Group #71789 — subgroups

Subgroups — #71789

Order 384.

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

normal subgroups only

This group has 292 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	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#836	C₂	1	4	yes	maximal
3	#4924	C₂	1	1	yes	maximal, characteristic
4	C₄×#208	C₂	1	1	yes	maximal, characteristic
5	#1966	—	3	1	no	maximal, Sylow
6	C₂²×C₃×Q₈	C₂²	1	2	yes	—
7	C₃×C₄⋊Q₈	—	2	1	no	—
8	C₃×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₄⋊Q₈	—	2	2	no	—
10	C₂×C₃×C₄⋊C₄	C₄	1	4	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×#208	C₂²	1	2	yes	—
13	C₂×#208	C₂²	1	2	yes	—
14	#836	—	2	8	no	—
15	C₄⋊(C₃⋊C₈)	—	2	2	no	—
16	#803	—	2	1	no	—
17	C₄×C₃⋊C₈	—	2	1	no	—
18	C₂×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	#423	—	3	1	no	—
20	C₂×#101	—	3	4	no	—
21	C₄×M₁₆	—	3	1	no	—
22	C₂×C₃×Q₈	—	2	4	no	—
23	C₂×C₃×Q₈	—	2	2	no	—
24	C₂×C₃×Q₈	—	2	2	no	—
25	C₂×C₃×Q₈	—	2	4	no	—
26	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₃×C₄⋊C₄	—	2	4	no	—
30	C₃×C₄⋊C₄	—	2	4	no	—
31	C₃×C₄²	D₄	1	1	yes	characteristic
32	C₃×C₄²	D₄	1	1	yes	characteristic
33	C₃×C₄²	D₄	1	2	yes	—
34	#208	—	2	2	no	—
35	#208	—	2	4	no	—
36	#208	D₄	1	4	yes	—
37	C₂×C₃⋊C₈	—	2	2	no	—
38	C₂×C₃⋊C₈	—	2	2	no	—
39	C₂²×Q₈	C₂×D₃	1	2	yes	—
40	C₂×M₁₆	—	3	2	no	—
41	C₂×M₁₆	—	3	2	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₄⋊Q₈	—	2	1	no	—
44	C₄⋊Q₈	—	2	2	no	—
45	C₂×C₄⋊C₄	Dic₃	1	4	yes	—
46	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₄⋊C₈	—	6	2	no	—
48	#101	—	6	8	no	—
49	#97	—	6	1	no	—
50	C₄×C₈	—	6	1	no	—
51	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
52	C₃×Q₈	—	4	4	no	—
53	C₃×Q₈	—	4	2	no	—
54	C₃×Q₈	—	4	2	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	—	2	4	no	—
57	C₂×C₃×C₄	—	2	8	no	—
58	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄	#41	1	2	yes	—
64	C₃⋊C₈	—	2	4	no	—
65	C₃⋊C₈	—	4	2	no	—
66	C₂×Q₈	—	2	4	no	—
67	C₂×Q₈	—	2	4	no	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂²×C₄	C₂×Dic₃	1	4	yes	—
71	C₂²×C₄	C₂×Dic₃	1	2	yes	—
72	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
73	M₁₆	—	3	4	no	—
74	M₁₆	—	6	2	no	—
75	M₁₆	—	6	4	no	—
76	C₂×C₈	—	6	2	no	—
77	C₂×C₈	—	6	2	no	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄²	#67	1	1	yes	characteristic
81	C₄²	#67	1	1	yes	characteristic
82	C₄²	#67	1	2	yes	—
83	C₂²×C₃	—	2	2	no	—
84	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₃	C₄×D₄	1	2	yes	—
86	C₃×C₄	—	2	2	no	—
87	C₃×C₄	—	4	4	no	—
88	C₃×C₄	#101	1	4	yes	complemented
89	C₃×C₄	#121	1	2	yes	—
90	C₃×C₄	#124	1	1	yes	characteristic
91	C₃×C₄	#127	1	1	yes	characteristic
92	C₂³	C₂²×Dic₃	1	1	yes	characteristic
93	Q₈	—	4	2	no	—
94	Q₈	—	4	4	no	—
95	Q₈	—	4	2	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	#217	1	2	yes	—
100	C₂×C₄	#217	1	2	yes	—
101	C₂×C₄	D₃×D₄	1	2	yes	—
102	C₂×C₄	#237	1	2	yes	—
103	C₂×C₄	C₂×#67	1	1	yes	characteristic
104	C₂×C₄	C₂×#67	1	1	yes	characteristic
105	C₈	—	6	4	no	—
106	C₈	—	12	2	no	—
107	C₂×C₃	—	2	2	no	—
108	C₂×C₃	#390	1	1	yes	characteristic
109	C₂×C₃	C₂×#101	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	D₄×Dic₃	1	2	yes	—
112	C₂²	C₂×#217	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	4	4	no	—
115	C₄	#836	1	4	yes	complemented
116	C₄	#936	1	1	yes	characteristic
117	C₄	#938	1	2	yes	—
118	C₄	#940	1	1	yes	characteristic
119	C₃	#1966	1	1	yes	characteristic, complemented, Sylow
120	C₂	—	2	2	no	—
121	C₂	#5221	1	1	yes	characteristic
122	C₂	C₂×#836	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#836

C₂

yes

maximal

#4924

C₂

yes

maximal, characteristic

C₄×#208

C₂

yes

maximal, characteristic

#1966

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₂×C₃×C₄⋊C₄

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

C₂×#208

C₂²

yes

—

#836

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₄×C₃⋊C₈

—

C₂×C₄⋊Q₈

D₃

yes

characteristic

#423

—

C₂×#101

—

C₄×M₁₆

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

—

#208

—

#208

—

#208

D₄

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

C₂×D₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

#101

—

#97

—

C₄×C₈

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

yes

C₄²

yes

C₄²

yes

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

C₄×D₄

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#101

yes

complemented

C₃×C₄

#121

yes

—

C₃×C₄

yes

C₃×C₄

yes

C₂³

C₂²×Dic₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#217

yes

—

100

C₂×C₄

#217

yes

—

101

C₂×C₄

D₃×D₄

yes

—

102

C₂×C₄

#237

yes

—

103

C₂×C₄

C₂×#67

yes

characteristic

104

C₂×C₄

C₂×#67

yes

characteristic

105

C₈

—

106

C₈

—

107

C₂×C₃

—

108

C₂×C₃

#390

yes

characteristic

109

C₂×C₃

C₂×#101

yes

—

110

C₂²

—

111

C₂²

D₄×Dic₃

yes

—

112

C₂²

C₂×#217

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

#836

yes

complemented

116

C₄

#936

yes

characteristic

117

C₄

#938

yes

—

118

C₄

#940

yes

characteristic

119

C₃

#1966

yes

characteristic, complemented, Sylow

120

C₂

—

121

C₂

#5221

yes

characteristic

122

C₂

C₂×#836

yes

—

	subgroup	G/H
center	C₂²	C₂×#217
commutator, prime radical	C₂³×C₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	#1966
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×#217

commutator, prime radical

C₂³×C₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×C₄⋊Q₈

C₂

socle

C₂²×C₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₂×C₄⋊Q₈

D₃

hypocenter

C₃

#1966

LCS

G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
88. C₃×C₄	#101	semidirect
115. C₄	#836	semidirect
119. C₃	#1966	semidirect

H ≅ G/N

type

88. C₃×C₄

#101

semidirect

115. C₄

#836

semidirect

119. C₃

#1966

semidirect

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

subgroup	index
C₂×C₃×C₄⋊Q₈	2
C₂×#836	2
#4924	2
C₄×#208	2
#1966	3

subgroup

index

C₂×C₃×C₄⋊Q₈

Subgroups — #71789

Conjugacy classes of subgroups

Complements

Maximal subgroups