Group #83329 — subgroups

Subgroups — #83329

Order 384.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 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	C₃×#533	C₂	1	1	yes	maximal, characteristic
2	#5024	C₂	1	8	yes	maximal, complemented
3	#4999	C₂	1	2	yes	maximal
4	#4997	C₂	1	2	yes	maximal
5	#4924	C₂	1	1	yes	maximal, characteristic
6	C₂×#803	C₂	1	1	yes	maximal, characteristic
7	#2961	—	3	1	no	maximal, Sylow
8	C₃×C₄⋊Q₈	C₄	1	2	yes	—
9	C₃×#125	C₄	1	2	yes	—
10	C₃×C₂²⋊Q₈	C₄	1	4	yes	—
11	C₃×C₄×Q₈	C₂²	1	4	yes	—
12	C₃×#117	C₂²	1	2	yes	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×#208	C₂²	1	2	yes	—
15	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
18	C₄⋊(C₃⋊C₈)	C₂²	1	8	yes	—
19	#803	C₂²	1	4	yes	—
20	#803	C₂²	1	4	yes	—
21	C₄×C₃⋊C₈	C₂²	1	4	yes	—
22	#533	D₃	1	1	yes	characteristic
23	#446	—	3	8	no	—
24	#424	—	3	2	no	—
25	#423	—	3	1	no	—
26	#405	—	3	2	no	—
27	C₂×#97	—	3	1	no	—
28	C₂×C₃×Q₈	C₂×C₄	1	4	yes	—
29	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
33	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
35	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²	C₂³	1	4	yes	—
38	C₃×C₄²	C₂³	1	2	yes	—
39	#208	—	2	4	no	—
40	C₂×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃⋊C₈	Q₈	1	4	yes	complemented
42	C₂×C₃⋊C₈	C₂³	1	4	yes	—
43	C₂×C₃⋊C₈	C₂³	1	2	yes	—
44	C₂×C₃⋊C₈	C₂³	1	2	yes	—
45	C₂×M₁₆	—	3	2	no	—
46	C₂²×C₈	—	3	1	no	—
47	C₂²×C₈	—	3	1	no	—
48	C₄⋊Q₈	Dic₃	1	2	yes	—
49	#125	Dic₃	1	2	yes	—
50	C₂²⋊Q₈	Dic₃	1	4	yes	—
51	C₄×Q₈	C₂×D₃	1	4	yes	—
52	#117	C₂×D₃	1	2	yes	—
53	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₄⋊C₈	—	3	4	no	—
55	C₄⋊C₈	—	3	8	no	—
56	#97	—	3	4	no	—
57	#97	—	3	4	no	—
58	C₄×C₈	—	3	4	no	—
59	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
60	C₃×Q₈	—	4	4	no	—
61	C₂×C₃×C₄	—	4	2	no	—
62	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
63	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
64	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
65	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₄	#41	1	2	yes	—
68	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
69	C₃⋊C₈	—	2	2	no	—
70	C₃⋊C₈	—	2	4	no	—
71	C₃⋊C₈	C₂×Q₈	1	4	yes	—
72	C₂×Q₈	C₂×Dic₃	1	4	yes	—
73	C₂²×C₄	C₂×Dic₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
75	M₁₆	—	6	4	no	—
76	C₂×C₈	—	3	4	no	—
77	C₂×C₈	—	3	2	no	—
78	C₂×C₈	—	3	4	no	—
79	C₂×C₈	—	3	2	no	—
80	C₂×C₈	—	6	2	no	—
81	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
82	C₄⋊C₄	C₂×Dic₃	1	8	yes	—
83	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
84	C₂²⋊C₄	C₂×Dic₃	1	4	yes	—
85	C₄²	C₂×Dic₃	1	1	yes	characteristic
86	C₄²	C₂×Dic₃	1	1	yes	characteristic
87	C₄²	C₂²×D₃	1	4	yes	—
88	C₄²	C₂²×D₃	1	2	yes	—
89	C₂²×C₃	—	2	2	no	—
90	C₂²×C₃	C₄×Q₈	1	2	yes	—
91	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
92	C₃×C₄	—	2	2	no	—
93	C₃×C₄	—	2	2	no	—
94	C₃×C₄	—	2	8	no	—
95	C₃×C₄	C₄×Q₈	1	2	yes	—
96	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
97	C₃×C₄	C₂×#41	1	1	yes	characteristic
98	C₂³	C₂²×Dic₃	1	1	yes	characteristic
99	Q₈	—	4	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	D₃×Q₈	1	2	yes	—
102	C₂×C₄	#239	1	2	yes	—
103	C₂×C₄	C₂²×Dic₃	1	2	yes	—
104	C₂×C₄	C₂²×Dic₃	1	2	yes	—
105	C₂×C₄	C₂²×Dic₃	1	8	yes	—
106	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
107	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
108	C₈	—	3	4	no	—
109	C₈	—	6	2	no	—
110	C₈	—	6	4	no	—
111	C₂×C₃	—	2	2	no	—
112	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₃	#568	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	Q₈×Dic₃	1	2	yes	—
116	C₂²	C₂³×Dic₃	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	8	no	—
120	C₄	Q₈×Dic₃	1	2	yes	—
121	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
122	C₄	C₂×#239	1	1	yes	characteristic
123	C₃	#2961	1	1	yes	characteristic, complemented, Sylow
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₃	1	1	yes	characteristic
126	C₂	#5819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#533

C₂

yes

maximal, characteristic

#5024

C₂

yes

maximal, complemented

#4999

C₂

yes

maximal

#4997

C₂

yes

maximal

#4924

C₂

yes

maximal, characteristic

C₂×#803

C₂

yes

maximal, characteristic

#2961

—

maximal, Sylow

C₃×C₄⋊Q₈

C₄

yes

—

C₃×#125

C₄

yes

—

C₃×C₂²⋊Q₈

C₄

yes

—

C₃×C₄×Q₈

C₂²

yes

—

C₃×#117

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

#803

C₂²

yes

—

#803

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

—

#533

D₃

yes

—

—

—

—

C₂×#97

—

C₂×C₃×Q₈

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₂²⋊C₄

C₂×C₄

yes

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂³

yes

—

C₃×C₄²

C₂³

yes

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

Q₈

yes

complemented

C₂×C₃⋊C₈

C₂³

yes

—

C₂×C₃⋊C₈

C₂³

yes

—

C₂×C₃⋊C₈

C₂³

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₃

yes

—

#125

Dic₃

yes

—

C₂²⋊Q₈

Dic₃

yes

—

C₄×Q₈

C₂×D₃

yes

—

#117

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

C₂×Q₈

yes

—

C₂×Q₈

C₂×Dic₃

yes

—

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₃

yes

—

C₄⋊C₄

C₂×Dic₃

yes

—

C₄⋊C₄

C₂×Dic₃

yes

—

C₂²⋊C₄

C₂×Dic₃

yes

—

C₄²

C₂×Dic₃

yes

characteristic

C₄²

C₂×Dic₃

yes

characteristic

C₄²

C₂²×D₃

yes

—

C₄²

C₂²×D₃

yes

—

C₂²×C₃

—

C₂²×C₃

C₄×Q₈

yes

—

C₂²×C₃

C₂³×C₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×Q₈

yes

—

C₃×C₄

C₂²×Q₈

yes

characteristic

C₃×C₄

C₂×#41

yes

characteristic

C₂³

C₂²×Dic₃

yes

characteristic

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

D₃×Q₈

yes

—

102

C₂×C₄

#239

yes

—

103

C₂×C₄

C₂²×Dic₃

yes

—

104

C₂×C₄

C₂²×Dic₃

yes

—

105

C₂×C₄

C₂²×Dic₃

yes

—

106

C₂×C₄

C₂²×Dic₃

yes

characteristic

107

C₂×C₄

C₂³×D₃

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂×C₃

—

112

C₂×C₃

C₂×C₄×Q₈

yes

characteristic

113

C₂×C₃

#568

yes

—

114

C₂²

—

115

C₂²

Q₈×Dic₃

yes

—

116

C₂²

C₂³×Dic₃

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

Q₈×Dic₃

yes

—

121

C₄

C₂×D₃×Q₈

yes

characteristic

122

C₄

C₂×#239

yes

characteristic

123

C₃

#2961

yes

characteristic, complemented, Sylow

124

C₂

—

125

C₂

C₂×Q₈×Dic₃

yes

characteristic

126

C₂

#5819

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#533	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#533	D₃
hypocenter	C₃	#2961
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃

commutator, socle

C₂²×C₃

C₂³×C₄

Fitting

C₃×#533

C₂

prime radical

C₂³×C₃

C₂²×C₄

cosocle

C₂×C₃×C₄

C₂⁴

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
2. #5024	C₂	semidirect
41. C₂×C₃⋊C₈	Q₈	semidirect
123. C₃	#2961	semidirect

H ≅ G/N

type

2. #5024

C₂

semidirect

41. C₂×C₃⋊C₈

Q₈

semidirect

123. C₃

#2961

semidirect

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

subgroup	index
C₃×#533	2
#5024	2
#4999	2
#4997	2
#4924	2
C₂×#803	2
#2961	3

subgroup

index

Subgroups — #83329

Conjugacy classes of subgroups

Complements

Maximal subgroups