Group #28384333 — subgroups

Subgroups — #28384333

Order 14272.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20297303	C₂	1	2	yes	maximal, complemented
2	#20297297	C₂	1	2	yes	maximal, complemented
3	#20297297	C₂	1	2	yes	maximal, complemented
4	#20297294	C₂	1	2	yes	maximal, complemented
5	#20297291	C₂	1	1	yes	maximal, characteristic
6	#20297285	C₂	1	1	yes	maximal, characteristic
7	#20297283	C₂	1	2	yes	maximal, complemented
8	#20297281	C₂	1	2	yes	maximal, complemented
9	#123×C₂₂₃	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₄₄₆	C₂²	1	2	yes	—
11	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
14	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
15	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
16	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
17	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
18	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
19	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
20	C₂²⋊Dic₂₂₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₂₂₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₂₂₃	C₂²	1	2	yes	—
23	C₂²⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₂₃	C₂²	1	2	yes	—
26	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
31	Dic₄₄₆	—	4	2	no	—
32	Dic₄₄₆	—	4	1	no	—
33	C₂×Dic₂₂₃	—	4	1	no	—
34	C₂×Dic₂₂₃	C₂³	1	2	yes	—
35	C₂×Dic₂₂₃	C₂³	1	2	yes	—
36	C₂×Dic₂₂₃	C₂³	1	2	yes	—
37	C₂×Dic₂₂₃	C₂³	1	2	yes	—
38	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
39	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
40	D₄×C₂₂₃	—	4	1	no	—
41	C₂×C₄×C₂₂₃	—	4	1	no	—
42	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
43	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
44	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₂₂₃	—	2	1	no	—
46	C₂²×C₂₂₃	—	2	1	no	—
47	C₂²×C₂₂₃	—	2	2	no	—
48	C₂²×C₂₂₃	—	2	2	no	—
49	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
50	C₄×C₂₂₃	—	2	2	no	—
51	C₄×C₂₂₃	—	2	1	no	—
52	C₄×C₂₂₃	—	2	2	no	—
53	Dic₂₂₃	—	2	2	no	—
54	Dic₂₂₃	—	2	2	no	—
55	Dic₂₂₃	—	2	2	no	—
56	Dic₂₂₃	—	2	2	no	—
57	C₂×C₂₂₃	—	4	1	no	—
58	C₂×C₂₂₃	—	4	1	no	—
59	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
61	C₂₂₃	#563	1	1	yes	characteristic, complemented, Sylow
62	#563	—	223	1	no	maximal, Sylow
63	C₄⋊Q₈	—	223	2	no	—
64	#126	—	223	2	no	—
65	#126	—	223	2	no	—
66	#125	—	223	2	no	—
67	#124	—	223	1	no	—
68	#123	—	223	1	no	—
69	#123	D₂₂₃	1	1	yes	characteristic
70	C₂²⋊Q₈	—	223	2	no	—
71	C₂²⋊Q₈	—	223	2	no	—
72	C₂×Q₈	—	223	2	no	—
73	C₂×Q₈	—	223	1	no	—
74	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
75	C₂²×C₄	—	223	1	no	—
76	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
77	C₄⋊C₄	—	223	2	no	—
78	C₄⋊C₄	—	223	2	no	—
79	C₄⋊C₄	—	223	2	no	—
80	C₄⋊C₄	—	223	2	no	—
81	C₄⋊C₄	—	223	2	no	—
82	C₄⋊C₄	—	223	2	no	—
83	C₄⋊C₄	—	223	2	no	—
84	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
85	C₂²⋊C₄	—	223	2	no	—
86	C₂²⋊C₄	—	223	2	no	—
87	C₂²⋊C₄	—	223	2	no	—
88	C₂²⋊C₄	—	223	1	no	—
89	C₂²⋊C₄	C₂×D₂₂₃	1	2	yes	—
90	C₂²⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
91	C₄²	—	223	1	no	—
92	C₄²	—	223	2	no	—
93	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
94	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
95	Q₈	—	892	2	no	—
96	Q₈	—	892	1	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	892	1	no	—
100	C₂×C₄	—	223	2	no	—
101	C₂×C₄	—	223	2	no	—
102	C₂×C₄	—	223	2	no	—
103	C₂×C₄	—	223	2	no	—
104	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
105	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
106	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	2	no	—
111	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	2	no	—
115	C₄	—	446	2	no	—
116	C₄	—	446	2	no	—
117	C₄	—	446	2	no	—
118	C₄	—	446	2	no	—
119	C₂	—	4	1	no	—
120	C₂	—	4	1	no	—
121	C₂	#20297349	1	1	yes	characteristic
122	C₂	#20297351	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20297303

C₂

yes

maximal, complemented

#20297297

C₂

yes

maximal, complemented

#20297297

C₂

yes

maximal, complemented

#20297294

C₂

yes

maximal, complemented

#20297291

C₂

yes

maximal, characteristic

#20297285

C₂

yes

maximal, characteristic

#20297283

C₂

yes

maximal, complemented

#20297281

C₂

yes

maximal, complemented

#123×C₂₂₃

C₂

yes

maximal, characteristic

C₂×Dic₄₄₆

C₂²

yes

—

C₂×Dic₄₄₆

C₂²

yes

characteristic

C₂²×Dic₂₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

C₂²⋊Dic₂₂₃

C₂²

yes

—

C₂²⋊Dic₂₂₃

C₂²

yes

—

C₂²⋊Dic₂₂₃

C₂²

yes

—

C₂²⋊Dic₂₂₃

C₂²

yes

characteristic

C₄×Dic₂₂₃

C₂²

yes

characteristic

C₄×Dic₂₂₃

C₂²

yes

—

C₂×D₄×C₂₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₂₃

C₂²

yes

characteristic

Dic₄₄₆

—

Dic₄₄₆

—

C₂×Dic₂₂₃

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂³×C₂₂₃

C₂³

yes

characteristic

C₂³×C₂₂₃

C₂³

yes

characteristic

D₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

H(5,𝔽₂)

yes

characteristic

C₂×C₂₂₃

2₋¹⁺⁴

yes

—

C₂₂₃

#563

yes

characteristic, complemented, Sylow

#563

—

223

maximal, Sylow

C₄⋊Q₈

—

223

—

#126

—

223

—

#126

—

223

—

#125

—

223

—

#124

—

223

—

#123

—

223

—

#123

D₂₂₃

yes

characteristic

C₂²⋊Q₈

—

223

—

C₂²⋊Q₈

—

223

—

C₂×Q₈

—

223

—

C₂×Q₈

—

223

—

C₂×D₄

C₂×D₂₂₃

yes

characteristic

C₂²×C₄

—

223

—

C₂²×C₄

C₂×D₂₂₃

yes

characteristic

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

C₂×D₂₂₃

yes

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

C₂×D₂₂₃

yes

—

C₂²⋊C₄

C₂×D₂₂₃

yes

characteristic

C₄²

—

223

—

C₄²

—

223

—

C₂³

C₂²×D₂₂₃

yes

characteristic

C₂³

C₂²×D₂₂₃

yes

characteristic

Q₈

—

892

—

Q₈

—

892

—

D₄

—

C₂×C₄

—

C₂×C₄

—

892

—

100

C₂×C₄

—

223

—

101

C₂×C₄

—

223

—

102

C₂×C₄

—

223

—

103

C₂×C₄

—

223

—

104

C₂×C₄

C₂²×D₂₂₃

yes

—

105

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

106

C₂×C₄

C₂²×D₂₂₃

yes

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×D₂₂₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

446

—

116

C₄

—

446

—

117

C₄

—

446

—

118

C₄

—

446

—

119

C₂

—

120

C₂

—

121

C₂

#20297349

yes

characteristic

122

C₂

#20297351

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	#123×C₂₂₃	C₂
prime radical	C₂×D₄×C₂₂₃	C₂²
hypercenter	#123	D₂₂₃
hypocenter	C₂₂₃	#563
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₂₃

commutator, socle, cosocle

C₂²×C₂₂₃

C₂⁴

Fitting

#123×C₂₂₃

C₂

prime radical

C₂×D₄×C₂₂₃

C₂²

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
1. #20297303	C₂	semidirect
2. #20297297	C₂	semidirect
3. #20297297	C₂	semidirect
4. #20297294	C₂	semidirect
7. #20297283	C₂	semidirect
8. #20297281	C₂	semidirect
61. C₂₂₃	#563	semidirect

H ≅ G/N

type

1. #20297303

C₂

semidirect

2. #20297297

C₂

semidirect

3. #20297297

C₂

semidirect

4. #20297294

C₂

semidirect

7. #20297283

C₂

semidirect

8. #20297281

C₂

semidirect

61. C₂₂₃

#563

semidirect

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

subgroup	index
#20297303	2
#20297297	2
#20297294	2
#20297291	2
#20297285	2
#20297283	2
#20297281	2
#123×C₂₂₃	2
#563	223

subgroup

index

#123×C₂₂₃

223

Subgroups — #28384333

Conjugacy classes of subgroups

Complements

Maximal subgroups