Group #15672121 — subgroups

Subgroups — #15672121

Order 2944.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic
2	#13092473	C₂	1	2	yes	maximal
3	#13092166	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092165	C₂	1	2	yes	maximal, complemented
5	C₂₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₂₃	C₄	1	2	yes	—
7	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	#10625115	C₄	1	2	yes	—
9	C₄⋊C₄×D₂₃	—	2	2	no	—
10	#10625059	—	2	2	no	—
11	C₄²×D₂₃	C₂²	1	1	yes	characteristic
12	#10625040	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	(C₂×D₂₃)⋊C₈	—	2	2	no	—
15	#10624997	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
18	Q₈×D₂₃	—	4	4	no	—
19	C₂×C₄×D₂₃	—	2	2	no	—
20	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₂₃	D₄	1	2	yes	—
22	C₂×Dic₄₆	—	2	2	no	—
23	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₂₃	—	2	2	no	—
28	Dic₂₃⋊C₄	—	2	4	no	—
29	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₂₃	D₄	1	2	yes	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂²×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₂₃	—	4	2	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
37	C₂×Dic₂₃	—	2	2	no	—
38	C₂×Dic₂₃	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×D₂₃	—	2	2	no	—
41	C₄×D₂₃	—	2	2	no	—
42	C₄×D₂₃	—	2	2	no	—
43	C₄×D₂₃	—	4	4	no	—
44	Dic₄₆	—	4	4	no	—
45	Dic₄₆	—	4	2	no	—
46	C₈×C₂₃	—	4	2	no	—
47	C₂₃⋊C₈	—	4	2	no	—
48	#1558	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₂₃	—	2	2	no	—
51	C₂×D₂₃	#100	1	2	yes	—
52	C₄×C₂₃	—	2	1	no	—
53	C₄×C₂₃	—	4	2	no	—
54	C₄×C₂₃	#137	1	2	yes	—
55	C₄×C₂₃	#137	1	2	yes	—
56	Dic₂₃	—	2	1	no	—
57	Dic₂₃	—	2	2	no	—
58	Dic₂₃	—	4	2	no	—
59	C₂×C₄⋊Q₈	—	23	1	no	—
60	#432	—	23	2	no	—
61	#333	—	23	1	no	—
62	#333	—	23	2	no	—
63	#333	D₂₃	1	1	yes	characteristic, complemented
64	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
65	C₂×C₂₃	#419	1	2	yes	—
66	D₂₃	—	2	2	no	—
67	C₂²×Q₈	—	23	2	no	—
68	C₄⋊Q₈	—	23	2	no	—
69	C₄⋊Q₈	—	23	1	no	—
70	C₄⋊Q₈	—	46	2	no	—
71	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
72	C₂×C₄⋊C₄	—	46	2	no	—
73	C₂×C₄²	—	23	1	no	—
74	C₄⋊C₈	—	23	2	no	—
75	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
76	C₂²⋊C₈	—	46	2	no	—
77	#97	—	46	2	no	—
78	C₂₃	#1558	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₈	—	23	2	no	—
80	C₂×Q₈	—	46	2	no	—
81	C₂×Q₈	—	92	4	no	—
82	C₂×Q₈	C₄×D₂₃	1	2	yes	—
83	C₂²×C₄	—	23	1	no	—
84	C₂²×C₄	—	23	2	no	—
85	C₂²×C₄	—	46	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	46	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	46	4	no	—
90	C₄⋊C₄	—	46	2	no	—
91	C₄²	—	23	1	no	—
92	C₄²	—	23	2	no	—
93	C₄²	C₂²×D₂₃	1	1	yes	characteristic
94	C₂³	—	23	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	92	2	no	—
97	Q₈	—	92	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	23	2	no	—
100	C₂×C₄	—	23	1	no	—
101	C₂×C₄	—	46	2	no	—
102	C₂×C₄	—	46	2	no	—
103	C₂×C₄	—	46	2	no	—
104	C₂×C₄	—	46	2	no	—
105	C₂×C₄	—	92	4	no	—
106	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₃	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	92	2	no	—
110	C₂²	—	23	2	no	—
111	C₂²	—	46	2	no	—
112	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	46	1	no	—
116	C₄	—	46	2	no	—
117	C₄	—	92	2	no	—
118	C₄	#10625086	1	2	yes	—
119	C₄	#10625089	1	2	yes	—
120	C₂	—	46	2	no	—
121	C₂	#100×D₂₃	1	1	yes	characteristic
122	C₂	#13092424	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₂₃

C₂

yes

maximal, characteristic

#13092473

C₂

yes

maximal

#13092166

C₂

yes

maximal, characteristic, complemented

#13092165

C₂

yes

maximal, complemented

C₂₃×#333

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₂₃

C₄

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

#10625115

C₄

yes

—

C₄⋊C₄×D₂₃

—

#10625059

—

C₄²×D₂₃

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

—

(C₂×D₂₃)⋊C₈

—

#10624997

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

Q₈×D₂₃

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

C₂×C₄

yes

characteristic

C₂×C₄×D₂₃

D₄

yes

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂²×D₂₃

C₂²⋊C₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²⋊C₄

yes

—

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1558

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

#100

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#137

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊Q₈

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

C₂×#100

yes

characteristic

C₂×C₂₃

#419

yes

—

D₂₃

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

—

C₂²⋊C₈

—

#97

—

C₂₃

#1558

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

107

C₂×C₄

D₄×D₂₃

yes

—

108

C₈

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂²⋊C₄×D₂₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#10625086

yes

—

119

C₄

#10625089

yes

—

120

C₂

—

121

C₂

#100×D₂₃

yes

characteristic

122

C₂

#13092424

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#333	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	#333	D₂₃
hypocenter	C₂₃	#1558
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#333

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₃

C₂²⋊C₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #13092166	C₂	semidirect
4. #13092165	C₂	semidirect
5. C₂₃×#333	C₂	semidirect
63. #333	D₂₃	semidirect
78. C₂₃	#1558	semidirect

H ≅ G/N

type

3. #13092166

C₂

semidirect

4. #13092165

C₂

semidirect

5. C₂₃×#333

C₂

semidirect

63. #333

D₂₃

semidirect

78. C₂₃

#1558

semidirect

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

subgroup	index
C₄⋊Q₈×D₂₃	2
#13092473	2
#13092166	2
#13092165	2
C₂₃×#333	2
#1558	23

subgroup

index

C₄⋊Q₈×D₂₃

C₂₃×#333

Subgroups — #15672121

Conjugacy classes of subgroups

Complements

Maximal subgroups