Group #18011244 — subgroups

Subgroups — #18011244

Order 4736.

484 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 204 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×#11924624	C₂	1	2	yes	maximal
2	#14558540	C₂	1	6	yes	maximal, complemented
3	#14558538	C₂	1	3	yes	maximal, complemented
4	D₃₇⋊(C₄⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₄×#10596966	C₂	1	1	yes	maximal, characteristic
6	#11924624	—	2	8	no	—
7	C₂×#10596953	C₂²	1	6	yes	—
8	C₂×D₃₇⋊C₈	C₂²	1	6	yes	—
9	#11924603	C₂²	1	1	yes	characteristic
10	#11924603	C₂²	1	3	yes	—
11	#11924603	C₂²	1	6	yes	—
12	C₄⋊#64438	C₂²	1	6	yes	—
13	C₂×#10596966	C₂²	1	1	yes	characteristic
14	Q₈×Dic₃₇	C₄	1	1	yes	characteristic
15	C₄×D₁₄₈	C₄	1	3	yes	—
16	#11924498	C₄	1	3	yes	—
17	#11924494	C₂²	1	3	yes	—
18	C₄²×D₃₇	C₂²	1	3	yes	—
19	C₄×Q₈×C₃₇	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
21	#10596966	D₄	1	4	yes	—
22	#10596966	Q₈	1	4	yes	—
23	C₂×D₁₄₈	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₃₇	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₃₇	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₃₇	C₂³	1	3	yes	—
27	C₂×#64438	C₂³	1	2	yes	—
28	C₂×#64438	C₂³	1	6	yes	—
29	#10596953	—	2	12	no	—
30	D₃₇⋊C₈	—	2	12	no	—
31	C₄⋊C₄×C₃₇	C₂×C₄	1	3	yes	—
32	C₄²×C₃₇	C₂×C₄	1	3	yes	—
33	#10596938	C₂×C₄	1	6	yes	—
34	C₄⋊Dic₃₇	C₂×C₄	1	3	yes	—
35	C₄×Dic₃₇	C₂×C₄	1	3	yes	—
36	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
37	C₄×Dic₃₇	C₂³	1	3	yes	—
38	C₂²×D₃₇	C₂²×C₄	1	3	yes	—
39	Q₈×C₃₇	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂²×C₄	1	3	yes	—
43	C₂×Dic₃₇	C₂²×C₄	1	3	yes	—
44	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
46	D₁₄₈	C₄⋊C₄	1	12	yes	—
47	C₄×D₃₇	—	2	6	no	—
48	C₄×D₃₇	—	4	3	no	—
49	C₄×D₃₇	C₂×D₄	1	6	yes	—
50	C₄×D₃₇	C₂×Q₈	1	6	yes	—
51	#64438	—	2	2	no	—
52	#64438	—	2	6	no	—
53	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
54	C₂×D₃₇	—	2	3	no	—
55	C₂×D₃₇	—	2	3	no	—
56	C₂×D₃₇	C₂×C₄⋊C₄	1	6	yes	—
57	C₄×C₃₇	—	2	3	no	—
58	C₄×C₃₇	—	2	1	no	—
59	C₄×C₃₇	C₂×C₄⋊C₄	1	6	yes	—
60	Dic₃₇	—	2	3	no	—
61	Dic₃₇	#131	1	2	yes	complemented
62	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
63	Dic₃₇	C₂²×Q₈	1	1	yes	characteristic
64	#2906	—	37	1	no	maximal, Sylow
65	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
67	C₂×C₃₇	#568	1	1	yes	characteristic
68	D₃₇	—	2	6	no	—
69	C₂×#131	—	37	2	no	—
70	C₄×#41	—	37	1	no	—
71	#424	—	37	6	no	—
72	#423	—	37	3	no	—
73	C₂×C₄⋊C₈	—	37	3	no	—
74	C₃₇	#2906	1	1	yes	characteristic, complemented, Sylow
75	C₂×#41	—	37	1	no	—
76	#131	—	74	8	no	—
77	C₂×M₁₆	—	37	6	no	—
78	C₂²×C₈	—	37	6	no	—
79	C₄×Q₈	—	37	1	no	—
80	C₄×Q₈	#3866	1	1	yes	characteristic
81	C₄×D₄	—	37	3	no	—
82	C₄×D₄	—	37	3	no	—
83	#117	—	37	3	no	—
84	C₂×C₄²	—	37	3	no	—
85	C₄⋊C₈	—	37	6	no	—
86	C₄⋊C₈	—	37	1	no	—
87	C₄⋊C₈	—	37	3	no	—
88	C₄⋊C₈	—	37	6	no	—
89	#41	—	37	4	no	—
90	#41	—	37	4	no	—
91	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
92	C₂×D₄	—	37	3	no	—
93	C₂²×C₄	—	37	3	no	—
94	C₂²×C₄	—	37	3	no	—
95	C₂²×C₄	—	37	3	no	—
96	M₁₆	—	74	12	no	—
97	C₂×C₈	—	37	2	no	—
98	C₂×C₈	—	37	6	no	—
99	C₂×C₈	—	74	12	no	—
100	C₄⋊C₄	—	37	3	no	—
101	C₄⋊C₄	C₂×#3866	1	3	yes	—
102	C₂²⋊C₄	—	37	6	no	—
103	C₄²	—	37	3	no	—
104	C₄²	—	37	1	no	—
105	C₄²	—	37	3	no	—
106	C₄²	C₂×#3866	1	3	yes	—
107	C₂³	—	37	3	no	—
108	Q₈	C₄⋊#3866	1	4	yes	—
109	D₄	—	37	12	no	—
110	C₂×C₄	—	37	1	no	—
111	C₂×C₄	—	37	1	no	—
112	C₂×C₄	—	37	6	no	—
113	C₂×C₄	—	37	6	no	—
114	C₂×C₄	—	37	3	no	—
115	C₂×C₄	—	74	6	no	—
116	C₂×C₄	—	148	3	no	—
117	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
118	C₂×C₄	C₂²×#3866	1	3	yes	—
119	C₂×C₄	C₂²×#3866	1	3	yes	—
120	C₈	—	74	2	no	—
121	C₈	—	74	6	no	—
122	C₂²	—	37	6	no	—
123	C₂²	—	74	3	no	—
124	C₂²	—	74	3	no	—
125	C₂²	C₂³×#3866	1	1	yes	characteristic
126	C₄	—	2	3	no	—
127	C₄	—	2	1	no	—
128	C₄	—	37	1	no	—
129	C₄	—	37	1	no	—
130	C₄	—	37	2	no	—
131	C₄	—	74	3	no	—
132	C₄	C₂×C₄⋊#3866	1	6	yes	—
133	C₂	—	74	6	no	—
134	C₂	C₂²×C₄⋊#3866	1	1	yes	characteristic
135	C₂	D₃₇⋊#131	1	1	yes	characteristic
136	C₂	#14558613	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924624

C₂

yes

maximal

#14558540

C₂

yes

maximal, complemented

#14558538

C₂

yes

maximal, complemented

D₃₇⋊(C₄⋊C₈)

C₂

yes

maximal, complemented

C₄×#10596966

C₂

yes

maximal, characteristic

#11924624

—

C₂×#10596953

C₂²

yes

—

C₂×D₃₇⋊C₈

C₂²

yes

—

#11924603

C₂²

yes

characteristic

#11924603

C₂²

yes

—

#11924603

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

C₂×#10596966

C₂²

yes

characteristic

Q₈×Dic₃₇

C₄

yes

characteristic

C₄×D₁₄₈

C₄

yes

—

#11924498

C₄

yes

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

—

C₄×Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

characteristic

#10596966

D₄

yes

—

#10596966

Q₈

yes

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

#10596953

—

D₃₇⋊C₈

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

#10596938

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₄×Dic₃₇

C₂³

yes

—

C₂²×D₃₇

C₂²×C₄

yes

—

Q₈×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂⁴

yes

characteristic

D₁₄₈

C₄⋊C₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×D₄

yes

—

C₄×D₃₇

C₂×Q₈

yes

—

#64438

—

#64438

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

—

Dic₃₇

—

Dic₃₇

#131

yes

complemented

Dic₃₇

C₂²×D₄

yes

characteristic

Dic₃₇

C₂²×Q₈

yes

characteristic

#2906

—

maximal, Sylow

C₂×C₃₇

C₂²×C₄⋊C₄

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

C₂×#131

—

C₄×#41

—

#424

—

#423

—

C₂×C₄⋊C₈

—

C₃₇

#2906

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

#3866

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₂×#3866

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂×#3866

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×#3866

yes

—

107

C₂³

—

108

Q₈

C₄⋊#3866

yes

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

148

—

117

C₂×C₄

C₂²×#3866

yes

characteristic

118

C₂×C₄

C₂²×#3866

yes

—

119

C₂×C₄

C₂²×#3866

yes

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂³×#3866

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂×C₄⋊#3866

yes

—

133

C₂

—

134

C₂

C₂²×C₄⋊#3866

yes

characteristic

135

C₂

D₃₇⋊#131

yes

characteristic

136

C₂

#14558613

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄×Q₈×C₃₇	C₄
prime radical	C₂×#10596966	C₂²
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄×Q₈	#3866
hypocenter	C₃₇	#2906
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#3866

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₄×Q₈×C₃₇

C₄

prime radical

C₂×#10596966

C₂²

cosocle

C₂×Dic₃₇

C₂⁴

C₄×Q₈

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
2. #14558540	C₂	semidirect
3. #14558538	C₂	semidirect
4. D₃₇⋊(C₄⋊C₈)	C₂	semidirect
61. Dic₃₇	#131	semidirect
74. C₃₇	#2906	semidirect

H ≅ G/N

type

2. #14558540

C₂

semidirect

3. #14558538

C₂

semidirect

4. D₃₇⋊(C₄⋊C₈)

C₂

semidirect

61. Dic₃₇

#131

semidirect

74. C₃₇

#2906

semidirect

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

subgroup	index
C₂×#11924624	2
#14558540	2
#14558538	2
D₃₇⋊(C₄⋊C₈)	2
C₄×#10596966	2
#2906	37

subgroup

index

D₃₇⋊(C₄⋊C₈)

Subgroups — #18011244

Conjugacy classes of subgroups

Complements

Maximal subgroups