Group #36468784 — subgroups

Subgroups — #36468784

Order 22784.

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

normal subgroups only

This group has 204 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	#26019643	C₂	1	2	yes	maximal
3	#26018324	C₂	1	1	yes	maximal, characteristic
4	#26018307	C₂	1	2	yes	maximal
5	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic
6	#19151805	—	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	#19151344	—	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	C₂×#13083564	—	4	2	no	—
18	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
19	C₂×#13083549	—	2	1	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	D₄	1	1	yes	characteristic
22	#15667941	—	2	1	no	—
23	#15667936	—	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	#13083564	—	8	4	no	—
33	C₂×#10623494	—	2	1	no	—
34	C₂×#10623494	—	4	2	no	—
35	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
36	#13083553	—	2	1	no	—
37	C₄×Dic₈₉	—	2	1	no	—
38	#13083549	—	4	1	no	—
39	C₂×Q₈×C₈₉	—	4	2	no	—
40	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₈₉	—	4	2	no	—
43	C₄⋊C₄×C₈₉	—	4	2	no	—
44	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₈₉	—	2	1	no	—
46	C₁₆×C₈₉	Q₁₆	1	2	yes	complemented
47	Q₈×C₈₉	—	8	2	no	—
48	Q₈×C₈₉	—	8	2	no	—
49	C₂×C₄×C₈₉	—	8	2	no	—
50	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
53	C₂×Dic₈₉	—	2	1	no	—
54	C₂×Dic₈₉	—	2	1	no	—
55	Dic₁₇₈	—	2	1	no	—
56	Dic₁₇₈	D₁₆	1	2	yes	—
57	C₈×C₈₉	—	2	1	no	—
58	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₈₉	#137	1	1	yes	characteristic
60	#10623494	—	4	1	no	—
61	#10623494	—	8	2	no	—
62	C₂²×C₈₉	#466	1	1	yes	characteristic
63	C₄×C₈₉	—	2	1	no	—
64	C₄×C₈₉	—	16	2	no	—
65	C₄×C₈₉	#462	1	1	yes	characteristic
66	C₄×C₈₉	#473	1	1	yes	characteristic
67	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
68	C₄×C₈₉	#508	1	1	yes	characteristic
69	Dic₈₉	—	2	2	no	—
70	Dic₈₉	—	4	1	no	—
71	#12457	—	89	1	no	maximal, Sylow
72	C₂×C₈₉	#1674	1	1	yes	characteristic
73	C₂×C₈₉	#2215	1	1	yes	characteristic
74	C₂×C₈₉	#2222	1	1	yes	characteristic
75	#2252	D₈₉	1	1	yes	characteristic
76	Q₈×C₁₆	—	89	1	no	—
77	#1674	—	89	2	no	—
78	#1371	—	89	1	no	—
79	#1364	—	89	2	no	—
80	C₈₉	#12457	1	1	yes	characteristic, complemented, Sylow
81	#500	C₂×D₈₉	1	2	yes	—
82	#462	—	178	2	no	—
83	C₈×Q₈	—	89	1	no	—
84	#366	—	2	1	no	—
85	#366	—	2	1	no	—
86	C₄⋊C₁₆	—	89	1	no	—
87	C₄⋊C₁₆	—	178	1	no	—
88	C₄×C₁₆	—	178	1	no	—
89	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
90	#335	—	89	2	no	—
91	#332	—	178	2	no	—
92	C₂×Q₁₆	—	356	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄×Q₈	—	89	1	no	—
95	C₂×C₁₆	—	178	1	no	—
96	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	C₄⋊C₈	—	89	1	no	—
101	C₄⋊C₈	—	178	1	no	—
102	C₄⋊C₈	—	178	2	no	—
103	#103	—	356	2	no	—
104	C₄×C₈	—	178	1	no	—
105	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	89	1	no	—
108	Q₁₆	—	712	4	no	—
109	C₂×C₈	—	356	2	no	—
110	C₂×C₈	—	178	1	no	—
111	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
112	C₂×C₈	#13083573	1	1	yes	characteristic
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	89	1	no	—
116	C₄⋊C₄	—	178	1	no	—
117	C₄²	—	178	1	no	—
118	C₄²	D₄×D₈₉	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	356	1	no	—
121	C₁₆	#13083564	1	2	yes	complemented
122	Q₈	—	8	2	no	—
123	Q₈	—	8	2	no	—
124	Q₈	—	89	2	no	—
125	Q₈	—	178	1	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	—	178	1	no	—
128	C₂×C₄	—	178	1	no	—
129	C₂×C₄	#15667983	1	1	yes	characteristic
130	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
131	C₂×C₄	#15668031	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	356	1	no	—
134	C₈	—	712	2	no	—
135	C₈	C₂×#13083564	1	1	yes	characteristic
136	C₈	#15668043	1	1	yes	characteristic
137	C₂²	#19151831	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	16	2	no	—
140	C₄	—	356	1	no	—
141	C₄	—	178	2	no	—
142	C₄	#19151805	1	1	yes	characteristic
143	C₄	#19151870	1	1	yes	characteristic
144	C₄	D₁₆×D₈₉	1	1	yes	characteristic
145	C₄	#19152027	1	1	yes	characteristic
146	C₂	#26019643	1	1	yes	characteristic
147	C₂	#26022335	1	1	yes	characteristic
148	C₂	#26022374	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₇₈

C₂

yes

maximal, characteristic

#26019643

C₂

yes

maximal

#26018324

C₂

yes

maximal, characteristic

#26018307

C₂

yes

maximal

C₈₉×#2252

C₂

yes

maximal, characteristic

#19151805

—

C₈×Dic₁₇₈

C₂²

yes

characteristic

C₄⋊#13083549

C₂²

yes

characteristic

#19151416

—

C₁₆×Dic₈₉

—

#19151351

C₂²

yes

—

#19151344

—

#500×C₈₉

C₂²

yes

—

#366×C₈₉

—

#366×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#13083564

—

C₄×Dic₁₇₈

D₄

yes

—

—

D₄

yes

characteristic

#15667941

—

#15667936

—

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

—

—

—

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

complemented

Q₈×C₈₉

—

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

D₁₆

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×D₁₆

yes

characteristic

C₄×C₈₉

#508

yes

characteristic

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

Q₈×C₁₆

—

#1674

—

#1371

—

#1364

—

C₈₉

#12457

yes

characteristic, complemented, Sylow

#500

C₂×D₈₉

yes

—

#462

—

178

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

C₄×C₁₆

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

#332

—

178

—

C₂×Q₁₆

—

356

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

#107

—

#107

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

178

—

102

C₄⋊C₈

—

178

—

103

#103

—

356

—

104

C₄×C₈

—

178

—

105

C₄×C₈

C₂²×D₈₉

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

108

Q₁₆

—

712

—

109

C₂×C₈

—

356

—

110

C₂×C₈

—

178

—

111

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

112

C₂×C₈

#13083573

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

178

—

117

C₄²

—

178

—

118

C₄²

D₄×D₈₉

yes

characteristic

119

C₁₆

—

120

C₁₆

—

356

—

121

C₁₆

#13083564

yes

complemented

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

178

—

126

C₂×C₄

—

127

C₂×C₄

—

178

—

128

C₂×C₄

—

178

—

129

C₂×C₄

#15667983

yes

characteristic

130

C₂×C₄

D₈×D₈₉

yes

characteristic

131

C₂×C₄

#15668031

yes

characteristic

132

C₈

—

133

C₈

—

356

—

134

C₈

—

712

—

135

C₈

C₂×#13083564

yes

characteristic

136

C₈

#15668043

yes

characteristic

137

C₂²

#19151831

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

356

—

141

C₄

—

178

—

142

C₄

#19151805

yes

characteristic

143

C₄

#19151870

yes

characteristic

144

C₄

D₁₆×D₈₉

yes

characteristic

145

C₄

#19152027

yes

characteristic

146

C₂

#26019643

yes

characteristic

147

C₂

#26022335

yes

characteristic

148

C₂

#26022374

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	#12457
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
46. C₁₆×C₈₉	Q₁₆	semidirect
80. C₈₉	#12457	semidirect
121. C₁₆	#13083564	semidirect

H ≅ G/N

type

46. C₁₆×C₈₉

Q₁₆

semidirect

80. C₈₉

#12457

semidirect

121. C₁₆

#13083564

semidirect

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

subgroup	index
C₁₆×Dic₁₇₈	2
#26019643	2
#26018324	2
#26018307	2
C₈₉×#2252	2
#12457	89

subgroup

index

C₁₆×Dic₁₇₈

C₈₉×#2252

Subgroups — #36468784

Conjugacy classes of subgroups

Complements

Maximal subgroups