Group #28384155 — subgroups

Subgroups — #28384155

Order 14272.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂×#16813171	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297302	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297293	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297292	C₂	1	1	yes	maximal, characteristic, complemented
6	#20297289	C₂	1	4	yes	maximal, complemented
7	#20297282	C₂	1	4	yes	maximal, complemented
8	#20297261	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#16813171	—	2	4	no	—
11	C₂×Dic₄₄₆	C₂²	1	2	yes	—
12	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
13	Q₈×D₂₂₃	—	2	4	no	—
14	C₂×D₈₉₂	C₂²	1	2	yes	—
15	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
17	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
18	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
19	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
20	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
21	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
23	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
24	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
28	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
29	C₂²×D₂₂₃	C₂³	1	2	yes	—
30	C₂²×D₂₂₃	C₂³	1	1	yes	characteristic
31	Dic₄₄₆	—	2	2	no	—
32	Dic₄₄₆	—	2	4	no	—
33	D₈₉₂	—	2	4	no	—
34	D₈₉₂	—	2	2	no	—
35	C₂×Dic₂₂₃	C₂³	1	2	yes	—
36	C₂×Dic₂₂₃	C₂³	1	2	yes	—
37	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
38	C₄×D₂₂₃	—	2	4	no	—
39	C₄×D₂₂₃	—	2	4	no	—
40	C₄×D₂₂₃	D₄	1	4	yes	—
41	Q₈×C₂₂₃	—	2	2	no	—
42	Q₈×C₂₂₃	—	2	2	no	—
43	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
44	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
45	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
46	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
48	C₂×D₂₂₃	—	2	2	no	—
49	C₂×D₂₂₃	—	2	2	no	—
50	C₂×D₂₂₃	—	2	4	no	—
51	C₂×D₂₂₃	C₂×D₄	1	2	yes	—
52	C₄×C₂₂₃	—	2	2	no	—
53	C₄×C₂₂₃	—	2	2	no	—
54	C₄×C₂₂₃	—	2	2	no	—
55	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
56	Dic₂₂₃	—	2	2	no	—
57	Dic₂₂₃	—	2	2	no	—
58	Dic₂₂₃	C₂×D₄	1	2	yes	—
59	C₂×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
61	D₂₂₃	—	2	2	no	—
62	D₂₂₃	—	4	2	no	—
63	C₂₂₃	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	223	1	no	maximal, Sylow
65	C₂×#41	—	223	1	no	—
66	C₂²×Q₈	—	223	1	no	—
67	C₄⋊Q₈	—	223	1	no	—
68	C₄⋊Q₈	D₂₂₃	1	1	yes	characteristic, complemented
69	#124	—	223	1	no	—
70	#124	—	223	1	no	—
71	#123	—	223	4	no	—
72	C₂²⋊Q₈	—	223	4	no	—
73	#117	—	223	1	no	—
74	#41	—	446	4	no	—
75	C₂×Q₈	—	446	4	no	—
76	C₂×Q₈	—	223	2	no	—
77	C₂×Q₈	—	223	1	no	—
78	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
80	C₂×D₄	—	223	2	no	—
81	C₂×D₄	—	223	1	no	—
82	C₂²×C₄	—	223	2	no	—
83	C₂²×C₄	—	223	2	no	—
84	C₂²×C₄	—	223	1	no	—
85	C₄⋊C₄	—	223	2	no	—
86	C₄⋊C₄	—	223	4	no	—
87	C₄⋊C₄	C₂×D₂₂₃	1	4	yes	—
88	C₂²⋊C₄	—	223	2	no	—
89	C₂²⋊C₄	—	223	4	no	—
90	C₂²⋊C₄	—	223	4	no	—
91	C₄²	—	223	1	no	—
92	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
93	C₂³	—	223	2	no	—
94	C₂³	—	223	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	446	4	no	—
98	Q₈	—	446	2	no	—
99	D₄	—	446	4	no	—
100	D₄	—	446	2	no	—
101	C₂×C₄	—	446	4	no	—
102	C₂×C₄	—	446	4	no	—
103	C₂×C₄	—	223	2	no	—
104	C₂×C₄	—	223	2	no	—
105	C₂×C₄	—	223	1	no	—
106	C₂×C₄	—	223	4	no	—
107	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
108	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
109	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
110	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
111	C₂²	—	446	2	no	—
112	C₂²	—	446	4	no	—
113	C₂²	—	446	2	no	—
114	C₂²	—	223	2	no	—
115	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	446	2	no	—
120	C₄	—	446	2	no	—
121	C₄	—	223	2	no	—
122	C₄	D₄×D₂₂₃	1	2	yes	—
123	C₂	—	892	2	no	—
124	C₂	—	446	2	no	—
125	C₂	C₂×D₄×D₂₂₃	1	1	yes	characteristic
126	C₂	#20297352	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813171

C₂

yes

maximal, characteristic

C₂×Q₈×D₂₂₃

C₂

yes

maximal, characteristic, complemented

#20297302

C₂

yes

maximal, characteristic, complemented

#20297293

C₂

yes

maximal, characteristic, complemented

#20297292

C₂

yes

maximal, characteristic, complemented

#20297289

C₂

yes

maximal, complemented

#20297282

C₂

yes

maximal, complemented

#20297261

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₂₂₃

C₂

yes

maximal, characteristic, complemented

#16813171

—

C₂×Dic₄₄₆

C₂²

yes

—

C₂×Dic₄₄₆

C₂²

yes

characteristic

Q₈×D₂₂₃

—

C₂×D₈₉₂

C₂²

yes

—

C₂×D₈₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₂₃

C₂²

yes

—

C₂×C₄×D₂₂₃

C₂²

yes

—

C₂×C₄×D₂₂₃

C₂²

yes

characteristic

Dic₂₂₃⋊C₄

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

C₄×Dic₂₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₄²×C₂₂₃

C₂²

yes

characteristic

C₂²×D₂₂₃

C₂³

yes

—

C₂²×D₂₂₃

C₂³

yes

characteristic

Dic₄₄₆

—

Dic₄₄₆

—

D₈₉₂

—

D₈₉₂

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

characteristic

C₄×D₂₂₃

—

C₄×D₂₂₃

—

C₄×D₂₂₃

D₄

yes

—

Q₈×C₂₂₃

—

Q₈×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

C₂×D₄

yes

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×D₄

yes

—

Dic₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

C₂×D₄

yes

—

C₂×C₂₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₂₃

2₋¹⁺⁴

yes

—

D₂₂₃

—

D₂₂₃

—

C₂₂₃

#536

yes

characteristic, complemented, Sylow

#536

—

223

maximal, Sylow

C₂×#41

—

223

—

C₂²×Q₈

—

223

—

C₄⋊Q₈

—

223

—

C₄⋊Q₈

D₂₂₃

yes

characteristic, complemented

#124

—

223

—

#124

—

223

—

#123

—

223

—

C₂²⋊Q₈

—

223

—

#117

—

223

—

#41

—

446

—

C₂×Q₈

—

446

—

C₂×Q₈

—

223

—

C₂×Q₈

—

223

—

C₂×Q₈

C₂×D₂₂₃

yes

characteristic

C₂×Q₈

C₂×D₂₂₃

yes

characteristic

C₂×D₄

—

223

—

C₂×D₄

—

223

—

C₂²×C₄

—

223

—

C₂²×C₄

—

223

—

C₂²×C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

C₂×D₂₂₃

yes

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₄²

—

223

—

C₄²

C₂×D₂₂₃

yes

characteristic

C₂³

—

223

—

C₂³

—

223

—

Q₈

—

Q₈

—

Q₈

—

446

—

Q₈

—

446

—

D₄

—

446

—

100

D₄

—

446

—

101

C₂×C₄

—

446

—

102

C₂×C₄

—

446

—

103

C₂×C₄

—

223

—

104

C₂×C₄

—

223

—

105

C₂×C₄

—

223

—

106

C₂×C₄

—

223

—

107

C₂×C₄

C₂²×D₂₂₃

yes

—

108

C₂×C₄

C₂²×D₂₂₃

yes

—

109

C₂×C₄

C₂²×D₂₂₃

yes

—

110

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

111

C₂²

—

446

—

112

C₂²

—

446

—

113

C₂²

—

446

—

114

C₂²

—

223

—

115

C₂²

C₂³×D₂₂₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

446

—

120

C₄

—

446

—

121

C₄

—

223

—

122

C₄

D₄×D₂₂₃

yes

—

123

C₂

—

892

—

124

C₂

—

446

—

125

C₂

C₂×D₄×D₂₂₃

yes

characteristic

126

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×#16813171	C₂
hypercenter	C₄⋊Q₈	D₂₂₃
hypocenter	C₂₂₃	#536
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×#16813171

C₂

hypercenter

C₄⋊Q₈

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₂₂₃	C₂	semidirect
3. #20297302	C₂	semidirect
4. #20297293	C₂	semidirect
5. #20297292	C₂	semidirect
6. #20297289	C₂	semidirect
7. #20297282	C₂	semidirect
8. #20297261	C₂	semidirect
9. C₄⋊Q₈×C₂₂₃	C₂	semidirect
63. C₂₂₃	#536	semidirect
68. C₄⋊Q₈	D₂₂₃	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₂₂₃

C₂

semidirect

3. #20297302

C₂

semidirect

4. #20297293

C₂

semidirect

5. #20297292

C₂

semidirect

6. #20297289

C₂

semidirect

7. #20297282

C₂

semidirect

8. #20297261

C₂

semidirect

9. C₄⋊Q₈×C₂₂₃

C₂

semidirect

63. C₂₂₃

#536

semidirect

68. C₄⋊Q₈

D₂₂₃

semidirect

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

subgroup	index
C₂×#16813171	2
C₂×Q₈×D₂₂₃	2
#20297302	2
#20297293	2
#20297292	2
#20297289	2
#20297282	2
#20297261	2
C₄⋊Q₈×C₂₂₃	2
#536	223

subgroup

index

C₂×Q₈×D₂₂₃

C₄⋊Q₈×C₂₂₃

223

Subgroups — #28384155

Conjugacy classes of subgroups

Complements

Maximal subgroups