Group #17993674 — subgroups

Subgroups — #17993674

Order 4736.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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	#14557907	C₂	1	1	yes	maximal, characteristic
2	#14557733	C₂	1	2	yes	maximal
3	#14557282	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#64436	C₂	1	2	yes	maximal, complemented
5	Q₈²×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596941	—	2	2	no	—
7	C₂×#10596941	—	2	2	no	—
8	#11924533	C₂²	1	1	yes	characteristic
9	C₄⋊#64436	C₂²	1	2	yes	—
10	#11924456	—	2	2	no	—
11	C₄×#64436	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₃₇	—	2	2	no	—
13	C₄⋊Q₈×C₃₇	—	2	2	no	—
14	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₃₇	—	2	2	no	—
16	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₃₇	—	2	2	no	—
18	C₂×Q₈×C₃₇	D₄	1	2	yes	—
19	C₂×Q₈×C₃₇	D₄	1	2	yes	—
20	C₂×D₁₄₈	—	2	1	no	—
21	C₂×D₁₄₈	—	2	2	no	—
22	C₄⋊C₄×C₃₇	—	2	4	no	—
23	C₄⋊C₄×C₃₇	—	2	2	no	—
24	C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₄⋊C₄×C₃₇	D₄	1	2	yes	—
26	C₄²×C₃₇	—	2	2	no	—
27	C₄²×C₃₇	—	2	2	no	—
28	C₄²×C₃₇	C₂³	1	1	yes	characteristic
29	#10596941	—	4	4	no	—
30	#10596941	—	4	4	no	—
31	C₂×#64436	—	2	1	no	—
32	C₂×#64436	—	2	2	no	—
33	C₂²×D₃₇	—	4	1	no	—
34	Q₈×C₃₇	—	2	2	no	—
35	Q₈×C₃₇	—	4	2	no	—
36	Q₈×C₃₇	—	8	2	no	—
37	Q₈×C₃₇	SD₁₆	1	4	yes	complemented
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	—	2	2	no	—
40	C₂×C₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
43	D₁₄₈	—	4	2	no	—
44	D₁₄₈	—	4	2	no	—
45	D₁₄₈	—	4	2	no	—
46	#64436	—	2	2	no	—
47	#64436	—	4	2	no	—
48	C₂²×C₃₇	#120	1	1	yes	characteristic
49	C₂×D₃₇	—	8	2	no	—
50	C₂×D₃₇	—	8	1	no	—
51	C₄×C₃₇	—	2	4	no	—
52	C₄×C₃₇	—	2	1	no	—
53	C₄×C₃₇	—	4	2	no	—
54	C₄×C₃₇	—	4	2	no	—
55	C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
56	C₄×C₃₇	#137	1	2	yes	—
57	#1623	—	37	1	no	maximal, Sylow
58	C₂×C₃₇	#448	1	2	yes	—
59	C₂×C₃₇	#453	1	1	yes	characteristic
60	D₃₇	—	16	1	no	—
61	Q₈²	D₃₇	1	1	yes	characteristic, complemented
62	#492	—	37	1	no	—
63	#460	—	37	2	no	—
64	#333	—	37	1	no	—
65	Q₈⋊C₈	—	37	2	no	—
66	C₃₇	#1623	1	1	yes	characteristic, complemented, Sylow
67	C₂×SD₁₆	—	74	2	no	—
68	C₂×SD₁₆	—	74	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
72	#127	—	37	1	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	C₂×D₃₇	1	2	yes	—
75	C₄⋊C₈	—	37	2	no	—
76	#102	—	74	2	no	—
77	C₄×C₈	—	37	1	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	C₂²⋊D₃₇	1	2	yes	—
80	C₂×Q₈	C₂²⋊D₃₇	1	2	yes	—
81	C₂×D₄	—	74	2	no	—
82	C₂×D₄	—	74	1	no	—
83	SD₁₆	—	148	4	no	—
84	SD₁₆	—	148	4	no	—
85	C₂×C₈	—	74	1	no	—
86	C₂×C₈	—	74	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	C₂²⋊D₃₇	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂²×D₃₇	1	1	yes	characteristic
94	C₂³	—	148	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	#10596941	1	4	yes	complemented
99	D₄	—	148	2	no	—
100	D₄	—	148	2	no	—
101	D₄	—	148	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
107	C₈	—	74	2	no	—
108	C₈	—	148	2	no	—
109	C₂²	—	296	1	no	—
110	C₂²	—	296	2	no	—
111	C₂²	#11924505	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	4	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂×#10596941	1	2	yes	—
117	C₄	#11924567	1	2	yes	—
118	C₂	—	592	1	no	—
119	C₂	#14557672	1	2	yes	—
120	C₂	#14557701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557907

C₂

yes

maximal, characteristic

#14557733

C₂

yes

maximal

#14557282

C₂

yes

maximal, characteristic, complemented

Q₈⋊#64436

C₂

yes

maximal, complemented

Q₈²×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596941

—

C₂×#10596941

—

#11924533

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

—

#11924456

—

C₄×#64436

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

—

C₄⋊Q₈×C₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄×Q₈×C₃₇

—

C₄×Q₈×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

—

C₄²×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂³

yes

—

—

—

—

C₂²×D₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

SD₁₆

yes

complemented

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

#64436

—

#64436

—

C₂²×C₃₇

#120

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×SD₁₆

yes

—

C₄×C₃₇

#137

yes

—

#1623

—

maximal, Sylow

C₂×C₃₇

#448

yes

—

C₂×C₃₇

#453

yes

characteristic

D₃₇

—

Q₈²

D₃₇

yes

characteristic, complemented

#492

—

#460

—

#333

—

Q₈⋊C₈

—

C₃₇

#1623

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃₇

yes

—

C₄⋊C₈

—

#102

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

—

C₂×Q₈

C₂²⋊D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

148

—

SD₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₃₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

148

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#10596941

yes

complemented

D₄

—

148

—

100

D₄

—

148

—

101

D₄

—

148

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

107

C₈

—

108

C₈

—

148

—

109

C₂²

—

296

—

110

C₂²

—

296

—

111

C₂²

#11924505

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×#10596941

yes

—

117

C₄

#11924567

yes

—

118

C₂

—

592

—

119

C₂

#14557672

yes

—

120

C₂

#14557701

yes

characteristic

	subgroup	G/H
center	C₂²	#11924505
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	Q₈²×C₃₇	C₂
socle	C₂²×C₃₇	#120
prime radical	#11924533	C₂²
hypercenter	Q₈²	D₃₇
hypocenter	C₃₇	#1623
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#11924505

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

Q₈²×C₃₇

C₂

C₂²×C₃₇

C₂²

Q₈²

D₃₇

hypocenter

C₃₇

#1623

LCS

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
3. #14557282	C₂	semidirect
4. Q₈⋊#64436	C₂	semidirect
5. Q₈²×C₃₇	C₂	semidirect
37. Q₈×C₃₇	SD₁₆	semidirect
61. Q₈²	D₃₇	semidirect
66. C₃₇	#1623	semidirect
98. Q₈	#10596941	semidirect

H ≅ G/N

type

3. #14557282

C₂

semidirect

4. Q₈⋊#64436

C₂

semidirect

5. Q₈²×C₃₇

C₂

semidirect

37. Q₈×C₃₇

SD₁₆

semidirect

61. Q₈²

D₃₇

semidirect

66. C₃₇

#1623

semidirect

98. Q₈

#10596941

semidirect

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

subgroup	index
#14557907	2
#14557733	2
#14557282	2
Q₈⋊#64436	2
Q₈²×C₃₇	2
#1623	37

subgroup

index

Q₈²×C₃₇

Subgroups — #17993674

Conjugacy classes of subgroups

Complements

Maximal subgroups