Group #37507290 — subgroups

Subgroups — #37507290

Order 22912.

208 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 121 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×#15668243	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#26038161	C₂	1	2	yes	maximal, complemented
4	#26038160	C₂	1	2	yes	maximal, complemented
5	#431×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₄	1	2	yes	—
7	C₂²×C₁₇₉⋊C₈	C₄	1	2	yes	—
8	#19153024	C₄	1	2	yes	—
9	C₄×C₁₇₉⋊C₈	C₄	1	2	yes	—
10	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₁₇₉⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₇₉	C₄	1	2	yes	—
15	C₂²⋊C₈×C₁₇₉	C₄	1	2	yes	—
16	C₄×C₈×C₁₇₉	C₂²	1	2	yes	—
17	#15668243	—	2	4	no	—
18	C₂×C₁₇₉⋊C₈	—	2	4	no	—
19	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₇₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₇₉	C₂×C₄	1	4	yes	—
26	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
30	C₁₇₉⋊C₈	—	2	2	no	—
31	C₁₇₉⋊C₈	—	2	4	no	—
32	C₁₇₉⋊C₈	M₁₆	1	4	yes	complemented
33	C₂³×C₁₇₉	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₇₉	—	2	2	no	—
35	C₂×C₄×C₁₇₉	—	2	2	no	—
36	C₂×C₄×C₁₇₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₁₇₉	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₇₉	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₁₇₉	M₁₆	1	4	yes	—
40	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₇₉	—	2	4	no	—
44	C₂²×C₁₇₉	—	2	1	no	—
45	C₂²×C₁₇₉	—	2	1	no	—
46	C₂²×C₁₇₉	#97	1	2	yes	—
47	C₂²×C₁₇₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₇₉	—	2	1	no	—
49	C₄×C₁₇₉	—	2	1	no	—
50	C₄×C₁₇₉	#97	1	2	yes	—
51	C₄×C₁₇₉	C₂×M₁₆	1	2	yes	—
52	C₄×C₁₇₉	C₂×M₁₆	1	2	yes	—
53	C₄×C₁₇₉	#131	1	2	yes	—
54	C₂×C₁₇₉	—	2	2	no	—
55	C₂×C₁₇₉	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₇₉	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₁₇₉	#405	1	1	yes	characteristic
58	C₁₇₉	#1451	1	1	yes	characteristic, complemented, Sylow
59	#1451	—	179	1	no	maximal, Sylow
60	#431	D₁₇₉	1	1	yes	characteristic
61	C₄×M₁₆	—	179	1	no	—
62	C₂×C₄×C₈	—	179	1	no	—
63	#322	—	179	2	no	—
64	#322	—	179	2	no	—
65	C₂×M₁₆	—	179	2	no	—
66	C₂²×C₈	—	179	2	no	—
67	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
68	C₄⋊C₈	Dic₁₇₉	1	2	yes	—
69	C₂²⋊C₈	Dic₁₇₉	1	2	yes	—
70	#97	—	179	2	no	—
71	C₄×C₈	—	179	2	no	—
72	C₄×C₈	—	179	1	no	—
73	C₄×C₈	—	179	1	no	—
74	C₄×C₈	—	179	2	no	—
75	C₄×C₈	C₂×D₁₇₉	1	2	yes	—
76	C₂²×C₄	C₄×D₁₇₉	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₁₇₉	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₁₇₉	1	1	yes	characteristic
79	M₁₆	—	358	4	no	—
80	C₂×C₈	—	358	4	no	—
81	C₂×C₈	—	179	2	no	—
82	C₂×C₈	—	179	4	no	—
83	C₂×C₈	—	179	2	no	—
84	C₂×C₈	C₂×Dic₁₇₉	1	4	yes	—
85	C₄²	C₄×D₁₇₉	1	1	yes	characteristic
86	C₄²	C₄×D₁₇₉	1	1	yes	characteristic
87	C₄²	C₂×Dic₁₇₉	1	1	yes	characteristic
88	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
89	C₂³	C₄×Dic₁₇₉	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄×Dic₁₇₉	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₁₇₉	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₁₇₉	1	1	yes	characteristic
95	C₂×C₄	#15668242	1	4	yes	—
96	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₁₇₉	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	—	358	2	no	—
101	C₈	—	358	4	no	—
102	C₈	—	179	4	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#19153023	1	2	yes	—
106	C₂²	C₂×C₄×Dic₁₇₉	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	#19153023	1	2	yes	—
110	C₄	M₁₆×D₁₇₉	1	2	yes	—
111	C₄	C₂×#15668242	1	2	yes	—
112	C₄	#19153120	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#19153023	1	1	yes	characteristic
115	C₂	M₁₆×Dic₁₇₉	1	1	yes	characteristic
116	C₂	#26038395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15668243

C₂

yes

maximal, characteristic

C₂×C₄×C₁₇₉⋊C₈

C₂

yes

maximal, characteristic

#26038161

C₂

yes

maximal, complemented

#26038160

C₂

yes

maximal, complemented

#431×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₄

yes

—

C₂²×C₁₇₉⋊C₈

C₄

yes

—

#19153024

C₄

yes

—

C₄×C₁₇₉⋊C₈

C₄

yes

—

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₇₉⋊C₈

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₉

C₄

yes

—

C₂²⋊C₈×C₁₇₉

C₄

yes

—

C₄×C₈×C₁₇₉

C₂²

yes

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

C₂³

yes

characteristic

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

M₁₆

yes

complemented

C₂³×C₁₇₉

C₄²

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₄²

yes

characteristic

C₂×C₄×C₁₇₉

C₄²

yes

characteristic

C₂×C₄×C₁₇₉

C₄²

yes

characteristic

C₂×C₄×C₁₇₉

M₁₆

yes

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#97

yes

—

C₂²×C₁₇₉

C₂×C₄²

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#97

yes

—

C₄×C₁₇₉

C₂×M₁₆

yes

—

C₄×C₁₇₉

C₂×M₁₆

yes

—

C₄×C₁₇₉

#131

yes

—

C₂×C₁₇₉

—

C₂×C₁₇₉

C₂×#97

yes

characteristic

C₂×C₁₇₉

C₄×M₁₆

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1451

—

179

maximal, Sylow

#431

D₁₇₉

yes

characteristic

C₄×M₁₆

—

179

—

C₂×C₄×C₈

—

179

—

#322

—

179

—

#322

—

179

—

C₂×M₁₆

—

179

—

C₂²×C₈

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

C₄⋊C₈

Dic₁₇₉

yes

—

C₂²⋊C₈

Dic₁₇₉

yes

—

#97

—

179

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

C₂×D₁₇₉

yes

—

C₂²×C₄

C₄×D₁₇₉

yes

characteristic

C₂²×C₄

C₄×D₁₇₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₉

yes

characteristic

M₁₆

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

C₂×Dic₁₇₉

yes

—

C₄²

C₄×D₁₇₉

yes

characteristic

C₄²

C₄×D₁₇₉

yes

characteristic

C₄²

C₂×Dic₁₇₉

yes

characteristic

C₄²

C₂²×D₁₇₉

yes

characteristic

C₂³

C₄×Dic₁₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₇₉

yes

characteristic

C₂×C₄

C₄×Dic₁₇₉

yes

characteristic

C₂×C₄

C₄×Dic₁₇₉

yes

characteristic

C₂×C₄

#15668242

yes

—

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

C₂×C₄

C₂²×Dic₁₇₉

yes

characteristic

C₈

—

100

C₈

—

358

—

101

C₈

—

358

—

102

C₈

—

179

—

103

C₂²

—

104

C₂²

—

105

C₂²

#19153023

yes

—

106

C₂²

C₂×C₄×Dic₁₇₉

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#19153023

yes

—

110

C₄

M₁₆×D₁₇₉

yes

—

111

C₄

C₂×#15668242

yes

—

112

C₄

#19153120

yes

—

113

C₂

—

114

C₂

C₂×#19153023

yes

characteristic

115

C₂

M₁₆×Dic₁₇₉

yes

characteristic

116

C₂

#26038395

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₇₉
commutator, socle	C₂²×C₁₇₉	C₂×C₄²
Fitting	#431×C₁₇₉	C₂
prime radical	C₂³×C₁₇₉	C₄²
cosocle	C₄²×C₁₇₉	C₂³
hypercenter	#431	D₁₇₉
hypocenter	C₁₇₉	#1451
LCS	G ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₇₉

commutator, socle

C₂²×C₁₇₉

C₂×C₄²

Fitting

#431×C₁₇₉

C₂

prime radical

C₂³×C₁₇₉

C₄²

cosocle

C₄²×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂²×C₁₇₉ ▷ C₁₇₉

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #26038161	C₂	semidirect
4. #26038160	C₂	semidirect
32. C₁₇₉⋊C₈	M₁₆	semidirect
58. C₁₇₉	#1451	semidirect

H ≅ G/N

type

3. #26038161

C₂

semidirect

4. #26038160

C₂

semidirect

32. C₁₇₉⋊C₈

M₁₆

semidirect

58. C₁₇₉

#1451

semidirect

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

subgroup	index
C₄×#15668243	2
C₂×C₄×C₁₇₉⋊C₈	2
#26038161	2
#26038160	2
#431×C₁₇₉	2
#1451	179

subgroup

index

C₂×C₄×C₁₇₉⋊C₈

#431×C₁₇₉

179

Subgroups — #37507290

Conjugacy classes of subgroups

Complements

Maximal subgroups