Group #69771 — subgroups

Subgroups — #69771

Order 384.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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×#869	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#816	C₂	1	4	yes	maximal
4	#4935	C₂	1	1	yes	maximal, characteristic
5	#1956	—	3	1	no	maximal, Sylow
6	C₂²×Dic₆	C₂²	1	2	yes	—
7	C₂²×C₃×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	#869	—	2	2	no	—
13	#869	C₄	1	4	yes	complemented
14	C₃×C₄×C₈	—	2	2	no	—
15	#816	—	2	8	no	—
16	C₂×C₄⋊Q₈	—	3	1	no	—
17	C₂×#103	—	3	4	no	—
18	C₂×C₄×C₈	D₃	1	1	yes	characteristic
19	#384	—	3	1	no	—
20	C₂²×C₃×C₄	D₄	1	2	yes	—
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃	—	2	2	no	—
23	C₂²×Dic₃	—	2	2	no	—
24	C₂×Dic₆	—	2	4	no	—
25	C₂×Dic₆	—	2	4	no	—
26	C₂×Dic₆	—	2	2	no	—
27	C₂×Dic₆	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₈	—	2	2	no	—
29	C₂×C₃×C₈	—	2	2	no	—
30	C₂×C₃×C₈	D₄	1	4	yes	—
31	C₃×C₄²	C₂×C₄	1	2	yes	—
32	C₃×C₄²	D₄	1	2	yes	—
33	C₄⋊Dic₃	—	2	8	no	—
34	C₄⋊Dic₃	—	2	4	no	—
35	C₄⋊Dic₃	—	4	4	no	—
36	C₂²×Q₈	—	3	2	no	—
37	C₂²×C₈	C₂×D₃	1	2	yes	—
38	C₄⋊Q₈	—	3	4	no	—
39	C₄⋊Q₈	—	6	2	no	—
40	C₂×C₄⋊C₄	—	3	2	no	—
41	C₂×C₄⋊C₄	—	6	2	no	—
42	C₂×C₄⋊C₄	—	6	2	no	—
43	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
44	#103	—	6	8	no	—
45	C₄×C₈	—	2	2	no	—
46	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
48	C₂×C₃×C₄	SD₁₆	1	4	yes	—
49	C₂×C₃×C₄	Q₁₆	1	4	yes	—
50	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
52	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄	#41	1	2	yes	—
54	C₂×Dic₃	—	2	4	no	—
55	C₂×Dic₃	—	4	2	no	—
56	C₂×Dic₃	—	4	2	no	—
57	C₂×Dic₃	—	4	2	no	—
58	C₂×Dic₃	—	4	4	no	—
59	Dic₆	—	2	8	no	—
60	Dic₆	—	4	4	no	—
61	C₃×C₈	—	2	4	no	—
62	C₂×Q₈	—	3	4	no	—
63	C₂×Q₈	—	6	2	no	—
64	C₂×Q₈	—	6	4	no	—
65	C₂×Q₈	—	6	4	no	—
66	C₂²×C₄	—	6	2	no	—
67	C₂²×C₄	—	6	2	no	—
68	C₂²×C₄	D₁₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	D₁₂	1	4	yes	—
73	C₄⋊C₄	—	6	8	no	—
74	C₄⋊C₄	—	6	4	no	—
75	C₄⋊C₄	—	12	4	no	—
76	C₄²	C₄×D₃	1	2	yes	—
77	C₄²	#67	1	2	yes	—
78	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₃	#124	1	1	yes	characteristic
80	C₂²×C₃	#127	1	1	yes	characteristic
81	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
82	C₂²×C₃	C₂×Q₁₆	1	2	yes	—
83	C₃×C₄	#103	1	8	yes	complemented
84	C₃×C₄	C₄×D₄	1	2	yes	—
85	C₃×C₄	#121	1	2	yes	—
86	Dic₃	—	4	4	no	—
87	Dic₃	—	8	2	no	—
88	C₂³	C₂×D₁₂	1	1	yes	characteristic
89	Q₈	—	6	8	no	—
90	Q₈	—	12	4	no	—
91	C₂×C₄	—	6	4	no	—
92	C₂×C₄	—	12	2	no	—
93	C₂×C₄	—	12	2	no	—
94	C₂×C₄	—	12	2	no	—
95	C₂×C₄	—	12	4	no	—
96	C₂×C₄	#204	1	4	yes	—
97	C₂×C₄	Dic₁₂	1	4	yes	—
98	C₂×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
100	C₂×C₄	C₂×D₁₂	1	2	yes	—
101	C₂×C₄	#235	1	2	yes	—
102	C₂×C₄	C₂×#67	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂×C₃	#390	1	1	yes	characteristic
105	C₂×C₃	C₂×#103	1	2	yes	—
106	C₂×C₃	#487	1	2	yes	—
107	C₂×C₃	#492	1	1	yes	characteristic
108	C₂×C₃	#494	1	1	yes	characteristic
109	C₂²	#874	1	1	yes	characteristic
110	C₂²	#875	1	1	yes	characteristic
111	C₂²	C₂×#204	1	2	yes	—
112	C₂²	C₂×Dic₁₂	1	2	yes	—
113	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
114	C₄	—	12	4	no	—
115	C₄	—	24	2	no	—
116	C₄	#816	1	8	yes	complemented
117	C₄	C₄×D₁₂	1	2	yes	—
118	C₄	#930	1	2	yes	—
119	C₃	#1956	1	1	yes	characteristic, complemented, Sylow
120	C₂	#4681	1	2	yes	—
121	C₂	#4693	1	1	yes	characteristic
122	C₂	#4699	1	1	yes	characteristic
123	C₂	#4939	1	1	yes	characteristic
124	C₂	C₂×#816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#869

C₂

yes

maximal, characteristic

C₂×C₃×C₄×C₈

C₂

yes

maximal, characteristic

C₂×#816

C₂

yes

maximal

#4935

C₂

yes

maximal, characteristic

#1956

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂²

yes

—

#869

—

#869

C₄

yes

complemented

C₃×C₄×C₈

—

#816

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

D₃

yes

characteristic

#384

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃

yes

characteristic

#103

—

C₄×C₈

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

Q₁₆

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

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃

yes

—

C₄²

#67

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

C₂×SD₁₆

yes

—

C₂²×C₃

C₂×Q₁₆

yes

—

C₃×C₄

#103

yes

complemented

C₃×C₄

C₄×D₄

yes

—

C₃×C₄

#121

yes

—

Dic₃

—

Dic₃

—

C₂³

C₂×D₁₂

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#204

yes

—

C₂×C₄

Dic₁₂

yes

—

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₃

yes

characteristic

100

C₂×C₄

C₂×D₁₂

yes

—

101

C₂×C₄

#235

yes

—

102

C₂×C₄

C₂×#67

yes

characteristic

103

C₈

—

104

C₂×C₃

#390

yes

characteristic

105

C₂×C₃

C₂×#103

yes

—

106

C₂×C₃

#487

yes

—

107

C₂×C₃

#492

yes

characteristic

108

C₂×C₃

#494

yes

characteristic

109

C₂²

#874

yes

characteristic

110

C₂²

#875

yes

characteristic

111

C₂²

C₂×#204

yes

—

112

C₂²

C₂×Dic₁₂

yes

—

113

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#816

yes

complemented

117

C₄

C₄×D₁₂

yes

—

118

C₄

#930

yes

—

119

C₃

#1956

yes

characteristic, complemented, Sylow

120

C₂

#4681

yes

—

121

C₂

#4693

yes

characteristic

122

C₂

#4699

yes

characteristic

123

C₂

#4939

yes

characteristic

124

C₂

C₂×#816

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1956
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×C₄×C₈

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₂×C₄×C₈

D₃

hypocenter

C₃

#1956

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
13. #869	C₄	semidirect
83. C₃×C₄	#103	semidirect
116. C₄	#816	semidirect
119. C₃	#1956	semidirect

H ≅ G/N

type

13. #869

C₄

semidirect

83. C₃×C₄

#103

semidirect

116. C₄

#816

semidirect

119. C₃

#1956

semidirect

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

subgroup	index
C₂×#869	2
C₂×C₃×C₄×C₈	2
C₂×#816	2
#4935	2
#1956	3

subgroup

index

C₂×C₃×C₄×C₈

Subgroups — #69771

Conjugacy classes of subgroups

Complements

Maximal subgroups