Group #36353681 — subgroups

Subgroups — #36353681

Order 22144.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#19149608	C₂	1	2	yes	maximal
2	#26011337	C₂	1	1	yes	maximal, characteristic
3	#26011246	C₂	1	1	yes	maximal, characteristic
4	#26011245	C₂	1	2	yes	maximal
5	#336×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13063436	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
9	#19149608	—	2	4	no	—
10	#19149586	C₄	1	2	yes	—
11	#19149586	C₄	1	2	yes	—
12	C₂²×C₈×C₁₇₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₇₃	C₂×C₄	1	2	yes	—
17	C₂×#13063436	—	2	2	no	—
18	C₂×#13063436	—	2	2	no	—
19	C₂×#13063436	—	2	2	no	—
20	C₄×Dic₁₇₃	—	2	2	no	—
21	C₄×Dic₁₇₃	D₄	1	2	yes	—
22	C₄×Dic₁₇₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₇₃	—	2	2	no	—
26	C₂×C₈×C₁₇₃	—	2	2	no	—
27	C₂×C₈×C₁₇₃	—	2	2	no	—
28	C₄²×C₁₇₃	—	2	2	no	—
29	C₂×Dic₁₇₃	—	2	2	no	—
30	C₂×Dic₁₇₃	—	2	2	no	—
31	C₂×Dic₁₇₃	—	2	1	no	—
32	C₂×Dic₁₇₃	—	2	1	no	—
33	C₂×Dic₁₇₃	—	2	2	no	—
34	C₂×Dic₁₇₃	—	2	2	no	—
35	C₂×Dic₁₇₃	C₄⋊C₄	1	4	yes	—
36	#13063436	—	4	2	no	—
37	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇₃	—	2	2	no	—
39	C₂×C₄×C₁₇₃	—	2	2	no	—
40	C₂×C₄×C₁₇₃	—	2	2	no	—
41	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₇₃	#41	1	2	yes	—
44	C₂×C₄×C₁₇₃	#41	1	2	yes	—
45	C₈×C₁₇₃	—	4	2	no	—
46	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₇₃	#117	1	1	yes	characteristic
48	C₂²×C₁₇₃	#117	1	1	yes	characteristic
49	C₂²×C₁₇₃	#131	1	2	yes	—
50	C₂²×C₁₇₃	#131	1	2	yes	—
51	C₄×C₁₇₃	—	4	2	no	—
52	C₄×C₁₇₃	#124	1	2	yes	—
53	C₄×C₁₇₃	#125	1	2	yes	—
54	Dic₁₇₃	—	2	4	no	—
55	Dic₁₇₃	—	4	1	no	—
56	Dic₁₇₃	—	4	1	no	—
57	C₂×C₁₇₃	#382	1	1	yes	characteristic
58	C₂×C₁₇₃	#424	1	2	yes	—
59	C₂×C₁₇₃	#433	1	2	yes	—
60	C₂×C₁₇₃	#433	1	1	yes	characteristic
61	C₂×C₁₇₃	#433	1	1	yes	characteristic
62	C₁₇₃	#1832	1	1	yes	characteristic, complemented, Sylow
63	#1832	—	173	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	173	2	no	—
65	#376	—	173	1	no	—
66	#336	—	173	2	no	—
67	#336	—	173	1	no	—
68	#336	D₁₇₃	1	1	yes	characteristic
69	C₂²×C₈	—	173	2	no	—
70	C₂²×C₈	C₂×D₁₇₃	1	2	yes	—
71	C₂×C₄²	—	173	1	no	—
72	C₂×C₄²	—	173	1	no	—
73	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
74	C₄⋊C₈	—	346	4	no	—
75	#95	—	173	2	no	—
76	#95	—	173	2	no	—
77	C₂²×C₄	—	173	1	no	—
78	C₂²×C₄	—	173	1	no	—
79	C₂²×C₄	—	173	2	no	—
80	C₂²×C₄	C₄×D₁₇₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	346	2	no	—
86	C₂×C₈	—	346	2	no	—
87	C₂×C₈	—	346	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	346	2	no	—
90	C₄²	—	173	2	no	—
91	C₄²	—	173	2	no	—
92	C₂³	C₂×C₄×D₁₇₃	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	346	2	no	—
97	C₂×C₄	—	346	2	no	—
98	C₂×C₄	—	346	1	no	—
99	C₂×C₄	—	346	1	no	—
100	C₂×C₄	—	346	2	no	—
101	C₂×C₄	—	346	2	no	—
102	C₂×C₄	—	173	4	no	—
103	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₁₇₃	1	1	yes	characteristic
105	C₂×C₄	#14583825	1	2	yes	—
106	C₂×C₄	#14583826	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	692	2	no	—
109	C₂²	C₄⋊C₄×D₁₇₃	1	1	yes	characteristic
110	C₂²	#19149635	1	1	yes	characteristic
111	C₂²	#19149636	1	1	yes	characteristic
112	C₂²	#19149685	1	2	yes	—
113	C₂²	#19149687	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	346	4	no	—
116	C₄	—	692	1	no	—
117	C₄	—	692	1	no	—
118	C₄	#19149665	1	2	yes	—
119	C₄	#19149669	1	2	yes	—
120	C₂	#26011355	1	1	yes	characteristic
121	C₂	#26011516	1	2	yes	—
122	C₂	#26011549	1	2	yes	—
123	C₂	#26011550	1	1	yes	characteristic
124	C₂	#26011553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149608

C₂

yes

maximal

#26011337

C₂

yes

maximal, characteristic

#26011246

C₂

yes

maximal, characteristic

#26011245

C₂

yes

maximal

#336×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×#13063436

C₂²

yes

—

C₂×C₄×Dic₁₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₃

C₂²

yes

characteristic

#19149608

—

#19149586

C₄

yes

—

#19149586

C₄

yes

—

C₂²×C₈×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₂²×Dic₁₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₇₃

C₂×C₄

yes

—

C₂×#13063436

—

C₂×#13063436

—

C₂×#13063436

—

C₄×Dic₁₇₃

—

C₄×Dic₁₇₃

D₄

yes

—

C₄×Dic₁₇₃

Q₈

yes

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₄²×C₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

C₄⋊C₄

yes

—

#13063436

—

C₂³×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₄⋊C₄

yes

C₂²×C₁₇₃

yes

C₂²×C₁₇₃

yes

C₂²×C₁₇₃

yes

—

C₂²×C₁₇₃

#131

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#124

yes

—

C₄×C₁₇₃

#125

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

#382

yes

characteristic

C₂×C₁₇₃

#424

yes

—

C₂×C₁₇₃

#433

yes

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₁₇₃

yes

characteristic, complemented, Sylow

#1832

—

173

maximal, Sylow

C₂×C₄⋊C₈

—

173

—

#376

—

173

—

#336

—

173

—

#336

—

173

—

#336

D₁₇₃

yes

characteristic

C₂²×C₈

—

173

—

C₂²×C₈

C₂×D₁₇₃

yes

—

C₂×C₄²

—

173

—

C₂×C₄²

—

173

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

C₄⋊C₈

—

346

—

#95

—

173

—

#95

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

C₄×D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₄²

—

C₄²

—

346

—

C₄²

—

173

—

C₄²

—

173

—

C₂³

C₂×C₄×D₁₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

346

—

C₂×C₄

—

346

—

C₂×C₄

—

346

—

C₂×C₄

—

346

—

100

C₂×C₄

—

346

—

101

C₂×C₄

—

346

—

102

C₂×C₄

—

173

—

103

C₂×C₄

D₄×D₁₇₃

yes

characteristic

104

C₂×C₄

Q₈×D₁₇₃

yes

characteristic

105

C₂×C₄

#14583825

yes

—

106

C₂×C₄

#14583826

yes

—

107

C₈

—

108

C₈

—

692

—

109

C₂²

C₄⋊C₄×D₁₇₃

yes

characteristic

110

C₂²

#19149635

yes

characteristic

111

C₂²

#19149636

yes

characteristic

112

C₂²

#19149685

yes

—

113

C₂²

#19149687

yes

—

114

C₄

—

115

C₄

—

346

—

116

C₄

—

692

—

117

C₄

—

692

—

118

C₄

#19149665

yes

—

119

C₄

#19149669

yes

—

120

C₂

#26011355

yes

characteristic

121

C₂

#26011516

yes

—

122

C₂

#26011549

yes

—

123

C₂

#26011550

yes

characteristic

124

C₂

#26011553

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₇₃
commutator, socle, prime radical	C₂³×C₁₇₃	C₂²×C₄
Fitting	#336×C₁₇₃	C₂
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#336	D₁₇₃
hypocenter	C₁₇₃	#1832
LCS	G ▷ C₂³×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₇₃

commutator, socle, prime radical

C₂³×C₁₇₃

C₂²×C₄

Fitting

#336×C₁₇₃

C₂

cosocle

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂³×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₁₇₃	#1832	semidirect

H ≅ G/N

type

62. C₁₇₃

#1832

semidirect

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

subgroup	index
C₂×#19149608	2
#26011337	2
#26011246	2
#26011245	2
#336×C₁₇₃	2
#1832	173

subgroup

index

#336×C₁₇₃

173

Subgroups — #36353681

Conjugacy classes of subgroups

Complements

Maximal subgroups