Group #36378357 — subgroups

Subgroups — #36378357

Order 22592.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#19150736	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#26014992	C₂	1	1	yes	maximal, characteristic, complemented
4	#26014990	C₂	1	1	yes	maximal, characteristic, complemented
5	#26014979	C₂	1	1	yes	maximal, characteristic, complemented
6	#26014974	C₂	1	4	yes	maximal, complemented
7	#26014969	C₂	1	4	yes	maximal, complemented
8	#26014949	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#19150736	—	2	4	no	—
11	C₂×Dic₇₀₆	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₀₆	C₂²	1	1	yes	characteristic
13	C₂×Dic₇₀₆	C₂²	1	2	yes	—
14	Q₈×D₃₅₃	—	2	4	no	—
15	C₂×C₂²⋊D₃₅₃	C₂²	1	2	yes	—
16	C₂²×Dic₃₅₃	C₂²	1	2	yes	—
17	C₂×C₄×D₃₅₃	C₂²	1	2	yes	—
18	C₂×C₄×D₃₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₃₅₃	C₂²	1	4	yes	—
20	#19150716	C₂²	1	4	yes	—
21	#19150716	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₅₃	C₂²	1	4	yes	—
23	#19150714	C₂²	1	2	yes	—
24	C₄×Dic₃₅₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₃₅₃	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₃₅₃	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₃₅₃	C₂²	1	4	yes	—
28	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
29	C₂²×D₃₅₃	C₂³	1	1	yes	characteristic
30	Dic₇₀₆	—	2	2	no	—
31	Dic₇₀₆	—	2	2	no	—
32	Dic₇₀₆	—	2	4	no	—
33	C₂²⋊D₃₅₃	—	2	4	no	—
34	C₂×Dic₃₅₃	—	2	4	no	—
35	C₂×Dic₃₅₃	C₂³	1	2	yes	—
36	C₂×Dic₃₅₃	C₂³	1	2	yes	—
37	C₂×Dic₃₅₃	C₂³	1	2	yes	—
38	C₂×Dic₃₅₃	C₂³	1	1	yes	characteristic
39	C₄×D₃₅₃	—	2	4	no	—
40	C₄×D₃₅₃	D₄	1	4	yes	—
41	C₂³×C₃₅₃	C₂³	1	2	yes	—
42	Q₈×C₃₅₃	—	2	2	no	—
43	D₄×C₃₅₃	—	2	2	no	—
44	C₂×C₄×C₃₅₃	C₂³	1	2	yes	—
45	C₂×C₄×C₃₅₃	C₂³	1	2	yes	—
46	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₃₅₃	—	2	2	no	—
48	C₂²×C₃₅₃	—	2	4	no	—
49	C₂²×C₃₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×D₃₅₃	—	2	2	no	—
51	C₂×D₃₅₃	C₂×D₄	1	2	yes	—
52	C₄×C₃₅₃	—	2	2	no	—
53	C₄×C₃₅₃	—	2	2	no	—
54	C₄×C₃₅₃	C₂×D₄	1	2	yes	—
55	Dic₃₅₃	—	2	2	no	—
56	Dic₃₅₃	—	2	2	no	—
57	Dic₃₅₃	—	2	2	no	—
58	Dic₃₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₃₅₃	—	4	2	no	—
60	C₂×C₃₅₃	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₃₅₃	2₋¹⁺⁴	1	2	yes	—
62	D₃₅₃	—	2	2	no	—
63	C₃₅₃	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	353	1	no	maximal, Sylow
65	C₂×#41	—	353	1	no	—
66	C₂²×Q₈	—	353	1	no	—
67	C₄⋊Q₈	—	353	1	no	—
68	C₄⋊Q₈	—	353	1	no	—
69	#124	—	353	1	no	—
70	#124	D₃₅₃	1	1	yes	characteristic, complemented
71	#123	—	353	4	no	—
72	C₂²⋊Q₈	—	353	4	no	—
73	#117	—	353	1	no	—
74	#41	—	706	4	no	—
75	C₂×Q₈	—	353	2	no	—
76	C₂×Q₈	—	353	1	no	—
77	C₂×Q₈	—	353	1	no	—
78	C₂×Q₈	—	706	4	no	—
79	C₂×Q₈	C₂×D₃₅₃	1	1	yes	characteristic
80	C₂×D₄	—	353	2	no	—
81	C₂×D₄	C₂×D₃₅₃	1	1	yes	characteristic
82	C₂²×C₄	—	353	1	no	—
83	C₂²×C₄	—	353	2	no	—
84	C₂²×C₄	—	353	2	no	—
85	C₄⋊C₄	—	353	2	no	—
86	C₄⋊C₄	—	353	4	no	—
87	C₄⋊C₄	—	353	4	no	—
88	C₂²⋊C₄	—	353	4	no	—
89	C₂²⋊C₄	—	353	2	no	—
90	C₂²⋊C₄	C₂×D₃₅₃	1	4	yes	—
91	C₄²	—	353	1	no	—
92	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
93	C₂³	—	353	1	no	—
94	C₂³	C₂²×D₃₅₃	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	706	4	no	—
97	Q₈	—	706	2	no	—
98	Q₈	—	706	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	706	4	no	—
101	C₂×C₄	—	353	1	no	—
102	C₂×C₄	—	353	4	no	—
103	C₂×C₄	—	353	2	no	—
104	C₂×C₄	—	353	2	no	—
105	C₂×C₄	—	353	2	no	—
106	C₂×C₄	—	706	4	no	—
107	C₂×C₄	—	706	4	no	—
108	C₂×C₄	C₂²×D₃₅₃	1	2	yes	—
109	C₂×C₄	C₂²×D₃₅₃	1	2	yes	—
110	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	353	2	no	—
114	C₂²	—	706	2	no	—
115	C₂²	C₂³×D₃₅₃	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	353	2	no	—
119	C₄	—	706	2	no	—
120	C₄	—	706	2	no	—
121	C₄	—	706	2	no	—
122	C₄	D₄×D₃₅₃	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	706	2	no	—
125	C₂	C₂×D₄×D₃₅₃	1	1	yes	characteristic
126	C₂	#26015039	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19150736

C₂

yes

maximal, characteristic

C₂×Q₈×D₃₅₃

C₂

yes

maximal, characteristic, complemented

#26014992

C₂

yes

maximal, characteristic, complemented

#26014990

C₂

yes

maximal, characteristic, complemented

#26014979

C₂

yes

maximal, characteristic, complemented

#26014974

C₂

yes

maximal, complemented

#26014969

C₂

yes

maximal, complemented

#26014949

C₂

yes

maximal, characteristic, complemented

#124×C₃₅₃

C₂

yes

maximal, characteristic, complemented

#19150736

—

C₂×Dic₇₀₆

C₂²

yes

characteristic

C₂×Dic₇₀₆

C₂²

yes

characteristic

C₂×Dic₇₀₆

C₂²

yes

—

Q₈×D₃₅₃

—

C₂×C₂²⋊D₃₅₃

C₂²

yes

—

C₂²×Dic₃₅₃

C₂²

yes

—

C₂×C₄×D₃₅₃

C₂²

yes

—

C₂×C₄×D₃₅₃

C₂²

yes

characteristic

C₄⋊Dic₃₅₃

C₂²

yes

—

#19150716

C₂²

yes

—

#19150716

C₂²

yes

—

C₂²⋊Dic₃₅₃

C₂²

yes

—

#19150714

C₂²

yes

—

C₄×Dic₃₅₃

C₂²

yes

characteristic

C₂×Q₈×C₃₅₃

C₂²

yes

characteristic

C₂×D₄×C₃₅₃

C₂²

yes

characteristic

C₂²⋊C₄×C₃₅₃

C₂²

yes

—

C₄²×C₃₅₃

C₂²

yes

characteristic

C₂²×D₃₅₃

C₂³

yes

characteristic

Dic₇₀₆

—

Dic₇₀₆

—

Dic₇₀₆

—

C₂²⋊D₃₅₃

—

C₂×Dic₃₅₃

—

C₂×Dic₃₅₃

C₂³

yes

—

C₂×Dic₃₅₃

C₂³

yes

—

C₂×Dic₃₅₃

C₂³

yes

—

C₂×Dic₃₅₃

C₂³

yes

characteristic

C₄×D₃₅₃

—

C₄×D₃₅₃

D₄

yes

—

C₂³×C₃₅₃

C₂³

yes

—

Q₈×C₃₅₃

—

D₄×C₃₅₃

—

C₂×C₄×C₃₅₃

C₂³

yes

—

C₂×C₄×C₃₅₃

C₂³

yes

—

C₂×C₄×C₃₅₃

C₂³

yes

characteristic

C₂²×C₃₅₃

—

C₂²×C₃₅₃

—

C₂²×C₃₅₃

C₂⁴

yes

characteristic

C₂×D₃₅₃

—

C₂×D₃₅₃

C₂×D₄

yes

—

C₄×C₃₅₃

—

C₄×C₃₅₃

—

C₄×C₃₅₃

C₂×D₄

yes

—

Dic₃₅₃

—

Dic₃₅₃

—

Dic₃₅₃

—

Dic₃₅₃

C₂×D₄

yes

—

C₂×C₃₅₃

—

C₂×C₃₅₃

C₂²×D₄

yes

characteristic

C₂×C₃₅₃

2₋¹⁺⁴

yes

—

D₃₅₃

—

C₃₅₃

#536

yes

characteristic, complemented, Sylow

#536

—

353

maximal, Sylow

C₂×#41

—

353

—

C₂²×Q₈

—

353

—

C₄⋊Q₈

—

353

—

C₄⋊Q₈

—

353

—

#124

—

353

—

#124

D₃₅₃

yes

characteristic, complemented

#123

—

353

—

C₂²⋊Q₈

—

353

—

#117

—

353

—

#41

—

706

—

C₂×Q₈

—

353

—

C₂×Q₈

—

353

—

C₂×Q₈

—

353

—

C₂×Q₈

—

706

—

C₂×Q₈

C₂×D₃₅₃

yes

characteristic

C₂×D₄

—

353

—

C₂×D₄

C₂×D₃₅₃

yes

characteristic

C₂²×C₄

—

353

—

C₂²×C₄

—

353

—

C₂²×C₄

—

353

—

C₄⋊C₄

—

353

—

C₄⋊C₄

—

353

—

C₄⋊C₄

—

353

—

C₂²⋊C₄

—

353

—

C₂²⋊C₄

—

353

—

C₂²⋊C₄

C₂×D₃₅₃

yes

—

C₄²

—

353

—

C₄²

C₂×D₃₅₃

yes

characteristic

C₂³

—

353

—

C₂³

C₂²×D₃₅₃

yes

—

Q₈

—

Q₈

—

706

—

Q₈

—

706

—

Q₈

—

706

—

D₄

—

100

D₄

—

706

—

101

C₂×C₄

—

353

—

102

C₂×C₄

—

353

—

103

C₂×C₄

—

353

—

104

C₂×C₄

—

353

—

105

C₂×C₄

—

353

—

106

C₂×C₄

—

706

—

107

C₂×C₄

—

706

—

108

C₂×C₄

C₂²×D₃₅₃

yes

—

109

C₂×C₄

C₂²×D₃₅₃

yes

—

110

C₂×C₄

C₂²×D₃₅₃

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

353

—

114

C₂²

—

706

—

115

C₂²

C₂³×D₃₅₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

353

—

119

C₄

—

706

—

120

C₄

—

706

—

121

C₄

—

706

—

122

C₄

D₄×D₃₅₃

yes

—

123

C₂

—

124

C₂

—

706

—

125

C₂

C₂×D₄×D₃₅₃

yes

characteristic

126

C₂

#26015039

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₃
commutator, socle, cosocle	C₂²×C₃₅₃	C₂⁴
Fitting	#124×C₃₅₃	C₂
prime radical	C₂×#19150736	C₂
hypercenter	#124	D₃₅₃
hypocenter	C₃₅₃	#536
LCS	G ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₅₃

commutator, socle, cosocle

C₂²×C₃₅₃

C₂⁴

Fitting

#124×C₃₅₃

C₂

C₂

D₃₅₃

C₃₅₃

G ▷ C₂²×C₃₅₃ ▷ C₃₅₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₃₅₃	C₂	semidirect
3. #26014992	C₂	semidirect
4. #26014990	C₂	semidirect
5. #26014979	C₂	semidirect
6. #26014974	C₂	semidirect
7. #26014969	C₂	semidirect
8. #26014949	C₂	semidirect
9. #124×C₃₅₃	C₂	semidirect
63. C₃₅₃	#536	semidirect
70. #124	D₃₅₃	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₃₅₃

C₂

semidirect

3. #26014992

C₂

semidirect

4. #26014990

C₂

semidirect

5. #26014979

C₂

semidirect

6. #26014974

C₂

semidirect

7. #26014969

C₂

semidirect

8. #26014949

C₂

semidirect

9. #124×C₃₅₃

C₂

semidirect

63. C₃₅₃

#536

semidirect

70. #124

D₃₅₃

semidirect

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

subgroup	index
C₂×#19150736	2
C₂×Q₈×D₃₅₃	2
#26014992	2
#26014990	2
#26014979	2
#26014974	2
#26014969	2
#26014949	2
#124×C₃₅₃	2
#536	353

subgroup

index

C₂×Q₈×D₃₅₃

#124×C₃₅₃

353

Subgroups — #36378357

Conjugacy classes of subgroups

Complements

Maximal subgroups