Group #37510576 — subgroups

Subgroups — #37510576

Order 22912.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#26038488	C₂	1	1	yes	maximal, characteristic
2	#26038462	C₂	1	1	yes	maximal, characteristic
3	#26038209	C₂	1	2	yes	maximal
4	#26038198	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₂²	1	2	yes	—
7	C₂×#15668243	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
9	#19153049	—	2	2	no	—
10	Dic₁₇₉⋊C₈	—	2	2	no	—
11	C₂×M₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
12	#117×C₁₇₉	C₂²	1	2	yes	—
13	C₂²×Dic₁₇₉	—	2	1	no	—
14	#15668243	—	2	2	no	—
15	#15668243	—	2	2	no	—
16	#15668243	—	2	2	no	—
17	C₂×C₁₇₉⋊C₈	—	2	1	no	—
18	C₂×C₁₇₉⋊C₈	—	2	2	no	—
19	C₄×Dic₁₇₉	—	2	1	no	—
20	C₄×Dic₁₇₉	Q₈	1	1	yes	characteristic
21	C₄×Dic₁₇₉	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₇₉	—	2	2	no	—
24	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₇₉	—	2	2	no	—
26	C₂²⋊C₄×C₁₇₉	—	2	2	no	—
27	C₄²×C₁₇₉	—	2	2	no	—
28	C₂×Dic₁₇₉	—	2	1	no	—
29	C₂×Dic₁₇₉	—	2	1	no	—
30	C₂×Dic₁₇₉	—	4	2	no	—
31	C₁₇₉⋊C₈	—	4	1	no	—
32	C₁₇₉⋊C₈	—	4	2	no	—
33	C₂³×C₁₇₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₇₉	—	4	2	no	—
35	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₇₉	#41	1	2	yes	—
38	C₂×C₄×C₁₇₉	#41	1	2	yes	—
39	C₈×C₁₇₉	—	2	2	no	—
40	C₂²×C₁₇₉	—	4	1	no	—
41	C₂²×C₁₇₉	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₇₉	#126	1	2	yes	—
43	C₄×C₁₇₉	—	8	2	no	—
44	C₄×C₁₇₉	#123	1	1	yes	characteristic
45	C₄×C₁₇₉	#124	1	1	yes	characteristic
46	C₄×C₁₇₉	#125	1	2	yes	—
47	Dic₁₇₉	—	2	2	no	—
48	Dic₁₇₉	—	4	1	no	—
49	C₂×C₁₇₉	—	2	1	no	—
50	C₂×C₁₇₉	—	2	2	no	—
51	C₂×C₁₇₉	#400	1	1	yes	characteristic
52	C₁₇₉	#2102	1	1	yes	characteristic, complemented, Sylow
53	#2102	—	179	1	no	maximal, Sylow
54	#430	—	179	1	no	—
55	#423	—	179	1	no	—
56	#344	—	179	2	no	—
57	#339	—	179	2	no	—
58	H(3,ℤ/4)	D₁₇₉	1	1	yes	characteristic
59	C₂×M₁₆	—	179	2	no	—
60	C₂×M₁₆	—	179	1	no	—
61	C₂×M₁₆	C₂×D₁₇₉	1	1	yes	characteristic
62	#117	C₂×D₁₇₉	1	2	yes	—
63	C₂×C₄²	—	179	1	no	—
64	#108	—	358	2	no	—
65	C₄⋊C₈	—	358	2	no	—
66	C₂²×C₄	—	358	1	no	—
67	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	358	2	no	—
70	M₁₆	—	358	2	no	—
71	M₁₆	—	358	2	no	—
72	C₂×C₈	—	358	1	no	—
73	C₂×C₈	—	358	2	no	—
74	C₂×C₈	D₇₁₆	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	358	1	no	—
79	C₄²	—	179	1	no	—
80	C₄²	—	179	1	no	—
81	C₂³	#15668258	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	358	1	no	—
84	C₂×C₄	—	358	1	no	—
85	C₂×C₄	—	716	2	no	—
86	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₇₉	1	1	yes	characteristic
88	C₂×C₄	#15668258	1	2	yes	—
89	C₂×C₄	#15668259	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	716	1	no	—
92	C₈	—	716	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#19153093	1	1	yes	characteristic
95	C₂²	#19153106	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	358	2	no	—
98	C₄	—	716	1	no	—
99	C₄	#19153097	1	1	yes	characteristic
100	C₄	#19153101	1	1	yes	characteristic
101	C₄	#19153103	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#26038379	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038488

C₂

yes

maximal, characteristic

#26038462

C₂

yes

maximal, characteristic

#26038209

C₂

yes

maximal

#26038198

C₂

yes

maximal

H(3,ℤ/4)×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

C₂×#15668243

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

#19153049

—

Dic₁₇₉⋊C₈

—

C₂×M₁₆×C₁₇₉

C₂²

yes

characteristic

#117×C₁₇₉

C₂²

yes

—

C₂²×Dic₁₇₉

—

#15668243

—

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

Q₈

yes

characteristic

C₄×Dic₁₇₉

Q₈

yes

characteristic

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇₉

#126

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#400

yes

characteristic

C₁₇₉

#2102

yes

characteristic, complemented, Sylow

#2102

—

179

maximal, Sylow

#430

—

179

—

#423

—

179

—

#344

—

179

—

#339

—

179

—

H(3,ℤ/4)

D₁₇₉

yes

characteristic

C₂×M₁₆

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

characteristic

#117

C₂×D₁₇₉

yes

—

C₂×C₄²

—

179

—

#108

—

358

—

C₄⋊C₈

—

358

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

358

—

M₁₆

—

358

—

M₁₆

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

D₇₁₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

358

—

C₄²

—

179

—

C₄²

—

179

—

C₂³

#15668258

yes

characteristic

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

—

358

—

C₂×C₄

—

716

—

C₂×C₄

C₂×D₇₁₆

yes

characteristic

C₂×C₄

Q₈×D₁₇₉

yes

characteristic

C₂×C₄

#15668258

yes

—

C₂×C₄

#15668259

yes

—

C₈

—

C₈

—

716

—

C₈

—

716

—

C₂²

—

C₂²

#19153093

yes

characteristic

C₂²

#19153106

yes

—

C₄

—

C₄

—

358

—

C₄

—

716

—

C₄

#19153097

yes

characteristic

100

C₄

#19153101

yes

characteristic

101

C₄

#19153103

yes

—

102

C₂

—

103

C₂

—

104

C₂

#26038379

yes

characteristic

	subgroup	G/H
center	C₄	#19153101
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	H(3,ℤ/4)×C₁₇₉	C₂
socle	C₂×C₁₇₉	#400
prime radical	C₂³×C₁₇₉	#41
hypercenter	H(3,ℤ/4)	D₁₇₉
hypocenter	C₁₇₉	#2102
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19153101

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

H(3,ℤ/4)×C₁₇₉

C₂

C₂×C₁₇₉

C₂³×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
52. C₁₇₉	#2102	semidirect

H ≅ G/N

type

52. C₁₇₉

#2102

semidirect

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

subgroup	index
#26038488	2
#26038462	2
#26038209	2
#26038198	2
H(3,ℤ/4)×C₁₇₉	2
#2102	179

subgroup

index

H(3,ℤ/4)×C₁₇₉

179

Subgroups — #37510576

Conjugacy classes of subgroups

Complements

Maximal subgroups