Group #45663398 — subgroups

Subgroups — #45663398

Order 32896.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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	#30657111	C₂	1	1	yes	maximal, characteristic
2	#30656937	C₂	1	2	yes	maximal
3	#30656486	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#14523339	C₂	1	2	yes	maximal, complemented
5	Q₈²×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16861605	—	2	2	no	—
7	C₂×#16861605	—	2	2	no	—
8	#22551589	C₂²	1	1	yes	characteristic
9	C₄⋊#14523339	C₂²	1	2	yes	—
10	#22551512	—	2	2	no	—
11	C₄×#14523339	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₂₅₇	—	2	2	no	—
13	C₄⋊Q₈×C₂₅₇	—	2	2	no	—
14	C₄⋊Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₂₅₇	—	2	2	no	—
16	C₄×Q₈×C₂₅₇	C₂²	1	2	yes	—
17	C₂×D₁₀₂₈	—	2	1	no	—
18	C₂×D₁₀₂₈	—	2	2	no	—
19	#16861605	—	4	4	no	—
20	#16861605	—	4	4	no	—
21	C₂×#14523339	—	2	1	no	—
22	C₂×#14523339	—	2	2	no	—
23	C₂×Q₈×C₂₅₇	—	2	2	no	—
24	C₂×Q₈×C₂₅₇	D₄	1	2	yes	—
25	C₂×Q₈×C₂₅₇	D₄	1	2	yes	—
26	C₄⋊C₄×C₂₅₇	—	2	4	no	—
27	C₄⋊C₄×C₂₅₇	—	2	2	no	—
28	C₄⋊C₄×C₂₅₇	—	2	2	no	—
29	C₄⋊C₄×C₂₅₇	D₄	1	2	yes	—
30	C₄²×C₂₅₇	—	2	2	no	—
31	C₄²×C₂₅₇	—	2	2	no	—
32	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
33	C₂²×D₂₅₇	—	4	1	no	—
34	D₁₀₂₈	—	4	2	no	—
35	D₁₀₂₈	—	4	2	no	—
36	D₁₀₂₈	—	4	2	no	—
37	#14523339	—	2	2	no	—
38	#14523339	—	4	2	no	—
39	Q₈×C₂₅₇	—	2	2	no	—
40	Q₈×C₂₅₇	—	4	2	no	—
41	Q₈×C₂₅₇	—	8	2	no	—
42	Q₈×C₂₅₇	SD₁₆	1	4	yes	complemented
43	C₂×C₄×C₂₅₇	—	2	2	no	—
44	C₂×C₄×C₂₅₇	—	2	2	no	—
45	C₂×C₄×C₂₅₇	—	2	2	no	—
46	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
48	C₂²×C₂₅₇	#120	1	1	yes	characteristic
49	C₂×D₂₅₇	—	8	2	no	—
50	C₂×D₂₅₇	—	8	1	no	—
51	C₄×C₂₅₇	—	2	4	no	—
52	C₄×C₂₅₇	—	2	1	no	—
53	C₄×C₂₅₇	—	4	2	no	—
54	C₄×C₂₅₇	—	4	2	no	—
55	C₄×C₂₅₇	C₂×SD₁₆	1	2	yes	—
56	C₄×C₂₅₇	#137	1	2	yes	—
57	C₂×C₂₅₇	#448	1	2	yes	—
58	C₂×C₂₅₇	#453	1	1	yes	characteristic
59	D₂₅₇	—	16	1	no	—
60	C₂₅₇	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	257	1	no	maximal, Sylow
62	Q₈²	D₂₅₇	1	1	yes	characteristic, complemented
63	#492	—	257	1	no	—
64	#460	—	257	2	no	—
65	#333	—	257	1	no	—
66	Q₈⋊C₈	—	257	2	no	—
67	C₂×SD₁₆	—	514	2	no	—
68	C₂×SD₁₆	—	514	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₂×D₂₅₇	1	1	yes	characteristic
72	#127	—	257	1	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	C₂×D₂₅₇	1	2	yes	—
75	C₄⋊C₈	—	257	2	no	—
76	#102	—	514	2	no	—
77	C₄×C₈	—	257	1	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	C₂²⋊D₂₅₇	1	2	yes	—
80	C₂×Q₈	C₂²⋊D₂₅₇	1	2	yes	—
81	C₂×D₄	—	514	2	no	—
82	C₂×D₄	—	514	1	no	—
83	SD₁₆	—	1028	4	no	—
84	SD₁₆	—	1028	4	no	—
85	C₂×C₈	—	514	1	no	—
86	C₂×C₈	—	514	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	C₂²⋊D₂₅₇	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
94	C₂³	—	1028	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	#16861605	1	4	yes	complemented
99	D₄	—	1028	2	no	—
100	D₄	—	1028	2	no	—
101	D₄	—	1028	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	2	yes	—
107	C₈	—	514	2	no	—
108	C₈	—	1028	2	no	—
109	C₂²	—	2056	2	no	—
110	C₂²	—	2056	1	no	—
111	C₂²	#22551561	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂×#16861605	1	2	yes	—
117	C₄	#22551623	1	2	yes	—
118	C₂	—	4112	1	no	—
119	C₂	#30656876	1	2	yes	—
120	C₂	#30656905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30657111

C₂

yes

maximal, characteristic

#30656937

C₂

yes

maximal

#30656486

C₂

yes

maximal, characteristic, complemented

Q₈⋊#14523339

C₂

yes

maximal, complemented

Q₈²×C₂₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#16861605

—

C₂×#16861605

—

#22551589

C₂²

yes

characteristic

C₄⋊#14523339

C₂²

yes

—

#22551512

—

C₄×#14523339

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₇

—

C₄⋊Q₈×C₂₅₇

—

C₄⋊Q₈×C₂₅₇

C₂²

yes

characteristic

C₄×Q₈×C₂₅₇

—

C₄×Q₈×C₂₅₇

C₂²

yes

—

C₂×D₁₀₂₈

—

C₂×D₁₀₂₈

—

—

—

—

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

D₄

yes

—

C₂×Q₈×C₂₅₇

D₄

yes

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

D₄

yes

—

C₄²×C₂₅₇

—

C₄²×C₂₅₇

—

C₄²×C₂₅₇

C₂³

yes

characteristic

C₂²×D₂₅₇

—

D₁₀₂₈

—

D₁₀₂₈

—

D₁₀₂₈

—

#14523339

—

#14523339

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

SD₁₆

yes

complemented

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂²×C₂₅₇

#120

yes

characteristic

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₂×SD₁₆

yes

—

C₄×C₂₅₇

#137

yes

—

C₂×C₂₅₇

#448

yes

—

C₂×C₂₅₇

#453

yes

characteristic

D₂₅₇

—

C₂₅₇

#1623

yes

characteristic, complemented, Sylow

#1623

—

257

maximal, Sylow

Q₈²

D₂₅₇

yes

characteristic, complemented

#492

—

257

—

#460

—

257

—

#333

—

257

—

Q₈⋊C₈

—

257

—

C₂×SD₁₆

—

514

—

C₂×SD₁₆

—

514

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₅₇

yes

characteristic

#127

—

257

—

C₄×Q₈

—

C₄×Q₈

C₂×D₂₅₇

yes

—

C₄⋊C₈

—

257

—

#102

—

514

—

C₄×C₈

—

257

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₅₇

yes

—

C₂×Q₈

C₂²⋊D₂₅₇

yes

—

C₂×D₄

—

514

—

C₂×D₄

—

514

—

SD₁₆

—

1028

—

SD₁₆

—

1028

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₂₅₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₅₇

yes

characteristic

C₂³

—

1028

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#16861605

yes

complemented

D₄

—

1028

—

100

D₄

—

1028

—

101

D₄

—

1028

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

—

107

C₈

—

514

—

108

C₈

—

1028

—

109

C₂²

—

2056

—

110

C₂²

—

2056

—

111

C₂²

#22551561

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×#16861605

yes

—

117

C₄

#22551623

yes

—

118

C₂

—

4112

—

119

C₂

#30656876

yes

—

120

C₂

#30656905

yes

characteristic

	subgroup	G/H
center	C₂²	#22551561
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	Q₈²×C₂₅₇	C₂
socle	C₂²×C₂₅₇	#120
prime radical	#22551589	C₂²
hypercenter	Q₈²	D₂₅₇
hypocenter	C₂₅₇	#1623
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#22551561

commutator, cosocle

C₄²×C₂₅₇

C₂³

Frattini

C₄²

C₂²×D₂₅₇

Fitting

Q₈²×C₂₅₇

C₂

C₂²×C₂₅₇

C₂²

Q₈²

D₂₅₇

hypocenter

C₂₅₇

#1623

LCS

G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
3. #30656486	C₂	semidirect
4. Q₈⋊#14523339	C₂	semidirect
5. Q₈²×C₂₅₇	C₂	semidirect
42. Q₈×C₂₅₇	SD₁₆	semidirect
60. C₂₅₇	#1623	semidirect
62. Q₈²	D₂₅₇	semidirect
98. Q₈	#16861605	semidirect

H ≅ G/N

type

3. #30656486

C₂

semidirect

4. Q₈⋊#14523339

C₂

semidirect

5. Q₈²×C₂₅₇

C₂

semidirect

42. Q₈×C₂₅₇

SD₁₆

semidirect

60. C₂₅₇

#1623

semidirect

62. Q₈²

D₂₅₇

semidirect

98. Q₈

#16861605

semidirect

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

subgroup	index
#30657111	2
#30656937	2
#30656486	2
Q₈⋊#14523339	2
Q₈²×C₂₅₇	2
#1623	257

subgroup

index

Q₈²×C₂₅₇

257

Subgroups — #45663398

Conjugacy classes of subgroups

Complements

Maximal subgroups