Group #36366241 — subgroups

Subgroups — #36366241

Order 22144.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#26012220	C₂	1	1	yes	maximal, characteristic
2	#26011843	C₂	1	3	yes	maximal
3	#26011783	C₂	1	3	yes	maximal
4	#26011701	C₂	1	3	yes	maximal
5	Q₈×#13063436	C₂	1	1	yes	maximal, characteristic
6	C₄×#14583816	C₂	1	3	yes	maximal
7	Q₈²×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14583816	C₂²	1	3	yes	—
9	#19149679	C₂²	1	3	yes	—
10	#19149670	—	2	3	no	—
11	Q₈×Dic₁₇₃	—	2	1	no	—
12	C₄×Dic₃₄₆	C₂²	1	3	yes	—
13	#19149612	C₂²	1	3	yes	—
14	C₄⋊#13063436	C₂²	1	3	yes	—
15	#19149603	C₂²	1	1	yes	characteristic
16	#19149602	C₂²	1	6	yes	—
17	#19149602	C₂²	1	3	yes	—
18	C₄×#13063436	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₁₇₃	—	2	3	no	—
20	C₄⋊Q₈×C₁₇₃	C₂²	1	3	yes	—
21	C₄×Q₈×C₁₇₃	—	2	1	no	—
22	C₄×Q₈×C₁₇₃	C₂²	1	3	yes	—
23	C₄×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
24	C₂×Dic₃₄₆	C₂³	1	3	yes	—
25	#14583816	—	2	6	no	—
26	C₂×#13063436	C₂³	1	1	yes	characteristic
27	C₂×#13063436	C₂³	1	3	yes	—
28	C₄⋊Dic₁₇₃	—	2	3	no	—
29	C₄⋊Dic₁₇₃	—	2	3	no	—
30	C₄⋊Dic₁₇₃	C₂³	1	1	yes	characteristic
31	#14583805	—	2	3	no	—
32	C₄×Dic₁₇₃	—	2	3	no	—
33	C₂×Q₈×C₁₇₃	—	2	3	no	—
34	C₂×Q₈×C₁₇₃	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₁₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₇₃	—	2	3	no	—
37	C₄⋊C₄×C₁₇₃	—	2	3	no	—
38	C₄⋊C₄×C₁₇₃	D₄	1	3	yes	—
39	C₄⋊C₄×C₁₇₃	C₂³	1	3	yes	—
40	C₄²×C₁₇₃	—	2	3	no	—
41	C₄²×C₁₇₃	C₂³	1	3	yes	—
42	Dic₃₄₆	—	2	3	no	—
43	Dic₃₄₆	—	2	3	no	—
44	C₂×Dic₁₇₃	—	2	3	no	—
45	C₂×Dic₁₇₃	—	2	1	no	—
46	#13063436	—	2	3	no	—
47	#13063436	—	2	1	no	—
48	Q₈×C₁₇₃	—	2	1	no	—
49	Q₈×C₁₇₃	—	8	3	no	—
50	Q₈×C₁₇₃	Q₁₆	1	4	yes	complemented
51	Q₈×C₁₇₃	#41	1	2	yes	—
52	C₂×C₄×C₁₇₃	—	2	1	no	—
53	C₂×C₄×C₁₇₃	—	2	3	no	—
54	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
57	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₇₃	—	2	2	no	—
59	C₄×C₁₇₃	—	2	3	no	—
60	C₄×C₁₇₃	—	4	3	no	—
61	C₄×C₁₇₃	C₂×Q₁₆	1	6	yes	—
62	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
63	C₄×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₁₇₃	—	4	3	no	—
65	Dic₁₇₃	—	4	1	no	—
66	C₂×C₁₇₃	#548	1	1	yes	characteristic
67	C₂×C₁₇₃	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₁₇₃	#577	1	1	yes	characteristic
69	C₁₇₃	#3363	1	1	yes	characteristic, complemented, Sylow
70	#3363	—	173	1	no	maximal, Sylow
71	Q₈²	D₁₇₃	1	1	yes	characteristic
72	#557	—	173	1	no	—
73	#487	—	173	3	no	—
74	#477	—	173	3	no	—
75	#462	—	173	3	no	—
76	C₈×Q₈	—	173	1	no	—
77	C₄×Q₁₆	—	173	3	no	—
78	C₂×Q₁₆	—	173	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	173	3	no	—
81	C₄⋊Q₈	C₂×D₁₇₃	1	3	yes	—
82	#125	—	346	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	346	1	no	—
85	C₄×Q₈	—	173	3	no	—
86	C₄×Q₈	C₂×D₁₇₃	1	3	yes	—
87	C₄×Q₈	C₂×D₁₇₃	1	1	yes	characteristic
88	#107	—	173	3	no	—
89	C₄⋊C₈	—	173	3	no	—
90	#103	—	173	1	no	—
91	#103	—	173	3	no	—
92	#103	—	173	6	no	—
93	C₄×C₈	—	173	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	173	3	no	—
96	C₂×Q₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₁₇₃	1	1	yes	characteristic
98	Q₁₆	—	346	6	no	—
99	C₂×C₈	—	173	1	no	—
100	C₂×C₈	—	173	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	346	3	no	—
104	C₄⋊C₄	—	346	3	no	—
105	C₄⋊C₄	—	346	3	no	—
106	C₄⋊C₄	—	173	1	no	—
107	C₄⋊C₄	C₂²⋊D₁₇₃	1	3	yes	—
108	C₄⋊C₄	C₂²×D₁₇₃	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	346	3	no	—
111	C₄²	C₂²×D₁₇₃	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	346	3	no	—
115	Q₈	—	346	3	no	—
116	Q₈	#14583816	1	4	yes	complemented
117	Q₈	#14583827	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	346	1	no	—
121	C₂×C₄	—	346	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	3	yes	—
124	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
125	C₈	—	346	3	no	—
126	C₈	—	346	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	3	no	—
131	C₄	—	692	1	no	—
132	C₄	—	692	3	no	—
133	C₄	C₂×#14583816	1	6	yes	—
134	C₄	C₂×#14583827	1	1	yes	characteristic
135	C₄	#19149727	1	1	yes	characteristic
136	C₂	#26012152	1	1	yes	characteristic
137	C₂	C₂²×#14583816	1	1	yes	characteristic
138	C₂	#26012302	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26012220

C₂

yes

maximal, characteristic

#26011843

C₂

yes

maximal

#26011783

C₂

yes

maximal

#26011701

C₂

yes

maximal

Q₈×#13063436

C₂

yes

maximal, characteristic

C₄×#14583816

C₂

yes

maximal

Q₈²×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583816

C₂²

yes

—

#19149679

C₂²

yes

—

#19149670

—

Q₈×Dic₁₇₃

—

C₄×Dic₃₄₆

C₂²

yes

—

#19149612

C₂²

yes

—

C₄⋊#13063436

C₂²

yes

—

#19149603

C₂²

yes

characteristic

#19149602

C₂²

yes

—

#19149602

C₂²

yes

—

C₄×#13063436

C₂²

yes

—

C₄⋊Q₈×C₁₇₃

—

C₄⋊Q₈×C₁₇₃

C₂²

yes

—

C₄×Q₈×C₁₇₃

—

C₄×Q₈×C₁₇₃

C₂²

yes

—

C₄×Q₈×C₁₇₃

C₂²

yes

characteristic

C₂×Dic₃₄₆

C₂³

yes

—

#14583816

—

C₂×#13063436

C₂³

yes

characteristic

C₂×#13063436

C₂³

yes

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

C₂³

yes

characteristic

#14583805

—

C₄×Dic₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂×Q₈×C₁₇₃

D₄

yes

characteristic

C₂×Q₈×C₁₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

D₄

yes

—

C₄⋊C₄×C₁₇₃

C₂³

yes

—

C₄²×C₁₇₃

—

C₄²×C₁₇₃

C₂³

yes

—

Dic₃₄₆

—

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

#13063436

—

Q₈×C₁₇₃

—

Q₈×C₁₇₃

—

Q₈×C₁₇₃

Q₁₆

yes

complemented

Q₈×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂⁴

yes

characteristic

C₂²×C₁₇₃

C₂²×D₄

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂×Q₁₆

yes

—

C₄×C₁₇₃

C₂×#41

yes

characteristic

C₄×C₁₇₃

2₋¹⁺⁴

yes

characteristic

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

#548

yes

characteristic

C₂×C₁₇₃

C₂²×Q₁₆

yes

C₂×C₁₇₃

yes

C₁₇₃

yes

characteristic, complemented, Sylow

#3363

—

173

maximal, Sylow

Q₈²

D₁₇₃

yes

characteristic

#557

—

173

—

#487

—

173

—

#477

—

173

—

#462

—

173

—

C₈×Q₈

—

173

—

C₄×Q₁₆

—

173

—

C₂×Q₁₆

—

173

—

C₄⋊Q₈

—

C₄⋊Q₈

—

173

—

C₄⋊Q₈

C₂×D₁₇₃

yes

—

#125

—

346

—

C₄×Q₈

—

C₄×Q₈

—

346

—

C₄×Q₈

—

173

—

C₄×Q₈

C₂×D₁₇₃

yes

—

C₄×Q₈

C₂×D₁₇₃

yes

characteristic

#107

—

173

—

C₄⋊C₈

—

173

—

#103

—

173

—

#103

—

173

—

#103

—

173

—

C₄×C₈

—

173

—

C₂×Q₈

—

C₂×Q₈

—

173

—

C₂×Q₈

C₂²⋊D₁₇₃

yes

characteristic

C₂×Q₈

C₂²×D₁₇₃

yes

characteristic

Q₁₆

—

346

—

C₂×C₈

—

173

—

100

C₂×C₈

—

173

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

346

—

104

C₄⋊C₄

—

346

—

105

C₄⋊C₄

—

346

—

106

C₄⋊C₄

—

173

—

107

C₄⋊C₄

C₂²⋊D₁₇₃

yes

—

108

C₄⋊C₄

C₂²×D₁₇₃

yes

—

109

C₄²

—

110

C₄²

—

346

—

111

C₄²

C₂²×D₁₇₃

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

346

—

115

Q₈

—

346

—

116

Q₈

#14583816

yes

complemented

117

Q₈

#14583827

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

346

—

121

C₂×C₄

—

346

—

122

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

124

C₂×C₄

C₂³×D₁₇₃

yes

characteristic

125

C₈

—

346

—

126

C₈

—

346

—

127

C₂²

C₂²×C₂²⋊D₁₇₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

692

—

132

C₄

—

692

—

133

C₄

C₂×#14583816

yes

—

134

C₄

C₂×#14583827

yes

characteristic

135

C₄

#19149727

yes

characteristic

136

C₂

#26012152

yes

characteristic

137

C₂

C₂²×#14583816

yes

characteristic

138

C₂

#26012302

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇₃
commutator, cosocle	C₂×C₄×C₁₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	Q₈²×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂²×D₄
hypercenter	Q₈²	D₁₇₃
hypocenter	C₁₇₃	#3363
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₇₃

commutator, cosocle

C₂×C₄×C₁₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇₃

Fitting

Q₈²×C₁₇₃

C₂

socle, prime radical

C₂²×C₁₇₃

C₂²×D₄

hypercenter

Q₈²

D₁₇₃

hypocenter

C₁₇₃

#3363

LCS

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
50. Q₈×C₁₇₃	Q₁₆	semidirect
69. C₁₇₃	#3363	semidirect
116. Q₈	#14583816	semidirect

H ≅ G/N

type

50. Q₈×C₁₇₃

Q₁₆

semidirect

69. C₁₇₃

#3363

semidirect

116. Q₈

#14583816

semidirect

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

subgroup	index
#26012220	2
#26011843	2
#26011783	2
#26011701	2
Q₈×#13063436	2
C₄×#14583816	2
Q₈²×C₁₇₃	2
#3363	173

subgroup

index

Q₈²×C₁₇₃

173

Subgroups — #36366241

Conjugacy classes of subgroups

Complements

Maximal subgroups