Group #19143361 — subgroups

Subgroups — C₂×C₄×Dic₁₇₂ #19143361

Order 5504.

488 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 221 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×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₈₆	C₂	1	2	yes	maximal
3	C₄×Dic₁₇₂	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13063259	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063250	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₈₆	C₂²	1	2	yes	—
8	C₂×Dic₁₇₂	C₄	1	8	yes	direct product factor
9	C₂×Dic₁₇₂	C₂²	1	4	yes	—
10	C₄×Dic₈₆	—	2	4	no	—
11	C₄×Dic₈₆	C₂²	1	8	yes	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
13	C₂×Dic₄₃⋊C₄	—	2	2	no	—
14	C₂×C₄×Dic₄₃	—	2	2	no	—
15	#13063259	C₂²	1	4	yes	—
16	#13063250	C₂²	1	8	yes	—
17	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₄₃	C₂²	1	4	yes	—
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₃	—	2	2	no	—
25	C₂²×Dic₄₃	—	2	2	no	—
26	C₂×Dic₈₆	—	2	2	no	—
27	C₂×Dic₈₆	—	2	4	no	—
28	C₂×Dic₈₆	C₂×C₄	1	8	yes	—
29	C₂×Dic₈₆	C₂³	1	4	yes	—
30	C₂×C₈×C₄₃	—	2	2	no	—
31	C₂×C₈×C₄₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₄₃	C₂³	1	2	yes	—
33	C₂×C₈×C₄₃	C₂³	1	2	yes	—
34	C₄²×C₄₃	D₄	1	2	yes	—
35	C₄²×C₄₃	C₂³	1	2	yes	—
36	C₄⋊Dic₄₃	—	2	2	no	—
37	C₄⋊Dic₄₃	C₂³	1	4	yes	—
38	Dic₄₃⋊C₄	—	2	8	no	—
39	C₄×Dic₄₃	—	2	8	no	—
40	Dic₁₇₂	C₂×C₄	1	16	yes	direct product factor
41	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	Q₁₆	1	4	yes	complemented
44	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₄₃	#41	1	2	yes	—
49	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₄₃	—	2	8	no	—
51	C₂×Dic₄₃	—	2	4	no	—
52	C₂×Dic₄₃	—	2	4	no	—
53	C₂×Dic₄₃	—	4	4	no	—
54	Dic₈₆	—	2	8	no	—
55	Dic₈₆	—	2	4	no	—
56	Dic₈₆	C₂²×C₄	1	8	yes	—
57	C₈×C₄₃	—	2	2	no	—
58	C₈×C₄₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄₃	C₄×D₄	1	2	yes	—
60	C₂²×C₄₃	C₂×Q₁₆	1	2	yes	—
61	C₂²×C₄₃	#135	1	2	yes	—
62	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₄₃	—	2	2	no	—
64	C₄×C₄₃	C₄×D₄	1	2	yes	—
65	C₄×C₄₃	C₂×Q₁₆	1	4	yes	complemented
66	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
68	Dic₄₃	—	2	8	no	—
69	Dic₄₃	—	4	4	no	—
70	C₂×C₄×Q₁₆	—	43	1	no	maximal, Sylow
71	C₂×C₄₃	C₄×Q₁₆	1	4	yes	complemented
72	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
73	C₂×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
74	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
75	C₂²×Q₁₆	—	43	1	no	—
76	C₂×C₄×Q₈	—	43	2	no	—
77	C₄×Q₁₆	—	43	8	no	—
78	C₂×#107	—	43	1	no	—
79	C₂×#103	—	43	2	no	—
80	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic
81	C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	43	2	no	—
83	C₂×Q₁₆	—	43	4	no	—
84	C₂×Q₁₆	—	43	8	no	—
85	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
87	C₄×Q₈	—	43	8	no	—
88	C₄×Q₈	—	86	4	no	—
89	C₂×C₄⋊C₄	—	43	2	no	—
90	C₂×C₄⋊C₄	—	86	2	no	—
91	C₂×C₄²	—	86	2	no	—
92	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
93	#107	—	43	4	no	—
94	#103	—	43	8	no	—
95	C₄×C₈	C₂×D₄₃	1	4	yes	—
96	C₂×Q₈	—	43	8	no	—
97	C₂×Q₈	—	43	4	no	—
98	C₂×Q₈	—	86	2	no	—
99	C₂×Q₈	—	86	4	no	—
100	C₂²×C₄	—	86	2	no	—
101	C₂²×C₄	—	86	2	no	—
102	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
103	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
105	Q₁₆	—	43	16	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	C₄×D₄₃	1	4	yes	—
108	C₂×C₈	C₂²×D₄₃	1	2	yes	—
109	C₂×C₈	C₂²×D₄₃	1	2	yes	—
110	C₄⋊C₄	—	43	4	no	—
111	C₄⋊C₄	—	86	2	no	—
112	C₄⋊C₄	—	86	8	no	—
113	C₄²	—	86	8	no	—
114	C₄²	D₁₇₂	1	2	yes	—
115	C₄²	C₂²×D₄₃	1	2	yes	—
116	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
117	Q₈	—	43	8	no	—
118	Q₈	—	86	8	no	—
119	Q₈	—	86	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	86	8	no	—
122	C₂×C₄	—	86	4	no	—
123	C₂×C₄	—	86	4	no	—
124	C₂×C₄	—	172	4	no	—
125	C₂×C₄	Dic₁₇₂	1	4	yes	direct product factor
126	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
127	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
128	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
129	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
130	C₂×C₄	#10621898	1	2	yes	—
131	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	C₂×C₄×D₄₃	1	4	yes	—
134	C₂²	C₄×D₁₇₂	1	2	yes	—
135	C₂²	C₂×Dic₁₇₂	1	2	yes	—
136	C₂²	#13063347	1	2	yes	—
137	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	86	8	no	—
140	C₄	—	172	4	no	—
141	C₄	C₄×D₁₇₂	1	2	yes	—
142	C₄	C₂×Dic₁₇₂	1	4	yes	direct product factor
143	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
144	C₄	C₂×#10621898	1	1	yes	characteristic
145	C₂	C₄×Dic₁₇₂	1	4	yes	direct product factor
146	C₂	C₂×C₄×D₁₇₂	1	1	yes	characteristic
147	C₂	C₂²×Dic₁₇₂	1	1	yes	characteristic
148	C₂	C₂×#13063347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₇₂

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₆

C₂

yes

maximal

C₄×Dic₁₇₂

C₂

yes

maximal, direct product factor

C₂×#13063259

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal

C₂×C₄×C₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

C₂×Dic₁₇₂

C₄

yes

direct product factor

C₂×Dic₁₇₂

C₂²

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

#13063259

C₂²

yes

—

#13063250

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×Dic₈₆

C₂³

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

C₂×C₄

yes

direct product factor

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

Q₁₆

yes

complemented

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

C₂²×C₄

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₂²×C₄

yes

—

C₂²×C₄₃

C₄×D₄

yes

—

C₂²×C₄₃

C₂×Q₁₆

yes

—

C₂²×C₄₃

#135

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

C₂×Q₁₆

yes

complemented

C₄×C₄₃

C₂³×C₄

yes

characteristic

C₄×C₄₃

C₂×#41

yes

characteristic

Dic₄₃

—

Dic₄₃

—

C₂×C₄×Q₁₆

—

maximal, Sylow

C₂×C₄₃

C₄×Q₁₆

yes

complemented

C₂×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂×C₄₃

C₂²×Q₁₆

yes

characteristic

C₂×C₄₃

C₂×#135

yes

characteristic

C₂²×Q₁₆

—

C₂×C₄×Q₈

—

C₄×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₄₃

yes

characteristic

C₄₃

C₂×C₄×Q₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#107

—

#103

—

C₄×C₈

C₂×D₄₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

D₁₇₂

yes

characteristic

103

C₂²×C₄

D₁₇₂

yes

characteristic

104

C₂²×C₄

C₂²×D₄₃

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

C₄×D₄₃

yes

—

108

C₂×C₈

C₂²×D₄₃

yes

—

109

C₂×C₈

C₂²×D₄₃

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

D₁₇₂

yes

—

115

C₄²

C₂²×D₄₃

yes

—

116

C₂³

C₂×D₁₇₂

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

172

—

125

C₂×C₄

Dic₁₇₂

yes

direct product factor

126

C₂×C₄

C₂×C₄×D₄₃

yes

—

127

C₂×C₄

C₂×D₁₇₂

yes

—

128

C₂×C₄

C₂×D₁₇₂

yes

characteristic

129

C₂×C₄

C₂×D₁₇₂

yes

—

130

C₂×C₄

#10621898

yes

—

131

C₂×C₄

C₂³×D₄₃

yes

characteristic

132

C₈

—

133

C₈

C₂×C₄×D₄₃

yes

—

134

C₂²

C₄×D₁₇₂

yes

—

135

C₂²

C₂×Dic₁₇₂

yes

—

136

C₂²

#13063347

yes

—

137

C₂²

C₂²×D₁₇₂

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

172

—

141

C₄

C₄×D₁₇₂

yes

—

142

C₄

C₂×Dic₁₇₂

yes

direct product factor

143

C₄

C₂²×C₄×D₄₃

yes

characteristic

144

C₄

C₂×#10621898

yes

characteristic

145

C₂

C₄×Dic₁₇₂

yes

direct product factor

146

C₂

C₂×C₄×D₁₇₂

yes

characteristic

147

C₂

C₂²×Dic₁₇₂

yes

characteristic

148

C₂

C₂×#13063347

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₇₂
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₁₇₂

commutator

C₄×C₄₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₂×C₄×C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₄₃

hypocenter

C₄₃

C₂×C₄×Q₁₆

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. C₄×Dic₁₇₂	C₂	direct (N × H)
8. C₂×Dic₁₇₂	C₄	direct (N × H)
40. Dic₁₇₂	C₂×C₄	direct (N × H)
43. C₂×C₄×C₄₃	Q₁₆	semidirect
65. C₄×C₄₃	C₂×Q₁₆	semidirect
71. C₂×C₄₃	C₄×Q₁₆	semidirect
81. C₄₃	C₂×C₄×Q₁₆	semidirect
125. C₂×C₄	Dic₁₇₂	direct (N × H)
142. C₄	C₂×Dic₁₇₂	direct (N × H)
145. C₂	C₄×Dic₁₇₂	direct (N × H)

H ≅ G/N

type

3. C₄×Dic₁₇₂

C₂

direct (N × H)

8. C₂×Dic₁₇₂

C₄

direct (N × H)

40. Dic₁₇₂

C₂×C₄

direct (N × H)

43. C₂×C₄×C₄₃

Q₁₆

semidirect

65. C₄×C₄₃

C₂×Q₁₆

semidirect

71. C₂×C₄₃

C₄×Q₁₆

semidirect

81. C₄₃

C₂×C₄×Q₁₆

semidirect

125. C₂×C₄

Dic₁₇₂

direct (N × H)

142. C₄

C₂×Dic₁₇₂

direct (N × H)

145. C₂

C₄×Dic₁₇₂

direct (N × H)

Subgroups — C₂×C₄×Dic₁₇₂ #19143361

Conjugacy classes of subgroups

Complements

Subgroups — C2×C4×Dic172 #19143361

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×Dic₁₇₂ #19143361