Group #37507909 — subgroups

Subgroups — #37507909

Order 22912.

148 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 71 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	#26038570	C₂	1	2	yes	maximal
2	#26038190	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₁₇₉⋊C₈)	C₂	1	2	yes	maximal
4	#26038160	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₇₉⋊C₈)	—	2	2	no	—
7	C₄⋊(C₁₇₉⋊C₈)	C₂²	1	2	yes	—
8	#19153024	—	2	2	no	—
9	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₇₉	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₁₇₉	C₂²	1	2	yes	—
13	#107×C₁₇₉	C₄	1	1	yes	characteristic
14	#103×C₁₇₉	C₄	1	2	yes	—
15	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₇₉⋊C₈	—	2	2	no	—
17	C₂×C₁₇₉⋊C₈	—	2	1	no	—
18	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
19	C₂×Q₈×C₁₇₉	C₂×C₄	1	2	yes	—
20	Q₁₆×C₁₇₉	—	2	2	no	—
21	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₇₉	—	2	2	no	—
24	C₄⋊C₄×C₁₇₉	C₂×C₄	1	2	yes	—
25	C₄²×C₁₇₉	—	2	2	no	—
26	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
27	C₁₇₉⋊C₈	—	2	2	no	—
28	C₁₇₉⋊C₈	—	4	1	no	—
29	C₁₇₉⋊C₈	—	4	2	no	—
30	Q₈×C₁₇₉	—	2	2	no	—
31	Q₈×C₁₇₉	—	2	2	no	—
32	C₂×C₄×C₁₇₉	—	2	2	no	—
33	C₂×C₄×C₁₇₉	—	2	2	no	—
34	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
37	C₈×C₁₇₉	—	2	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	—	4	2	no	—
42	C₄×C₁₇₉	—	4	2	no	—
43	C₄×C₁₇₉	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₁₇₉	#131	1	1	yes	characteristic
45	C₄×C₁₇₉	#137	1	2	yes	—
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	D₁₇₉	1	1	yes	characteristic
53	#334	—	179	1	no	—
54	Q₈⋊C₈	—	179	2	no	—
55	#322	—	179	1	no	—
56	C₂×Q₁₆	Dic₁₇₉	1	1	yes	characteristic
57	C₄×Q₈	C₂×D₁₇₉	1	2	yes	—
58	#107	Dic₁₇₉	1	1	yes	characteristic
59	C₄⋊C₈	—	358	2	no	—
60	C₄⋊C₈	—	179	2	no	—
61	#103	Dic₁₇₉	1	2	yes	—
62	#97	—	358	2	no	—
63	C₄×C₈	—	179	1	no	—
64	C₄×C₈	—	179	1	no	—
65	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
66	C₂×Q₈	C₂×Dic₁₇₉	1	2	yes	—
67	Q₁₆	—	2	2	no	—
68	C₂×C₈	—	358	2	no	—
69	C₂×C₈	—	358	1	no	—
70	C₂×C₈	—	179	2	no	—
71	C₂×C₈	C₂×Dic₁₇₉	1	1	yes	characteristic
72	C₂×C₈	C₂×Dic₁₇₉	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂×Dic₁₇₉	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
77	Q₈	—	2	2	no	—
78	Q₈	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂²×Dic₁₇₉	1	1	yes	characteristic
82	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
83	C₂×C₄	#15668258	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	358	2	no	—
86	C₈	—	716	1	no	—
87	C₈	—	716	2	no	—
88	C₈	#15668243	1	2	yes	—
89	C₂²	D₄×Dic₁₇₉	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	4	2	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂×#15668243	1	1	yes	characteristic
94	C₄	#19153120	1	1	yes	characteristic
95	C₄	#19153145	1	2	yes	—
96	C₂	#26038517	1	1	yes	characteristic
97	C₂	#26038546	1	1	yes	characteristic
98	C₂	#26038561	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038570

C₂

yes

maximal

#26038190

C₂

yes

maximal, characteristic

Q₈⋊(C₁₇₉⋊C₈)

C₂

yes

maximal

#26038160

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₁₇₉

C₂

yes

maximal, characteristic

C₄⋊(C₁₇₉⋊C₈)

—

C₄⋊(C₁₇₉⋊C₈)

C₂²

yes

—

#19153024

—

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇₉

C₄

yes

characteristic

C₄×Q₈×C₁₇₉

C₂²

yes

—

#107×C₁₇₉

C₄

yes

characteristic

#103×C₁₇₉

C₄

yes

—

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×Q₈×C₁₇₉

C₂×C₄

yes

—

Q₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

—

C₄²×C₁₇₉

C₂³

yes

characteristic

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

Q₈×C₁₇₉

—

Q₈×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

—

C₈×C₁₇₉

M₁₆

yes

—

C₂²×C₁₇₉

C₄×D₄

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×M₁₆

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1593

—

179

maximal, Sylow

#446

—

179

—

C₄×Q₁₆

D₁₇₉

yes

characteristic

#334

—

179

—

Q₈⋊C₈

—

179

—

#322

—

179

—

C₂×Q₁₆

Dic₁₇₉

yes

characteristic

C₄×Q₈

C₂×D₁₇₉

yes

—

#107

Dic₁₇₉

yes

characteristic

C₄⋊C₈

—

358

—

C₄⋊C₈

—

179

—

#103

Dic₁₇₉

yes

—

#97

—

358

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

C₂×Dic₁₇₉

yes

—

Q₁₆

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

179

—

C₂×C₈

C₂×Dic₁₇₉

yes

characteristic

C₂×C₈

C₂×Dic₁₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₇₉

yes

—

C₄²

—

C₄²

C₂²×D₁₇₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₇₉

yes

characteristic

C₂×C₄

D₄×D₁₇₉

yes

characteristic

C₂×C₄

#15668258

yes

characteristic

C₈

—

C₈

—

358

—

C₈

—

716

—

C₈

—

716

—

C₈

#15668243

yes

—

C₂²

D₄×Dic₁₇₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇₉
Fitting	C₄×Q₁₆×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	C₄×D₄
cosocle	C₄²×C₁₇₉	C₂³
hypercenter	C₄×Q₁₆	D₁₇₉
hypocenter	C₁₇₉	#1593
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇₉

Fitting

C₄×Q₁₆×C₁₇₉

C₂

socle, prime radical

C₂²×C₁₇₉

C₄×D₄

cosocle

C₄²×C₁₇₉

C₂³

hypercenter

C₄×Q₁₆

D₁₇₉

hypocenter

C₁₇₉

#1593

LCS

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

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
#26038570	2
#26038190	2
Q₈⋊(C₁₇₉⋊C₈)	2
#26038160	2
C₄×Q₁₆×C₁₇₉	2
#1593	179

subgroup

index

Q₈⋊(C₁₇₉⋊C₈)

C₄×Q₁₆×C₁₇₉

179

Subgroups — #37507909

Conjugacy classes of subgroups

Complements

Maximal subgroups