Group #37510420 — subgroups

Subgroups — #37510420

Order 22912.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#19153038	C₂	1	2	yes	maximal
2	#26038345	C₂	1	1	yes	maximal, characteristic
3	#26038305	C₂	1	1	yes	maximal, characteristic
4	#26038202	C₂	1	2	yes	maximal
5	#336×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₇₉	C₂²	1	1	yes	characteristic
9	#19153038	—	2	4	no	—
10	C₁₇₉⋊#95	—	2	2	no	—
11	C₁₇₉⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₇₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₅₈	—	2	1	no	—
15	C₂×Dic₃₅₈	—	2	2	no	—
16	C₂×Dic₃₅₈	—	2	2	no	—
17	C₂×Dic₃₅₈	D₄	1	2	yes	—
18	C₂²×Dic₁₇₉	—	2	1	no	—
19	C₂²×Dic₁₇₉	—	2	2	no	—
20	C₂²×Dic₁₇₉	—	2	1	no	—
21	C₄⋊Dic₁₇₉	—	2	2	no	—
22	C₄⋊Dic₁₇₉	—	2	2	no	—
23	C₄⋊Dic₁₇₉	—	2	2	no	—
24	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₇₉	—	2	2	no	—
27	C₂×C₈×C₁₇₉	—	2	2	no	—
28	C₂×C₈×C₁₇₉	—	2	2	no	—
29	C₄²×C₁₇₉	—	2	2	no	—
30	Dic₃₅₈	—	2	4	no	—
31	Dic₃₅₈	—	4	2	no	—
32	C₂×Dic₁₇₉	—	2	2	no	—
33	C₂×Dic₁₇₉	—	4	1	no	—
34	C₂×Dic₁₇₉	—	4	1	no	—
35	C₂×Dic₁₇₉	—	4	2	no	—
36	C₂×Dic₁₇₉	—	4	1	no	—
37	C₂×Dic₁₇₉	—	4	2	no	—
38	C₂×Dic₁₇₉	—	4	2	no	—
39	C₂×Dic₁₇₉	—	4	2	no	—
40	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	—	2	2	no	—
42	C₂×C₄×C₁₇₉	—	2	2	no	—
43	C₂×C₄×C₁₇₉	—	2	2	no	—
44	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	#41	1	2	yes	—
46	C₂×C₄×C₁₇₉	#41	1	2	yes	—
47	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
48	C₈×C₁₇₉	—	4	2	no	—
49	C₂²×C₁₇₉	#121	1	1	yes	characteristic
50	C₂²×C₁₇₉	#123	1	1	yes	characteristic
51	C₂²×C₁₇₉	#124	1	1	yes	characteristic
52	C₂²×C₁₇₉	#135	1	2	yes	—
53	C₂²×C₁₇₉	#137	1	2	yes	—
54	C₄×C₁₇₉	—	4	2	no	—
55	C₄×C₁₇₉	#123	1	2	yes	—
56	C₄×C₁₇₉	#126	1	2	yes	—
57	Dic₁₇₉	—	4	2	no	—
58	Dic₁₇₉	—	8	2	no	—
59	Dic₁₇₉	—	8	1	no	—
60	C₂×C₁₇₉	#397	1	1	yes	characteristic
61	C₂×C₁₇₉	#464	1	2	yes	—
62	C₂×C₁₇₉	#485	1	2	yes	—
63	C₂×C₁₇₉	#488	1	1	yes	characteristic
64	C₂×C₁₇₉	#491	1	1	yes	characteristic
65	C₁₇₉	#2074	1	1	yes	characteristic, complemented, Sylow
66	#2074	—	179	1	no	maximal, Sylow
67	C₂×#103	—	179	2	no	—
68	#391	—	179	1	no	—
69	#382	—	179	1	no	—
70	#340	—	179	2	no	—
71	#336	D₁₇₉	1	1	yes	characteristic
72	C₂²×Q₈	—	179	1	no	—
73	C₂²×C₈	C₂×D₁₇₉	1	2	yes	—
74	C₂×C₄⋊C₄	—	179	1	no	—
75	C₂×C₄⋊C₄	—	179	2	no	—
76	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
77	#103	—	358	4	no	—
78	#95	—	358	2	no	—
79	#95	—	358	2	no	—
80	C₂×Q₈	—	358	1	no	—
81	C₂×Q₈	—	358	2	no	—
82	C₂×Q₈	—	358	2	no	—
83	C₂×Q₈	—	179	2	no	—
84	C₂²×C₄	—	358	1	no	—
85	C₂²×C₄	—	358	2	no	—
86	C₂²×C₄	—	358	1	no	—
87	C₂²×C₄	D₇₁₆	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	358	2	no	—
93	C₄⋊C₄	—	358	2	no	—
94	C₄⋊C₄	—	358	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₇₁₆	1	1	yes	characteristic
97	Q₈	—	358	4	no	—
98	Q₈	—	716	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	358	2	no	—
103	C₂×C₄	—	716	2	no	—
104	C₂×C₄	—	716	1	no	—
105	C₂×C₄	—	716	2	no	—
106	C₂×C₄	—	716	2	no	—
107	C₂×C₄	—	716	2	no	—
108	C₂×C₄	—	716	1	no	—
109	C₂×C₄	—	716	1	no	—
110	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
111	C₂×C₄	#15668258	1	2	yes	—
112	C₂×C₄	#15668259	1	2	yes	—
113	C₂×C₄	#15668260	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	#19153088	1	1	yes	characteristic
116	C₂²	#19153097	1	1	yes	characteristic
117	C₂²	#19153101	1	1	yes	characteristic
118	C₂²	#19153135	1	2	yes	—
119	C₂²	#19153146	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	1432	1	no	—
122	C₄	—	1432	2	no	—
123	C₄	—	716	2	no	—
124	C₄	#19153095	1	2	yes	—
125	C₄	#19153108	1	2	yes	—
126	C₂	#26038366	1	1	yes	characteristic
127	C₂	#26038662	1	2	yes	—
128	C₂	#26038781	1	2	yes	—
129	C₂	#26038794	1	1	yes	characteristic
130	C₂	#26038808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153038

C₂

yes

maximal

#26038345

C₂

yes

maximal, characteristic

#26038305

C₂

yes

maximal, characteristic

#26038202

C₂

yes

maximal

#336×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

characteristic

#19153038

—

C₁₇₉⋊#95

—

C₁₇₉⋊#95

—

C₂²×C₈×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

D₄

yes

—

C₂²×Dic₁₇₉

—

C₂²×Dic₁₇₉

—

C₂²×Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₄²×C₁₇₉

—

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

—

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

—

C₂²×C₁₇₉

#137

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#123

yes

—

C₄×C₁₇₉

#126

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

#397

yes

characteristic

C₂×C₁₇₉

#464

yes

—

C₂×C₁₇₉

#485

yes

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#2074

—

179

maximal, Sylow

C₂×#103

—

179

—

#391

—

179

—

#382

—

179

—

#340

—

179

—

#336

D₁₇₉

yes

characteristic

C₂²×Q₈

—

179

—

C₂²×C₈

C₂×D₁₇₉

yes

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#103

—

358

—

#95

—

358

—

#95

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

C₂²×C₄

—

358

—

C₂²×C₄

—

358

—

C₂²×C₄

—

358

—

C₂²×C₄

D₇₁₆

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄²

—

C₂³

C₂×D₇₁₆

yes

characteristic

Q₈

—

358

—

Q₈

—

716

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

358

—

103

C₂×C₄

—

716

—

104

C₂×C₄

—

716

—

105

C₂×C₄

—

716

—

106

C₂×C₄

—

716

—

107

C₂×C₄

—

716

—

108

C₂×C₄

—

716

—

109

C₂×C₄

—

716

—

110

C₂×C₄

D₄×D₁₇₉

yes

characteristic

111

C₂×C₄

#15668258

yes

—

112

C₂×C₄

#15668259

yes

—

113

C₂×C₄

#15668260

yes

characteristic

114

C₈

—

115

C₂²

#19153088

yes

characteristic

116

C₂²

#19153097

yes

characteristic

117

C₂²

#19153101

yes

characteristic

118

C₂²

#19153135

yes

—

119

C₂²

#19153146

yes

—

120

C₄

—

121

C₄

—

1432

—

122

C₄

—

1432

—

123

C₄

—

716

—

124

C₄

#19153095

yes

—

125

C₄

#19153108

yes

—

126

C₂

#26038366

yes

characteristic

127

C₂

#26038662

yes

—

128

C₂

#26038781

yes

—

129

C₂

#26038794

yes

characteristic

130

C₂

#26038808

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₆
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#336×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
hypercenter	#336	D₁₇₉
hypocenter	C₁₇₉	#2074
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₇₁₆

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#336×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
65. C₁₇₉	#2074	semidirect

H ≅ G/N

type

65. C₁₇₉

#2074

semidirect

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

subgroup	index
C₂×#19153038	2
#26038345	2
#26038305	2
#26038202	2
#336×C₁₇₉	2
#2074	179

subgroup

index

#336×C₁₇₉

179

Subgroups — #37510420

Conjugacy classes of subgroups

Complements

Maximal subgroups