Group #36362349 — subgroups

Subgroups — C₂×#26011597 #36362349

Order 22144.

480 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×Dic₆₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19149644	C₂	1	2	yes	maximal
3	#26011597	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19149602	C₂	1	2	yes	maximal
5	C₂×#19149589	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₁₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₄₆	C₂²	1	2	yes	—
8	C₂×Dic₆₉₂	C₄	1	8	yes	complemented
9	C₂×Dic₆₉₂	C₂²	1	4	yes	—
10	C₂²×#13063436	C₂²	1	1	yes	characteristic
11	#19149644	—	2	4	no	—
12	#19149644	C₂²	1	8	yes	—
13	C₂×#14583805	—	2	2	no	—
14	C₂×C₄×Dic₁₇₃	—	2	2	no	—
15	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
16	#19149602	C₂²	1	8	yes	—
17	#19149589	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
20	#106×C₁₇₃	C₂²	1	4	yes	—
21	C₂×Dic₃₄₆	—	2	2	no	—
22	C₂×Dic₃₄₆	—	2	4	no	—
23	C₂×Dic₃₄₆	C₂×C₄	1	8	yes	—
24	C₂×Dic₃₄₆	C₂³	1	4	yes	—
25	C₂²×Dic₁₇₃	—	2	2	no	—
26	C₂²×Dic₁₇₃	D₄	1	2	yes	—
27	Dic₆₉₂	C₂×C₄	1	16	yes	complemented
28	C₂×#13063436	—	2	2	no	—
29	C₂×#13063436	C₂³	1	2	yes	—
30	#14583805	—	2	8	no	—
31	C₄×Dic₁₇₃	—	2	8	no	—
32	C₄×Dic₁₇₃	D₄	1	2	yes	—
33	C₄×Dic₁₇₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₇₃	—	2	2	no	—
35	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₇₃	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₇₃	—	2	2	no	—
39	C₄⋊C₄×C₁₇₃	C₂³	1	4	yes	—
40	Dic₃₄₆	—	2	8	no	—
41	Dic₃₄₆	—	2	4	no	—
42	Dic₃₄₆	C₂²×C₄	1	8	yes	—
43	C₂×Dic₁₇₃	—	2	8	no	—
44	C₂×Dic₁₇₃	—	2	4	no	—
45	C₂×Dic₁₇₃	—	2	4	no	—
46	C₂×Dic₁₇₃	C₂×D₄	1	4	yes	—
47	#13063436	—	2	2	no	—
48	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇₃	—	2	4	no	—
50	C₂×C₄×C₁₇₃	—	4	4	no	—
51	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇₃	#41	1	2	yes	—
54	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
55	C₈×C₁₇₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₁₇₃	C₄×D₄	1	2	yes	—
57	C₂²×C₁₇₃	#137	1	4	yes	—
58	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₇₃	—	4	4	no	—
60	C₄×C₁₇₃	C₄×D₄	1	2	yes	—
61	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
63	Dic₁₇₃	—	2	8	no	—
64	Dic₁₇₃	—	2	4	no	—
65	C₂×C₁₇₃	#441	1	4	yes	complemented
66	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
67	C₂×C₁₇₃	C₂×#137	1	2	yes	—
68	C₁₇₃	C₂×#441	1	1	yes	characteristic, complemented, Sylow
69	C₂×#441	—	173	1	no	maximal, Sylow
70	C₂²×Q₁₆	—	173	1	no	—
71	C₂×C₄×Q₈	—	173	2	no	—
72	#441	—	173	8	no	—
73	C₂×#106	D₁₇₃	1	1	yes	characteristic
74	C₂×#103	—	173	2	no	—
75	C₂×#97	—	173	1	no	—
76	C₂²×Q₈	—	173	2	no	—
77	C₂×Q₁₆	—	173	8	no	—
78	C₂×Q₁₆	—	173	4	no	—
79	C₂²×C₈	—	173	1	no	—
80	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
81	C₄×Q₈	—	346	4	no	—
82	C₄×Q₈	—	173	8	no	—
83	C₂×C₄⋊C₄	—	346	2	no	—
84	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
85	C₂×C₄²	—	346	2	no	—
86	C₂×C₄²	—	173	1	no	—
87	#106	C₂×D₁₇₃	1	4	yes	—
88	#103	—	173	8	no	—
89	#97	—	173	4	no	—
90	C₂×Q₈	—	346	2	no	—
91	C₂×Q₈	—	346	4	no	—
92	C₂×Q₈	—	173	8	no	—
93	C₂×Q₈	—	173	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	346	2	no	—
96	C₂²×C₄	—	173	2	no	—
97	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
98	Q₁₆	—	173	16	no	—
99	C₂×C₈	—	346	2	no	—
100	C₂×C₈	—	173	2	no	—
101	C₂×C₈	C₄×D₁₇₃	1	4	yes	—
102	C₂×C₈	C₂²×D₁₇₃	1	2	yes	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	346	8	no	—
105	C₄⋊C₄	C₂²×D₁₇₃	1	4	yes	—
106	C₄²	—	346	8	no	—
107	C₄²	—	173	2	no	—
108	C₄²	—	173	2	no	—
109	C₂³	D₄×D₁₇₃	1	1	yes	characteristic
110	Q₈	—	346	8	no	—
111	Q₈	—	346	4	no	—
112	Q₈	—	173	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	346	8	no	—
116	C₂×C₄	—	346	4	no	—
117	C₂×C₄	—	346	4	no	—
118	C₂×C₄	—	173	4	no	—
119	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
120	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
121	C₂×C₄	#14583827	1	2	yes	—
122	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
123	C₈	—	346	2	no	—
124	C₈	C₂×C₄×D₁₇₃	1	4	yes	—
125	C₂²	#19149641	1	2	yes	—
126	C₂²	#19149708	1	4	yes	—
127	C₂²	C₂×D₄×D₁₇₃	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	346	8	no	—
130	C₄	—	346	4	no	—
131	C₄	#19149641	1	2	yes	—
132	C₄	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
133	C₄	C₂×#14583827	1	1	yes	characteristic
134	C₂	#26011597	1	4	yes	direct product factor
135	C₂	C₂×#19149641	1	1	yes	characteristic
136	C₂	C₂×#19149708	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₉₂

C₂

yes

maximal, characteristic

C₂×#19149644

C₂

yes

maximal

#26011597

C₂

yes

maximal, direct product factor

C₂×#19149602

C₂

yes

maximal

C₂×#19149589

C₂

yes

maximal, characteristic

C₂×#106×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₄₆

C₂²

yes

—

C₂×Dic₆₉₂

C₄

yes

complemented

C₂×Dic₆₉₂

C₂²

yes

—

C₂²×#13063436

C₂²

yes

characteristic

#19149644

—

#19149644

C₂²

yes

—

C₂×#14583805

—

C₂×C₄×Dic₁₇₃

—

C₂×C₄×Dic₁₇₃

C₂²

yes

characteristic

#19149602

C₂²

yes

—

#19149589

C₂²

yes

—

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

—

#106×C₁₇₃

C₂²

yes

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

C₂×C₄

yes

—

C₂×Dic₃₄₆

C₂³

yes

—

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

D₄

yes

—

Dic₆₉₂

C₂×C₄

yes

complemented

C₂×#13063436

—

C₂×#13063436

C₂³

yes

—

#14583805

—

C₄×Dic₁₇₃

—

C₄×Dic₁₇₃

D₄

yes

—

C₄×Dic₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

C₂³

yes

—

Dic₃₄₆

—

Dic₃₄₆

—

Dic₃₄₆

C₂²×C₄

yes

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

C₂×D₄

yes

—

#13063436

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

C₂⁴

yes

characteristic

C₈×C₁₇₃

C₂²×C₄

yes

—

C₂²×C₁₇₃

C₄×D₄

yes

—

C₂²×C₁₇₃

#137

yes

—

C₂²×C₁₇₃

C₂²×D₄

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×D₄

yes

—

C₄×C₁₇₃

C₂³×C₄

yes

characteristic

C₄×C₁₇₃

C₂×#41

yes

characteristic

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

#441

yes

complemented

C₂×C₁₇₃

C₂×C₄×D₄

yes

characteristic

C₂×C₁₇₃

C₂×#137

yes

—

C₁₇₃

C₂×#441

yes

characteristic, complemented, Sylow

C₂×#441

—

173

maximal, Sylow

C₂²×Q₁₆

—

173

—

C₂×C₄×Q₈

—

173

—

#441

—

173

—

C₂×#106

D₁₇₃

yes

characteristic

C₂×#103

—

173

—

C₂×#97

—

173

—

C₂²×Q₈

—

173

—

C₂×Q₁₆

—

173

—

C₂×Q₁₆

—

173

—

C₂²×C₈

—

173

—

C₂²×C₈

C₂×D₁₇₃

yes

characteristic

C₄×Q₈

—

346

—

C₄×Q₈

—

173

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

C₂×D₁₇₃

yes

—

C₂×C₄²

—

346

—

C₂×C₄²

—

173

—

#106

C₂×D₁₇₃

yes

—

#103

—

173

—

#97

—

173

—

C₂×Q₈

—

346

—

C₂×Q₈

—

346

—

C₂×Q₈

—

173

—

C₂×Q₈

—

173

—

C₂²×C₄

—

C₂²×C₄

—

346

—

C₂²×C₄

—

173

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

Q₁₆

—

173

—

C₂×C₈

—

346

—

100

C₂×C₈

—

173

—

101

C₂×C₈

C₄×D₁₇₃

yes

—

102

C₂×C₈

C₂²×D₁₇₃

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

346

—

105

C₄⋊C₄

C₂²×D₁₇₃

yes

—

106

C₄²

—

346

—

107

C₄²

—

173

—

108

C₄²

—

173

—

109

C₂³

D₄×D₁₇₃

yes

characteristic

110

Q₈

—

346

—

111

Q₈

—

346

—

112

Q₈

—

173

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

346

—

116

C₂×C₄

—

346

—

117

C₂×C₄

—

346

—

118

C₂×C₄

—

173

—

119

C₂×C₄

C₂×C₄×D₁₇₃

yes

—

120

C₂×C₄

D₄×D₁₇₃

yes

characteristic

121

C₂×C₄

#14583827

yes

—

122

C₂×C₄

C₂³×D₁₇₃

yes

characteristic

123

C₈

—

346

—

124

C₈

C₂×C₄×D₁₇₃

yes

—

125

C₂²

#19149641

yes

—

126

C₂²

#19149708

yes

—

127

C₂²

C₂×D₄×D₁₇₃

yes

characteristic

128

C₄

—

129

C₄

—

346

—

130

C₄

—

346

—

131

C₄

#19149641

yes

—

132

C₄

C₂²×C₄×D₁₇₃

yes

characteristic

133

C₄

C₂×#14583827

yes

characteristic

134

C₂

#26011597

yes

direct product factor

135

C₂

C₂×#19149641

yes

characteristic

136

C₂

C₂×#19149708

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₇₃
commutator	C₄×C₁₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	C₂×#106×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂×D₄
cosocle	C₂×C₄×C₁₇₃	C₂⁴
hypercenter	C₂×#106	D₁₇₃
hypocenter	C₁₇₃	C₂×#441
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₇₃

commutator

C₄×C₁₇₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₇₃

Fitting

C₂×#106×C₁₇₃

C₂

socle, prime radical

C₂³×C₁₇₃

C₂×D₄

cosocle

C₂×C₄×C₁₇₃

C₂⁴

D₁₇₃

C₁₇₃

G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #26011597	C₂	direct (N × H)
8. C₂×Dic₆₉₂	C₄	semidirect
27. Dic₆₉₂	C₂×C₄	semidirect
65. C₂×C₁₇₃	#441	semidirect
68. C₁₇₃	C₂×#441	semidirect
134. C₂	#26011597	direct (N × H)

H ≅ G/N

type

3. #26011597

C₂

direct (N × H)

8. C₂×Dic₆₉₂

C₄

semidirect

27. Dic₆₉₂

C₂×C₄

semidirect

65. C₂×C₁₇₃

#441

semidirect

68. C₁₇₃

C₂×#441

semidirect

134. C₂

#26011597

direct (N × H)

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

subgroup	index
C₂²×Dic₆₉₂	2
C₂×#19149644	2
#26011597	2
C₂×#19149602	2
C₂×#19149589	2
C₂×#106×C₁₇₃	2
C₂×#441	173

subgroup

index

C₂²×Dic₆₉₂

C₂×#106×C₁₇₃

173

Subgroups — C₂×#26011597 #36362349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26011597 #36362349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26011597 #36362349