Group #28384135 — subgroups

Subgroups — #28384135

Order 14272.

222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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	#20297304	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297295	C₂	1	2	yes	maximal, complemented
3	#20297284	C₂	1	2	yes	maximal
4	#20297280	C₂	1	2	yes	maximal
5	Q₈×Dic₂₂₃	C₂	1	2	yes	maximal, complemented
6	#20297269	C₂	1	2	yes	maximal, complemented
7	#20297262	C₂	1	2	yes	maximal
8	C₄²×D₂₂₃	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₄₄₆	C₂²	1	2	yes	—
11	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
15	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
16	C₄⋊Dic₂₂₃	C₂²	1	4	yes	—
17	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₂₃	C₂²	1	4	yes	—
23	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
24	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
25	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
26	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
27	C₂²×D₂₂₃	C₂³	1	1	yes	characteristic
28	Dic₄₄₆	—	2	4	no	—
29	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₂₂₃	C₂³	1	4	yes	—
32	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
33	C₄×D₂₂₃	—	2	2	no	—
34	C₄×D₂₂₃	—	2	2	no	—
35	C₄×D₂₂₃	Q₈	1	4	yes	complemented
36	Q₈×C₂₂₃	—	2	4	no	—
37	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
38	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
42	C₂×D₂₂₃	—	2	1	no	—
43	C₂×D₂₂₃	—	2	1	no	—
44	C₂×D₂₂₃	C₂×Q₈	1	2	yes	—
45	C₄×C₂₂₃	—	2	4	no	—
46	C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
47	C₄×C₂₂₃	#41	1	2	yes	—
48	C₄×C₂₂₃	#41	1	2	yes	—
49	Dic₂₂₃	—	2	1	no	—
50	Dic₂₂₃	—	2	1	no	—
51	Dic₂₂₃	—	2	4	no	—
52	Dic₂₂₃	C₂×Q₈	1	2	yes	—
53	C₂×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
55	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
56	D₂₂₃	—	2	2	no	—
57	C₂₂₃	#533	1	1	yes	characteristic, complemented, Sylow
58	#533	—	223	1	no	maximal, Sylow
59	C₄⋊Q₈	—	223	1	no	—
60	C₄⋊Q₈	D₂₂₃	1	1	yes	characteristic, complemented
61	#125	—	223	2	no	—
62	C₂²⋊Q₈	—	223	2	no	—
63	C₂²⋊Q₈	—	223	2	no	—
64	C₄×Q₈	—	223	2	no	—
65	C₄×Q₈	—	223	2	no	—
66	#117	—	223	2	no	—
67	C₂×C₄²	—	223	1	no	—
68	C₂×Q₈	—	223	2	no	—
69	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
70	C₂²×C₄	—	223	1	no	—
71	C₂²×C₄	—	223	1	no	—
72	C₂²×C₄	—	223	1	no	—
73	C₄⋊C₄	—	223	4	no	—
74	C₄⋊C₄	—	223	2	no	—
75	C₄⋊C₄	—	223	4	no	—
76	C₄⋊C₄	—	223	2	no	—
77	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
78	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
79	C₂²⋊C₄	—	223	4	no	—
80	C₄²	—	223	1	no	—
81	C₄²	—	223	1	no	—
82	C₄²	—	223	1	no	—
83	C₄²	—	223	4	no	—
84	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
85	C₂³	—	223	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	446	4	no	—
88	C₂×C₄	—	446	2	no	—
89	C₂×C₄	—	446	2	no	—
90	C₂×C₄	—	223	1	no	—
91	C₂×C₄	—	223	1	no	—
92	C₂×C₄	—	223	4	no	—
93	C₂×C₄	—	223	1	no	—
94	C₂×C₄	—	223	4	no	—
95	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
96	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
99	C₂²	—	446	1	no	—
100	C₂²	—	446	1	no	—
101	C₂²	—	223	2	no	—
102	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	446	1	no	—
105	C₄	—	446	1	no	—
106	C₄	—	446	4	no	—
107	C₄	—	223	2	no	—
108	C₄	Q₈×D₂₂₃	1	2	yes	—
109	C₄	#16813169	1	2	yes	—
110	C₄	#16813171	1	2	yes	—
111	C₂	—	446	2	no	—
112	C₂	C₂×Q₈×D₂₂₃	1	1	yes	characteristic
113	C₂	C₂×#16813169	1	1	yes	characteristic
114	C₂	C₂×#16813171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20297304

C₂

yes

maximal, characteristic, complemented

#20297295

C₂

yes

maximal, complemented

#20297284

C₂

yes

maximal

#20297280

C₂

yes

maximal

Q₈×Dic₂₂₃

C₂

yes

maximal, complemented

#20297269

C₂

yes

maximal, complemented

#20297262

C₂

yes

maximal

C₄²×D₂₂₃

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₂₂₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₄₆

C₂²

yes

—

C₂×C₄×D₂₂₃

C₂²

yes

characteristic

C₂×C₄×D₂₂₃

C₂²

yes

characteristic

C₂×C₄×D₂₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

C₄×Dic₂₂₃

C₂²

yes

characteristic

C₄×Dic₂₂₃

C₂²

yes

characteristic

C₄×Dic₂₂₃

C₂²

yes

characteristic

C₄×Dic₂₂₃

C₂²

yes

—

C₂×Q₈×C₂₂₃

C₂²

yes

—

C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₄²×C₂₂₃

C₂²

yes

characteristic

C₂²×D₂₂₃

C₂³

yes

characteristic

Dic₄₄₆

—

C₂×Dic₂₂₃

C₂³

yes

characteristic

C₂×Dic₂₂₃

C₂³

yes

characteristic

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

characteristic

C₄×D₂₂₃

—

C₄×D₂₂₃

—

C₄×D₂₂₃

Q₈

yes

complemented

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

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

C₂×Q₈

yes

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×Q₈

yes

—

C₄×C₂₂₃

#41

yes

—

C₄×C₂₂₃

#41

yes

—

Dic₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

C₂×Q₈

yes

—

C₂×C₂₂₃

C₂²×Q₈

yes

characteristic

C₂×C₂₂₃

C₂×#41

yes

characteristic

C₂×C₂₂₃

C₂×#41

yes

characteristic

D₂₂₃

—

C₂₂₃

#533

yes

characteristic, complemented, Sylow

#533

—

223

maximal, Sylow

C₄⋊Q₈

—

223

—

C₄⋊Q₈

D₂₂₃

yes

characteristic, complemented

#125

—

223

—

C₂²⋊Q₈

—

223

—

C₂²⋊Q₈

—

223

—

C₄×Q₈

—

223

—

C₄×Q₈

—

223

—

#117

—

223

—

C₂×C₄²

—

223

—

C₂×Q₈

—

223

—

C₂×Q₈

C₂×D₂₂₃

yes

—

C₂²×C₄

—

223

—

C₂²×C₄

—

223

—

C₂²×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

—

C₄²

—

223

—

C₄²

—

223

—

C₄²

—

223

—

C₄²

—

223

—

C₄²

C₂×D₂₂₃

yes

characteristic

C₂³

—

223

—

Q₈

—

Q₈

—

446

—

C₂×C₄

—

446

—

C₂×C₄

—

446

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

C₂×C₄

C₂²×D₂₂₃

yes

—

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

C₂²

—

446

—

100

C₂²

—

446

—

101

C₂²

—

223

—

102

C₂²

C₂³×D₂₂₃

yes

characteristic

103

C₄

—

104

C₄

—

446

—

105

C₄

—

446

—

106

C₄

—

446

—

107

C₄

—

223

—

108

C₄

Q₈×D₂₂₃

yes

—

109

C₄

#16813169

yes

—

110

C₄

#16813171

yes

—

111

C₂

—

446

—

112

C₂

C₂×Q₈×D₂₂₃

yes

characteristic

113

C₂

C₂×#16813169

yes

characteristic

114

C₂

C₂×#16813171

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	C₄⋊Q₈×C₂₂₃	C₂
prime radical	C₂²×D₂₂₃	C₂³
hypercenter	C₄⋊Q₈	D₂₂₃
hypocenter	C₂₂₃	#533
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

C₂²×D₂₂₃

C₂³

hypercenter

C₄⋊Q₈

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #20297304	C₂	semidirect
2. #20297295	C₂	semidirect
5. Q₈×Dic₂₂₃	C₂	semidirect
6. #20297269	C₂	semidirect
9. C₄⋊Q₈×C₂₂₃	C₂	semidirect
35. C₄×D₂₂₃	Q₈	semidirect
57. C₂₂₃	#533	semidirect
60. C₄⋊Q₈	D₂₂₃	semidirect

H ≅ G/N

type

1. #20297304

C₂

semidirect

2. #20297295

C₂

semidirect

5. Q₈×Dic₂₂₃

C₂

semidirect

6. #20297269

C₂

semidirect

9. C₄⋊Q₈×C₂₂₃

C₂

semidirect

35. C₄×D₂₂₃

Q₈

semidirect

57. C₂₂₃

#533

semidirect

60. C₄⋊Q₈

D₂₂₃

semidirect

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

subgroup	index
#20297304	2
#20297295	2
#20297284	2
#20297280	2
Q₈×Dic₂₂₃	2
#20297269	2
#20297262	2
C₄²×D₂₂₃	2
C₄⋊Q₈×C₂₂₃	2
#533	223

subgroup

index

Q₈×Dic₂₂₃

C₄²×D₂₂₃

C₄⋊Q₈×C₂₂₃

#533

223

Subgroups — #28384135

Conjugacy classes of subgroups

Complements

Maximal subgroups