Group #28384295 — subgroups

Subgroups — #28384295

Order 14272.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20297290	C₂	1	4	yes	maximal, complemented
3	#20297278	C₂	1	2	yes	maximal, complemented
4	#20297275	C₂	1	2	yes	maximal, complemented
5	C₄×D₈₉₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#20297263	C₂	1	2	yes	maximal, complemented
7	#20297262	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₄₄₆	C₂²	1	2	yes	—
10	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₂₂₃	C₂²	1	4	yes	—
12	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
13	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
14	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
16	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	—
17	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
19	C₄×Dic₂₂₃	C₂²	1	4	yes	—
20	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
22	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
24	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
25	C₂²×D₂₂₃	C₂³	1	2	yes	—
26	Dic₄₄₆	—	4	2	no	—
27	D₈₉₂	—	2	2	no	—
28	C₂²⋊D₂₂₃	—	4	4	no	—
29	C₂×Dic₂₂₃	C₂³	1	2	yes	—
30	C₂×Dic₂₂₃	C₂³	1	4	yes	—
31	C₄×D₂₂₃	—	2	4	no	—
32	C₂³×C₂₂₃	C₂³	1	2	yes	—
33	D₄×C₂₂₃	—	2	2	no	—
34	C₂×C₄×C₂₂₃	—	2	4	no	—
35	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
37	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₂₂₃	—	2	2	no	—
40	C₂²×C₂₂₃	—	2	2	no	—
41	C₂²×C₂₂₃	—	2	2	no	—
42	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
43	C₂×D₂₂₃	—	2	2	no	—
44	C₂×D₂₂₃	—	2	2	no	—
45	C₂×D₂₂₃	—	2	2	no	—
46	C₄×C₂₂₃	—	2	2	no	—
47	C₄×C₂₂₃	—	2	1	no	—
48	C₄×C₂₂₃	#41	1	2	yes	—
49	C₄×C₂₂₃	#41	1	2	yes	—
50	Dic₂₂₃	—	2	2	no	—
51	Dic₂₂₃	—	2	4	no	—
52	C₂×C₂₂₃	—	4	2	no	—
53	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
54	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
55	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
56	D₂₂₃	—	4	2	no	—
57	C₂₂₃	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	223	1	no	maximal, Sylow
59	C₄⋊Q₈	—	223	1	no	—
60	#124	—	223	4	no	—
61	#121	—	223	2	no	—
62	#121	—	223	2	no	—
63	C₄×D₄	—	223	1	no	—
64	C₄×D₄	D₂₂₃	1	1	yes	characteristic, complemented
65	#117	—	223	2	no	—
66	#117	—	223	2	no	—
67	C₂×Q₈	—	223	2	no	—
68	C₂×D₄	—	223	4	no	—
69	C₂×D₄	—	223	1	no	—
70	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
71	C₂²×C₄	—	223	2	no	—
72	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
73	C₄⋊C₄	—	223	1	no	—
74	C₄⋊C₄	—	223	2	no	—
75	C₄⋊C₄	—	223	2	no	—
76	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
77	C₂²⋊C₄	—	223	4	no	—
78	C₂²⋊C₄	—	223	4	no	—
79	C₂²⋊C₄	—	223	2	no	—
80	C₂²⋊C₄	C₂×D₂₂₃	1	2	yes	—
81	C₄²	—	223	4	no	—
82	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
83	C₂³	—	223	2	no	—
84	C₂³	C₂²×D₂₂₃	1	2	yes	—
85	Q₈	—	892	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	892	4	no	—
88	D₄	—	446	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	446	4	no	—
91	C₂×C₄	—	223	2	no	—
92	C₂×C₄	—	223	4	no	—
93	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
94	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	446	2	no	—
101	C₂²	—	446	2	no	—
102	C₂²	—	446	2	no	—
103	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	446	2	no	—
107	C₄	—	446	4	no	—
108	C₄	#16813169	1	2	yes	—
109	C₄	#16813170	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	892	2	no	—
112	C₂	C₂×#16813169	1	1	yes	characteristic
113	C₂	C₂×#16813170	1	1	yes	characteristic
114	C₂	#20297350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20297304

C₂

yes

maximal, characteristic, complemented

#20297290

C₂

yes

maximal, complemented

#20297278

C₂

yes

maximal, complemented

#20297275

C₂

yes

maximal, complemented

C₄×D₈₉₂

C₂

yes

maximal, characteristic, complemented

#20297263

C₂

yes

maximal, complemented

#20297262

C₂

yes

maximal, complemented

C₄×D₄×C₂₂₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₄₆

C₂²

yes

—

C₂×D₈₉₂

C₂²

yes

characteristic

C₂×C₂²⋊D₂₂₃

C₂²

yes

—

C₂×C₄×D₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₂₃

C₂²

yes

—

C₂²⋊Dic₂₂₃

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

C₄×Dic₂₂₃

C₂²

yes

—

C₂×D₄×C₂₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₂₃

C₂²

yes

—

C₄⋊C₄×C₂₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₂₃

C₂²

yes

—

C₄²×C₂₂₃

C₂²

yes

characteristic

C₂²×D₂₂₃

C₂³

yes

—

Dic₄₄₆

—

D₈₉₂

—

C₂²⋊D₂₂₃

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₄×D₂₂₃

—

C₂³×C₂₂₃

C₂³

yes

—

D₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

#41

yes

—

C₄×C₂₂₃

#41

yes

—

Dic₂₂₃

—

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

C₂×#41

yes

characteristic

C₂×C₂₂₃

C₂×#41

yes

characteristic

C₂×C₂₂₃

H(5,𝔽₂)

yes

characteristic

D₂₂₃

—

C₂₂₃

#555

yes

characteristic, complemented, Sylow

#555

—

223

maximal, Sylow

C₄⋊Q₈

—

223

—

#124

—

223

—

#121

—

223

—

#121

—

223

—

C₄×D₄

—

223

—

C₄×D₄

D₂₂₃

yes

characteristic, complemented

#117

—

223

—

#117

—

223

—

C₂×Q₈

—

223

—

C₂×D₄

—

223

—

C₂×D₄

—

223

—

C₂×D₄

C₂×D₂₂₃

yes

characteristic

C₂²×C₄

—

223

—

C₂²×C₄

C₂×D₂₂₃

yes

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

C₂×D₂₂₃

yes

characteristic

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

C₂×D₂₂₃

yes

—

C₄²

—

223

—

C₄²

C₂×D₂₂₃

yes

characteristic

C₂³

—

223

—

C₂³

C₂²×D₂₂₃

yes

—

Q₈

—

892

—

D₄

—

D₄

—

892

—

D₄

—

446

—

C₂×C₄

—

C₂×C₄

—

446

—

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

—

C₂²

—

C₂²

—

100

C₂²

—

446

—

101

C₂²

—

446

—

102

C₂²

—

446

—

103

C₂²

C₂³×D₂₂₃

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

446

—

107

C₄

—

446

—

108

C₄

#16813169

yes

—

109

C₄

#16813170

yes

—

110

C₂

—

111

C₂

—

892

—

112

C₂

C₂×#16813169

yes

characteristic

113

C₂

C₂×#16813170

yes

characteristic

114

C₂

#20297350

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	C₄×D₄×C₂₂₃	C₂
hypercenter	C₄×D₄	D₂₂₃
hypocenter	C₂₂₃	#555
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₂₃

commutator, socle, cosocle

C₂²×C₂₂₃

C₂⁴

Fitting

C₄×D₄×C₂₂₃

C₂

hypercenter

C₄×D₄

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #20297304	C₂	semidirect
2. #20297290	C₂	semidirect
3. #20297278	C₂	semidirect
4. #20297275	C₂	semidirect
5. C₄×D₈₉₂	C₂	semidirect
6. #20297263	C₂	semidirect
7. #20297262	C₂	semidirect
8. C₄×D₄×C₂₂₃	C₂	semidirect
57. C₂₂₃	#555	semidirect
64. C₄×D₄	D₂₂₃	semidirect

H ≅ G/N

type

1. #20297304

C₂

semidirect

2. #20297290

C₂

semidirect

3. #20297278

C₂

semidirect

4. #20297275

C₂

semidirect

5. C₄×D₈₉₂

C₂

semidirect

6. #20297263

C₂

semidirect

7. #20297262

C₂

semidirect

8. C₄×D₄×C₂₂₃

C₂

semidirect

57. C₂₂₃

#555

semidirect

64. C₄×D₄

D₂₂₃

semidirect

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

subgroup	index
#20297304	2
#20297290	2
#20297278	2
#20297275	2
C₄×D₈₉₂	2
#20297263	2
#20297262	2
C₄×D₄×C₂₂₃	2
#555	223

subgroup

index

C₄×D₈₉₂

C₄×D₄×C₂₂₃

223

Subgroups — #28384295

Conjugacy classes of subgroups

Complements

Maximal subgroups