Group #28384329 — subgroups

Subgroups — #28384329

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	#20297302	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297299	C₂	1	2	yes	maximal, complemented
3	#20297298	C₂	1	2	yes	maximal, complemented
4	#20297294	C₂	1	2	yes	maximal, complemented
5	#20297293	C₂	1	1	yes	maximal, characteristic
6	#20297289	C₂	1	2	yes	maximal
7	#20297282	C₂	1	2	yes	maximal, complemented
8	#20297280	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
11	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
13	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
14	Dic₂₂₃⋊C₄	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	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
24	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
29	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
30	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
31	C₂²×D₂₂₃	C₂³	1	2	yes	—
32	Dic₄₄₆	—	4	1	no	—
33	D₈₉₂	—	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	C₄×D₂₂₃	—	4	2	no	—
38	Q₈×C₂₂₃	—	4	1	no	—
39	Q₈×C₂₂₃	—	4	1	no	—
40	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
41	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
45	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
46	C₂×D₂₂₃	—	2	2	no	—
47	C₂×D₂₂₃	—	2	4	no	—
48	C₄×C₂₂₃	—	2	1	no	—
49	C₄×C₂₂₃	—	2	2	no	—
50	C₄×C₂₂₃	—	2	1	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	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
58	D₂₂₃	—	4	2	no	—
59	C₂₂₃	#563	1	1	yes	characteristic, complemented, Sylow
60	#563	—	223	1	no	maximal, Sylow
61	C₄⋊Q₈	—	223	1	no	—
62	C₄⋊Q₈	D₂₂₃	1	1	yes	characteristic, complemented
63	#126	—	223	2	no	—
64	#126	—	223	2	no	—
65	#125	—	223	2	no	—
66	#124	—	223	1	no	—
67	#123	—	223	2	no	—
68	C₂²⋊Q₈	—	223	2	no	—
69	C₂²⋊Q₈	—	223	2	no	—
70	C₂×Q₈	—	223	1	no	—
71	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
73	C₂×D₄	—	223	1	no	—
74	C₂²×C₄	—	223	2	no	—
75	C₄⋊C₄	—	223	2	no	—
76	C₄⋊C₄	—	223	2	no	—
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	C₂×D₂₂₃	1	2	yes	—
82	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
83	C₂²⋊C₄	—	223	2	no	—
84	C₂²⋊C₄	—	223	2	no	—
85	C₂²⋊C₄	—	223	2	no	—
86	C₂²⋊C₄	—	223	2	no	—
87	C₂²⋊C₄	—	223	2	no	—
88	C₄²	—	223	1	no	—
89	C₄²	—	223	1	no	—
90	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
91	C₂³	—	223	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	892	1	no	—
95	D₄	—	892	1	no	—
96	C₂×C₄	—	892	2	no	—
97	C₂×C₄	—	223	2	no	—
98	C₂×C₄	—	223	2	no	—
99	C₂×C₄	—	223	2	no	—
100	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
103	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
105	C₂²	—	446	2	no	—
106	C₂²	—	446	4	no	—
107	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	446	2	no	—
114	C₄	—	446	2	no	—
115	C₄	—	446	2	no	—
116	C₂	—	892	2	no	—
117	C₂	#20297349	1	1	yes	characteristic
118	C₂	#20297352	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20297302

C₂

yes

maximal, characteristic, complemented

#20297299

C₂

yes

maximal, complemented

#20297298

C₂

yes

maximal, complemented

#20297294

C₂

yes

maximal, complemented

#20297293

C₂

yes

maximal, characteristic

#20297289

C₂

yes

maximal

#20297282

C₂

yes

maximal, complemented

#20297280

C₂

yes

maximal, complemented

C₄⋊Q₈×C₂₂₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₄₆

C₂²

yes

characteristic

C₂×D₈₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₂₃

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

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

C₄×Dic₂₂₃

C₂²

yes

characteristic

C₄×Dic₂₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₄²×C₂₂₃

C₂²

yes

characteristic

C₂²×D₂₂₃

C₂³

yes

—

Dic₄₄₆

—

D₈₉₂

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₄×D₂₂₃

—

Q₈×C₂₂₃

—

Q₈×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

C₂×C₂₂₃

H(5,𝔽₂)

yes

characteristic

C₂×C₂₂₃

2₋¹⁺⁴

yes

—

D₂₂₃

—

C₂₂₃

#563

yes

characteristic, complemented, Sylow

#563

—

223

maximal, Sylow

C₄⋊Q₈

—

223

—

C₄⋊Q₈

D₂₂₃

yes

characteristic, complemented

#126

—

223

—

#126

—

223

—

#125

—

223

—

#124

—

223

—

#123

—

223

—

C₂²⋊Q₈

—

223

—

C₂²⋊Q₈

—

223

—

C₂×Q₈

—

223

—

C₂×Q₈

C₂×D₂₂₃

yes

characteristic

C₂×Q₈

C₂×D₂₂₃

yes

characteristic

C₂×D₄

—

223

—

C₂²×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

—

C₄⋊C₄

C₂×D₂₂₃

yes

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₄²

—

223

—

C₄²

—

223

—

C₄²

C₂×D₂₂₃

yes

characteristic

C₂³

—

223

—

Q₈

—

Q₈

—

Q₈

—

892

—

D₄

—

892

—

C₂×C₄

—

892

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

100

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

101

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

102

C₂×C₄

C₂²×D₂₂₃

yes

—

103

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

104

C₂×C₄

C₂²×D₂₂₃

yes

—

105

C₂²

—

446

—

106

C₂²

—

446

—

107

C₂²

C₂³×D₂₂₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

446

—

114

C₄

—

446

—

115

C₄

—

446

—

116

C₂

—

892

—

117

C₂

#20297349

yes

characteristic

118

C₂

#20297352

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	C₄⋊Q₈×C₂₂₃	C₂
prime radical	C₂×D₈₉₂	C₂²
hypercenter	C₄⋊Q₈	D₂₂₃
hypocenter	C₂₂₃	#563
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₂₃

commutator, socle, cosocle

C₂²×C₂₂₃

C₂⁴

Fitting

C₄⋊Q₈×C₂₂₃

C₂

prime radical

C₂×D₈₉₂

C₂²

hypercenter

C₄⋊Q₈

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #20297302	C₂	semidirect
2. #20297299	C₂	semidirect
3. #20297298	C₂	semidirect
4. #20297294	C₂	semidirect
7. #20297282	C₂	semidirect
8. #20297280	C₂	semidirect
9. C₄⋊Q₈×C₂₂₃	C₂	semidirect
59. C₂₂₃	#563	semidirect
62. C₄⋊Q₈	D₂₂₃	semidirect

H ≅ G/N

type

1. #20297302

C₂

semidirect

2. #20297299

C₂

semidirect

3. #20297298

C₂

semidirect

4. #20297294

C₂

semidirect

7. #20297282

C₂

semidirect

8. #20297280

C₂

semidirect

9. C₄⋊Q₈×C₂₂₃

C₂

semidirect

59. C₂₂₃

#563

semidirect

62. C₄⋊Q₈

D₂₂₃

semidirect

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

subgroup	index
#20297302	2
#20297299	2
#20297298	2
#20297294	2
#20297293	2
#20297289	2
#20297282	2
#20297280	2
C₄⋊Q₈×C₂₂₃	2
#563	223

subgroup

index

C₄⋊Q₈×C₂₂₃

223

Subgroups — #28384329

Conjugacy classes of subgroups

Complements

Maximal subgroups