Group #18003382 — subgroups

Subgroups — C₂×#14557639 #18003382

Order 4736.

480 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×Dic₁₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924501	C₂	1	2	yes	maximal
3	#14557639	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11924459	C₂	1	2	yes	maximal
5	C₂×#11924446	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₇₄	C₂²	1	2	yes	—
8	C₂×Dic₁₄₈	C₄	1	8	yes	complemented
9	C₂×Dic₁₄₈	C₂²	1	4	yes	—
10	C₂²×#64436	C₂²	1	1	yes	characteristic
11	#11924501	—	2	4	no	—
12	#11924501	C₂²	1	8	yes	—
13	C₂×#10596936	—	2	2	no	—
14	C₂×C₄×Dic₃₇	—	2	2	no	—
15	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
16	#11924459	C₂²	1	8	yes	—
17	#11924446	C₂²	1	4	yes	—
18	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
20	#106×C₃₇	C₂²	1	4	yes	—
21	C₂²×C₄×C₃₇	—	2	2	no	—
22	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₇	—	2	2	no	—
24	C₂²×Dic₃₇	D₄	1	2	yes	—
25	C₂×Dic₇₄	—	2	2	no	—
26	C₂×Dic₇₄	—	2	4	no	—
27	C₂×Dic₇₄	C₂×C₄	1	8	yes	—
28	C₂×Dic₇₄	C₂³	1	4	yes	—
29	C₂×C₈×C₃₇	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₃₇	C₂³	1	2	yes	—
31	C₄⋊C₄×C₃₇	—	2	2	no	—
32	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
33	#10596936	—	2	8	no	—
34	C₄×Dic₃₇	—	2	8	no	—
35	C₄×Dic₃₇	D₄	1	2	yes	—
36	C₄×Dic₃₇	C₂³	1	2	yes	—
37	C₂×#64436	—	2	2	no	—
38	C₂×#64436	C₂³	1	2	yes	—
39	Dic₁₄₈	C₂×C₄	1	16	yes	complemented
40	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	—	2	4	no	—
42	C₂×C₄×C₃₇	—	4	4	no	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	#41	1	2	yes	—
46	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₃₇	—	2	8	no	—
48	C₂×Dic₃₇	—	2	4	no	—
49	C₂×Dic₃₇	—	2	4	no	—
50	C₂×Dic₃₇	C₂×D₄	1	4	yes	—
51	Dic₇₄	—	2	8	no	—
52	Dic₇₄	—	2	4	no	—
53	Dic₇₄	C₂²×C₄	1	8	yes	—
54	C₈×C₃₇	C₂²×C₄	1	4	yes	—
55	#64436	—	2	2	no	—
56	C₂²×C₃₇	C₄×D₄	1	2	yes	—
57	C₂²×C₃₇	#137	1	4	yes	—
58	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₃₇	—	4	4	no	—
60	C₄×C₃₇	C₄×D₄	1	2	yes	—
61	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
63	Dic₃₇	—	2	8	no	—
64	Dic₃₇	—	2	4	no	—
65	C₂×#441	—	37	1	no	maximal, Sylow
66	C₂×C₃₇	#441	1	4	yes	complemented
67	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
68	C₂×C₃₇	C₂×#137	1	2	yes	—
69	C₂²×Q₁₆	—	37	1	no	—
70	C₂×C₄×Q₈	—	37	2	no	—
71	#441	—	37	8	no	—
72	C₂×#106	D₃₇	1	1	yes	characteristic
73	C₂×#103	—	37	2	no	—
74	C₂×#97	—	37	1	no	—
75	C₃₇	C₂×#441	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	37	2	no	—
77	C₂×Q₁₆	—	37	8	no	—
78	C₂×Q₁₆	—	37	4	no	—
79	C₂²×C₈	—	37	1	no	—
80	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
81	C₄×Q₈	—	37	8	no	—
82	C₄×Q₈	—	74	4	no	—
83	C₂×C₄⋊C₄	—	74	2	no	—
84	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
85	C₂×C₄²	—	37	1	no	—
86	C₂×C₄²	—	74	2	no	—
87	#106	C₂×D₃₇	1	4	yes	—
88	#103	—	37	8	no	—
89	#97	—	37	4	no	—
90	C₂×Q₈	—	37	8	no	—
91	C₂×Q₈	—	37	4	no	—
92	C₂×Q₈	—	74	2	no	—
93	C₂×Q₈	—	74	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	37	2	no	—
96	C₂²×C₄	—	74	2	no	—
97	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
98	Q₁₆	—	37	16	no	—
99	C₂×C₈	—	37	2	no	—
100	C₂×C₈	—	74	2	no	—
101	C₂×C₈	C₄×D₃₇	1	4	yes	—
102	C₂×C₈	C₂²×D₃₇	1	2	yes	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	74	8	no	—
105	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
106	C₄²	—	37	2	no	—
107	C₄²	—	37	2	no	—
108	C₄²	—	74	8	no	—
109	C₂³	D₄×D₃₇	1	1	yes	characteristic
110	Q₈	—	37	8	no	—
111	Q₈	—	74	8	no	—
112	Q₈	—	74	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	37	4	no	—
116	C₂×C₄	—	74	8	no	—
117	C₂×C₄	—	74	4	no	—
118	C₂×C₄	—	74	4	no	—
119	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
120	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
121	C₂×C₄	#10596966	1	2	yes	—
122	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
123	C₈	—	74	2	no	—
124	C₈	C₂×C₄×D₃₇	1	4	yes	—
125	C₂²	#11924498	1	2	yes	—
126	C₂²	#11924565	1	4	yes	—
127	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	74	8	no	—
130	C₄	—	74	4	no	—
131	C₄	#11924498	1	2	yes	—
132	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
133	C₄	C₂×#10596966	1	1	yes	characteristic
134	C₂	#14557639	1	4	yes	direct product factor
135	C₂	C₂×#11924498	1	1	yes	characteristic
136	C₂	C₂×#11924565	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₄₈

C₂

yes

maximal, characteristic

C₂×#11924501

C₂

yes

maximal

#14557639

C₂

yes

maximal, direct product factor

C₂×#11924459

C₂

yes

maximal

C₂×#11924446

C₂

yes

maximal, characteristic

C₂×#106×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

C₂×Dic₁₄₈

C₄

yes

complemented

C₂×Dic₁₄₈

C₂²

yes

—

C₂²×#64436

C₂²

yes

characteristic

#11924501

—

#11924501

C₂²

yes

—

C₂×#10596936

—

C₂×C₄×Dic₃₇

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924459

C₂²

yes

—

#11924446

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

#106×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×Dic₇₄

C₂³

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂×#64436

—

C₂×#64436

C₂³

yes

—

Dic₁₄₈

C₂×C₄

yes

complemented

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

C₂²×C₄

yes

—

C₈×C₃₇

C₂²×C₄

yes

—

#64436

—

C₂²×C₃₇

C₄×D₄

yes

—

C₂²×C₃₇

#137

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

C₂³×C₄

yes

characteristic

C₄×C₃₇

C₂×#41

yes

characteristic

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×C₄×D₄

yes

characteristic

C₂×C₃₇

C₂×#137

yes

—

C₂²×Q₁₆

—

C₂×C₄×Q₈

—

#441

—

C₂×#106

D₃₇

yes

characteristic

C₂×#103

—

C₂×#97

—

C₃₇

C₂×#441

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

#106

C₂×D₃₇

yes

—

#103

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₄×D₃₇

yes

—

102

C₂×C₈

C₂²×D₃₇

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

C₂²×D₃₇

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂³

D₄×D₃₇

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×C₄×D₃₇

yes

—

120

C₂×C₄

D₄×D₃₇

yes

characteristic

121

C₂×C₄

#10596966

yes

—

122

C₂×C₄

C₂³×D₃₇

yes

characteristic

123

C₈

—

124

C₈

C₂×C₄×D₃₇

yes

—

125

C₂²

#11924498

yes

—

126

C₂²

#11924565

yes

—

127

C₂²

C₂×D₄×D₃₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#11924498

yes

—

132

C₄

C₂²×C₄×D₃₇

yes

characteristic

133

C₄

C₂×#10596966

yes

characteristic

134

C₂

#14557639

yes

direct product factor

135

C₂

C₂×#11924498

yes

characteristic

136

C₂

C₂×#11924565

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#106×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×#106	D₃₇
hypocenter	C₃₇	C₂×#441
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₃₇

commutator

C₄×C₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#106×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂×C₄×C₃₇

C₂⁴

D₃₇

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #14557639	C₂	direct (N × H)
8. C₂×Dic₁₄₈	C₄	semidirect
39. Dic₁₄₈	C₂×C₄	semidirect
66. C₂×C₃₇	#441	semidirect
75. C₃₇	C₂×#441	semidirect
134. C₂	#14557639	direct (N × H)

H ≅ G/N

type

3. #14557639

C₂

direct (N × H)

8. C₂×Dic₁₄₈

C₄

semidirect

39. Dic₁₄₈

C₂×C₄

semidirect

66. C₂×C₃₇

#441

semidirect

75. C₃₇

C₂×#441

semidirect

134. C₂

#14557639

direct (N × H)

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

subgroup	index
C₂²×Dic₁₄₈	2
C₂×#11924501	2
#14557639	2
C₂×#11924459	2
C₂×#11924446	2
C₂×#106×C₃₇	2
C₂×#441	37

subgroup

index

C₂²×Dic₁₄₈

C₂×#106×C₃₇

Subgroups — C₂×#14557639 #18003382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557639 #18003382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557639 #18003382