Group #15674373 — subgroups

Subgroups — #15674373

Order 2944.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13092974	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092178	C₂	1	4	yes	maximal
4	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	#10625097	C₂²	1	2	yes	—
8	#10625095	—	2	2	no	—
9	C₂×C₄×Dic₂₃	—	2	2	no	—
10	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
11	#10625041	C₄	1	2	yes	complemented
12	#10625040	C₄	1	2	yes	complemented
13	C₄×Dic₄₆	—	2	2	no	—
14	C₂₃×C₄⋊C₈	—	2	2	no	—
15	C₂²×C₄×C₂₃	D₄	1	2	yes	—
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	—	2	2	no	—
18	C₂×Dic₄₆	—	2	2	no	—
19	M₁₆×C₂₃	—	2	4	no	—
20	C₂×C₈×C₂₃	—	2	2	no	—
21	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₃	D₄	1	2	yes	—
24	C₂²⋊Dic₂₃	—	2	2	no	—
25	C₄⋊Dic₂₃	—	2	2	no	—
26	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
27	Dic₂₃⋊C₄	—	2	4	no	—
28	C₄×Dic₂₃	—	2	8	no	—
29	C₄×Dic₂₃	—	2	2	no	—
30	C₄×Dic₂₃	—	2	2	no	—
31	C₄×Dic₂₃	Q₈	1	4	yes	complemented
32	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	4	no	—
34	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₂₃	#41	1	2	yes	—
40	C₂×Dic₂₃	—	2	8	no	—
41	C₂×Dic₂₃	—	2	4	no	—
42	C₂×Dic₂₃	—	2	4	no	—
43	Dic₄₆	—	4	4	no	—
44	C₈×C₂₃	—	4	2	no	—
45	#1994	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₂₃	—	2	2	no	—
50	C₄×C₂₃	#104	1	4	yes	—
51	C₄×C₂₃	C₄×Q₈	1	2	yes	—
52	C₄×C₂₃	#124	1	1	yes	characteristic
53	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₂₃	—	2	8	no	—
55	Dic₂₃	—	4	4	no	—
56	#533	—	23	1	no	—
57	#423	D₂₃	1	1	yes	characteristic
58	C₄³	—	23	1	no	—
59	#339	—	23	4	no	—
60	C₂×C₂₃	—	2	2	no	—
61	C₂×C₂₃	#391	1	1	yes	characteristic
62	C₂×C₂₃	C₂×#104	1	2	yes	—
63	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
64	C₄⋊Q₈	—	23	2	no	—
65	#125	—	23	2	no	—
66	C₂²⋊Q₈	—	46	2	no	—
67	C₄×Q₈	—	46	2	no	—
68	#117	—	23	2	no	—
69	C₂×C₄²	—	23	2	no	—
70	C₂×C₄²	—	46	2	no	—
71	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
72	C₄⋊C₈	—	2	2	no	—
73	C₂₃	#1994	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	46	2	no	—
75	C₂²×C₄	—	46	2	no	—
76	C₂²×C₄	D₉₂	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	23	4	no	—
81	C₄⋊C₄	—	46	4	no	—
82	C₄⋊C₄	—	46	2	no	—
83	C₂²⋊C₄	—	46	2	no	—
84	C₄²	—	23	4	no	—
85	C₄²	—	46	8	no	—
86	C₄²	—	46	2	no	—
87	C₄²	—	46	2	no	—
88	C₄²	C₄×D₂₃	1	1	yes	characteristic
89	C₄²	C₄×D₂₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₂₃	1	2	yes	—
91	C₂³	C₂×D₉₂	1	1	yes	characteristic
92	Q₈	—	92	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	46	8	no	—
95	C₂×C₄	—	46	4	no	—
96	C₂×C₄	—	46	4	no	—
97	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
100	C₂×C₄	#67290	1	2	yes	—
101	C₂×C₄	Q₈×D₂₃	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	#10625068	1	2	yes	—
106	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	46	8	no	—
109	C₄	—	92	4	no	—
110	C₄	#10625007	1	4	yes	—
111	C₄	#10625058	1	2	yes	—
112	C₄	#10625107	1	1	yes	characteristic
113	C₄	#10625115	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#13092322	1	1	yes	characteristic
116	C₂	C₂×#10625007	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic

C₄²×Dic₂₃

C₂

yes

maximal, characteristic

#13092178

C₂

yes

maximal

C₂₃×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625095

—

C₂×C₄×Dic₂₃

—

C₂×C₄×Dic₂₃

C₂²

yes

—

#10625041

C₄

yes

complemented

#10625040

C₄

yes

complemented

C₄×Dic₄₆

—

C₂₃×C₄⋊C₈

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

Q₈

yes

complemented

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

C₈×C₂₃

—

#1994

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

#104

yes

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

#533

—

#423

D₂₃

yes

characteristic

C₄³

—

#339

—

C₂×C₂₃

—

C₂×C₂₃

#391

yes

characteristic

C₂×C₂₃

C₂×#104

yes

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂₃

#1994

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₂₃

yes

characteristic

C₄²

C₄×D₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×D₉₂

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

100

C₂×C₄

#67290

yes

—

101

C₂×C₄

Q₈×D₂₃

yes

—

102

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

#10625068

yes

—

106

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#10625007

yes

—

111

C₄

#10625058

yes

—

112

C₄

#10625107

yes

characteristic

113

C₄

#10625115

yes

characteristic

114

C₂

—

115

C₂

#13092322

yes

characteristic

116

C₂

C₂×#10625007

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#423	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#423	D₂₃
hypocenter	C₂₃	#1994
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#423

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
11. #10625041	C₄	semidirect
12. #10625040	C₄	semidirect
31. C₄×Dic₂₃	Q₈	semidirect
73. C₂₃	#1994	semidirect

H ≅ G/N

type

11. #10625041

C₄

semidirect

12. #10625040

C₄

semidirect

31. C₄×Dic₂₃

Q₈

semidirect

73. C₂₃

#1994

semidirect

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

subgroup	index
#13092974	2
C₄²×Dic₂₃	2
#13092178	2
C₂₃×#423	2
#1994	23

subgroup

index

#13092974

C₄²×Dic₂₃

#13092178

C₂₃×#423

#1994

Subgroups — #15674373

Conjugacy classes of subgroups

Complements

Maximal subgroups