Group #23765101 — subgroups

Subgroups — C₂×#19098366 #23765101

Order 10112.

232 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 101 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×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#19098366	C₂	1	4	yes	maximal, direct product factor
4	C₂×#107×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₅₈	—	2	1	no	—
6	C₂×Dic₃₁₆	—	2	1	no	—
7	C₂×Dic₃₁₆	—	2	2	no	—
8	C₂×Dic₃₁₆	C₄	1	4	yes	complemented
9	C₂×Dic₃₁₆	C₂²	1	2	yes	—
10	C₂×C₇₉⋊C₁₆	—	2	2	no	—
11	C₂×C₇₉⋊C₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
14	#107×C₇₉	—	2	1	no	—
15	#107×C₇₉	C₂²	1	2	yes	—
16	C₂×Dic₁₅₈	—	2	4	no	—
17	C₂×Dic₁₅₈	—	2	2	no	—
18	C₂×Dic₁₅₈	—	4	1	no	—
19	C₂×Dic₁₅₈	—	4	2	no	—
20	C₂²×Dic₇₉	—	4	1	no	—
21	Dic₃₁₆	—	2	4	no	—
22	Dic₃₁₆	—	2	2	no	—
23	Dic₃₁₆	C₂×C₄	1	4	yes	complemented
24	C₇₉⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×C₇₉	—	4	1	no	—
26	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₇₉	D₄	1	2	yes	—
29	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₇₉	—	2	2	no	—
32	C₄⋊C₄×C₇₉	—	4	1	no	—
33	C₂³×C₇₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₇₉	—	4	2	no	—
35	C₂×C₄×C₇₉	—	8	2	no	—
36	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
38	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
39	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₇₉	—	4	4	no	—
41	C₂×Dic₇₉	—	4	2	no	—
42	Dic₁₅₈	—	2	4	no	—
43	Dic₁₅₈	—	4	4	no	—
44	Dic₁₅₈	—	4	2	no	—
45	C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
46	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₇₉	#102	1	2	yes	—
49	C₂²×C₇₉	SD₃₂	1	2	yes	—
50	C₂²×C₇₉	Q₃₂	1	2	yes	—
51	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
52	C₄×C₇₉	—	8	2	no	—
53	C₄×C₇₉	#102	1	2	yes	—
54	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₇₉	—	4	4	no	—
57	C₂×C₇₉	#358	1	4	yes	complemented
58	C₂×C₇₉	C₂×#102	1	1	yes	characteristic
59	C₂×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
60	C₂×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
61	C₂×#358	—	79	1	no	maximal, Sylow
62	C₇₉	C₂×#358	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₁₆	—	79	1	no	—
64	C₂²×C₁₆	—	79	1	no	—
65	C₂×#107	D₇₉	1	1	yes	characteristic
66	#358	—	79	4	no	—
67	C₂²×Q₈	—	158	1	no	—
68	C₂×Q₁₆	—	79	2	no	—
69	C₂×Q₁₆	—	79	4	no	—
70	C₂×Q₁₆	—	158	1	no	—
71	C₂×Q₁₆	—	158	2	no	—
72	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	2	1	no	—
74	C₂×C₁₆	—	79	2	no	—
75	C₂×C₁₆	—	158	2	no	—
76	#107	—	2	1	no	—
77	#107	C₂×D₇₉	1	2	yes	—
78	C₂×Q₈	—	316	1	no	—
79	C₂×Q₈	—	316	2	no	—
80	C₂×Q₈	—	158	4	no	—
81	C₂×Q₈	—	158	2	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	—	316	1	no	—
84	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
85	Q₁₆	—	79	4	no	—
86	Q₁₆	—	158	4	no	—
87	Q₁₆	—	158	2	no	—
88	C₂×C₈	C₄×D₇₉	1	2	yes	—
89	C₂×C₈	D₃₁₆	1	2	yes	—
90	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
91	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	4	1	no	—
94	C₁₆	—	158	2	no	—
95	C₂³	#11929609	1	1	yes	characteristic
96	Q₈	—	316	4	no	—
97	Q₈	—	316	2	no	—
98	Q₈	—	158	4	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	316	4	no	—
102	C₂×C₄	—	316	2	no	—
103	C₂×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
104	C₂×C₄	#11929609	1	1	yes	characteristic
105	C₂×C₄	#11929611	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
107	C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
108	C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
109	C₈	C₂×D₃₁₆	1	1	yes	characteristic
110	C₂²	#14579081	1	2	yes	—
111	C₂²	#14579105	1	2	yes	—
112	C₂²	#14579107	1	2	yes	—
113	C₂²	C₂×#11929609	1	1	yes	characteristic
114	C₄	—	8	2	no	—
115	C₄	—	316	4	no	—
116	C₄	#14579081	1	2	yes	—
117	C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
118	C₄	C₂×#11929611	1	1	yes	characteristic
119	C₂	#19098366	1	4	yes	direct product factor
120	C₂	C₂×#14579081	1	1	yes	characteristic
121	C₂	C₂×#14579105	1	1	yes	characteristic
122	C₂	C₂×#14579107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₁₆

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂

yes

maximal, characteristic

#19098366

C₂

yes

maximal, direct product factor

C₂×#107×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

C₄

yes

complemented

C₂×Dic₃₁₆

C₂²

yes

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

#107×C₇₉

—

#107×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

C₂×C₄

yes

complemented

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂³×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂²⋊C₄

yes

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₇₉

#102

yes

—

C₂²×C₇₉

SD₃₂

yes

—

C₂²×C₇₉

Q₃₂

yes

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#102

yes

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

C₂×SD₃₂

yes

characteristic

C₂×C₇₉

C₂×Q₃₂

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂²×C₁₆

—

C₂×#107

D₇₉

yes

characteristic

#358

—

C₂²×Q₈

—

158

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₁₆

—

C₂×C₁₆

—

158

—

#107

—

#107

C₂×D₇₉

yes

—

C₂×Q₈

—

316

—

C₂×Q₈

—

316

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂²×C₄

—

C₂²×C₄

—

316

—

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

Q₁₆

—

Q₁₆

—

158

—

Q₁₆

—

158

—

C₂×C₈

C₄×D₇₉

yes

—

C₂×C₈

D₃₁₆

yes

—

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₁₆

—

158

—

C₂³

#11929609

yes

characteristic

Q₈

—

316

—

Q₈

—

316

—

Q₈

—

158

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

316

—

102

C₂×C₄

—

316

—

103

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

104

C₂×C₄

#11929609

yes

characteristic

105

C₂×C₄

#11929611

yes

—

106

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

107

C₈

(C₂×D₇₉)⋊C₄

yes

—

108

C₈

C₂×C₄×D₇₉

yes

characteristic

109

C₈

C₂×D₃₁₆

yes

characteristic

110

C₂²

#14579081

yes

—

111

C₂²

#14579105

yes

—

112

C₂²

#14579107

yes

—

113

C₂²

C₂×#11929609

yes

characteristic

114

C₄

—

115

C₄

—

316

—

116

C₄

#14579081

yes

—

117

C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

118

C₄

C₂×#11929611

yes

characteristic

119

C₂

#19098366

yes

direct product factor

120

C₂

C₂×#14579081

yes

characteristic

121

C₂

C₂×#14579105

yes

characteristic

122

C₂

C₂×#14579107

yes

characteristic

	subgroup	G/H
center	C₂³	#11929609
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	C₂×#107×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	D₈
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	C₂×#107	D₇₉
hypocenter	C₇₉	C₂×#358
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

subgroup

G/H

center

C₂³

#11929609

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

C₂×#107×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

D₈

cosocle

C₂×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

N	H ≅ G/N	type
3. #19098366	C₂	direct (N × H)
8. C₂×Dic₃₁₆	C₄	semidirect
23. Dic₃₁₆	C₂×C₄	semidirect
57. C₂×C₇₉	#358	semidirect
62. C₇₉	C₂×#358	semidirect
119. C₂	#19098366	direct (N × H)

H ≅ G/N

type

3. #19098366

C₂

direct (N × H)

8. C₂×Dic₃₁₆

C₄

semidirect

23. Dic₃₁₆

C₂×C₄

semidirect

57. C₂×C₇₉

#358

semidirect

62. C₇₉

C₂×#358

semidirect

119. C₂

#19098366

direct (N × H)

Subgroups — C₂×#19098366 #23765101

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098366 #23765101

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098366 #23765101