Group #36355370 — subgroups

Subgroups — #36355370

Order 22144.

138 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄⋊#14583801	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#14583801	C₂	1	2	yes	maximal
3	#26011267	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14583801	C₂	1	1	yes	maximal, characteristic, complemented
5	#405×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583801	C₂²	1	2	yes	—
7	C₂×#14583801	C₂²	1	2	yes	—
8	C₂×M₁₆×C₁₇₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	#117×C₁₇₃	C₄	1	1	yes	characteristic
11	#97×C₁₇₃	C₄	1	1	yes	characteristic
12	#97×C₁₇₃	C₄	1	1	yes	characteristic
13	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
15	#14583801	—	2	2	no	—
16	#14583801	—	2	2	no	—
17	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₁₇₃	—	2	2	no	—
19	C₂×C₈×C₁₇₃	—	2	2	no	—
20	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₇₃	C₈	1	2	yes	—
25	C₂²⋊C₄×C₁₇₃	C₈	1	2	yes	—
26	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₇₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇₃	—	2	1	no	—
30	C₂×C₄×C₁₇₃	—	2	1	no	—
31	C₂×C₄×C₁₇₃	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₇₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₇₃	—	2	1	no	—
36	C₈×C₁₇₃	—	2	1	no	—
37	C₈×C₁₇₃	M₁₆	1	2	yes	—
38	C₈×C₁₇₃	#41	1	2	yes	—
39	C₂²×C₁₇₃	—	2	1	no	—
40	C₂²×C₁₇₃	—	2	2	no	—
41	C₂²×C₁₇₃	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₇₃	—	2	1	no	—
43	C₄×C₁₇₃	—	2	2	no	—
44	C₄×C₁₇₃	—	2	2	no	—
45	C₄×C₁₇₃	#117	1	1	yes	characteristic
46	C₄×C₁₇₃	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₁₇₃	—	4	1	no	—
48	C₂×C₁₇₃	#431	1	1	yes	characteristic
49	C₂×C₁₇₃	#504	1	2	yes	—
50	C₁₇₃	#2164	1	1	yes	characteristic, complemented, Sylow
51	#2164	—	173	1	no	maximal, Sylow
52	#405	D₁₇₃	1	1	yes	characteristic
53	C₄⋊C₁₆	—	173	2	no	—
54	C₂²⋊C₁₆	—	173	2	no	—
55	#346	—	173	1	no	—
56	C₄×C₁₆	—	173	1	no	—
57	C₂×M₁₆	Dic₁₇₃	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
59	#117	Dic₁₇₃	1	1	yes	characteristic
60	C₂×C₁₆	—	173	2	no	—
61	C₂×C₁₆	—	173	2	no	—
62	#97	Dic₁₇₃	1	1	yes	characteristic
63	#97	Dic₁₇₃	1	1	yes	characteristic
64	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
65	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
66	C₂²×C₄	C₂×Dic₁₇₃	1	1	yes	characteristic
67	M₁₆	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
70	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
71	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
73	C₄⋊C₄	#13063436	1	2	yes	—
74	C₂²⋊C₄	#13063436	1	2	yes	—
75	C₄²	C₂×Dic₁₇₃	1	1	yes	characteristic
76	C₄²	C₂×Dic₁₇₃	1	1	yes	characteristic
77	C₁₆	—	346	2	no	—
78	C₁₆	—	346	2	no	—
79	C₂³	C₂×#13063436	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×#13063436	1	2	yes	—
83	C₂×C₄	C₂×#13063436	1	2	yes	—
84	C₂×C₄	C₂×#13063436	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₁₇₃	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	#14583810	1	2	yes	—
89	C₈	#14583826	1	2	yes	—
90	C₂²	—	2	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	C₂²×#13063436	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	#19149638	1	1	yes	characteristic
97	C₄	C₂×#14583810	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#26011544	1	1	yes	characteristic
100	C₂	#26011908	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#14583801

C₂

yes

maximal, complemented

C₂²⋊#14583801

C₂

yes

maximal

#26011267

C₂

yes

maximal, characteristic, complemented

C₄×#14583801

C₂

yes

maximal, characteristic, complemented

#405×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583801

C₂²

yes

—

C₂×#14583801

C₂²

yes

—

C₂×M₁₆×C₁₇₃

C₄

yes

characteristic

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

#117×C₁₇₃

C₄

yes

characteristic

#97×C₁₇₃

C₄

yes

characteristic

#97×C₁₇₃

C₄

yes

characteristic

C₄×C₈×C₁₇₃

C₂²

yes

characteristic

C₄×C₈×C₁₇₃

C₂²

yes

characteristic

#14583801

—

#14583801

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

characteristic

M₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₃

C₈

yes

—

C₂²⋊C₄×C₁₇₃

C₈

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

C₂³×C₁₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×C₈

yes

—

C₂×C₄×C₁₇₃

C₂×C₈

yes

—

C₂×C₄×C₁₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₈×C₁₇₃

—

C₈×C₁₇₃

—

C₈×C₁₇₃

M₁₆

yes

—

C₈×C₁₇₃

#41

yes

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂²×C₈

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#117

yes

characteristic

C₄×C₁₇₃

C₂×M₁₆

yes

characteristic

C₂×C₁₇₃

—

C₂×C₁₇₃

#431

yes

characteristic

C₂×C₁₇₃

#504

yes

—

C₁₇₃

#2164

yes

characteristic, complemented, Sylow

#2164

—

173

maximal, Sylow

#405

D₁₇₃

yes

characteristic

C₄⋊C₁₆

—

173

—

C₂²⋊C₁₆

—

173

—

#346

—

173

—

C₄×C₁₆

—

173

—

C₂×M₁₆

Dic₁₇₃

yes

characteristic

C₂²×C₈

C₂×D₁₇₃

yes

characteristic

#117

Dic₁₇₃

yes

characteristic

C₂×C₁₆

—

173

—

C₂×C₁₆

—

173

—

#97

Dic₁₇₃

yes

characteristic

#97

Dic₁₇₃

yes

characteristic

C₄×C₈

C₂×D₁₇₃

yes

characteristic

C₄×C₈

C₂×D₁₇₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₇₃

yes

characteristic

C₂×C₈

C₂×Dic₁₇₃

yes

characteristic

C₂×C₈

C₂×Dic₁₇₃

yes

characteristic

C₂×C₈

C₂²×D₁₇₃

yes

characteristic

C₄⋊C₄

#13063436

yes

—

C₂²⋊C₄

#13063436

yes

—

C₄²

C₂×Dic₁₇₃

yes

characteristic

C₄²

C₂×Dic₁₇₃

yes

characteristic

C₁₆

—

346

—

C₁₆

—

346

—

C₂³

C₂×#13063436

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#13063436

yes

—

C₂×C₄

C₂×#13063436

yes

—

C₂×C₄

C₂×#13063436

yes

characteristic

C₂×C₄

C₂²×Dic₁₇₃

yes

characteristic

C₈

—

C₈

—

C₈

#14583810

yes

—

C₈

#14583826

yes

—

C₂²

—

C₂²

—

C₂²

C₂²×#13063436

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

#26011544

yes

characteristic

100

C₂

#26011908

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₇₃
commutator, socle	C₂²×C₁₇₃	C₂²×C₈
Fitting	#405×C₁₇₃	C₂
prime radical	C₂³×C₁₇₃	C₂×C₈
cosocle	C₂×C₈×C₁₇₃	C₂³
hypercenter	#405	D₁₇₃
hypocenter	C₁₇₃	#2164
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₇₃

commutator, socle

C₂²×C₁₇₃

C₂²×C₈

Fitting

#405×C₁₇₃

C₂

prime radical

C₂³×C₁₇₃

C₂×C₈

cosocle

C₂×C₈×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂²×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂×C₈ ◁ #405

N	H ≅ G/N	type
1. C₄⋊#14583801	C₂	semidirect
3. #26011267	C₂	semidirect
4. C₄×#14583801	C₂	semidirect
50. C₁₇₃	#2164	semidirect

H ≅ G/N

type

1. C₄⋊#14583801

C₂

semidirect

3. #26011267

C₂

semidirect

4. C₄×#14583801

C₂

semidirect

50. C₁₇₃

#2164

semidirect

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

subgroup	index
C₄⋊#14583801	2
C₂²⋊#14583801	2
#26011267	2
C₄×#14583801	2
#405×C₁₇₃	2
#2164	173

subgroup

index

#405×C₁₇₃

173

Subgroups — #36355370

Conjugacy classes of subgroups

Complements

Maximal subgroups