Group #28384154 — subgroups

Subgroups — #28384154

Order 14272.

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

normal subgroups only

This group has 270 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	C₂×#16813170	C₂	1	1	yes	maximal, characteristic
2	#20297302	C₂	1	2	yes	maximal, complemented
3	#20297293	C₂	1	2	yes	maximal, complemented
4	#20297287	C₂	1	4	yes	maximal, complemented
5	#20297280	C₂	1	4	yes	maximal, complemented
6	#20297263	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16813170	—	2	4	no	—
9	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
10	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₂₂₃	C₂²	1	2	yes	—
12	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
13	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
14	Dic₂₂₃⋊C₄	C₂²	1	8	yes	—
15	C₂²⋊Dic₂₂₃	C₂²	1	2	yes	—
16	(C₂×D₂₂₃)⋊C₄	C₂²	1	8	yes	—
17	C₄×Dic₂₂₃	C₂²	1	2	yes	—
18	C₂×Q₈×C₂₂₃	—	2	4	no	—
19	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
20	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
21	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
22	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
23	C₂²×D₂₂₃	C₂³	1	2	yes	—
24	Dic₄₄₆	—	2	2	no	—
25	D₈₉₂	—	2	2	no	—
26	C₂²⋊D₂₂₃	—	2	4	no	—
27	C₂×Dic₂₂₃	C₂³	1	4	yes	—
28	C₂×Dic₂₂₃	C₂³	1	2	yes	—
29	C₄×D₂₂₃	—	2	4	no	—
30	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
31	Q₈×C₂₂₃	—	2	4	no	—
32	Q₈×C₂₂₃	—	2	4	no	—
33	C₂×C₄×C₂₂₃	—	2	4	no	—
34	C₂×C₄×C₂₂₃	D₄	1	4	yes	—
35	C₂×C₄×C₂₂₃	C₂³	1	4	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	C₂×D₄	1	2	yes	—
40	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
41	C₂×D₂₂₃	—	2	2	no	—
42	C₂×D₂₂₃	—	2	4	no	—
43	C₄×C₂₂₃	—	2	4	no	—
44	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
45	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
46	Dic₂₂₃	—	2	2	no	—
47	Dic₂₂₃	—	2	4	no	—
48	C₂×C₂₂₃	—	2	2	no	—
49	C₂×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
51	D₂₂₃	—	4	2	no	—
52	C₂₂₃	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	223	1	no	maximal, Sylow
54	C₂×#41	—	223	1	no	—
55	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	223	2	no	—
57	#124	—	223	2	no	—
58	#123	—	223	4	no	—
59	C₂²⋊Q₈	—	223	4	no	—
60	#117	—	223	1	no	—
61	#41	—	446	4	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	223	1	no	—
64	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
65	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
66	C₂×D₄	—	223	2	no	—
67	C₂×D₄	—	223	1	no	—
68	C₂²×C₄	—	223	2	no	—
69	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
70	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
71	C₄⋊C₄	—	223	8	no	—
72	C₄⋊C₄	—	223	2	no	—
73	C₂²⋊C₄	—	223	2	no	—
74	C₂²⋊C₄	—	223	8	no	—
75	C₄²	—	223	2	no	—
76	C₂³	—	223	2	no	—
77	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
78	Q₈	—	2	4	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	446	2	no	—
81	D₄	—	446	4	no	—
82	D₄	—	446	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	446	4	no	—
85	C₂×C₄	—	223	4	no	—
86	C₂×C₄	—	223	2	no	—
87	C₂×C₄	C₂²⋊D₂₂₃	1	4	yes	—
88	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
89	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	446	2	no	—
93	C₂²	—	446	4	no	—
94	C₂²	C₂×C₂²⋊D₂₂₃	1	2	yes	—
95	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	446	2	no	—
98	C₄	—	446	4	no	—
99	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
100	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	892	2	no	—
103	C₂	C₂²×C₂²⋊D₂₂₃	1	1	yes	characteristic
104	C₂	#20297352	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813170

C₂

yes

maximal, characteristic

#20297302

C₂

yes

maximal, complemented

#20297293

C₂

yes

maximal, complemented

#20297287

C₂

yes

maximal, complemented

#20297280

C₂

yes

maximal, complemented

#20297263

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₂₂₃

C₂

yes

maximal, characteristic, complemented

#16813170

—

C₂×Dic₄₄₆

C₂²

yes

characteristic

C₂×D₈₉₂

C₂²

yes

characteristic

C₂×C₂²⋊D₂₂₃

C₂²

yes

—

C₂×C₄×D₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

C₂²⋊Dic₂₂₃

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

C₄×Dic₂₂₃

C₂²

yes

—

C₂×Q₈×C₂₂₃

—

C₂×Q₈×C₂₂₃

C₂²

yes

—

C₂×Q₈×C₂₂₃

C₂²

yes

—

C₂²×C₄×C₂₂₃

C₂²

yes

—

C₂²×C₄×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

characteristic

Q₈×C₂₂₃

—

Q₈×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

D₄

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂×D₄

yes

—

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×D₄

yes

—

C₄×C₂₂₃

C₂×D₄

yes

—

Dic₂₂₃

—

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₂₃

2₋¹⁺⁴

yes

—

D₂₂₃

—

C₂₂₃

#536

yes

characteristic, complemented, Sylow

#536

—

223

maximal, Sylow

C₂×#41

—

223

—

C₂²×Q₈

D₂₂₃

yes

characteristic, complemented

C₄⋊Q₈

—

223

—

#124

—

223

—

#123

—

223

—

C₂²⋊Q₈

—

223

—

#117

—

223

—

#41

—

446

—

C₂×Q₈

—

C₂×Q₈

—

223

—

C₂×Q₈

C₂×D₂₂₃

yes

—

C₂×Q₈

C₂×D₂₂₃

yes

—

C₂×D₄

—

223

—

C₂×D₄

—

223

—

C₂²×C₄

—

223

—

C₂²×C₄

C₂×D₂₂₃

yes

—

C₂²×C₄

C₂×D₂₂₃

yes

characteristic

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₄²

—

223

—

C₂³

—

223

—

C₂³

C₂²×D₂₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

446

—

D₄

—

446

—

D₄

—

446

—

C₂×C₄

—

C₂×C₄

—

446

—

C₂×C₄

—

223

—

C₂×C₄

—

223

—

C₂×C₄

C₂²⋊D₂₂₃

yes

—

C₂×C₄

C₂²×D₂₂₃

yes

—

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

C₂²

—

C₂²

—

446

—

C₂²

—

446

—

C₂²

C₂×C₂²⋊D₂₂₃

yes

—

C₂²

C₂³×D₂₂₃

yes

characteristic

C₄

—

C₄

—

446

—

C₄

—

446

—

C₄

C₂×C₂²⋊D₂₂₃

yes

—

100

C₄

C₂×C₂²⋊D₂₂₃

yes

—

101

C₂

—

102

C₂

—

892

—

103

C₂

C₂²×C₂²⋊D₂₂₃

yes

characteristic

104

C₂

#20297352

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	C₂²×Q₈×C₂₂₃	C₂
prime radical	C₂×#16813170	C₂
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	#536
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₂₃

commutator, socle, cosocle

C₂²×C₂₂₃

C₂⁴

Fitting

C₂²×Q₈×C₂₂₃

C₂

prime radical

C₂×#16813170

C₂

hypercenter

C₂²×Q₈

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #20297302	C₂	semidirect
3. #20297293	C₂	semidirect
4. #20297287	C₂	semidirect
5. #20297280	C₂	semidirect
6. #20297263	C₂	semidirect
7. C₂²×Q₈×C₂₂₃	C₂	semidirect
52. C₂₂₃	#536	semidirect
55. C₂²×Q₈	D₂₂₃	semidirect

H ≅ G/N

type

2. #20297302

C₂

semidirect

3. #20297293

C₂

semidirect

4. #20297287

C₂

semidirect

5. #20297280

C₂

semidirect

6. #20297263

C₂

semidirect

7. C₂²×Q₈×C₂₂₃

C₂

semidirect

52. C₂₂₃

#536

semidirect

55. C₂²×Q₈

D₂₂₃

semidirect

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

subgroup	index
C₂×#16813170	2
#20297302	2
#20297293	2
#20297287	2
#20297280	2
#20297263	2
C₂²×Q₈×C₂₂₃	2
#536	223

subgroup

index

C₂²×Q₈×C₂₂₃

223

Subgroups — #28384154

Conjugacy classes of subgroups

Complements

Maximal subgroups