Group #21445432 — subgroups

Subgroups — #21445432

Order 7808.

178 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840114	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840112	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840110	C₂	1	1	yes	maximal, characteristic
4	#16840110	C₂	1	1	yes	maximal, characteristic
5	#16840109	C₂	1	1	yes	maximal, characteristic
6	C₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519777	C₂²	1	1	yes	characteristic
9	#14519774	C₂²	1	1	yes	characteristic
10	#14519701	—	2	2	no	—
11	#14519700	—	2	2	no	—
12	#14519699	—	2	2	no	—
13	#14519697	—	2	2	no	—
14	#14519697	—	2	2	no	—
15	C₄×#92418	C₂²	1	1	yes	characteristic
16	C₄×#92418	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	#103×C₆₁	—	2	2	no	—
20	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₆₁	—	2	1	no	—
22	C₂×Q₈×C₆₁	—	2	1	no	—
23	C₂×D₂₄₄	—	2	1	no	—
24	C₂×D₂₄₄	—	2	1	no	—
25	C₂×D₂₄₄	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	1	no	—
27	C₂×C₈×C₆₁	D₄	1	2	yes	—
28	C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₂×#92418	D₄	1	2	yes	—
36	C₂×#92418	D₄	1	2	yes	—
37	C₂²×D₆₁	—	4	1	no	—
38	Q₈×C₆₁	—	4	2	no	—
39	Q₈×C₆₁	—	4	2	no	—
40	C₂×C₄×C₆₁	—	4	1	no	—
41	C₂×C₄×C₆₁	—	4	1	no	—
42	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₆₁	—	4	1	no	—
46	D₂₄₄	—	4	2	no	—
47	D₂₄₄	—	4	2	no	—
48	D₂₄₄	—	4	2	no	—
49	Dic₁₂₂	—	4	2	no	—
50	C₈×C₆₁	—	2	2	no	—
51	#92418	—	2	2	no	—
52	#92418	—	2	2	no	—
53	C₂²×C₆₁	#127	1	1	yes	characteristic
54	C₂×D₆₁	—	8	1	no	—
55	C₂×D₆₁	—	8	1	no	—
56	C₂×D₆₁	—	8	1	no	—
57	C₄×C₆₁	—	8	1	no	—
58	C₄×C₆₁	—	8	1	no	—
59	C₄×C₆₁	#135	1	2	yes	—
60	C₄×C₆₁	#135	1	2	yes	—
61	C₄×C₆₁	#135	1	2	yes	—
62	Dic₆₁	—	8	1	no	—
63	#1711	—	61	1	no	maximal, Sylow
64	C₂×C₆₁	#495	1	1	yes	characteristic
65	C₂×C₆₁	#495	1	1	yes	characteristic
66	C₂×C₆₁	#495	1	1	yes	characteristic
67	D₆₁	—	16	1	no	—
68	#487	—	61	1	no	—
69	#487	—	61	1	no	—
70	#487	D₆₁	1	1	yes	characteristic, complemented
71	#486	—	61	1	no	—
72	#486	—	61	1	no	—
73	#486	—	61	1	no	—
74	C₈²	—	61	1	no	—
75	C₆₁	#1711	1	1	yes	characteristic, complemented, Sylow
76	C₄⋊Q₈	—	61	1	no	—
77	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
78	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
79	#127	—	61	1	no	—
80	#103	—	2	2	no	—
81	#103	—	122	2	no	—
82	#103	—	122	2	no	—
83	#102	—	122	2	no	—
84	#102	—	122	2	no	—
85	#102	—	122	2	no	—
86	C₄×C₈	—	61	1	no	—
87	C₄×C₈	—	61	1	no	—
88	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	122	1	no	—
92	C₂×D₄	—	122	1	no	—
93	C₂×D₄	—	122	1	no	—
94	C₂×D₄	—	122	1	no	—
95	C₂×C₈	—	61	2	no	—
96	C₂×C₈	—	61	2	no	—
97	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	122	1	no	—
103	C₄⋊C₄	—	122	1	no	—
104	C₄²	C₂²×D₆₁	1	1	yes	characteristic
105	C₂³	—	244	1	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	244	2	no	—
109	D₄	—	244	2	no	—
110	D₄	—	244	2	no	—
111	D₄	—	244	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	244	1	no	—
115	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
116	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	122	2	no	—
120	C₈	—	122	2	no	—
121	C₂²	—	488	1	no	—
122	C₂²	—	488	1	no	—
123	C₂²	—	488	1	no	—
124	C₂²	#14519773	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	488	1	no	—
128	C₄	#14519797	1	2	yes	—
129	C₄	#14519798	1	2	yes	—
130	C₄	#14519800	1	2	yes	—
131	C₂	—	976	1	no	—
132	C₂	#16840143	1	1	yes	characteristic
133	C₂	#16840144	1	1	yes	characteristic
134	C₂	#16840147	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840114

C₂

yes

maximal, characteristic, complemented

#16840112

C₂

yes

maximal, characteristic, complemented

#16840110

C₂

yes

maximal, characteristic

#16840110

C₂

yes

maximal, characteristic

#16840109

C₂

yes

maximal, characteristic

C₈×#92418

C₂

yes

maximal, characteristic, complemented

C₆₁×#487

C₂

yes

maximal, characteristic, complemented

#14519777

C₂²

yes

characteristic

#14519774

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#103×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

D₄

yes

—

C₂×#92418

D₄

yes

—

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

#92418

—

C₂²×C₆₁

#127

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

#135

yes

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#487

—

#487

—

#487

D₆₁

yes

characteristic, complemented

#486

—

#486

—

#486

—

C₈²

—

C₆₁

#1711

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

C₂×D₆₁

yes

—

—

—

122

—

#103

—

122

—

#102

—

122

—

#102

—

122

—

#102

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

122

—

103

C₄⋊C₄

—

122

—

104

C₄²

C₂²×D₆₁

yes

characteristic

105

C₂³

—

244

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

244

—

109

D₄

—

244

—

110

D₄

—

244

—

111

D₄

—

244

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

244

—

115

C₂×C₄

D₄×D₆₁

yes

characteristic

116

C₂×C₄

D₄×D₆₁

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

118

C₈

—

119

C₈

—

122

—

120

C₈

—

122

—

121

C₂²

—

488

—

122

C₂²

—

488

—

123

C₂²

—

488

—

124

C₂²

#14519773

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

488

—

128

C₄

#14519797

yes

—

129

C₄

#14519798

yes

—

130

C₄

#14519800

yes

—

131

C₂

—

976

—

132

C₂

#16840143

yes

characteristic

133

C₂

#16840144

yes

characteristic

134

C₂

#16840147

yes

characteristic

	subgroup	G/H
center	C₂²	#14519773
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#487	C₂
socle	C₂²×C₆₁	#127
hypercenter	#487	D₆₁
hypocenter	C₆₁	#1711
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#14519773

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#487

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
1. #16840114	C₂	semidirect
2. #16840112	C₂	semidirect
6. C₈×#92418	C₂	semidirect
7. C₆₁×#487	C₂	semidirect
70. #487	D₆₁	semidirect
75. C₆₁	#1711	semidirect

H ≅ G/N

type

1. #16840114

C₂

semidirect

2. #16840112

C₂

semidirect

6. C₈×#92418

C₂

semidirect

7. C₆₁×#487

C₂

semidirect

70. #487

D₆₁

semidirect

75. C₆₁

#1711

semidirect

Subgroups — #21445432

Conjugacy classes of subgroups

Complements