Group #36369489 — subgroups

Subgroups — #36369489

Order 22144.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#26012496	C₂	1	1	yes	maximal, characteristic
2	#26012495	C₂	1	1	yes	maximal, characteristic
3	#26012367	C₂	1	4	yes	maximal
4	#97×D₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14583836	C₂²	1	2	yes	—
6	C₂×#14583836	C₂²	1	2	yes	—
7	#19149746	—	2	2	no	—
8	C₄⋊#13063443	—	2	2	no	—
9	C₂×C₈×D₁₇₃	C₂²	1	2	yes	—
10	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
11	#19149590	C₄	1	1	yes	characteristic
12	#19149589	C₄	1	2	yes	—
13	#97×C₁₇₃	C₄	1	1	yes	characteristic
14	#14583836	—	2	4	no	—
15	#14583836	—	2	4	no	—
16	C₂×#13063443	—	2	2	no	—
17	C₂×#13063443	—	2	2	no	—
18	C₂×C₄×D₁₇₃	Q₈	1	2	yes	—
19	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
20	C₂×#13063436	C₂×C₄	1	2	yes	—
21	C₈×D₁₇₃	—	2	2	no	—
22	C₈×D₁₇₃	—	2	2	no	—
23	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₇₃	D₄	1	2	yes	—
25	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
26	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
27	#13063443	—	4	2	no	—
28	#13063443	—	4	2	no	—
29	C₂²×D₁₇₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×Dic₁₇₃	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
32	C₄×D₁₇₃	—	4	2	no	—
33	C₄×D₁₇₃	#41	1	2	yes	—
34	C₄×D₁₇₃	#41	1	2	yes	—
35	#13063436	—	2	2	no	—
36	C₂×C₄×C₁₇₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₇₃	—	2	2	no	—
39	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×D₁₇₃	—	2	2	no	—
41	C₂×D₁₇₃	#125	1	2	yes	—
42	C₄×C₁₇₃	—	2	2	no	—
43	C₄×C₁₇₃	#117	1	2	yes	—
44	Dic₁₇₃	—	2	2	no	—
45	Dic₁₇₃	#124	1	1	yes	characteristic
46	Dic₁₇₃	#124	1	1	yes	characteristic
47	C₂×C₁₇₃	#382	1	1	yes	characteristic
48	C₂×C₁₇₃	#430	1	2	yes	—
49	D₁₇₃	—	2	2	no	—
50	C₁₇₃	#1838	1	1	yes	characteristic, complemented, Sylow
51	#1838	—	173	1	no	maximal, Sylow
52	#423	—	173	1	no	—
53	#423	—	173	1	no	—
54	C₂×#97	—	173	1	no	—
55	#341	—	173	4	no	—
56	C₂×M₁₆	—	173	2	no	—
57	C₂×M₁₆	—	173	2	no	—
58	C₂²×C₈	—	173	2	no	—
59	C₂×C₄²	—	173	1	no	—
60	C₄⋊C₈	—	346	2	no	—
61	C₄⋊C₈	—	346	2	no	—
62	#97	—	173	2	no	—
63	#97	—	173	1	no	—
64	#97	#10621924	1	1	yes	characteristic
65	C₂²×C₄	—	173	1	no	—
66	C₂²×C₄	—	173	2	no	—
67	M₁₆	—	346	4	no	—
68	M₁₆	—	346	4	no	—
69	C₂×C₈	—	346	2	no	—
70	C₂×C₈	—	346	2	no	—
71	C₂×C₈	—	346	2	no	—
72	C₂×C₈	—	346	2	no	—
73	C₂×C₈	—	173	2	no	—
74	C₂×C₈	C₂×#10621924	1	2	yes	—
75	C₄²	—	173	2	no	—
76	C₄²	—	173	1	no	—
77	C₄²	C₂×#10621924	1	1	yes	characteristic
78	C₂³	—	173	1	no	—
79	C₂×C₄	—	173	2	no	—
80	C₂×C₄	—	173	1	no	—
81	C₂×C₄	—	173	2	no	—
82	C₂×C₄	—	173	2	no	—
83	C₂×C₄	—	692	2	no	—
84	C₂×C₄	C₄⋊#10621924	1	2	yes	—
85	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	346	2	no	—
88	C₈	—	692	2	no	—
89	C₈	—	692	2	no	—
90	C₂²	—	346	2	no	—
91	C₂²	—	173	2	no	—
92	C₂²	C₂×C₄⋊#10621924	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	346	2	no	—
95	C₄	—	173	1	no	—
96	C₄	—	173	1	no	—
97	C₄	#19149758	1	2	yes	—
98	C₂	—	346	2	no	—
99	C₂	#26012435	1	1	yes	characteristic
100	C₂	#26012506	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012496

C₂

yes

maximal, characteristic

#26012495

C₂

yes

maximal, characteristic

#26012367

C₂

yes

maximal

#97×D₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583836

C₂²

yes

—

C₂×#14583836

C₂²

yes

—

#19149746

—

C₄⋊#13063443

—

C₂×C₈×D₁₇₃

C₂²

yes

—

C₄²×D₁₇₃

C₂²

yes

characteristic

#19149590

C₄

yes

characteristic

#19149589

C₄

yes

—

#97×C₁₇₃

C₄

yes

—

—

—

—

C₂×C₄×D₁₇₃

Q₈

yes

—

C₂×C₄×D₁₇₃

C₂³

yes

characteristic

C₂×#13063436

C₂×C₄

yes

—

C₈×D₁₇₃

—

C₈×D₁₇₃

—

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

D₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

#13063443

—

#13063443

—

C₂²×D₁₇₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₇₃

C₄⋊C₄

yes

—

C₂×Dic₁₇₃

C₂×D₄

yes

characteristic

C₄×D₁₇₃

—

C₄×D₁₇₃

#41

yes

—

C₄×D₁₇₃

#41

yes

—

#13063436

—

C₂×C₄×C₁₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₈×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

#125

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#117

yes

—

Dic₁₇₃

—

Dic₁₇₃

yes

Dic₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

—

D₁₇₃

—

C₁₇₃

#1838

yes

characteristic, complemented, Sylow

#1838

—

173

maximal, Sylow

#423

—

173

—

#423

—

173

—

C₂×#97

—

173

—

#341

—

173

—

C₂×M₁₆

—

173

—

C₂×M₁₆

—

173

—

C₂²×C₈

—

173

—

C₂×C₄²

—

173

—

C₄⋊C₈

—

346

—

C₄⋊C₈

—

346

—

#97

—

173

—

#97

—

173

—

#97

#10621924

yes

characteristic

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

M₁₆

—

346

—

M₁₆

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

173

—

C₂×C₈

C₂×#10621924

yes

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

C₂×#10621924

yes

characteristic

C₂³

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

692

—

C₂×C₄

C₄⋊#10621924

yes

—

C₂×C₄

C₂²×#10621924

yes

characteristic

C₈

—

C₈

—

346

—

C₈

—

692

—

C₈

—

692

—

C₂²

—

346

—

C₂²

—

173

—

C₂²

C₂×C₄⋊#10621924

yes

characteristic

C₄

—

C₄

—

346

—

C₄

—

173

—

C₄

—

173

—

C₄

#19149758

yes

—

C₂

—

346

—

C₂

#26012435

yes

characteristic

100

C₂

#26012506

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10621924
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10621924
Fitting	#97×C₁₇₃	C₄
socle	C₂²×C₁₇₃	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇₃	C₂×Q₈
cosocle	C₂×C₄×D₁₇₃	C₂³
hypercenter	#97	#10621924
hypocenter	C₁₇₃	#1838
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#10621924

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10621924

Fitting

#97×C₁₇₃

C₄

socle

C₂²×C₁₇₃

C₂×C₄⋊C₄

prime radical

C₂²×D₁₇₃

C₂×Q₈

cosocle

C₂×C₄×D₁₇₃

C₂³

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
50. C₁₇₃	#1838	semidirect

H ≅ G/N

type

50. C₁₇₃

#1838

semidirect

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

subgroup	index
#26012496	2
#26012495	2
#26012367	2
#97×D₁₇₃	2
#1838	173

subgroup

index

#97×D₁₇₃

173

Subgroups — #36369489

Conjugacy classes of subgroups

Complements

Maximal subgroups