Group #21456199 — subgroups

Subgroups — #21456199

Order 7808.

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

normal subgroups only

This group has 380 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	#16840319	C₂	1	1	yes	maximal, characteristic
2	#16840130	C₂	1	4	yes	maximal, complemented
3	#16840112	C₂	1	4	yes	maximal, complemented
4	#16839771	C₂	1	4	yes	maximal
5	C₄×#11758194	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#11758194	C₂²	1	2	yes	—
8	#14519777	C₂²	1	2	yes	—
9	#14519769	C₂²	1	2	yes	—
10	#14519756	—	2	2	no	—
11	C₄×Dic₁₂₂	—	2	2	no	—
12	#14519736	C₂²	1	2	yes	—
13	#14519701	C₂²	1	16	yes	—
14	#14519688	C₂²	1	2	yes	—
15	C₄×#92418	C₂²	1	2	yes	—
16	C₂²×Q₈×C₆₁	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₆₁	—	2	2	no	—
18	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
21	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₆₁	—	2	2	no	—
23	C₂×Q₈×C₆₁	—	2	8	no	—
24	C₂×Q₈×C₆₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₆₁	—	2	2	no	—
26	C₂²×C₄×C₆₁	D₄	1	2	yes	—
27	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₂₂	—	2	2	no	—
29	C₄⋊C₄×C₆₁	—	2	4	no	—
30	C₄⋊C₄×C₆₁	—	2	4	no	—
31	C₄²×C₆₁	D₄	1	2	yes	—
32	C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₆₁	—	2	2	no	—
35	C₄⋊Dic₆₁	—	2	2	no	—
36	C₄⋊Dic₆₁	C₂³	1	4	yes	—
37	#11758196	—	2	4	no	—
38	C₄×Dic₆₁	—	2	2	no	—
39	#11758194	—	2	4	no	—
40	C₂×#92418	C₂³	1	4	yes	—
41	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₆₁	—	2	8	no	—
43	Q₈×C₆₁	—	4	4	no	—
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	—	2	4	no	—
46	C₂×C₄×C₆₁	—	4	4	no	—
47	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₁	#41	1	4	yes	—
51	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₆₁	—	2	4	no	—
53	Dic₁₂₂	—	4	4	no	—
54	#92418	—	2	4	no	—
55	C₂²×C₆₁	—	2	2	no	—
56	C₂²×C₆₁	#124	1	2	yes	—
57	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₆₁	—	2	2	no	—
59	C₄×C₆₁	—	4	4	no	—
60	C₄×C₆₁	#124	1	2	yes	—
61	C₄×C₆₁	#137	1	4	yes	—
62	C₄×C₆₁	C₂×#41	1	2	yes	—
63	Dic₆₁	—	4	4	no	—
64	#3139	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	C₂×#124	1	1	yes	characteristic
67	C₂×C₆₁	C₂×#137	1	2	yes	—
68	#533	—	61	1	no	—
69	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
70	#491	—	61	4	no	—
71	#487	—	61	4	no	—
72	#419	—	61	4	no	—
73	C₄×M₁₆	—	61	1	no	—
74	C₆₁	#3139	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	C₂×D₆₁	1	2	yes	—
76	C₂×M₁₆	—	61	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	61	2	no	—
79	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
80	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
81	#125	—	61	2	no	—
82	C₂²⋊Q₈	—	122	2	no	—
83	C₄×Q₈	—	122	2	no	—
84	#117	—	61	2	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
87	#103	—	61	16	no	—
88	#97	—	61	2	no	—
89	C₄×C₈	—	61	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	122	2	no	—
93	C₂×Q₈	C₂²×D₆₁	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
96	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
97	M₁₆	—	122	4	no	—
98	C₂×C₈	—	61	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	61	4	no	—
102	C₄⋊C₄	—	122	4	no	—
103	C₄⋊C₄	—	122	2	no	—
104	C₂²⋊C₄	—	122	2	no	—
105	C₄²	—	122	2	no	—
106	C₄²	C₂²⋊D₆₁	1	2	yes	—
107	C₄²	C₂²×D₆₁	1	1	yes	characteristic
108	C₄²	C₂²×D₆₁	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	244	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	122	4	no	—
117	C₂×C₄	#11758224	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
121	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
122	C₈	—	122	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#14519764	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	244	4	no	—
129	C₄	#14519764	1	2	yes	—
130	C₄	#14519808	1	4	yes	—
131	C₄	C₂×#11758224	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#14519764	1	1	yes	characteristic
134	C₂	C₂×#14519808	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16840130

C₂

yes

maximal, complemented

#16840112

C₂

yes

maximal, complemented

#16839771

C₂

yes

maximal

C₄×#11758194

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

#14519777

C₂²

yes

—

#14519769

C₂²

yes

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

C₂²

yes

—

#14519701

C₂²

yes

—

#14519688

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₂²×Q₈×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

#11758194

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

#124

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#124

yes

—

C₄×C₆₁

#137

yes

—

C₄×C₆₁

C₂×#41

yes

—

Dic₆₁

—

#3139

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×#124

yes

characteristic

C₂×C₆₁

C₂×#137

yes

—

#533

—

C₂×C₄⋊Q₈

D₆₁

yes

—

—

—

C₄×M₁₆

—

C₆₁

#3139

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₆₁

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

—

C₄⋊Q₈

C₂×D₆₁

yes

—

#125

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂²×D₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

122

—

103

C₄⋊C₄

—

122

—

104

C₂²⋊C₄

—

122

—

105

C₄²

—

122

—

106

C₄²

C₂²⋊D₆₁

yes

—

107

C₄²

C₂²×D₆₁

yes

characteristic

108

C₄²

C₂²×D₆₁

yes

characteristic

109

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

244

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

122

—

117

C₂×C₄

#11758224

yes

—

118

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

120

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

121

C₂×C₄

C₂³×D₆₁

yes

characteristic

122

C₈

—

122

—

123

C₂²

—

124

C₂²

#14519764

yes

—

125

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

244

—

129

C₄

#14519764

yes

—

130

C₄

#14519808

yes

—

131

C₄

C₂×#11758224

yes

—

132

C₂

—

133

C₂

C₂×#14519764

yes

characteristic

134

C₂

C₂×#14519808

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#3139
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₁

hypocenter

C₆₁

#3139

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16840130	C₂	semidirect
3. #16840112	C₂	semidirect
74. C₆₁	#3139	semidirect

H ≅ G/N

type

2. #16840130

C₂

semidirect

3. #16840112

C₂

semidirect

74. C₆₁

#3139

semidirect

Subgroups — #21456199

Conjugacy classes of subgroups

Complements