Group #31779513 — subgroups

Subgroups — #31779513

Order 17536.

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×D₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#22577459	C₂	1	2	yes	maximal
3	#22577152	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577151	C₂	1	2	yes	maximal, complemented
5	#333×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₁₃₇	C₄	1	2	yes	—
7	#17984088	C₂²	1	1	yes	characteristic
8	#17984087	—	2	2	no	—
9	#17984086	C₄	1	2	yes	—
10	C₄⋊C₄×D₁₃₇	—	2	2	no	—
11	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊#11764040	C₂²	1	2	yes	—
13	#17984000	—	2	2	no	—
14	#17983998	—	2	2	no	—
15	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×Dic₂₇₄	—	2	2	no	—
18	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
19	Q₈×D₁₃₇	—	4	4	no	—
20	C₂×C₄×D₁₃₇	—	2	2	no	—
21	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₃₇	D₄	1	2	yes	—
23	C₂×#11764040	—	2	2	no	—
24	C₄⋊Dic₁₃₇	—	2	2	no	—
25	#14555489	—	2	4	no	—
26	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₃₇	D₄	1	2	yes	—
28	C₂×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₃₇	—	2	2	no	—
30	C₄⋊C₄×C₁₃₇	—	2	2	no	—
31	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₂₇₄	—	4	4	no	—
34	Dic₂₇₄	—	4	2	no	—
35	C₂×Dic₁₃₇	—	2	2	no	—
36	C₂×Dic₁₃₇	C₂²⋊C₄	1	2	yes	—
37	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×D₁₃₇	—	2	2	no	—
39	C₄×D₁₃₇	—	2	2	no	—
40	C₄×D₁₃₇	—	2	2	no	—
41	C₄×D₁₃₇	—	4	4	no	—
42	#11764040	—	4	2	no	—
43	Q₈×C₁₃₇	—	4	2	no	—
44	C₂×C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
47	C₈×C₁₃₇	—	4	2	no	—
48	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₃₇	—	2	2	no	—
50	C₂×D₁₃₇	#100	1	2	yes	—
51	C₄×C₁₃₇	—	2	1	no	—
52	C₄×C₁₃₇	—	4	2	no	—
53	C₄×C₁₃₇	#137	1	2	yes	—
54	C₄×C₁₃₇	#137	1	2	yes	—
55	Dic₁₃₇	—	2	1	no	—
56	Dic₁₃₇	—	2	2	no	—
57	Dic₁₃₇	—	4	2	no	—
58	C₂×C₁₃₇	C₂×#100	1	1	yes	characteristic
59	C₂×C₁₃₇	#419	1	2	yes	—
60	D₁₃₇	—	2	2	no	—
61	C₁₃₇	#1558	1	1	yes	characteristic, complemented, Sylow
62	#1558	—	137	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	—	137	1	no	—
64	#432	—	137	2	no	—
65	#333	—	137	1	no	—
66	#333	—	137	2	no	—
67	#333	D₁₃₇	1	1	yes	characteristic, complemented
68	C₂²×Q₈	—	137	2	no	—
69	C₄⋊Q₈	—	274	2	no	—
70	C₄⋊Q₈	—	137	2	no	—
71	C₄⋊Q₈	—	137	1	no	—
72	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	274	2	no	—
74	C₂×C₄²	—	137	1	no	—
75	C₄⋊C₈	—	137	2	no	—
76	C₄⋊C₈	C₂×D₁₃₇	1	2	yes	—
77	C₂²⋊C₈	—	274	2	no	—
78	#97	—	274	2	no	—
79	C₂×Q₈	—	274	2	no	—
80	C₂×Q₈	—	548	4	no	—
81	C₂×Q₈	—	137	2	no	—
82	C₂×Q₈	C₄×D₁₃₇	1	2	yes	—
83	C₂²×C₄	—	274	2	no	—
84	C₂²×C₄	—	137	1	no	—
85	C₂²×C₄	—	137	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	274	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	274	4	no	—
90	C₄⋊C₄	—	274	2	no	—
91	C₄²	—	137	1	no	—
92	C₄²	—	137	2	no	—
93	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
94	C₂³	—	137	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	548	2	no	—
97	Q₈	—	548	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	274	2	no	—
100	C₂×C₄	—	274	2	no	—
101	C₂×C₄	—	274	2	no	—
102	C₂×C₄	—	274	2	no	—
103	C₂×C₄	—	548	4	no	—
104	C₂×C₄	—	137	2	no	—
105	C₂×C₄	—	137	1	no	—
106	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
107	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	548	2	no	—
110	C₂²	—	274	2	no	—
111	C₂²	—	137	2	no	—
112	C₂²	C₂²⋊C₄×D₁₃₇	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	274	2	no	—
116	C₄	—	274	1	no	—
117	C₄	—	548	2	no	—
118	C₄	#17984117	1	2	yes	—
119	C₄	#17984120	1	2	yes	—
120	C₂	—	274	2	no	—
121	C₂	#100×D₁₃₇	1	1	yes	characteristic
122	C₂	#22577410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₁₃₇

C₂

yes

maximal, characteristic

#22577459

C₂

yes

maximal

#22577152

C₂

yes

maximal, characteristic, complemented

#22577151

C₂

yes

maximal, complemented

#333×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₃₇

C₄

yes

—

#17984088

C₂²

yes

characteristic

#17984087

—

#17984086

C₄

yes

—

C₄⋊C₄×D₁₃₇

—

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄⋊#11764040

C₂²

yes

—

#17984000

—

#17983998

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

Q₈×D₁₃₇

—

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₇

D₄

yes

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

—

#14555489

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

—

C₂×Q₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂²×D₁₃₇

C₂²⋊C₄

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂²⋊C₄

yes

—

C₂×Dic₁₃₇

C₂²⋊C₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

#100

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#137

yes

—

C₄×C₁₃₇

#137

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

C₂×#100

yes

characteristic

C₂×C₁₃₇

#419

yes

—

D₁₃₇

—

C₁₃₇

#1558

yes

characteristic, complemented, Sylow

#1558

—

137

maximal, Sylow

C₂×C₄⋊Q₈

—

137

—

#432

—

137

—

#333

—

137

—

#333

—

137

—

#333

D₁₃₇

yes

characteristic, complemented

C₂²×Q₈

—

137

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

C₂×C₄⋊C₄

—

274

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

C₂×D₁₃₇

yes

—

C₂²⋊C₈

—

274

—

#97

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

548

—

C₂×Q₈

—

137

—

C₂×Q₈

C₄×D₁₃₇

yes

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

C₂×C₈

—

274

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

—

137

—

Q₈

—

Q₈

—

548

—

Q₈

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

100

C₂×C₄

—

274

—

101

C₂×C₄

—

274

—

102

C₂×C₄

—

274

—

103

C₂×C₄

—

548

—

104

C₂×C₄

—

137

—

105

C₂×C₄

—

137

—

106

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

107

C₂×C₄

D₄×D₁₃₇

yes

—

108

C₈

—

109

C₈

—

548

—

110

C₂²

—

274

—

111

C₂²

—

137

—

112

C₂²

C₂²⋊C₄×D₁₃₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

274

—

116

C₄

—

274

—

117

C₄

—

548

—

118

C₄

#17984117

yes

—

119

C₄

#17984120

yes

—

120

C₂

—

274

—

121

C₂

#100×D₁₃₇

yes

characteristic

122

C₂

#22577410

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#333×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃₇	C₂²⋊C₄
cosocle	C₄²×C₁₃₇	C₂³
hypercenter	#333	D₁₃₇
hypocenter	C₁₃₇	#1558
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#333×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃₇

C₂²⋊C₄

cosocle

C₄²×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #22577152	C₂	semidirect
4. #22577151	C₂	semidirect
5. #333×C₁₃₇	C₂	semidirect
61. C₁₃₇	#1558	semidirect
67. #333	D₁₃₇	semidirect

H ≅ G/N

type

3. #22577152

C₂

semidirect

4. #22577151

C₂

semidirect

5. #333×C₁₃₇

C₂

semidirect

61. C₁₃₇

#1558

semidirect

67. #333

D₁₃₇

semidirect

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

subgroup	index
C₄⋊Q₈×D₁₃₇	2
#22577459	2
#22577152	2
#22577151	2
#333×C₁₃₇	2
#1558	137

subgroup

index

C₄⋊Q₈×D₁₃₇

#333×C₁₃₇

137

Subgroups — #31779513

Conjugacy classes of subgroups

Complements

Maximal subgroups