Group #68724 — subgroups

Subgroups — #68724

Order 384.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#907	C₂	1	1	yes	maximal, characteristic
2	#5132	C₂	1	2	yes	maximal
3	C₂×#822	C₂	1	2	yes	maximal, complemented
4	C₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
6	#1813	—	3	1	no	maximal, Sylow
7	C₂×#235	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×#208	C₂²	1	2	yes	—
11	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	#907	—	2	4	no	—
13	C₂×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
14	C₂×D₃×C₈	C₄	1	2	yes	—
15	#824	—	2	2	no	—
16	#823	—	2	2	no	—
17	#822	—	2	4	no	—
18	C₂×#131	—	3	1	no	—
19	C₂×#108	—	3	2	no	—
20	#413	—	3	2	no	—
21	#338	—	3	1	no	—
22	#338	D₃	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂×#67	Q₈	1	2	yes	—
25	#235	—	2	2	no	—
26	#235	—	2	2	no	—
27	C₂×D₁₂	D₄	1	1	yes	characteristic
28	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
29	C₂×Dic₆	D₄	1	1	yes	characteristic
30	C₃×M₁₆	—	2	2	no	—
31	C₃×M₁₆	—	2	2	no	—
32	C₃×M₁₆	—	2	2	no	—
33	C₂×C₃×C₈	—	2	2	no	—
34	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
35	#208	—	2	2	no	—
36	#208	—	2	2	no	—
37	C₂×C₃⋊C₈	—	2	2	no	—
38	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃⋊C₈	D₄	1	4	yes	—
41	ΓL(1,𝔽₂₅)	—	2	4	no	—
42	D₃×C₈	—	2	4	no	—
43	C₂×#41	—	3	1	no	—
44	#131	—	6	4	no	—
45	C₂×M₁₆	—	3	2	no	—
46	C₂×M₁₆	—	3	2	no	—
47	C₂×M₁₆	C₂×D₃	1	2	yes	—
48	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₈	—	3	1	no	—
50	C₂²×C₈	—	3	2	no	—
51	#108	—	6	4	no	—
52	#101	—	6	2	no	—
53	#100	—	6	2	no	—
54	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×D₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄	#41	1	2	yes	—
60	#67	—	2	4	no	—
61	C₂×Dic₃	C₄⋊C₄	1	2	yes	—
62	D₁₂	—	2	2	no	—
63	C₄×D₃	—	2	2	no	—
64	C₄×D₃	—	2	2	no	—
65	Dic₆	—	2	2	no	—
66	C₃×C₈	—	2	2	no	—
67	C₃×C₈	—	4	2	no	—
68	C₃⋊C₈	—	4	2	no	—
69	C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
71	#41	—	6	2	no	—
72	#41	—	6	2	no	—
73	C₂×Q₈	—	3	1	no	—
74	C₂×D₄	—	3	2	no	—
75	C₂×D₄	—	3	1	no	—
76	C₂²×C₄	—	3	2	no	—
77	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
78	M₁₆	—	2	2	no	—
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	6	4	no	—
82	M₁₆	—	6	2	no	—
83	M₁₆	—	6	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	3	1	no	—
86	C₂×C₈	—	3	1	no	—
87	C₂×C₈	—	3	4	no	—
88	C₂×C₈	—	6	4	no	—
89	C₂×C₈	—	6	2	no	—
90	C₂×C₈	C₄×D₃	1	2	yes	—
91	C₂²×C₃	—	4	1	no	—
92	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
94	C₂×D₃	—	2	2	no	—
95	C₂×D₃	—	4	2	no	—
96	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₃×C₄	#117	1	1	yes	characteristic
98	C₃×C₄	#121	1	2	yes	—
99	Dic₃	—	2	2	no	—
100	C₂³	—	3	2	no	—
101	C₂³	D₃×Q₈	1	1	yes	characteristic
102	Q₈	—	6	2	no	—
103	D₄	—	6	4	no	—
104	D₄	—	6	2	no	—
105	C₂×C₄	—	3	2	no	—
106	C₂×C₄	—	6	2	no	—
107	C₂×C₄	—	6	2	no	—
108	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
109	C₂×C₄	#235	1	2	yes	—
110	C₂×C₄	D₃×D₄	1	2	yes	—
111	C₂×C₄	D₃×D₄	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	3	2	no	—
114	C₈	—	3	2	no	—
115	C₈	—	4	2	no	—
116	C₈	—	12	2	no	—
117	C₂×C₃	—	2	1	no	—
118	C₂×C₃	—	2	2	no	—
119	C₂×C₃	#380	1	1	yes	characteristic
120	D₃	—	4	2	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	6	2	no	—
123	C₂²	—	12	2	no	—
124	C₂²	D₃×C₄⋊C₄	1	1	yes	characteristic
125	C₂²	#896	1	2	yes	—
126	C₄	—	6	2	no	—
127	C₄	#872	1	1	yes	characteristic
128	C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
129	C₄	#884	1	2	yes	—
130	C₃	#1813	1	1	yes	characteristic, complemented, Sylow
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	12	2	no	—
134	C₂	#4667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#907

C₂

yes

maximal, characteristic

#5132

C₂

yes

maximal

C₂×#822

C₂

yes

maximal, complemented

C₃×#338

C₂

yes

maximal, characteristic, complemented

C₃⋊#338

C₂

yes

maximal, characteristic, complemented

#1813

—

maximal, Sylow

C₂×#235

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

—

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

characteristic

#907

—

C₂×ΓL(1,𝔽₂₅)

C₄

yes

—

C₂×D₃×C₈

C₄

yes

—

—

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×#67

Q₈

yes

—

#235

—

#235

—

C₂×D₁₂

D₄

yes

characteristic

C₂×C₄×D₃

C₂×C₄

yes

—

C₂×Dic₆

D₄

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃⋊C₈

D₄

yes

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

—

C₂×M₁₆

C₂×D₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

#101

—

#100

—

C₂³×C₃

C₂×Q₈

yes

characteristic

C₂²×D₃

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

#67

—

C₂×Dic₃

C₄⋊C₄

yes

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

C₂²⋊C₄

yes

—

C₃⋊C₈

C₂²⋊C₄

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃

yes

—

C₂²×C₃

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

C₂×C₂²⋊C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

Dic₃

—

100

C₂³

—

101

C₂³

D₃×Q₈

yes

characteristic

102

Q₈

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₄×D₃

yes

characteristic

109

C₂×C₄

#235

yes

—

110

C₂×C₄

D₃×D₄

yes

—

111

C₂×C₄

D₃×D₄

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

#380

yes

characteristic

120

D₃

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

D₃×C₄⋊C₄

yes

characteristic

125

C₂²

#896

yes

—

126

C₄

—

127

C₄

#872

yes

characteristic

128

C₄

D₃×C₂²⋊C₄

yes

characteristic

129

C₄

#884

yes

—

130

C₃

#1813

yes

characteristic, complemented, Sylow

131

C₂

—

132

C₂

—

133

C₂

—

134

C₂

#4667

yes

characteristic

	subgroup	G/H
center	C₄	#872
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#338	C₂
socle	C₂×C₃	#380
prime radical	C₂×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#338	D₃
hypocenter	C₃	#1813
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#872

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂×C₃

C₂²

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
3. C₂×#822	C₂	semidirect
4. C₃×#338	C₂	semidirect
5. C₃⋊#338	C₂	semidirect
22. #338	D₃	semidirect
130. C₃	#1813	semidirect

H ≅ G/N

type

3. C₂×#822

C₂

semidirect

4. C₃×#338

C₂

semidirect

5. C₃⋊#338

C₂

semidirect

22. #338

D₃

semidirect

130. C₃

#1813

semidirect

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

subgroup	index
C₂×#907	2
#5132	2
C₂×#822	2
C₃×#338	2
C₃⋊#338	2
#1813	3

subgroup

index

Subgroups — #68724

Conjugacy classes of subgroups

Complements

Maximal subgroups