Group #28417106 — subgroups

Subgroups — #28417106

Order 14912.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16836594	C₂	1	2	yes	maximal, complemented
2	#21418532	C₂	1	1	yes	maximal, characteristic, complemented
3	#21418531	C₂	1	2	yes	maximal, complemented
4	#21418509	C₂	1	2	yes	maximal, complemented
5	#21418508	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₄₆₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#21418497	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#16836594	—	2	8	no	—
10	C₂×Dic₄₆₆	C₂²	1	1	yes	characteristic
11	C₂×D₉₃₂	C₂²	1	2	yes	—
12	C₂×D₉₃₂	C₂²	1	2	yes	—
13	C₂×D₉₃₂	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₂₃₃	C₂²	1	4	yes	—
15	C₂×C₂²⋊D₂₃₃	C₂²	1	4	yes	—
16	C₂×C₄×D₂₃₃	C₂²	1	4	yes	—
17	C₄⋊Dic₂₃₃	C₂²	1	1	yes	characteristic
18	#16836573	C₂²	1	2	yes	—
19	#16836571	C₂²	1	4	yes	—
20	C₄×Dic₂₃₃	C₂²	1	2	yes	—
21	C₂×D₄×C₂₃₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₂₃₃	C₂²	1	2	yes	—
23	C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
25	C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
26	C₂²×D₂₃₃	C₂³	1	4	yes	—
27	Dic₄₆₆	D₄	1	4	yes	complemented
28	D₉₃₂	—	2	4	no	—
29	D₉₃₂	—	2	4	no	—
30	D₉₃₂	—	4	2	no	—
31	C₂²⋊D₂₃₃	—	2	8	no	—
32	C₂²⋊D₂₃₃	—	4	4	no	—
33	C₂×Dic₂₃₃	C₂³	1	2	yes	—
34	C₂×Dic₂₃₃	C₂³	1	2	yes	—
35	C₄×D₂₃₃	—	2	8	no	—
36	C₂³×C₂₃₃	C₂³	1	2	yes	—
37	D₄×C₂₃₃	—	2	2	no	—
38	C₂×C₄×C₂₃₃	—	2	4	no	—
39	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₃₃	C₂³	1	2	yes	—
41	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
43	C₂²×C₂₃₃	—	2	2	no	—
44	C₂²×C₂₃₃	—	2	2	no	—
45	C₂²×C₂₃₃	—	2	2	no	—
46	C₂²×C₂₃₃	C₂⁴	1	1	yes	characteristic
47	C₂×D₂₃₃	—	2	4	no	—
48	C₂×D₂₃₃	—	2	4	no	—
49	C₂×D₂₃₃	—	2	4	no	—
50	C₄×C₂₃₃	—	2	2	no	—
51	C₄×C₂₃₃	—	2	1	no	—
52	C₄×C₂₃₃	C₂×D₄	1	2	yes	—
53	C₄×C₂₃₃	#41	1	2	yes	complemented
54	Dic₂₃₃	—	2	2	no	—
55	Dic₂₃₃	C₂×D₄	1	4	yes	—
56	C₂×C₂₃₃	—	4	2	no	—
57	C₂×C₂₃₃	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₂₃₃	C₂×#41	1	1	yes	characteristic
59	C₂×C₂₃₃	H(5,𝔽₂)	1	1	yes	characteristic
60	D₂₃₃	—	4	4	no	—
61	C₂₃₃	#550	1	1	yes	characteristic, complemented, Sylow
62	#550	—	233	1	no	maximal, Sylow
63	C₂×#41	—	233	2	no	—
64	#127	—	233	2	no	—
65	#127	—	233	1	no	—
66	#121	—	233	2	no	—
67	#121	—	233	4	no	—
68	C₄×Q₈	—	233	1	no	—
69	C₄×D₄	—	233	2	no	—
70	C₄×D₄	D₂₃₃	1	1	yes	characteristic, complemented
71	#41	—	466	8	no	—
72	C₂×Q₈	—	233	1	no	—
73	C₂×D₄	—	233	4	no	—
74	C₂×D₄	—	233	4	no	—
75	C₂×D₄	—	233	2	no	—
76	C₂×D₄	—	233	2	no	—
77	C₂×D₄	—	233	2	no	—
78	C₂×D₄	C₂×D₂₃₃	1	1	yes	characteristic
79	C₂²×C₄	—	233	4	no	—
80	C₂²×C₄	C₂×D₂₃₃	1	2	yes	—
81	C₄⋊C₄	—	233	2	no	—
82	C₄⋊C₄	—	233	1	no	—
83	C₄⋊C₄	C₂×D₂₃₃	1	1	yes	characteristic
84	C₂²⋊C₄	—	233	4	no	—
85	C₂²⋊C₄	C₂×D₂₃₃	1	2	yes	—
86	C₄²	—	233	2	no	—
87	C₄²	C₂×D₂₃₃	1	1	yes	characteristic
88	C₂³	—	233	4	no	—
89	C₂³	C₂²×D₂₃₃	1	2	yes	—
90	Q₈	—	233	4	no	—
91	D₄	—	2	2	no	—
92	D₄	—	932	2	no	—
93	D₄	—	932	4	no	—
94	D₄	—	466	8	no	—
95	D₄	—	466	4	no	—
96	D₄	—	466	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	466	8	no	—
99	C₂×C₄	—	233	2	no	—
100	C₂×C₄	—	233	2	no	—
101	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₂₃₃	1	2	yes	—
103	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	466	4	no	—
109	C₂²	—	466	4	no	—
110	C₂²	—	466	4	no	—
111	C₂²	C₂³×D₂₃₃	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	466	2	no	—
115	C₄	—	233	4	no	—
116	C₄	D₄×D₂₃₃	1	2	yes	—
117	C₄	#16836594	1	2	yes	complemented
118	C₂	—	4	2	no	—
119	C₂	—	932	4	no	—
120	C₂	C₂×D₄×D₂₃₃	1	1	yes	characteristic
121	C₂	C₂×#16836594	1	1	yes	characteristic
122	C₂	#21418581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16836594

C₂

yes

maximal, complemented

#21418532

C₂

yes

maximal, characteristic, complemented

#21418531

C₂

yes

maximal, complemented

#21418509

C₂

yes

maximal, complemented

#21418508

C₂

yes

maximal, complemented

C₄×Dic₄₆₆

C₂

yes

maximal, characteristic, complemented

#21418497

C₂

yes

maximal, complemented

C₄×D₄×C₂₃₃

C₂

yes

maximal, characteristic, complemented

#16836594

—

C₂×Dic₄₆₆

C₂²

yes

characteristic

C₂×D₉₃₂

C₂²

yes

—

C₂×D₉₃₂

C₂²

yes

—

C₂×D₉₃₂

C₂²

yes

—

C₂×C₂²⋊D₂₃₃

C₂²

yes

—

C₂×C₂²⋊D₂₃₃

C₂²

yes

—

C₂×C₄×D₂₃₃

C₂²

yes

—

C₄⋊Dic₂₃₃

C₂²

yes

characteristic

#16836573

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

D₄

yes

complemented

D₉₃₂

—

D₉₃₂

—

D₉₃₂

—

C₂²⋊D₂₃₃

—

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

C₂×D₄

yes

—

C₄×C₂₃₃

#41

yes

complemented

Dic₂₃₃

—

Dic₂₃₃

C₂×D₄

yes

—

C₂×C₂₃₃

—

C₂×C₂₃₃

C₂²×D₄

yes

characteristic

C₂×C₂₃₃

C₂×#41

yes

characteristic

C₂×C₂₃₃

H(5,𝔽₂)

yes

characteristic

D₂₃₃

—

C₂₃₃

#550

yes

characteristic, complemented, Sylow

#550

—

233

maximal, Sylow

C₂×#41

—

233

—

#127

—

233

—

#127

—

233

—

#121

—

233

—

#121

—

233

—

C₄×Q₈

—

233

—

C₄×D₄

—

233

—

C₄×D₄

D₂₃₃

yes

characteristic, complemented

#41

—

466

—

C₂×Q₈

—

233

—

C₂×D₄

—

233

—

C₂×D₄

—

233

—

C₂×D₄

—

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

C₂×D₂₃₃

yes

characteristic

C₂²⋊C₄

—

233

—

C₂²⋊C₄

C₂×D₂₃₃

yes

—

C₄²

—

233

—

C₄²

C₂×D₂₃₃

yes

characteristic

C₂³

—

233

—

C₂³

C₂²×D₂₃₃

yes

—

Q₈

—

233

—

D₄

—

D₄

—

932

—

D₄

—

932

—

D₄

—

466

—

D₄

—

466

—

D₄

—

466

—

C₂×C₄

—

C₂×C₄

—

466

—

C₂×C₄

—

233

—

100

C₂×C₄

—

233

—

101

C₂×C₄

C₂²×D₂₃₃

yes

characteristic

102

C₂×C₄

C₂²×D₂₃₃

yes

—

103

C₂×C₄

C₂²×D₂₃₃

yes

characteristic

104

C₂×C₄

C₂²×D₂₃₃

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

466

—

109

C₂²

—

466

—

110

C₂²

—

466

—

111

C₂²

C₂³×D₂₃₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

466

—

115

C₄

—

233

—

116

C₄

D₄×D₂₃₃

yes

—

117

C₄

#16836594

yes

complemented

118

C₂

—

119

C₂

—

932

—

120

C₂

C₂×D₄×D₂₃₃

yes

characteristic

121

C₂

C₂×#16836594

yes

characteristic

122

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	#550
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. C₂×#16836594	C₂	semidirect
2. #21418532	C₂	semidirect
3. #21418531	C₂	semidirect
4. #21418509	C₂	semidirect
5. #21418508	C₂	semidirect
6. C₄×Dic₄₆₆	C₂	semidirect
7. #21418497	C₂	semidirect
8. C₄×D₄×C₂₃₃	C₂	semidirect
27. Dic₄₆₆	D₄	semidirect
53. C₄×C₂₃₃	#41	semidirect
61. C₂₃₃	#550	semidirect
70. C₄×D₄	D₂₃₃	semidirect
117. C₄	#16836594	semidirect

H ≅ G/N

type

1. C₂×#16836594

C₂

semidirect

2. #21418532

C₂

semidirect

3. #21418531

C₂

semidirect

4. #21418509

C₂

semidirect

5. #21418508

C₂

semidirect

6. C₄×Dic₄₆₆

C₂

semidirect

7. #21418497

C₂

semidirect

8. C₄×D₄×C₂₃₃

C₂

semidirect

27. Dic₄₆₆

D₄

semidirect

53. C₄×C₂₃₃

#41

semidirect

61. C₂₃₃

#550

semidirect

70. C₄×D₄

D₂₃₃

semidirect

117. C₄

#16836594

semidirect

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

subgroup	index
C₂×#16836594	2
#21418532	2
#21418531	2
#21418509	2
#21418508	2
C₄×Dic₄₆₆	2
#21418497	2
C₄×D₄×C₂₃₃	2
#550	233

subgroup

index

C₄×Dic₄₆₆

C₄×D₄×C₂₃₃

233

Subgroups — #28417106

Conjugacy classes of subgroups

Complements

Maximal subgroups