Group #37509749 — subgroups

Subgroups — #37509749

Order 22912.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19153049	C₂	1	1	yes	maximal, characteristic
2	#26038466	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₇₉	C₂	1	1	yes	maximal, characteristic
4	#26038198	C₂	1	2	yes	maximal
5	#427×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₂²	1	2	yes	—
7	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
9	#19153049	—	2	2	no	—
10	C₄⋊(C₁₇₉⋊C₈)	—	2	4	no	—
11	C₈×Dic₁₇₉	—	2	2	no	—
12	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
13	#117×C₁₇₉	C₂²	1	2	yes	—
14	#107×C₁₇₉	C₄	1	2	yes	complemented
15	#106×C₁₇₉	C₄	1	2	yes	complemented
16	C₂²×Dic₁₇₉	—	2	1	no	—
17	#15668243	—	2	2	no	—
18	#15668243	—	2	2	no	—
19	C₂×C₁₇₉⋊C₈	—	2	1	no	—
20	C₂×C₁₇₉⋊C₈	—	2	2	no	—
21	C₂×C₁₇₉⋊C₈	—	2	1	no	—
22	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
23	C₄×Dic₁₇₉	—	2	1	no	—
24	C₄×Dic₁₇₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₇₉	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₇₉	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₇₉	—	2	2	no	—
32	C₄²×C₁₇₉	—	2	2	no	—
33	C₂×Dic₁₇₉	—	2	4	no	—
34	C₂×Dic₁₇₉	—	2	2	no	—
35	C₁₇₉⋊C₈	—	2	2	no	—
36	C₁₇₉⋊C₈	—	4	2	no	—
37	C₂³×C₁₇₉	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₇₉	—	2	4	no	—
39	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
45	C₈×C₁₇₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₇₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₇₉	—	2	2	no	—
48	C₂²×C₁₇₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₇₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₇₉	#125	1	1	yes	characteristic
51	C₄×C₁₇₉	—	4	4	no	—
52	C₄×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₇₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₇₉	#121	1	1	yes	characteristic
55	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₇₉	—	2	4	no	—
57	C₂×C₁₇₉	—	2	1	no	—
58	C₂×C₁₇₉	—	2	1	no	—
59	C₂×C₁₇₉	#389	1	1	yes	characteristic
60	C₂×C₁₇₉	#443	1	2	yes	—
61	C₁₇₉	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	179	1	no	maximal, Sylow
63	C₂×#108	—	179	1	no	—
64	#427	D₁₇₉	1	1	yes	characteristic
65	#424	—	179	2	no	—
66	C₂×C₄×C₈	—	179	1	no	—
67	#339	—	179	2	no	—
68	C₂×M₁₆	—	179	2	no	—
69	C₂²×C₈	—	179	1	no	—
70	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
71	#117	C₂×D₁₇₉	1	2	yes	—
72	C₂×C₄²	—	179	1	no	—
73	#108	—	358	2	no	—
74	#107	Dic₁₇₉	1	2	yes	complemented
75	#106	Dic₁₇₉	1	2	yes	complemented
76	C₄⋊C₈	—	358	4	no	—
77	C₄×C₈	—	358	2	no	—
78	C₂²×C₄	—	358	1	no	—
79	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
80	M₁₆	—	358	2	no	—
81	M₁₆	—	358	2	no	—
82	C₂×C₈	—	358	1	no	—
83	C₂×C₈	—	358	1	no	—
84	C₂×C₈	—	358	2	no	—
85	C₂×C₈	—	179	2	no	—
86	C₂×C₈	C₂×Dic₁₇₉	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₁₇₉	1	1	yes	characteristic
88	C₂×C₈	D₇₁₆	1	2	yes	—
89	C₂×C₈	Dic₃₅₈	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₁₇₉	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	358	1	no	—
94	C₄²	—	179	2	no	—
95	C₂³	#15668258	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	358	4	no	—
98	C₂×C₄	—	358	2	no	—
99	C₂×C₄	C₂²×Dic₁₇₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₇₉	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₃₅₈	1	1	yes	characteristic
104	C₂×C₄	#15668260	1	1	yes	characteristic
105	C₈	—	358	2	no	—
106	C₈	—	716	2	no	—
107	C₈	C₄⋊Dic₁₇₉	1	2	yes	—
108	C₈	C₄⋊Dic₁₇₉	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₇₉	1	1	yes	characteristic
111	C₂²	#19153093	1	1	yes	characteristic
112	C₂²	#19153104	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	358	4	no	—
115	C₄	C₂×C₄⋊Dic₁₇₉	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₇₉	1	1	yes	characteristic
117	C₄	#19153088	1	1	yes	characteristic
118	C₄	#19153112	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#26038338	1	1	yes	characteristic
122	C₂	#26038556	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153049

C₂

yes

maximal, characteristic

#26038466

C₂

yes

maximal

C₂×C₈×Dic₁₇₉

C₂

yes

maximal, characteristic

#26038198

C₂

yes

maximal

#427×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

#19153049

—

C₄⋊(C₁₇₉⋊C₈)

—

C₈×Dic₁₇₉

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

#117×C₁₇₉

C₂²

yes

—

#107×C₁₇₉

C₄

yes

complemented

#106×C₁₇₉

C₄

yes

complemented

C₂²×Dic₁₇₉

—

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

Q₈

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

D₄

yes

—

C₂×C₈×C₁₇₉

Q₈

yes

—

C₄⋊C₄×C₁₇₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

C₄⋊C₄

yes

—

C₈×C₁₇₉

C₄⋊C₄

yes

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₄×D₄

yes

characteristic

C₂²×C₁₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇₉

#125

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇₉

C₄×Q₈

yes

characteristic

C₄×C₁₇₉

#121

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#389

yes

characteristic

C₂×C₁₇₉

#443

yes

—

C₁₇₉

#1945

yes

characteristic, complemented, Sylow

#1945

—

179

maximal, Sylow

C₂×#108

—

179

—

#427

D₁₇₉

yes

characteristic

#424

—

179

—

C₂×C₄×C₈

—

179

—

#339

—

179

—

C₂×M₁₆

—

179

—

C₂²×C₈

—

179

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

#117

C₂×D₁₇₉

yes

—

C₂×C₄²

—

179

—

#108

—

358

—

#107

Dic₁₇₉

yes

complemented

#106

Dic₁₇₉

yes

complemented

C₄⋊C₈

—

358

—

C₄×C₈

—

358

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

358

—

M₁₆

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

179

—

C₂×C₈

C₂×Dic₁₇₉

yes

characteristic

C₂×C₈

C₂×Dic₁₇₉

yes

characteristic

C₂×C₈

D₇₁₆

yes

—

C₂×C₈

Dic₃₅₈

yes

—

C₄⋊C₄

C₂×Dic₁₇₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

358

—

C₄²

—

179

—

C₂³

#15668258

yes

characteristic

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

—

358

—

C₂×C₄

C₂²×Dic₁₇₉

yes

characteristic

100

C₂×C₄

D₄×D₁₇₉

yes

characteristic

101

C₂×C₄

C₂×D₇₁₆

yes

characteristic

102

C₂×C₄

Q₈×D₁₇₉

yes

characteristic

103

C₂×C₄

C₂×Dic₃₅₈

yes

characteristic

104

C₂×C₄

#15668260

yes

characteristic

105

C₈

—

358

—

106

C₈

—

716

—

107

C₈

C₄⋊Dic₁₇₉

yes

—

108

C₈

C₄⋊Dic₁₇₉

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₇₉

yes

characteristic

111

C₂²

#19153093

yes

characteristic

112

C₂²

#19153104

yes

characteristic

113

C₄

—

114

C₄

—

358

—

115

C₄

C₂×C₄⋊Dic₁₇₉

yes

characteristic

116

C₄

Q₈×Dic₁₇₉

yes

characteristic

117

C₄

#19153088

yes

characteristic

118

C₄

#19153112

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#26038338

yes

characteristic

122

C₂

#26038556

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#427×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₄×D₄
prime radical	C₂³×C₁₇₉	#41
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#427	D₁₇₉
hypocenter	C₁₇₉	#1945
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#427×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₄×D₄

prime radical

C₂³×C₁₇₉

#41

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₇₉	C₄	semidirect
15. #106×C₁₇₉	C₄	semidirect
61. C₁₇₉	#1945	semidirect
74. #107	Dic₁₇₉	semidirect
75. #106	Dic₁₇₉	semidirect

H ≅ G/N

type

14. #107×C₁₇₉

C₄

semidirect

15. #106×C₁₇₉

C₄

semidirect

61. C₁₇₉

#1945

semidirect

74. #107

Dic₁₇₉

semidirect

75. #106

Dic₁₇₉

semidirect

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

subgroup	index
C₂×#19153049	2
#26038466	2
C₂×C₈×Dic₁₇₉	2
#26038198	2
#427×C₁₇₉	2
#1945	179

subgroup

index

C₂×C₈×Dic₁₇₉

#427×C₁₇₉

179

Subgroups — #37509749

Conjugacy classes of subgroups

Complements

Maximal subgroups