Group #5685 — subgroups

Subgroups — #5685

Order 192.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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×#125	C₂	1	1	yes	maximal, characteristic
2	#890	C₂	1	1	yes	maximal, characteristic
3	#889	C₂	1	2	yes	maximal
4	#889	C₂	1	2	yes	maximal
5	#888	C₂	1	1	yes	maximal, characteristic
6	#888	C₂	1	2	yes	maximal
7	#887	C₂	1	1	yes	maximal, characteristic
8	#887	C₂	1	2	yes	maximal
9	#887	C₂	1	1	yes	maximal, characteristic
10	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
12	#557	—	3	1	no	maximal, Sylow
13	C₂×Dic₆	C₂²	1	1	yes	characteristic
14	C₂×Dic₆	C₂²	1	2	yes	—
15	C₂×Dic₆	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₃	C₂²	1	2	yes	—
24	Dic₃⋊C₄	C₂²	1	1	yes	characteristic
25	Dic₃⋊C₄	C₂²	1	2	yes	—
26	Dic₃⋊C₄	C₂²	1	1	yes	characteristic
27	Dic₃⋊C₄	C₂²	1	2	yes	—
28	Dic₃⋊C₄	C₂²	1	2	yes	—
29	Dic₃⋊C₄	C₂²	1	2	yes	—
30	Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₄×Dic₃	C₂²	1	1	yes	characteristic
32	C₄×Dic₃	C₂²	1	2	yes	—
33	C₄×Dic₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₃	C₂²	1	2	yes	—
35	C₄×Dic₃	C₂²	1	2	yes	—
36	C₄⋊Q₈	—	3	2	no	—
37	C₄⋊Q₈	—	3	1	no	—
38	#125	—	3	2	no	—
39	#125	—	3	2	no	—
40	#125	—	3	1	no	—
41	#125	D₃	1	1	yes	characteristic
42	C₄×Q₈	—	3	1	no	—
43	C₄×Q₈	—	3	2	no	—
44	C₄×Q₈	—	3	1	no	—
45	C₄×Q₈	—	3	1	no	—
46	C₄×Q₈	—	3	1	no	—
47	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂³	1	2	yes	—
49	C₂×C₃×C₄	C₂³	1	2	yes	—
50	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂×Dic₃	C₂³	1	1	yes	characteristic
53	C₂×Dic₃	C₂³	1	1	yes	characteristic
54	C₂×Dic₃	C₂³	1	2	yes	—
55	C₂×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×Dic₃	C₂³	1	2	yes	—
57	C₂×Dic₃	C₂³	1	1	yes	characteristic
58	Dic₆	—	2	2	no	—
59	Dic₆	—	2	4	no	—
60	Dic₆	Q₈	1	4	yes	complemented
61	C₂×Q₈	—	3	1	no	—
62	C₂×Q₈	—	3	2	no	—
63	C₂×Q₈	—	3	1	no	—
64	C₄⋊C₄	—	3	1	no	—
65	C₄⋊C₄	—	3	2	no	—
66	C₄⋊C₄	—	3	2	no	—
67	C₄⋊C₄	—	3	1	no	—
68	C₄⋊C₄	—	3	1	no	—
69	C₄⋊C₄	—	3	2	no	—
70	C₄⋊C₄	—	3	2	no	—
71	C₄⋊C₄	—	3	1	no	—
72	C₄⋊C₄	—	3	2	no	—
73	C₄⋊C₄	—	3	2	no	—
74	C₄⋊C₄	C₂×D₃	1	2	yes	—
75	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
76	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₃	1	2	yes	—
78	C₄²	—	3	2	no	—
79	C₄²	—	3	2	no	—
80	C₄²	—	3	1	no	—
81	C₄²	—	3	2	no	—
82	C₄²	—	3	1	no	—
83	C₄²	C₂×D₃	1	1	yes	characteristic
84	C₂²×C₃	C₂⁴	1	1	yes	characteristic
85	C₃×C₄	—	2	2	no	—
86	C₃×C₄	—	2	1	no	—
87	C₃×C₄	—	2	1	no	—
88	C₃×C₄	—	2	2	no	—
89	C₃×C₄	C₂×Q₈	1	2	yes	—
90	Dic₃	—	2	1	no	—
91	Dic₃	—	2	2	no	—
92	Dic₃	—	2	1	no	—
93	Dic₃	—	2	1	no	—
94	Dic₃	C₂×Q₈	1	4	yes	—
95	Dic₃	#41	1	2	yes	—
96	Q₈	—	3	4	no	—
97	Q₈	—	6	4	no	—
98	Q₈	—	6	2	no	—
99	C₂×C₄	—	3	2	no	—
100	C₂×C₄	—	3	1	no	—
101	C₂×C₄	—	3	1	no	—
102	C₂×C₄	—	3	1	no	—
103	C₂×C₄	—	3	1	no	—
104	C₂×C₄	—	3	2	no	—
105	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₄	C₂²×D₃	1	2	yes	—
107	C₂×C₄	C₂²×D₃	1	2	yes	—
108	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂²	C₂³×D₃	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	2	no	—
118	C₄	—	3	4	no	—
119	C₄	—	3	2	no	—
120	C₄	—	6	1	no	—
121	C₄	—	6	2	no	—
122	C₄	—	6	1	no	—
123	C₄	—	6	1	no	—
124	C₄	D₃×Q₈	1	2	yes	—
125	C₃	#557	1	1	yes	characteristic, complemented, Sylow
126	C₂	C₂×D₃×Q₈	1	1	yes	characteristic
127	C₂	D₃×#41	1	1	yes	characteristic
128	C₂	#1010	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#125

C₂

yes

maximal, characteristic

#890

C₂

yes

maximal, characteristic

#889

C₂

yes

maximal

#889

C₂

yes

maximal

#888

C₂

yes

maximal, characteristic

#888

C₂

yes

maximal

#887

C₂

yes

maximal, characteristic

#887

C₂

yes

maximal

#887

C₂

yes

maximal, characteristic

C₄×Dic₆

C₂

yes

maximal, characteristic

C₄×Dic₆

C₂

yes

maximal, characteristic

#557

—

maximal, Sylow

C₂×Dic₆

C₂²

yes

characteristic

C₂×Dic₆

C₂²

yes

—

C₂×Dic₆

C₂²

yes

characteristic

C₃×C₄⋊C₄

C₂²

yes

characteristic

C₃×C₄⋊C₄

C₂²

yes

—

C₃×C₄⋊C₄

C₂²

yes

characteristic

C₃×C₄⋊C₄

C₂²

yes

—

C₃×C₄²

C₂²

yes

characteristic

C₄⋊Dic₃

C₂²

yes

characteristic

C₄⋊Dic₃

C₂²

yes

characteristic

C₄⋊Dic₃

C₂²

yes

—

Dic₃⋊C₄

C₂²

yes

characteristic

Dic₃⋊C₄

C₂²

yes

—

Dic₃⋊C₄

C₂²

yes

characteristic

Dic₃⋊C₄

C₂²

yes

—

Dic₃⋊C₄

C₂²

yes

—

Dic₃⋊C₄

C₂²

yes

—

Dic₃⋊C₄

C₂²

yes

—

C₄×Dic₃

C₂²

yes

characteristic

C₄×Dic₃

C₂²

yes

—

C₄×Dic₃

C₂²

yes

characteristic

C₄×Dic₃

C₂²

yes

—

C₄×Dic₃

C₂²

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

D₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₄

C₂³

yes

—

C₂×C₃×C₄

C₂³

yes

—

C₂×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₄

C₂³

yes

characteristic

C₂×Dic₃

C₂³

yes

characteristic

C₂×Dic₃

C₂³

yes

characteristic

C₂×Dic₃

C₂³

yes

—

C₂×Dic₃

C₂³

yes

characteristic

C₂×Dic₃

C₂³

yes

—

C₂×Dic₃

C₂³

yes

characteristic

Dic₆

—

Dic₆

—

Dic₆

Q₈

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃

yes

—

C₄⋊C₄

C₂×D₃

yes

characteristic

C₄⋊C₄

C₂×D₃

yes

characteristic

C₄⋊C₄

C₂×D₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₃

yes

characteristic

C₂²×C₃

C₂⁴

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×Q₈

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

C₂×Q₈

yes

—

Dic₃

#41

yes

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×D₃

yes

characteristic

106

C₂×C₄

C₂²×D₃

yes

—

107

C₂×C₄

C₂²×D₃

yes

—

108

C₂×C₄

C₂²×D₃

yes

characteristic

109

C₂×C₄

C₂²×D₃

yes

characteristic

110

C₂×C₃

C₂²×Q₈

yes

characteristic

111

C₂×C₃

C₂×#41

yes

characteristic

112

C₂×C₃

2₋¹⁺⁴

yes

characteristic

113

C₂²

C₂³×D₃

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

D₃×Q₈

yes

—

125

C₃

#557

yes

characteristic, complemented, Sylow

126

C₂

C₂×D₃×Q₈

yes

characteristic

127

C₂

D₃×#41

yes

characteristic

128

C₂

#1010

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#125	C₂
hypercenter	#125	D₃
hypocenter	C₃	#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂²×C₃

C₂⁴

C₂

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
60. Dic₆	Q₈	semidirect
125. C₃	#557	semidirect

H ≅ G/N

type

60. Dic₆

Q₈

semidirect

125. C₃

#557

semidirect

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

subgroup	index
C₃×#125	2
#890	2
#889	2
#888	2
#887	2
C₄×Dic₆	2
#557	3

subgroup

index

C₄×Dic₆

Subgroups — #5685

Conjugacy classes of subgroups

Complements

Maximal subgroups