Group #44449707 — subgroups

Subgroups — #44449707

Order 29824.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#28416488	C₂	1	1	yes	maximal, characteristic
2	#28416459	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₂₃₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16836578	—	2	1	no	—
5	C₄⋊#14272338	—	2	1	no	—
6	C₄⋊#14272338	—	2	1	no	—
7	C₄⋊#14272338	C₄	1	2	yes	—
8	C₂²⋊#14272338	C₄	1	2	yes	—
9	C₄×#14272338	—	2	1	no	—
10	C₂²×C₈×C₂₃₃	—	2	1	no	—
11	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₃₃	—	2	1	no	—
13	C₄⋊C₈×C₂₃₃	C₄	1	2	yes	—
14	#16836578	—	4	1	no	—
15	#16836578	—	4	1	no	—
16	C₂×#14272338	—	2	2	no	—
17	C₂×#14272338	—	2	2	no	—
18	C₂²×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₂₃₃	—	2	2	no	—
22	C₂×C₈×C₂₃₃	—	4	1	no	—
23	C₂×C₈×C₂₃₃	—	4	1	no	—
24	C₄²×C₂₃₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₂₃₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
27	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
28	#14272338	—	4	2	no	—
29	#14272338	—	4	2	no	—
30	C₂³×C₂₃₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃₃	—	2	2	no	—
32	C₂×C₄×C₂₃₃	—	4	1	no	—
33	C₂×C₄×C₂₃₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃₃	D₈	1	1	yes	characteristic
40	C₂×C₄×C₂₃₃	SD₁₆	1	1	yes	characteristic
41	C₂×C₄×C₂₃₃	SD₁₆	1	1	yes	characteristic
42	C₂×C₄×C₂₃₃	Q₁₆	1	1	yes	characteristic
43	C₈×C₂₃₃	—	4	2	no	—
44	C₂²×C₂₃₃	—	2	1	no	—
45	C₂²×C₂₃₃	—	2	1	no	—
46	C₂²×C₂₃₃	#95	1	1	yes	characteristic
47	C₂²×C₂₃₃	#106	1	1	yes	characteristic
48	C₂²×C₂₃₃	#107	1	1	yes	characteristic
49	C₄×C₂₃₃	—	2	1	no	—
50	C₄×C₂₃₃	—	2	1	no	—
51	C₄×C₂₃₃	—	2	1	no	—
52	C₄×C₂₃₃	#102	1	1	yes	characteristic
53	C₄×C₂₃₃	#102	1	1	yes	characteristic
54	C₄×C₂₃₃	#103	1	1	yes	characteristic
55	C₄×C₂₃₃	#103	1	1	yes	characteristic
56	C₄×C₂₃₃	#108	1	2	yes	—
57	C₂×C₂₃₃	—	2	1	no	—
58	C₂×C₂₃₃	—	2	1	no	—
59	C₂×C₂₃₃	#340	1	1	yes	characteristic
60	C₂×C₂₃₃	#341	1	1	yes	characteristic
61	C₂×C₂₃₃	#344	1	1	yes	characteristic
62	C₂₃₃	#1301	1	1	yes	characteristic, complemented, Sylow
63	#1301	—	233	1	no	maximal, Sylow
64	#431	—	233	1	no	—
65	#423	—	233	1	no	—
66	C₂×C₄⋊C₈	D₂₃₃	1	1	yes	characteristic
67	C₂×M₁₆	—	466	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂×C₄²	C₂×D₂₃₃	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	466	1	no	—
72	C₄⋊C₈	—	466	1	no	—
73	C₄⋊C₈	—	233	2	no	—
74	C₄⋊C₈	Dic₂₃₃	1	2	yes	—
75	C₂²⋊C₈	—	233	2	no	—
76	C₄×C₈	—	466	1	no	—
77	C₂²×C₄	C₄×D₂₃₃	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
79	C₂²×C₄	Dic₄₆₆	1	1	yes	characteristic
80	M₁₆	—	932	1	no	—
81	M₁₆	—	932	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	466	2	no	—
86	C₂×C₈	—	466	2	no	—
87	C₄²	C₄×D₂₃₃	1	1	yes	characteristic
88	C₄²	C₂×Dic₂₃₃	1	1	yes	characteristic
89	C₄²	C₂²⋊D₂₃₃	1	1	yes	characteristic
90	C₄²	D₉₃₂	1	1	yes	characteristic
91	C₂³	#16836573	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₂₃₃	1	1	yes	characteristic
95	C₂×C₄	#16836571	1	1	yes	characteristic
96	C₂×C₄	#16836571	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₂₃₃	1	1	yes	characteristic
98	C₂×C₄	#16836573	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₂₃₃	1	1	yes	characteristic
100	C₂×C₄	#16836579	1	1	yes	characteristic
101	C₂×C₄	#16836581	1	1	yes	characteristic
102	C₂×C₄	#16836582	1	1	yes	characteristic
103	C₂×C₄	#16836584	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	932	2	no	—
106	C₈	—	932	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#21418442	1	1	yes	characteristic
110	C₂²	#21418466	1	1	yes	characteristic
111	C₂²	#21418468	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#21418455	1	1	yes	characteristic
116	C₄	#21418456	1	1	yes	characteristic
117	C₄	#21418458	1	1	yes	characteristic
118	C₄	#21418459	1	1	yes	characteristic
119	C₄	#21418471	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#28416197	1	1	yes	characteristic
123	C₂	#28416200	1	1	yes	characteristic
124	C₂	#28416207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28416488

C₂

yes

maximal, characteristic

#28416459

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₃₃

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

C₂²⋊#14272338

C₄

yes

—

C₄×#14272338

—

C₂²×C₈×C₂₃₃

—

C₂×C₄²×C₂₃₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₃₃

—

C₄⋊C₈×C₂₃₃

C₄

yes

—

—

—

—

—

C₂²×C₄×C₂₃₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₃

D₄

yes

characteristic

C₂²×C₄×C₂₃₃

Q₈

yes

characteristic

C₂×C₈×C₂₃₃

—

C₂×C₈×C₂₃₃

—

C₂×C₈×C₂₃₃

—

C₄²×C₂₃₃

C₂×C₄

yes

characteristic

C₄²×C₂₃₃

C₂×C₄

yes

characteristic

C₄²×C₂₃₃

D₄

yes

characteristic

C₄²×C₂₃₃

D₄

yes

characteristic

#14272338

—

#14272338

—

C₂³×C₂₃₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

C₄²

yes

characteristic

C₂×C₄×C₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

D₈

yes

characteristic

C₂×C₄×C₂₃₃

SD₁₆

yes

characteristic

C₂×C₄×C₂₃₃

SD₁₆

yes

characteristic

C₂×C₄×C₂₃₃

Q₁₆

yes

characteristic

C₈×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

yes

C₂²×C₂₃₃

yes

C₂²×C₂₃₃

yes

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

yes

C₄×C₂₃₃

yes

C₄×C₂₃₃

yes

C₄×C₂₃₃

yes

C₄×C₂₃₃

yes

—

C₂×C₂₃₃

—

C₂×C₂₃₃

—

C₂×C₂₃₃

yes

C₂×C₂₃₃

yes

C₂×C₂₃₃

yes

C₂₃₃

yes

characteristic, complemented, Sylow

#1301

—

233

maximal, Sylow

#431

—

233

—

#423

—

233

—

C₂×C₄⋊C₈

D₂₃₃

yes

characteristic

C₂×M₁₆

—

466

—

C₂²×C₈

—

C₂×C₄²

C₂×D₂₃₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

466

—

C₄⋊C₈

—

466

—

C₄⋊C₈

—

233

—

C₄⋊C₈

Dic₂₃₃

yes

—

C₂²⋊C₈

—

233

—

C₄×C₈

—

466

—

C₂²×C₄

C₄×D₂₃₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃₃

yes

characteristic

C₂²×C₄

Dic₄₆₆

yes

characteristic

M₁₆

—

932

—

M₁₆

—

932

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

466

—

C₂×C₈

—

466

—

C₄²

C₄×D₂₃₃

yes

characteristic

C₄²

C₂×Dic₂₃₃

yes

characteristic

C₄²

C₂²⋊D₂₃₃

yes

characteristic

C₄²

D₉₃₂

yes

characteristic

C₂³

#16836573

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₃₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₂₃₃

yes

characteristic

C₂×C₄

#16836573

yes

characteristic

C₂×C₄

C₄⋊Dic₂₃₃

yes

characteristic

100

C₂×C₄

#16836579

yes

characteristic

101

C₂×C₄

#16836581

yes

characteristic

102

C₂×C₄

#16836582

yes

characteristic

103

C₂×C₄

#16836584

yes

characteristic

104

C₈

—

105

C₈

—

932

—

106

C₈

—

932

—

107

C₂²

—

108

C₂²

—

109

C₂²

#21418442

yes

characteristic

110

C₂²

#21418466

yes

characteristic

111

C₂²

#21418468

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#21418455

yes

characteristic

116

C₄

#21418456

yes

characteristic

117

C₄

#21418458

yes

characteristic

118

C₄

#21418459

yes

characteristic

119

C₄

#21418471

yes

—

120

C₂

—

121

C₂

—

122

C₂

#28416197

yes

characteristic

123

C₂

#28416200

yes

characteristic

124

C₂

#28416207

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#16836571
commutator	C₂×C₄×C₂₃₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₃₃
Fitting	C₂×C₄⋊C₈×C₂₃₃	C₂
socle	C₂²×C₂₃₃	#95
prime radical	C₂³×C₂₃₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₃₃	C₂²
hypercenter	C₂×C₄⋊C₈	D₂₃₃
hypocenter	C₂₃₃	#1301
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

#16836571

commutator

C₂×C₄×C₂₃₃

C₄²

Frattini

C₂×C₄²

C₂×D₂₃₃

Fitting

C₂×C₄⋊C₈×C₂₃₃

C₂

socle

C₂²×C₂₃₃

#95

prime radical

C₂³×C₂₃₃

C₄⋊C₄

cosocle

C₂×C₄²×C₂₃₃

C₂²

hypercenter

C₂×C₄⋊C₈

D₂₃₃

hypocenter

C₂₃₃

#1301

LCS

G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
62. C₂₃₃	#1301	semidirect

H ≅ G/N

type

62. C₂₃₃

#1301

semidirect

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

subgroup	index
#28416488	2
#28416459	2
C₂×C₄⋊C₈×C₂₃₃	2
#1301	233

subgroup

index

#28416488

#28416459

C₂×C₄⋊C₈×C₂₃₃

#1301

233

Subgroups — #44449707

Conjugacy classes of subgroups

Complements

Maximal subgroups