Group #36468789 — subgroups

Subgroups — #36468789

Order 22784.

192 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26019653	C₂	1	2	yes	maximal
3	#26018324	C₂	1	1	yes	maximal, characteristic
4	#26018310	C₂	1	2	yes	maximal
5	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic
6	#19151890	—	2	2	no	—
7	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
8	C₄⋊#13083549	C₂²	1	1	yes	characteristic
9	#19151416	—	2	1	no	—
10	C₁₆×Dic₈₉	—	2	1	no	—
11	#19151351	C₂²	1	2	yes	—
12	#19151347	—	2	2	no	—
13	#500×C₈₉	C₂²	1	2	yes	—
14	#366×C₈₉	—	2	1	no	—
15	#366×C₈₉	—	2	1	no	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
18	C₂×#13083549	—	2	1	no	—
19	#15667945	—	4	2	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	D₄	1	1	yes	characteristic
22	#15667941	—	2	1	no	—
23	#15667935	—	4	2	no	—
24	C₈×Dic₈₉	—	2	1	no	—
25	C₄⋊Q₈×C₈₉	—	2	2	no	—
26	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
28	#107×C₈₉	—	2	2	no	—
29	#107×C₈₉	—	2	2	no	—
30	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
32	C₂×#10623494	—	2	1	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
35	#13083553	—	2	1	no	—
36	C₄×Dic₈₉	—	2	1	no	—
37	#13083549	—	4	1	no	—
38	C₂×Q₈×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₈₉	—	4	2	no	—
42	C₄⋊C₄×C₈₉	—	4	2	no	—
43	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₈₉	—	2	1	no	—
45	C₁₆×C₈₉	Q₁₆	1	2	yes	—
46	Q₈×C₈₉	—	16	2	no	—
47	C₂×C₄×C₈₉	—	8	2	no	—
48	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
51	C₂×Dic₈₉	—	2	1	no	—
52	C₂×Dic₈₉	—	2	1	no	—
53	Dic₁₇₈	—	2	1	no	—
54	Dic₁₇₈	Q₃₂	1	2	yes	—
55	C₈×C₈₉	—	2	1	no	—
56	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₈₉	#137	1	1	yes	characteristic
58	#10623494	—	4	1	no	—
59	#10623494	—	8	2	no	—
60	C₂²×C₈₉	#466	1	1	yes	characteristic
61	C₄×C₈₉	—	2	1	no	—
62	C₄×C₈₉	—	16	2	no	—
63	C₄×C₈₉	#462	1	1	yes	characteristic
64	C₄×C₈₉	#473	1	1	yes	characteristic
65	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
66	C₄×C₈₉	#508	1	1	yes	characteristic
67	Dic₈₉	—	2	2	no	—
68	Dic₈₉	—	4	1	no	—
69	#12458	—	89	1	no	maximal, Sylow
70	C₂×C₈₉	#1674	1	1	yes	characteristic
71	C₂×C₈₉	#2216	1	1	yes	characteristic
72	C₂×C₈₉	#2221	1	1	yes	characteristic
73	#2252	D₈₉	1	1	yes	characteristic
74	Q₈×C₁₆	—	89	1	no	—
75	#1676	—	89	2	no	—
76	#1371	—	89	1	no	—
77	#1365	—	89	2	no	—
78	C₈₉	#12458	1	1	yes	characteristic, complemented, Sylow
79	#500	C₂×D₈₉	1	2	yes	—
80	#477	—	178	2	no	—
81	C₈×Q₈	—	89	1	no	—
82	#366	—	2	1	no	—
83	#366	—	2	1	no	—
84	C₄⋊C₁₆	—	89	1	no	—
85	C₄⋊C₁₆	—	178	1	no	—
86	C₄×C₁₆	—	178	1	no	—
87	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
88	#335	—	89	2	no	—
89	#333	—	178	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	89	1	no	—
92	C₂×C₁₆	—	178	1	no	—
93	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	356	2	no	—
98	C₄⋊C₈	—	89	1	no	—
99	C₄⋊C₈	—	178	1	no	—
100	C₄⋊C₈	—	178	2	no	—
101	#103	—	356	2	no	—
102	C₄×C₈	—	178	1	no	—
103	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	89	1	no	—
106	C₂×C₈	—	356	2	no	—
107	C₂×C₈	—	178	1	no	—
108	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
109	C₂×C₈	#13083573	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	89	1	no	—
113	C₄⋊C₄	—	178	1	no	—
114	C₄²	—	178	1	no	—
115	C₄²	D₄×D₈₉	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	356	1	no	—
118	C₁₆	#13083564	1	2	yes	—
119	Q₈	—	16	2	no	—
120	Q₈	—	89	2	no	—
121	Q₈	—	178	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	178	1	no	—
124	C₂×C₄	—	178	1	no	—
125	C₂×C₄	#15667983	1	1	yes	characteristic
126	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
127	C₂×C₄	#15668031	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	356	1	no	—
130	C₈	—	712	2	no	—
131	C₈	C₂×#13083564	1	1	yes	characteristic
132	C₈	#15668043	1	1	yes	characteristic
133	C₂²	#19151831	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	16	2	no	—
136	C₄	—	356	1	no	—
137	C₄	—	178	2	no	—
138	C₄	#19151805	1	1	yes	characteristic
139	C₄	#19151870	1	1	yes	characteristic
140	C₄	Q₃₂×D₈₉	1	1	yes	characteristic
141	C₄	#19152031	1	1	yes	characteristic
142	C₂	#26019643	1	1	yes	characteristic
143	C₂	#26022340	1	1	yes	characteristic
144	C₂	#26022369	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₇₈

C₂

yes

maximal, characteristic

#26019653

C₂

yes

maximal

#26018324

C₂

yes

maximal, characteristic

#26018310

C₂

yes

maximal

C₈₉×#2252

C₂

yes

maximal, characteristic

#19151890

—

C₈×Dic₁₇₈

C₂²

yes

characteristic

C₄⋊#13083549

C₂²

yes

characteristic

#19151416

—

C₁₆×Dic₈₉

—

#19151351

C₂²

yes

—

#19151347

—

#500×C₈₉

C₂²

yes

—

#366×C₈₉

—

#366×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×Dic₁₇₈

D₄

yes

—

—

—

D₄

yes

characteristic

#15667941

—

#15667935

—

C₈×Dic₈₉

—

C₄⋊Q₈×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

D₈

yes

characteristic

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₈

yes

characteristic

#13083553

—

C₄×Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

Q₁₆

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

Q₃₂

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

C₈×C₈₉

yes

—

—

C₂²×C₈₉

#466

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×Q₃₂

yes

characteristic

C₄×C₈₉

#508

yes

characteristic

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

Q₈×C₁₆

—

#1676

—

#1371

—

#1365

—

C₈₉

#12458

yes

characteristic, complemented, Sylow

#500

C₂×D₈₉

yes

—

#477

—

178

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

C₄×C₁₆

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

#333

—

178

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₈₉

yes

—

—

—

356

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

100

C₄⋊C₈

—

178

—

101

#103

—

356

—

102

C₄×C₈

—

178

—

103

C₄×C₈

C₂²×D₈₉

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×C₈

—

356

—

107

C₂×C₈

—

178

—

108

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

109

C₂×C₈

#13083573

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

178

—

114

C₄²

—

178

—

115

C₄²

D₄×D₈₉

yes

characteristic

116

C₁₆

—

117

C₁₆

—

356

—

118

C₁₆

#13083564

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

178

—

122

C₂×C₄

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

#15667983

yes

characteristic

126

C₂×C₄

D₈×D₈₉

yes

characteristic

127

C₂×C₄

#15668031

yes

characteristic

128

C₈

—

129

C₈

—

356

—

130

C₈

—

712

—

131

C₈

C₂×#13083564

yes

characteristic

132

C₈

#15668043

yes

characteristic

133

C₂²

#19151831

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

356

—

137

C₄

—

178

—

138

C₄

#19151805

yes

characteristic

139

C₄

#19151870

yes

characteristic

140

C₄

Q₃₂×D₈₉

yes

characteristic

141

C₄

#19152031

yes

characteristic

142

C₂

#26019643

yes

characteristic

143

C₂

#26022340

yes

characteristic

144

C₂

#26022369

yes

characteristic

	subgroup	G/H
center	C₂²	#19151831
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2252	C₂
socle, prime radical	C₂²×C₈₉	#466
hypercenter	#2252	D₈₉
hypocenter	C₈₉	#12458
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#19151831

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2252

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
78. C₈₉	#12458	semidirect

H ≅ G/N

type

78. C₈₉

#12458

semidirect

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

subgroup	index
C₁₆×Dic₁₇₈	2
#26019653	2
#26018324	2
#26018310	2
C₈₉×#2252	2
#12458	89

subgroup

index

C₁₆×Dic₁₇₈

C₈₉×#2252

Subgroups — #36468789

Conjugacy classes of subgroups

Complements

Maximal subgroups