Group #37508079 — subgroups

Subgroups — #37508079

Order 22912.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#26039169	C₂	1	1	yes	maximal, characteristic, complemented
2	#26038813	C₂	1	1	yes	maximal, characteristic
3	#26038640	C₂	1	2	yes	maximal
4	#26038169	C₂	1	2	yes	maximal, complemented
5	#333×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668248	—	2	2	no	—
7	C₂×#15668246	—	2	2	no	—
8	#19153112	—	2	2	no	—
9	#19153111	—	2	2	no	—
10	#19153110	C₂²	1	1	yes	characteristic
11	C₄×Dic₃₅₈	C₂²	1	2	yes	—
12	#19153078	—	2	2	no	—
13	#19153035	—	2	2	no	—
14	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₇₉	C₂²	1	2	yes	—
17	C₂×Dic₃₅₈	—	2	2	no	—
18	C₂×Dic₃₅₈	D₄	1	2	yes	—
19	C₂×D₇₁₆	—	2	2	no	—
20	C₂×D₇₁₆	—	2	1	no	—
21	#15668248	—	4	4	no	—
22	#15668246	—	4	4	no	—
23	C₂×C₁₇₉⋊C₈	—	2	1	no	—
24	C₄⋊Dic₁₇₉	D₄	1	2	yes	—
25	Dic₁₇₉⋊C₄	—	2	4	no	—
26	Dic₁₇₉⋊C₄	—	2	2	no	—
27	C₄×Dic₁₇₉	—	2	2	no	—
28	C₄×Dic₁₇₉	—	2	2	no	—
29	C₂×Q₈×C₁₇₉	D₄	1	2	yes	—
30	C₂×C₈×C₁₇₉	—	2	2	no	—
31	C₄⋊C₄×C₁₇₉	—	2	2	no	—
32	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
33	C₂²×D₁₇₉	—	4	1	no	—
34	Dic₃₅₈	—	2	2	no	—
35	Dic₃₅₈	—	8	2	no	—
36	Dic₃₅₈	SD₁₆	1	4	yes	complemented
37	D₇₁₆	—	4	2	no	—
38	D₇₁₆	—	4	2	no	—
39	D₇₁₆	—	4	2	no	—
40	C₂×Dic₁₇₉	—	2	2	no	—
41	C₂×Dic₁₇₉	—	2	2	no	—
42	C₁₇₉⋊C₈	—	2	2	no	—
43	Q₈×C₁₇₉	—	4	2	no	—
44	C₂×C₄×C₁₇₉	—	2	2	no	—
45	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₇₉	—	4	2	no	—
48	C₂²×C₁₇₉	#120	1	1	yes	characteristic
49	C₂×D₁₇₉	—	8	2	no	—
50	C₂×D₁₇₉	—	8	1	no	—
51	C₄×C₁₇₉	—	2	1	no	—
52	C₄×C₁₇₉	—	4	2	no	—
53	C₄×C₁₇₉	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₇₉	#137	1	2	yes	—
55	Dic₁₇₉	—	2	4	no	—
56	Dic₁₇₉	—	4	2	no	—
57	C₂×C₁₇₉	#448	1	2	yes	—
58	C₂×C₁₇₉	#453	1	1	yes	characteristic
59	D₁₇₉	—	16	1	no	—
60	C₁₇₉	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	179	1	no	maximal, Sylow
62	Q₈²	—	179	1	no	—
63	#492	—	179	1	no	—
64	#460	—	179	2	no	—
65	#333	D₁₇₉	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	179	2	no	—
67	C₂×SD₁₆	—	358	2	no	—
68	C₂×SD₁₆	—	358	2	no	—
69	C₄⋊Q₈	—	358	2	no	—
70	C₄⋊Q₈	—	358	2	no	—
71	C₄⋊Q₈	C₂×D₁₇₉	1	1	yes	characteristic
72	#127	—	179	1	no	—
73	C₄×Q₈	—	358	2	no	—
74	C₄×Q₈	—	179	2	no	—
75	C₄⋊C₈	C₂×D₁₇₉	1	2	yes	—
76	#102	—	358	2	no	—
77	C₄×C₈	—	179	1	no	—
78	C₂×Q₈	—	358	2	no	—
79	C₂×Q₈	—	179	2	no	—
80	C₂×Q₈	D₇₁₆	1	2	yes	—
81	C₂×D₄	—	358	2	no	—
82	C₂×D₄	—	358	1	no	—
83	SD₁₆	—	716	4	no	—
84	SD₁₆	—	716	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	358	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	358	4	no	—
89	C₄⋊C₄	—	358	2	no	—
90	C₄⋊C₄	—	179	2	no	—
91	C₄²	—	358	2	no	—
92	C₄²	—	358	2	no	—
93	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
94	C₂³	—	716	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	358	2	no	—
97	Q₈	—	1432	2	no	—
98	Q₈	—	179	4	no	—
99	D₄	—	716	2	no	—
100	D₄	—	716	2	no	—
101	D₄	—	716	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	358	2	no	—
104	C₂×C₄	—	358	2	no	—
105	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
106	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	358	2	no	—
109	C₂²	—	1432	2	no	—
110	C₂²	—	1432	1	no	—
111	C₂²	#19153083	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	358	4	no	—
115	C₄	—	716	2	no	—
116	C₄	SD₁₆×D₁₇₉	1	2	yes	—
117	C₄	#19153145	1	2	yes	—
118	C₂	—	2864	1	no	—
119	C₂	#26038579	1	2	yes	—
120	C₂	#26038609	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26039169

C₂

yes

maximal, characteristic, complemented

#26038813

C₂

yes

maximal, characteristic

#26038640

C₂

yes

maximal

#26038169

C₂

yes

maximal, complemented

#333×C₁₇₉

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₃₅₈

C₂²

yes

—

#19153078

—

#19153035

—

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₉

C₂²

yes

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

D₄

yes

—

C₂×D₇₁₆

—

C₂×D₇₁₆

—

#15668248

—

#15668246

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

D₄

yes

—

Dic₁₇₉⋊C₄

—

Dic₁₇₉⋊C₄

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₂×Q₈×C₁₇₉

D₄

yes

—

C₂×C₈×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂³

yes

characteristic

C₂²×D₁₇₉

—

Dic₃₅₈

—

Dic₃₅₈

—

Dic₃₅₈

SD₁₆

yes

complemented

D₇₁₆

—

D₇₁₆

—

D₇₁₆

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

Q₈×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₈×C₁₇₉

—

C₂²×C₁₇₉

#120

yes

characteristic

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×SD₁₆

yes

—

C₄×C₁₇₉

#137

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

#448

yes

—

C₂×C₁₇₉

#453

yes

characteristic

D₁₇₉

—

C₁₇₉

#1623

yes

characteristic, complemented, Sylow

#1623

—

179

maximal, Sylow

Q₈²

—

179

—

#492

—

179

—

#460

—

179

—

#333

D₁₇₉

yes

characteristic, complemented

Q₈⋊C₈

—

179

—

C₂×SD₁₆

—

358

—

C₂×SD₁₆

—

358

—

C₄⋊Q₈

—

358

—

C₄⋊Q₈

—

358

—

C₄⋊Q₈

C₂×D₁₇₉

yes

characteristic

#127

—

179

—

C₄×Q₈

—

358

—

C₄×Q₈

—

179

—

C₄⋊C₈

C₂×D₁₇₉

yes

—

#102

—

358

—

C₄×C₈

—

179

—

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

C₂×Q₈

D₇₁₆

yes

—

C₂×D₄

—

358

—

C₂×D₄

—

358

—

SD₁₆

—

716

—

SD₁₆

—

716

—

C₂×C₈

—

C₂×C₈

—

358

—

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

179

—

C₄²

—

358

—

C₄²

—

358

—

C₄²

C₂²×D₁₇₉

yes

characteristic

C₂³

—

716

—

Q₈

—

Q₈

—

358

—

Q₈

—

1432

—

Q₈

—

179

—

D₄

—

716

—

100

D₄

—

716

—

101

D₄

—

716

—

102

C₂×C₄

—

103

C₂×C₄

—

358

—

104

C₂×C₄

—

358

—

105

C₂×C₄

D₄×D₁₇₉

yes

—

106

C₂×C₄

C₂×D₇₁₆

yes

characteristic

107

C₈

—

108

C₈

—

358

—

109

C₂²

—

1432

—

110

C₂²

—

1432

—

111

C₂²

#19153083

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

358

—

115

C₄

—

716

—

116

C₄

SD₁₆×D₁₇₉

yes

—

117

C₄

#19153145

yes

—

118

C₂

—

2864

—

119

C₂

#26038579

yes

—

120

C₂

#26038609

yes

characteristic

	subgroup	G/H
center	C₂²	#19153083
commutator, cosocle	C₄²×C₁₇₉	C₂³
Frattini	C₄²	C₂²×D₁₇₉
Fitting	#333×C₁₇₉	C₂
socle	C₂²×C₁₇₉	#120
prime radical	#19153110	C₂²
hypercenter	#333	D₁₇₉
hypocenter	C₁₇₉	#1623
LCS	G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#19153083

commutator, cosocle

C₄²×C₁₇₉

C₂³

Frattini

C₄²

C₂²×D₁₇₉

Fitting

#333×C₁₇₉

C₂

C₂²×C₁₇₉

C₂²

D₁₇₉

C₁₇₉

G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #26039169	C₂	semidirect
4. #26038169	C₂	semidirect
5. #333×C₁₇₉	C₂	semidirect
36. Dic₃₅₈	SD₁₆	semidirect
60. C₁₇₉	#1623	semidirect
65. #333	D₁₇₉	semidirect

H ≅ G/N

type

1. #26039169

C₂

semidirect

4. #26038169

C₂

semidirect

5. #333×C₁₇₉

C₂

semidirect

36. Dic₃₅₈

SD₁₆

semidirect

60. C₁₇₉

#1623

semidirect

65. #333

D₁₇₉

semidirect

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

subgroup	index
#26039169	2
#26038813	2
#26038640	2
#26038169	2
#333×C₁₇₉	2
#1623	179

subgroup

index

#333×C₁₇₉

179

Subgroups — #37508079

Conjugacy classes of subgroups

Complements

Maximal subgroups