929608	—	2	3	no	—
6	C₂×#11929608	C₂²	1	6	yes	—
7	C₂²×C₇₉⋊C₈	—	2	1	no	—
8	C₂²×C₇₉⋊C₈	C₄	1	6	yes	—
9	C₂²×C₇₉⋊C₈	C₄	1	2	yes	—
10	C₂²×C₇₉⋊C₈	C₂²	1	6	yes	—
11	#14579096	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₇₉	—	2	3	no	—
14	C₂×M₁₆×C₇₉	C₂²	1	6	yes	—
15	C₂²×C₈×C₇₉	—	2	1	no	—
16	#11929608	—	2	6	no	—
17	#11929608	C₂³	1	4	yes	—
18	C₂×C₇₉⋊C₈	—	2	6	no	—
19	C₂×C₇₉⋊C₈	C₂×C₄	1	12	yes	—
20	C₂×C₇₉⋊C₈	C₂×C₄	1	12	yes	—
21	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
22	C₂⁴×C₇₉	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₇₉	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	D₄	1	3	yes	—
26	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₉	Q₈	1	3	yes	—
28	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
29	M₁₆×C₇₉	—	2	6	no	—
30	M₁₆×C₇₉	C₂³	1	4	yes	—
31	C₂×C₈×C₇₉	—	2	6	no	—
32	C₂³×C₇₉	—	2	4	no	—
33	C₂³×C₇₉	C₄⋊C₄	1	3	yes	—
34	C₂³×C₇₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂³×C₇₉	C₂×Q₈	1	3	yes	—
36	C₂×C₄×C₇₉	C₄⋊C₄	1	6	yes	—
37	C₂×C₄×C₇₉	C₄⋊C₄	1	3	yes	—
38	C₂×C₄×C₇₉	C₄⋊C₄	1	3	yes	—
39	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
40	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
42	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
43	C₂×C₄×C₇₉	C₂×Q₈	1	3	yes	—
44	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
45	C₈×C₇₉	—	2	4	no	—
46	C₇₉⋊C₈	—	2	4	no	—
47	C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
48	C₂²×C₇₉	—	2	6	no	—
49	C₂²×C₇₉	—	2	6	no	—
50	C₂²×C₇₉	#108	1	4	yes	complemented
51	C₂²×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
52	C₂²×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
53	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
54	C₄×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
55	C₄×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
56	C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
57	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₇₉	—	2	3	no	—
59	C₂×C₇₉	—	2	1	no	—
60	C₂×C₇₉	C₂×#108	1	6	yes	complemented
61	C₂×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×#108	—	79	1	no	maximal, Sylow
63	C₇₉	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
64	C₂²×M₁₆	—	79	1	no	—
65	C₂²×M₁₆	D₇₉	1	1	yes	characteristic
66	C₂³×C₈	—	79	1	no	—
67	C₂×#108	—	79	12	no	—
68	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
69	C₂×M₁₆	—	2	3	no	—
70	C₂×M₁₆	—	79	6	no	—
71	C₂×M₁₆	—	158	3	no	—
72	C₂×M₁₆	C₂×D₇₉	1	6	yes	—
73	C₂²×C₈	—	2	1	no	—
74	C₂²×C₈	—	79	6	no	—
75	C₂²×C₈	—	79	2	no	—
76	C₂²×C₈	—	79	6	no	—
77	C₂²×C₈	—	158	1	no	—
78	#108	—	79	16	no	—
79	C₂⁴	Dic₁₅₈	1	1	yes	characteristic
80	C₂²×C₄	Dic₁₅₈	1	3	yes	—
81	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
82	C₂²×C₄	C₄×D₇₉	1	3	yes	—
83	C₂²×C₄	C₂²⋊D₇₉	1	3	yes	—
84	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
85	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
86	M₁₆	—	2	6	no	—
87	M₁₆	—	79	4	no	—
88	M₁₆	—	158	6	no	—
89	M₁₆	C₂²×D₇₉	1	4	yes	—
90	C₂×C₈	—	2	6	no	—
91	C₂×C₈	—	79	12	no	—
92	C₂×C₈	—	79	12	no	—
93	C₂×C₈	—	79	4	no	—
94	C₂×C₈	—	158	6	no	—
95	C₂³	—	2	4	no	—
96	C₂³	Dic₇₉⋊C₄	1	3	yes	—
97	C₂³	Dic₇₉⋊C₄	1	1	yes	characteristic
98	C₂³	C₂×Dic₁₅₈	1	3	yes	—
99	C₂×C₄	Dic₇₉⋊C₄	1	6	yes	—
100	C₂×C₄	Dic₇₉⋊C₄	1	3	yes	—
101	C₂×C₄	Dic₇₉⋊C₄	1	3	yes	—
102	C₂×C₄	C₂×C₄×D₇₉	1	3	yes	—
103	C₂×C₄	C₂×C₄×D₇₉	1	3	yes	—
104	C₂×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
105	C₂×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
106	C₂×C₄	C₂×Dic₁₅₈	1	3	yes	—
107	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
108	C₈	—	2	4	no	—
109	C₈	—	79	8	no	—
110	C₈	—	158	4	no	—
111	C₂²	—	2	6	no	—
112	C₂²	—	2	6	no	—
113	C₂²	#14579096	1	4	yes	direct product factor
114	C₂²	C₂×Dic₇₉⋊C₄	1	3	yes	—
115	C₂²	C₂×Dic₇₉⋊C₄	1	3	yes	—
116	C₂²	C₂²×Dic₁₅₈	1	1	yes	characteristic
117	C₄	C₂×Dic₇₉⋊C₄	1	3	yes	—
118	C₄	C₂×Dic₇₉⋊C₄	1	3	yes	—
119	C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
120	C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	3	no	—
123	C₂	C₂×#14579096	1	6	yes	direct product factor
124	C₂	C₂²×Dic₇₉⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11929608

C₂

yes

maximal, characteristic

C₂³×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₂×#14579096

C₂

yes

maximal, direct product factor

C₂²×M₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

—

C₂×#11929608

C₂²

yes

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

C₄

yes

—

C₂²×C₇₉⋊C₈

C₄

yes

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

#14579096

C₂²

yes

direct product factor

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₂²

yes

—

C₂²×C₈×C₇₉

—

#11929608

—

#11929608

C₂³

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂⁴×C₇₉

Q₈

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

Q₈

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₄⋊C₄

yes

—

C₂³×C₇₉

C₄⋊C₄

yes

characteristic

C₂³×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

C₂²×C₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#108

yes

complemented

C₂²×C₇₉

C₂×C₄⋊C₄

yes

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

—

C₂²×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₄×C₇₉

C₂³×C₄

yes

characteristic

C₄×C₇₉

C₂²×D₄

yes

characteristic

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#108

yes

complemented

C₂×C₇₉

C₂²×C₄⋊C₄

yes

characteristic

C₂²×#108

—

maximal, Sylow

C₇₉

C₂²×#108

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×M₁₆

D₇₉

yes

characteristic

C₂³×C₈

—

C₂×#108

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

158

—

C₂×M₁₆

C₂×D₇₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

158

—

#108

—

C₂⁴

Dic₁₅₈

yes

characteristic

C₂²×C₄

Dic₁₅₈

yes

—

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₄×D₇₉

yes

—

C₂²×C₄

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

C₂²×C₄

C₂²×D₇₉

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

158

—

M₁₆

C₂²×D₇₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂³

—

C₂³

Dic₇₉⋊C₄

yes

—

C₂³

Dic₇₉⋊C₄

yes

characteristic

C₂³

C₂×Dic₁₅₈

yes

—

C₂×C₄

Dic₇₉⋊C₄

yes

—

100

C₂×C₄

Dic₇₉⋊C₄

yes

—

101

C₂×C₄

Dic₇₉⋊C₄

yes

—

102

C₂×C₄

C₂×C₄×D₇₉

yes

—

103

C₂×C₄

C₂×C₄×D₇₉

yes

—

104

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

105

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

106

C₂×C₄

C₂×Dic₁₅₈

yes

—

107

C₂×C₄

C₂³×D₇₉

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

158

—

111

C₂²

—

112

C₂²

—

113

C₂²

#14579096

yes

direct product factor

114

C₂²

C₂×Dic₇₉⋊C₄

yes

—

115

C₂²

C₂×Dic₇₉⋊C₄

yes

—

116

C₂²

C₂²×Dic₁₅₈

yes

characteristic

117

C₄

C₂×Dic₇₉⋊C₄

yes

—

118

C₄

C₂×Dic₇₉⋊C₄

yes

—

119

C₄

C₂²×C₄×D₇₉

yes

characteristic

120

C₄

C₂²×C₂²⋊D₇₉

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₂×#14579096

yes

direct product factor

124

C₂

C₂²×Dic₇₉⋊C₄

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂²×M₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₄⋊C₄
prime radical	C₂⁴×C₇₉	Q₈
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	C₂²×M₁₆	D₇₉
hypocenter	C₇₉	C₂²×#108
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₇₉

commutator

C₄×C₇₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂²×M₁₆×C₇₉

C₂

socle

C₂³×C₇₉

C₄⋊C₄

prime radical

C₂⁴×C₇₉

Q₈

cosocle

C₂×C₄×C₇₉

C₂⁴

hypercenter

C₂²×M₁₆

D₇₉

hypocenter

C₇₉

C₂²×#108

LCS

G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
3. C₂×#14579096	C₂	direct (N × H)
11. #14579096	C₂²	direct (N × H)
50. C₂²×C₇₉	#108	semidirect
60. C₂×C₇₉	C₂×#108	semidirect
63. C₇₉	C₂²×#108	semidirect
113. C₂²	#14579096	direct (N × H)
123. C₂	C₂×#14579096	direct (N × H)

H ≅ G/N

type

3. C₂×#14579096

C₂

direct (N × H)

11. #14579096

C₂²

direct (N × H)

50. C₂²×C₇₉

#108

semidirect

60. C₂×C₇₉

C₂×#108

semidirect

63. C₇₉

C₂²×#108

semidirect

113. C₂²

#14579096

direct (N × H)

123. C₂

C₂×#14579096

direct (N × H)

Subgroups — C₂²×#14579096 #23772002

Conjugacy classes of subgroups

Complements

Subgroups — C22×#14579096 #23772002

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14579096 #23772002