Group #28417150 — subgroups

Subgroups — #28417150

Order 14912.

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	#21418535	C₂	1	1	yes	maximal, characteristic, complemented
2	#21418521	C₂	1	4	yes	maximal, complemented
3	#21418509	C₂	1	2	yes	maximal, complemented
4	#21418506	C₂	1	2	yes	maximal, complemented
5	C₄×D₉₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#21418494	C₂	1	2	yes	maximal, complemented
7	#21418493	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₄₆₆	C₂²	1	2	yes	—
10	C₂×D₉₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₂₃₃	C₂²	1	4	yes	—
12	C₂×C₄×D₂₃₃	C₂²	1	2	yes	—
13	C₄⋊Dic₂₃₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₂₃₃	C₂²	1	2	yes	—
15	C₄⋊Dic₂₃₃	C₂²	1	2	yes	—
16	C₂²⋊Dic₂₃₃	C₂²	1	4	yes	—
17	#16836571	C₂²	1	2	yes	—
18	#16836571	C₂²	1	4	yes	—
19	C₄×Dic₂₃₃	C₂²	1	4	yes	—
20	C₂×D₄×C₂₃₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₂₃₃	C₂²	1	2	yes	—
22	C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
24	C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
25	C₂²×D₂₃₃	C₂³	1	2	yes	—
26	Dic₄₆₆	—	4	2	no	—
27	D₉₃₂	—	2	2	no	—
28	C₂²⋊D₂₃₃	—	4	4	no	—
29	C₂×Dic₂₃₃	C₂³	1	2	yes	—
30	C₂×Dic₂₃₃	C₂³	1	4	yes	—
31	C₄×D₂₃₃	—	2	4	no	—
32	C₂³×C₂₃₃	C₂³	1	2	yes	—
33	D₄×C₂₃₃	—	2	2	no	—
34	C₂×C₄×C₂₃₃	—	2	4	no	—
35	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₂₃₃	C₂³	1	2	yes	—
37	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
39	C₂²×C₂₃₃	—	2	2	no	—
40	C₂²×C₂₃₃	—	2	2	no	—
41	C₂²×C₂₃₃	—	2	2	no	—
42	C₂²×C₂₃₃	C₂⁴	1	1	yes	characteristic
43	C₂×D₂₃₃	—	2	2	no	—
44	C₂×D₂₃₃	—	2	2	no	—
45	C₂×D₂₃₃	—	2	2	no	—
46	C₄×C₂₃₃	—	2	2	no	—
47	C₄×C₂₃₃	—	2	1	no	—
48	C₄×C₂₃₃	#41	1	2	yes	—
49	C₄×C₂₃₃	#41	1	2	yes	—
50	Dic₂₃₃	—	2	2	no	—
51	Dic₂₃₃	—	2	4	no	—
52	C₂×C₂₃₃	—	4	2	no	—
53	C₂×C₂₃₃	C₂×#41	1	1	yes	characteristic
54	C₂×C₂₃₃	C₂×#41	1	1	yes	characteristic
55	C₂×C₂₃₃	H(5,𝔽₂)	1	1	yes	characteristic
56	D₂₃₃	—	4	2	no	—
57	C₂₃₃	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	233	1	no	maximal, Sylow
59	C₄⋊Q₈	—	233	1	no	—
60	#124	—	233	4	no	—
61	#121	—	233	2	no	—
62	#121	—	233	2	no	—
63	C₄×D₄	—	233	1	no	—
64	C₄×D₄	D₂₃₃	1	1	yes	characteristic, complemented
65	#117	—	233	2	no	—
66	#117	—	233	2	no	—
67	C₂×Q₈	—	233	2	no	—
68	C₂×D₄	—	233	4	no	—
69	C₂×D₄	—	233	1	no	—
70	C₂×D₄	C₂×D₂₃₃	1	1	yes	characteristic
71	C₂²×C₄	—	233	2	no	—
72	C₂²×C₄	C₂×D₂₃₃	1	2	yes	—
73	C₄⋊C₄	—	233	1	no	—
74	C₄⋊C₄	—	233	2	no	—
75	C₄⋊C₄	—	233	2	no	—
76	C₄⋊C₄	C₂×D₂₃₃	1	1	yes	characteristic
77	C₂²⋊C₄	—	233	4	no	—
78	C₂²⋊C₄	—	233	4	no	—
79	C₂²⋊C₄	—	233	2	no	—
80	C₂²⋊C₄	C₂×D₂₃₃	1	2	yes	—
81	C₄²	—	233	4	no	—
82	C₄²	C₂×D₂₃₃	1	1	yes	characteristic
83	C₂³	—	233	2	no	—
84	C₂³	C₂²×D₂₃₃	1	2	yes	—
85	Q₈	—	932	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	932	4	no	—
88	D₄	—	466	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	466	4	no	—
91	C₂×C₄	—	233	2	no	—
92	C₂×C₄	—	233	4	no	—
93	C₂×C₄	C₂²×D₂₃₃	1	2	yes	—
94	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	466	2	no	—
101	C₂²	—	466	2	no	—
102	C₂²	—	466	2	no	—
103	C₂²	C₂³×D₂₃₃	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	466	2	no	—
107	C₄	—	466	4	no	—
108	C₄	#16836593	1	2	yes	—
109	C₄	#16836594	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	932	2	no	—
112	C₂	C₂×#16836593	1	1	yes	characteristic
113	C₂	C₂×#16836594	1	1	yes	characteristic
114	C₂	#21418581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21418535

C₂

yes

maximal, characteristic, complemented

#21418521

C₂

yes

maximal, complemented

#21418509

C₂

yes

maximal, complemented

#21418506

C₂

yes

maximal, complemented

C₄×D₉₃₂

C₂

yes

maximal, characteristic, complemented

#21418494

C₂

yes

maximal, complemented

#21418493

C₂

yes

maximal, complemented

C₄×D₄×C₂₃₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₆₆

C₂²

yes

—

C₂×D₉₃₂

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃₃

C₂²

yes

—

C₂×C₄×D₂₃₃

C₂²

yes

—

C₄⋊Dic₂₃₃

C₂²

yes

characteristic

C₄⋊Dic₂₃₃

C₂²

yes

—

C₄⋊Dic₂₃₃

C₂²

yes

—

C₂²⋊Dic₂₃₃

C₂²

yes

—

#16836571

C₂²

yes

—

#16836571

C₂²

yes

—

C₄×Dic₂₃₃

C₂²

yes

—

C₂×D₄×C₂₃₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃₃

C₂²

yes

—

C₄⋊C₄×C₂₃₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃₃

C₂²

yes

—

C₄²×C₂₃₃

C₂²

yes

characteristic

C₂²×D₂₃₃

C₂³

yes

—

Dic₄₆₆

—

D₉₃₂

—

C₂²⋊D₂₃₃

—

C₂×Dic₂₃₃

C₂³

yes

—

C₂×Dic₂₃₃

C₂³

yes

—

C₄×D₂₃₃

—

C₂³×C₂₃₃

C₂³

yes

—

D₄×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

C₂³

yes

characteristic

C₂×C₄×C₂₃₃

C₂³

yes

—

C₂×C₄×C₂₃₃

C₂³

yes

characteristic

C₂×C₄×C₂₃₃

C₂³

yes

characteristic

C₂²×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

C₂⁴

yes

characteristic

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

#41

yes

—

C₄×C₂₃₃

#41

yes

—

Dic₂₃₃

—

Dic₂₃₃

—

C₂×C₂₃₃

—

C₂×C₂₃₃

C₂×#41

yes

characteristic

C₂×C₂₃₃

C₂×#41

yes

characteristic

C₂×C₂₃₃

H(5,𝔽₂)

yes

characteristic

D₂₃₃

—

C₂₃₃

#555

yes

characteristic, complemented, Sylow

#555

—

233

maximal, Sylow

C₄⋊Q₈

—

233

—

#124

—

233

—

#121

—

233

—

#121

—

233

—

C₄×D₄

—

233

—

C₄×D₄

D₂₃₃

yes

characteristic, complemented

#117

—

233

—

#117

—

233

—

C₂×Q₈

—

233

—

C₂×D₄

—

233

—

C₂×D₄

—

233

—

C₂×D₄

C₂×D₂₃₃

yes

characteristic

C₂²×C₄

—

233

—

C₂²×C₄

C₂×D₂₃₃

yes

—

C₄⋊C₄

—

233

—

C₄⋊C₄

—

233

—

C₄⋊C₄

—

233

—

C₄⋊C₄

C₂×D₂₃₃

yes

characteristic

C₂²⋊C₄

—

233

—

C₂²⋊C₄

—

233

—

C₂²⋊C₄

—

233

—

C₂²⋊C₄

C₂×D₂₃₃

yes

—

C₄²

—

233

—

C₄²

C₂×D₂₃₃

yes

characteristic

C₂³

—

233

—

C₂³

C₂²×D₂₃₃

yes

—

Q₈

—

932

—

D₄

—

D₄

—

932

—

D₄

—

466

—

C₂×C₄

—

C₂×C₄

—

466

—

C₂×C₄

—

233

—

C₂×C₄

—

233

—

C₂×C₄

C₂²×D₂₃₃

yes

—

C₂×C₄

C₂²×D₂₃₃

yes

characteristic

C₂×C₄

C₂²×D₂₃₃

yes

characteristic

C₂×C₄

C₂²×D₂₃₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

466

—

101

C₂²

—

466

—

102

C₂²

—

466

—

103

C₂²

C₂³×D₂₃₃

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

466

—

107

C₄

—

466

—

108

C₄

#16836593

yes

—

109

C₄

#16836594

yes

—

110

C₂

—

111

C₂

—

932

—

112

C₂

C₂×#16836593

yes

characteristic

113

C₂

C₂×#16836594

yes

characteristic

114

C₂

#21418581

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₃
commutator, socle, cosocle	C₂²×C₂₃₃	C₂⁴
Fitting	C₄×D₄×C₂₃₃	C₂
hypercenter	C₄×D₄	D₂₃₃
hypocenter	C₂₃₃	#555
LCS	G ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃₃

commutator, socle, cosocle

C₂²×C₂₃₃

C₂⁴

Fitting

C₄×D₄×C₂₃₃

C₂

hypercenter

C₄×D₄

D₂₃₃

C₂₃₃

G ▷ C₂²×C₂₃₃ ▷ C₂₃₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #21418535	C₂	semidirect
2. #21418521	C₂	semidirect
3. #21418509	C₂	semidirect
4. #21418506	C₂	semidirect
5. C₄×D₉₃₂	C₂	semidirect
6. #21418494	C₂	semidirect
7. #21418493	C₂	semidirect
8. C₄×D₄×C₂₃₃	C₂	semidirect
57. C₂₃₃	#555	semidirect
64. C₄×D₄	D₂₃₃	semidirect

H ≅ G/N

type

1. #21418535

C₂

semidirect

2. #21418521

C₂

semidirect

3. #21418509

C₂

semidirect

4. #21418506

C₂

semidirect

5. C₄×D₉₃₂

C₂

semidirect

6. #21418494

C₂

semidirect

7. #21418493

C₂

semidirect

8. C₄×D₄×C₂₃₃

C₂

semidirect

57. C₂₃₃

#555

semidirect

64. C₄×D₄

D₂₃₃

semidirect

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

subgroup	index
#21418535	2
#21418521	2
#21418509	2
#21418506	2
C₄×D₉₃₂	2
#21418494	2
#21418493	2
C₄×D₄×C₂₃₃	2
#555	233

subgroup

index

C₄×D₉₃₂

C₄×D₄×C₂₃₃

233

Subgroups — #28417150

Conjugacy classes of subgroups

Complements

Maximal subgroups