Group #28384287 — subgroups

Subgroups — #28384287

Order 14272.

228 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20297283	C₂	1	2	yes	maximal, complemented
4	#20297281	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₄₄₆	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₂₂₃	C₂	1	2	yes	maximal
7	C₂×C₄⋊Dic₂₂₃	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₄₆	C₂²	1	2	yes	—
11	C₂²×Dic₂₂₃	C₂²	1	4	yes	—
12	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₂₂₃	C₂²	1	4	yes	—
14	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
15	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
16	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
17	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
18	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	—
19	C₄×Dic₂₂₃	C₂²	1	2	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	Dic₄₄₆	—	2	2	no	—
26	Dic₄₄₆	—	4	2	no	—
27	C₂×Dic₂₂₃	—	4	4	no	—
28	C₂×Dic₂₂₃	C₂³	1	2	yes	—
29	C₂×Dic₂₂₃	C₂³	1	2	yes	—
30	C₂×Dic₂₂₃	C₂³	1	4	yes	—
31	C₂³×C₂₂₃	C₂³	1	2	yes	—
32	D₄×C₂₂₃	Q₈	1	4	yes	complemented
33	C₂×C₄×C₂₂₃	—	2	4	no	—
34	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
36	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₂₂₃	—	2	2	no	—
39	C₂²×C₂₂₃	—	2	2	no	—
40	C₂²×C₂₂₃	C₂×Q₈	1	4	yes	—
41	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
42	C₄×C₂₂₃	—	2	2	no	—
43	C₄×C₂₂₃	—	2	1	no	—
44	C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
45	C₄×C₂₂₃	#41	1	2	yes	—
46	Dic₂₂₃	—	2	2	no	—
47	Dic₂₂₃	—	2	2	no	—
48	Dic₂₂₃	—	2	4	no	—
49	C₂×C₂₂₃	—	2	4	no	—
50	C₂×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
52	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂₂₃	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	223	1	no	maximal, Sylow
55	C₄⋊Q₈	—	223	1	no	—
56	#125	—	223	2	no	—
57	C₂²⋊Q₈	—	223	2	no	—
58	C₂²⋊Q₈	—	223	4	no	—
59	C₄×Q₈	—	223	1	no	—
60	C₄×D₄	—	223	2	no	—
61	C₄×D₄	D₂₂₃	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	223	2	no	—
63	C₂×Q₈	—	223	2	no	—
64	C₂×Q₈	—	223	1	no	—
65	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
66	C₂²×C₄	—	223	4	no	—
67	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
68	C₄⋊C₄	—	223	2	no	—
69	C₄⋊C₄	—	223	4	no	—
70	C₄⋊C₄	—	223	1	no	—
71	C₄⋊C₄	—	223	4	no	—
72	C₄⋊C₄	—	223	2	no	—
73	C₄⋊C₄	—	223	2	no	—
74	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
75	C₂²⋊C₄	—	223	4	no	—
76	C₂²⋊C₄	C₂×D₂₂₃	1	2	yes	—
77	C₄²	—	223	2	no	—
78	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
79	C₂³	C₂²×D₂₂₃	1	2	yes	—
80	Q₈	—	892	2	no	—
81	Q₈	—	446	2	no	—
82	D₄	Dic₄₄₆	1	4	yes	complemented
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	892	4	no	—
85	C₂×C₄	—	223	2	no	—
86	C₂×C₄	—	223	2	no	—
87	C₂×C₄	—	223	4	no	—
88	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
90	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	—	2	2	no	—
94	C₂²	C₂×Dic₄₄₆	1	4	yes	—
95	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	1	no	—
98	C₄	—	446	2	no	—
99	C₄	—	446	2	no	—
100	C₄	—	446	4	no	—
101	C₄	C₂×Dic₄₄₆	1	2	yes	—
102	C₄	#16813169	1	2	yes	—
103	C₂	—	2	4	no	—
104	C₂	C₂²×Dic₄₄₆	1	1	yes	characteristic
105	C₂	C₂×#16813169	1	1	yes	characteristic
106	C₂	#20297350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20297304

C₂

yes

maximal, characteristic, complemented

#20297295

C₂

yes

maximal, complemented

#20297283

C₂

yes

maximal, complemented

#20297281

C₂

yes

maximal, complemented

C₄×Dic₄₄₆

C₂

yes

maximal, characteristic, complemented

D₄×Dic₂₂₃

C₂

yes

maximal

C₂×C₄⋊Dic₂₂₃

C₂

yes

maximal, complemented

C₄×D₄×C₂₂₃

C₂

yes

maximal, characteristic

C₂×Dic₄₄₆

C₂²

yes

characteristic

C₂×Dic₄₄₆

C₂²

yes

—

C₂²×Dic₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

C₂²⋊Dic₂₂₃

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

Dic₄₄₆

—

Dic₄₄₆

—

C₂×Dic₂₂₃

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂³×C₂₂₃

C₂³

yes

—

D₄×C₂₂₃

Q₈

yes

complemented

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

C₂×Q₈

yes

—

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×Q₈

yes

—

C₄×C₂₂₃

#41

yes

—

Dic₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

C₂²×Q₈

yes

characteristic

C₂×C₂₂₃

C₂×#41

yes

C₂×C₂₂₃

yes

C₂₂₃

yes

characteristic, complemented, Sylow

#554

—

223

maximal, Sylow

C₄⋊Q₈

—

223

—

#125

—

223

—

C₂²⋊Q₈

—

223

—

C₂²⋊Q₈

—

223

—

C₄×Q₈

—

223

—

C₄×D₄

—

223

—

C₄×D₄

D₂₂₃

yes

characteristic

C₂×C₄⋊C₄

—

223

—

C₂×Q₈

—

223

—

C₂×Q₈

—

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

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

C₂×D₂₂₃

yes

characteristic

C₂²⋊C₄

—

223

—

C₂²⋊C₄

C₂×D₂₂₃

yes

—

C₄²

—

223

—

C₄²

C₂×D₂₂₃

yes

characteristic

C₂³

C₂²×D₂₂₃

yes

—

Q₈

—

892

—

Q₈

—

446

—

D₄

Dic₄₄₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

892

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

C₂×C₄

C₂²×D₂₂₃

yes

—

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₄₄₆

yes

—

C₂²

C₂³×D₂₂₃

yes

characteristic

C₄

—

C₄

—

C₄

—

446

—

C₄

—

446

—

100

C₄

—

446

—

101

C₄

C₂×Dic₄₄₆

yes

—

102

C₄

#16813169

yes

—

103

C₂

—

104

C₂

C₂²×Dic₄₄₆

yes

characteristic

105

C₂

C₂×#16813169

yes

characteristic

106

C₂

#20297350

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	C₄×D₄×C₂₂₃	C₂
prime radical	C₂×D₄×C₂₂₃	C₂²
hypercenter	C₄×D₄	D₂₂₃
hypocenter	C₂₂₃	#554
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

prime radical

C₂×D₄×C₂₂₃

C₂²

hypercenter

C₄×D₄

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #20297304	C₂	semidirect
2. #20297295	C₂	semidirect
3. #20297283	C₂	semidirect
4. #20297281	C₂	semidirect
5. C₄×Dic₄₄₆	C₂	semidirect
7. C₂×C₄⋊Dic₂₂₃	C₂	semidirect
32. D₄×C₂₂₃	Q₈	semidirect
53. C₂₂₃	#554	semidirect
82. D₄	Dic₄₄₆	semidirect

H ≅ G/N

type

1. #20297304

C₂

semidirect

2. #20297295

C₂

semidirect

3. #20297283

C₂

semidirect

4. #20297281

C₂

semidirect

5. C₄×Dic₄₄₆

C₂

semidirect

7. C₂×C₄⋊Dic₂₂₃

C₂

semidirect

32. D₄×C₂₂₃

Q₈

semidirect

53. C₂₂₃

#554

semidirect

82. D₄

Dic₄₄₆

semidirect

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

subgroup	index
#20297304	2
#20297295	2
#20297283	2
#20297281	2
C₄×Dic₄₄₆	2
D₄×Dic₂₂₃	2
C₂×C₄⋊Dic₂₂₃	2
C₄×D₄×C₂₂₃	2
#554	223

subgroup

index

C₄×Dic₄₄₆

D₄×Dic₂₂₃

C₂×C₄⋊Dic₂₂₃

C₄×D₄×C₂₂₃

#554

223

Subgroups — #28384287

Conjugacy classes of subgroups

Complements

Maximal subgroups