Group #17993320 — subgroups

Subgroups — #17993320

Order 4736.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
2	#14557589	C₂	1	2	yes	maximal
3	#14557282	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557281	C₂	1	2	yes	maximal, complemented
5	C₃₇×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₃₇	C₄	1	2	yes	—
7	#11924536	C₂²	1	1	yes	characteristic
8	#11924535	—	2	2	no	—
9	#11924534	C₄	1	2	yes	—
10	C₄⋊C₄×D₃₇	—	2	2	no	—
11	C₄²×D₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊#64436	C₂²	1	2	yes	—
13	#11924448	—	2	2	no	—
14	#11924446	—	2	2	no	—
15	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
18	Q₈×D₃₇	—	4	4	no	—
19	C₂×C₄×D₃₇	—	2	2	no	—
20	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₃₇	D₄	1	2	yes	—
22	C₂×Dic₇₄	—	2	2	no	—
23	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₃₇	—	2	2	no	—
25	C₄⋊C₄×C₃₇	—	2	2	no	—
26	C₄²×C₃₇	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₃₇	—	2	2	no	—
28	#10596936	—	2	4	no	—
29	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₃₇	D₄	1	2	yes	—
31	C₂×#64436	—	2	2	no	—
32	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₃₇	—	4	2	no	—
34	C₂×C₄×C₃₇	—	2	2	no	—
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
37	C₂×Dic₃₇	—	2	2	no	—
38	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×D₃₇	—	2	2	no	—
41	C₄×D₃₇	—	2	2	no	—
42	C₄×D₃₇	—	2	2	no	—
43	C₄×D₃₇	—	4	4	no	—
44	Dic₇₄	—	4	4	no	—
45	Dic₇₄	—	4	2	no	—
46	C₈×C₃₇	—	4	2	no	—
47	#64436	—	4	2	no	—
48	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₃₇	—	2	2	no	—
50	C₂×D₃₇	#100	1	2	yes	—
51	C₄×C₃₇	—	2	1	no	—
52	C₄×C₃₇	—	4	2	no	—
53	C₄×C₃₇	#137	1	2	yes	—
54	C₄×C₃₇	#137	1	2	yes	—
55	Dic₃₇	—	2	1	no	—
56	Dic₃₇	—	2	2	no	—
57	Dic₃₇	—	4	2	no	—
58	#1558	—	37	1	no	maximal, Sylow
59	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
60	C₂×C₃₇	#419	1	2	yes	—
61	D₃₇	—	2	2	no	—
62	C₂×C₄⋊Q₈	—	37	1	no	—
63	#432	—	37	2	no	—
64	#333	—	37	1	no	—
65	#333	—	37	2	no	—
66	#333	D₃₇	1	1	yes	characteristic, complemented
67	C₃₇	#1558	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	—	37	2	no	—
69	C₄⋊Q₈	—	37	2	no	—
70	C₄⋊Q₈	—	37	1	no	—
71	C₄⋊Q₈	—	74	2	no	—
72	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	74	2	no	—
74	C₂×C₄²	—	37	1	no	—
75	C₄⋊C₈	—	37	2	no	—
76	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
77	C₂²⋊C₈	—	74	2	no	—
78	#97	—	74	2	no	—
79	C₂×Q₈	—	37	2	no	—
80	C₂×Q₈	—	74	2	no	—
81	C₂×Q₈	—	148	4	no	—
82	C₂×Q₈	C₄×D₃₇	1	2	yes	—
83	C₂²×C₄	—	37	1	no	—
84	C₂²×C₄	—	37	2	no	—
85	C₂²×C₄	—	74	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	74	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	74	4	no	—
90	C₄⋊C₄	—	74	2	no	—
91	C₄²	—	37	1	no	—
92	C₄²	—	37	2	no	—
93	C₄²	C₂²×D₃₇	1	1	yes	characteristic
94	C₂³	—	37	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	148	2	no	—
97	Q₈	—	148	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	37	2	no	—
100	C₂×C₄	—	37	1	no	—
101	C₂×C₄	—	74	2	no	—
102	C₂×C₄	—	74	2	no	—
103	C₂×C₄	—	74	2	no	—
104	C₂×C₄	—	74	2	no	—
105	C₂×C₄	—	148	4	no	—
106	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
107	C₂×C₄	D₄×D₃₇	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	148	2	no	—
110	C₂²	—	37	2	no	—
111	C₂²	—	74	2	no	—
112	C₂²	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	74	1	no	—
116	C₄	—	74	2	no	—
117	C₄	—	148	2	no	—
118	C₄	#11924565	1	2	yes	—
119	C₄	#11924568	1	2	yes	—
120	C₂	—	74	2	no	—
121	C₂	#100×D₃₇	1	1	yes	characteristic
122	C₂	#14557540	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₃₇

C₂

yes

maximal, characteristic

#14557589

C₂

yes

maximal

#14557282

C₂

yes

maximal, characteristic, complemented

#14557281

C₂

yes

maximal, complemented

C₃₇×#333

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₃₇

C₄

yes

—

#11924536

C₂²

yes

characteristic

#11924535

—

#11924534

C₄

yes

—

C₄⋊C₄×D₃₇

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

—

#11924448

—

#11924446

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

Q₈×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂×#64436

—

C₂²×D₃₇

C₂²⋊C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

#100

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#137

yes

—

C₄×C₃₇

#137

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

C₂×C₄⋊Q₈

—

—

—

—

D₃₇

yes

characteristic, complemented

C₃₇

#1558

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₇

yes

—

C₂²⋊C₈

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂×Q₈

C₄×D₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

148

—

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

148

—

106

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

107

C₂×C₄

D₄×D₃₇

yes

—

108

C₈

—

109

C₈

—

148

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂²⋊C₄×D₃₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

148

—

118

C₄

#11924565

yes

—

119

C₄

#11924568

yes

—

120

C₂

—

121

C₂

#100×D₃₇

yes

characteristic

122

C₂

#14557540

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#333	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₃₇	C₂²⋊C₄
cosocle	C₄²×C₃₇	C₂³
hypercenter	#333	D₃₇
hypocenter	C₃₇	#1558
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#333

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₃₇

C₂²⋊C₄

cosocle

C₄²×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #14557282	C₂	semidirect
4. #14557281	C₂	semidirect
5. C₃₇×#333	C₂	semidirect
66. #333	D₃₇	semidirect
67. C₃₇	#1558	semidirect

H ≅ G/N

type

3. #14557282

C₂

semidirect

4. #14557281

C₂

semidirect

5. C₃₇×#333

C₂

semidirect

66. #333

D₃₇

semidirect

67. C₃₇

#1558

semidirect

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

subgroup	index
C₄⋊Q₈×D₃₇	2
#14557589	2
#14557282	2
#14557281	2
C₃₇×#333	2
#1558	37

subgroup

index

C₄⋊Q₈×D₃₇

C₃₇×#333

Subgroups — #17993320

Conjugacy classes of subgroups

Complements

Maximal subgroups