Group #37507910 — subgroups

Subgroups — #37507910

Order 22912.

148 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 65 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	#26038569	C₂	1	2	yes	maximal
2	C₄×Dic₇₁₆	C₂	1	1	yes	maximal, characteristic
3	#26038190	C₂	1	1	yes	maximal, characteristic
4	#26038171	C₂	1	2	yes	maximal
5	#322×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₆	C₄	1	1	yes	characteristic
7	C₄×Dic₃₅₈	C₂²	1	2	yes	—
8	#19153047	C₄	1	1	yes	characteristic
9	C₄⋊(C₁₇₉⋊C₈)	C₂²	1	2	yes	—
10	Dic₁₇₉⋊C₈	—	2	2	no	—
11	#19153038	C₄	1	2	yes	—
12	#19153023	—	2	2	no	—
13	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₈	C₂×C₄	1	2	yes	—
17	Dic₇₁₆	—	2	2	no	—
18	C₂×C₁₇₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₇₉	C₂×C₄	1	2	yes	—
20	Dic₁₇₉⋊C₄	—	2	2	no	—
21	C₄×Dic₁₇₉	—	2	2	no	—
22	C₂×C₈×C₁₇₉	—	2	1	no	—
23	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
26	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
27	Dic₃₅₈	—	2	2	no	—
28	Dic₃₅₈	—	2	2	no	—
29	C₂×Dic₁₇₉	—	2	2	no	—
30	C₂×Dic₁₇₉	—	2	2	no	—
31	C₁₇₉⋊C₈	—	4	2	no	—
32	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
35	C₈×C₁₇₉	—	2	1	no	—
36	C₈×C₁₇₉	—	2	2	no	—
37	C₈×C₁₇₉	—	4	1	no	—
38	C₈×C₁₇₉	M₁₆	1	2	yes	—
39	C₂²×C₁₇₉	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₇₉	—	2	1	no	—
41	C₄×C₁₇₉	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₁₇₉	#131	1	1	yes	characteristic
43	C₄×C₁₇₉	#137	1	2	yes	—
44	Dic₁₇₉	—	4	2	no	—
45	Dic₁₇₉	—	4	2	no	—
46	C₂×C₁₇₉	#436	1	1	yes	characteristic
47	C₂×C₁₇₉	#441	1	1	yes	characteristic
48	C₂×C₁₇₉	#444	1	1	yes	characteristic
49	C₁₇₉	#1593	1	1	yes	characteristic, complemented, Sylow
50	#1593	—	179	1	no	maximal, Sylow
51	#446	—	179	2	no	—
52	C₄×Q₁₆	—	179	1	no	—
53	#334	—	179	1	no	—
54	Q₈⋊C₈	—	179	2	no	—
55	#322	D₁₇₉	1	1	yes	characteristic
56	C₂×Q₁₆	—	179	1	no	—
57	C₄×Q₈	—	179	2	no	—
58	#107	—	179	1	no	—
59	C₄⋊C₈	—	358	2	no	—
60	C₄⋊C₈	—	179	2	no	—
61	#103	—	179	2	no	—
62	#97	—	358	2	no	—
63	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
64	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
65	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
66	C₂×Q₈	—	179	2	no	—
67	Q₁₆	—	358	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	358	2	no	—
70	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
71	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
72	C₂×C₈	D₇₁₆	1	2	yes	—
73	C₄⋊C₄	—	358	2	no	—
74	C₄⋊C₄	—	179	2	no	—
75	C₄²	—	358	2	no	—
76	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
77	Q₈	—	358	2	no	—
78	Q₈	—	358	2	no	—
79	C₂×C₄	—	358	2	no	—
80	C₂×C₄	—	358	2	no	—
81	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
82	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
83	C₂×C₄	#15668259	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	716	2	no	—
88	C₈	#15668242	1	2	yes	—
89	C₂²	C₄×D₇₁₆	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	716	2	no	—
92	C₄	—	716	2	no	—
93	C₄	C₂×#15668242	1	1	yes	characteristic
94	C₄	#19153119	1	1	yes	characteristic
95	C₄	#19153146	1	2	yes	—
96	C₂	#26038518	1	1	yes	characteristic
97	C₂	#26038547	1	1	yes	characteristic
98	C₂	#26038562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038569

C₂

yes

maximal

C₄×Dic₇₁₆

C₂

yes

maximal, characteristic

#26038190

C₂

yes

maximal, characteristic

#26038171

C₂

yes

maximal

#322×C₁₇₉

C₂

yes

maximal, characteristic

C₂×Dic₇₁₆

C₄

yes

characteristic

C₄×Dic₃₅₈

C₂²

yes

—

#19153047

C₄

yes

characteristic

C₄⋊(C₁₇₉⋊C₈)

C₂²

yes

—

Dic₁₇₉⋊C₈

—

#19153038

C₄

yes

—

#19153023

—

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

C₂×C₄

yes

—

Dic₇₁₆

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

C₂×C₄

yes

—

Dic₁₇₉⋊C₄

—

C₄×Dic₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

D₄

yes

—

C₄²×C₁₇₉

C₂³

yes

characteristic

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

—

C₈×C₁₇₉

—

C₈×C₁₇₉

—

C₈×C₁₇₉

M₁₆

yes

—

C₂²×C₁₇₉

C₄×D₄

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×M₁₆

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1593

—

179

maximal, Sylow

#446

—

179

—

C₄×Q₁₆

—

179

—

#334

—

179

—

Q₈⋊C₈

—

179

—

#322

D₁₇₉

yes

characteristic

C₂×Q₁₆

—

179

—

C₄×Q₈

—

179

—

#107

—

179

—

C₄⋊C₈

—

358

—

C₄⋊C₈

—

179

—

#103

—

179

—

#97

—

358

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

—

179

—

Q₁₆

—

358

—

C₂×C₈

—

C₂×C₈

—

358

—

C₂×C₈

C₄×D₁₇₉

yes

characteristic

C₂×C₈

C₄×D₁₇₉

yes

characteristic

C₂×C₈

D₇₁₆

yes

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

179

—

C₄²

—

358

—

C₄²

C₂²×D₁₇₉

yes

characteristic

Q₈

—

358

—

Q₈

—

358

—

C₂×C₄

—

358

—

C₂×C₄

—

358

—

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

C₂×C₄

C₂×D₇₁₆

yes

characteristic

C₂×C₄

#15668259

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

716

—

C₈

#15668242

yes

—

C₂²

C₄×D₇₁₆

yes

characteristic

C₄

—

C₄

—

716

—

C₄

—

716

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₆
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇₉
Fitting	#322×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	C₄×D₄
cosocle	C₄²×C₁₇₉	C₂³
hypercenter	#322	D₁₇₉
hypocenter	C₁₇₉	#1593
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₇₁₆

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇₉

Fitting

#322×C₁₇₉

C₂

socle, prime radical

C₂²×C₁₇₉

C₄×D₄

cosocle

C₄²×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
49. C₁₇₉	#1593	semidirect

H ≅ G/N

type

49. C₁₇₉

#1593

semidirect

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

subgroup	index
#26038569	2
C₄×Dic₇₁₆	2
#26038190	2
#26038171	2
#322×C₁₇₉	2
#1593	179

subgroup

index

C₄×Dic₇₁₆

#322×C₁₇₉

179

Subgroups — #37507910

Conjugacy classes of subgroups

Complements

Maximal subgroups