Group #21456368 — subgroups

Subgroups — #21456368

Order 7808.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16840172	C₂	1	4	yes	maximal, complemented
3	#16840161	C₂	1	4	yes	maximal, complemented
4	#16839802	C₂	1	4	yes	maximal
5	C₄×#11758194	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×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	#14519710	C₂²	1	8	yes	—
14	#14519708	C₂²	1	8	yes	—
15	#14519688	C₂²	1	2	yes	—
16	C₄×#92418	C₂²	1	2	yes	—
17	#125×C₆₁	—	2	2	no	—
18	#125×C₆₁	C₂²	1	2	yes	—
19	#125×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
21	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	D₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₄⋊C₄×C₆₁	—	2	4	no	—
28	C₄⋊C₄×C₆₁	—	2	4	no	—
29	C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
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	Q₈	1	8	yes	complemented
40	C₂×#92418	C₂³	1	4	yes	—
41	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	4	no	—
43	C₂×C₄×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	—	4	4	no	—
45	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	4	no	—
51	Dic₁₂₂	—	4	4	no	—
52	#92418	C₂×Q₈	1	8	yes	—
53	C₂²×C₆₁	—	2	2	no	—
54	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	—	2	2	no	—
58	C₄×C₆₁	—	4	4	no	—
59	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
60	C₄×C₆₁	C₂²×Q₈	1	2	yes	—
61	Dic₆₁	—	4	4	no	—
62	#3166	—	61	1	no	maximal, Sylow
63	C₂×C₆₁	—	2	2	no	—
64	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₆₁	#575	1	2	yes	—
66	#533	—	61	1	no	—
67	C₂×#125	D₆₁	1	1	yes	characteristic
68	#501	—	61	4	no	—
69	#499	—	61	4	no	—
70	#428	—	61	4	no	—
71	C₄×M₁₆	—	61	1	no	—
72	C₆₁	#3166	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	—	61	2	no	—
74	C₄⋊Q₈	—	61	2	no	—
75	#125	—	2	2	no	—
76	#125	—	61	2	no	—
77	#125	C₂×D₆₁	1	2	yes	—
78	#125	C₂×D₆₁	1	2	yes	—
79	C₂²⋊Q₈	—	122	2	no	—
80	C₄×Q₈	—	122	2	no	—
81	#117	—	61	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
84	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
85	#107	—	61	8	no	—
86	#106	—	61	8	no	—
87	#97	—	61	2	no	—
88	C₄×C₈	—	61	2	no	—
89	C₂×Q₈	—	122	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
92	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
93	M₁₆	—	61	8	no	—
94	C₂×C₈	—	61	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	61	4	no	—
99	C₄⋊C₄	—	122	4	no	—
100	C₄⋊C₄	—	122	2	no	—
101	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
102	C₂²⋊C₄	—	122	2	no	—
103	C₄²	—	122	2	no	—
104	C₄²	C₂²⋊D₆₁	1	2	yes	—
105	C₄²	C₂²×D₆₁	1	1	yes	characteristic
106	C₄²	C₂²×D₆₁	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
108	Q₈	—	244	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	122	4	no	—
113	C₂×C₄	Q₈×D₆₁	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
118	C₈	—	61	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#14519775	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	244	4	no	—
126	C₄	#14519775	1	2	yes	—
127	C₄	C₂×Q₈×D₆₁	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#14519775	1	1	yes	characteristic
130	C₂	#16840560	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16840172

C₂

yes

maximal, complemented

#16840161

C₂

yes

maximal, complemented

#16839802

C₂

yes

maximal

C₄×#11758194

C₂

yes

maximal, characteristic

C₂×#125×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

#14519777

C₂²

yes

—

#14519769

C₂²

yes

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

C₂²

yes

—

#14519710

C₂²

yes

—

#14519708

C₂²

yes

—

#14519688

C₂²

yes

—

C₄×#92418

C₂²

yes

—

#125×C₆₁

—

#125×C₆₁

C₂²

yes

—

#125×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

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

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

—

Dic₆₁

—

#3166

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

#575

yes

—

#533

—

C₂×#125

D₆₁

yes

—

—

—

C₄×M₁₆

—

C₆₁

#3166

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

—

#125

C₂×D₆₁

yes

—

#125

C₂×D₆₁

yes

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

—

—

—

C₄×C₈

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

100

C₄⋊C₄

—

122

—

101

C₄⋊C₄

C₂²×D₆₁

yes

—

102

C₂²⋊C₄

—

122

—

103

C₄²

—

122

—

104

C₄²

C₂²⋊D₆₁

yes

—

105

C₄²

C₂²×D₆₁

yes

characteristic

106

C₄²

C₂²×D₆₁

yes

characteristic

107

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

108

Q₈

—

244

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

122

—

113

C₂×C₄

Q₈×D₆₁

yes

—

114

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

115

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

116

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

117

C₂×C₄

C₂³×D₆₁

yes

characteristic

118

C₈

—

119

C₂²

—

120

C₂²

#14519775

yes

—

121

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

244

—

126

C₄

#14519775

yes

—

127

C₄

C₂×Q₈×D₆₁

yes

—

128

C₂

—

129

C₂

C₂×#14519775

yes

characteristic

130

C₂

#16840560

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#125×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₂×#125	D₆₁
hypocenter	C₆₁	#3166
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#125×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #16840172	C₂	semidirect
3. #16840161	C₂	semidirect
39. #11758194	Q₈	semidirect
72. C₆₁	#3166	semidirect

H ≅ G/N

type

2. #16840172

C₂

semidirect

3. #16840161

C₂

semidirect

39. #11758194

Q₈

semidirect

72. C₆₁

#3166

semidirect

Subgroups — #21456368

Conjugacy classes of subgroups

Complements