Finite Group Databasebeta

← C89×#32706 #36410554

Subgroups — C89×#32706 #36410554

Order 22784.

1140 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2847C212yesmaximal, complemented
2C89×#2075C213yesmaximal, complemented
3C89×#2033C216yesmaximal, complemented
4C89×#1999C213yesmaximal, complemented
5C89×#1779C211yesmaximal, characteristic, complemented
6C24×C4×C89C2211yescharacteristic
7C22×D8×C89C2211yescharacteristic, complemented
8C2×#121×C89C2216yes—
9C2×#120×C89—22no—
10C2×#107×C89C2213yescomplemented
11C2×#102×C89C2216yescomplemented
12C2×C22⋊C8×C89C2213yes—
13C2×C22⋊C8×C89C2213yes—
14#386×C89—26no—
15#380×C89C2216yes—
16#340×C89C2213yescomplemented
17#340×C89C2213yescomplemented
18C25×C89D411yescharacteristic
19C22×D4×C89—26no—
20C22×D4×C89C2312yes—
21C23×C4×C89—43no—
22C23×C4×C89—43no—
23C23×C4×C89D413yes—
24C23×C4×C89C2313yes—
25C2×D8×C89—43no—
26C22×C8×C89C2313yes—
27C22×C8×C89C2311yescharacteristic
28#121×C89—46no—
29#121×C89—46no—
30#120×C89—48no—
31C2×C4⋊C4×C89C2316yes—
32C2×C22⋊C4×C89—26no—
33C2×C22⋊C4×C89—26no—
34#107×C89—43no—
35#102×C89—46no—
36C22⋊C8×C89—43no—
37C22⋊C8×C89—43no—
38#95×C89—26no—
39C24×C89—22no—
40C24×C89—43no—
41C24×C89—43no—
42C24×C89D814yescomplemented
43C24×C89C2×D413yes—
44C2×D4×C89—26no—
45C2×D4×C89—26no—
46C2×D4×C89—412no—
47C2×D4×C89—412no—
48C2×D4×C89—412no—
49C22×C4×C89—23no—
50C22×C4×C89—23no—
51C22×C4×C89—26no—
52C22×C4×C89—26no—
53C22×C4×C89—26no—
54C22×C4×C89—43no—
55C22×C4×C89—46no—
56C22×C4×C89—412no—
57C22×C4×C89—44no—
58C22×C4×C89C2×D413yes—
59C22×C4×C89C2411yescharacteristic
60D8×C89—44no—
61C2×C8×C89—26no—
62C2×C8×C89—23no—
63C2×C8×C89—23no—
64C4⋊C4×C89—26no—
65C4⋊C4×C89—26no—
66C22⋊C4×C89—412no—
67C22⋊C4×C89—412no—
68C23×C89—26no—
69C23×C89—23no—
70C23×C89—26no—
71C23×C89—23no—
72C23×C89—412no—
73C23×C89—44no—
74C23×C89—412no—
75C23×C89—46no—
76C23×C89—42no—
77C23×C89—46no—
78C23×C89C2×D816yes—
79C23×C89C22×D411yescharacteristic
80D4×C89—46no—
81D4×C89—42no—
82D4×C89—824no—
83C2×C4×C89—26no—
84C2×C4×C89—23no—
85C2×C4×C89—412no—
86C2×C4×C89—46no—
87C2×C4×C89—43no—
88C2×C4×C89—43no—
89C2×C4×C89—46no—
90C2×C4×C89—46no—
91C2×C4×C89—46no—
92C2×C4×C89H(5,𝔽2)13yes—
93C2×C4×C89H(5,𝔽2)13yes—
94C8×C89—43no—
95C8×C89—41no—
96C22×C89—212no—
97C22×C89—26no—
98C22×C89—416no—
99C22×C89—46no—
100C22×C89—43no—
101C22×C89—43no—
102C22×C89—86no—
103C22×C89—86no—
104C22×C89—82no—
105C22×C89#53413yes—
106C22×C89C22×D811yescharacteristic
107C22×C89#57513yes—
108C4×C89—43no—
109C4×C89—86no—
110C4×C89#56013yes—
111C4×C89H(3,𝔽4)11yescharacteristic
112#32706C8911yesmaximal, characteristic, direct product factor, Sylow
113C2×C89—46no—
114C2×C89—162no—
115C2×C89#281211yescharacteristic
116C2×C89#318613yes—
117C2×C89#319713yes—
118#2847C2×C8912yescomplemented
119#2075C2×C8913yescomplemented
120#2033C2×C8916yescomplemented
121#1999C2×C8913yescomplemented
122#1779C2×C8911yescharacteristic, complemented
123C89#3270611yescharacteristic, direct product factor, Sylow
124C24×C4C22×C8911yescharacteristic
125C22×D8C22×C8911yescharacteristic, complemented
126C2×#121C22×C8916yes—
127C2×#120—22no—
128C2×#107C22×C8913yescomplemented
129C2×#102C22×C8916yescomplemented
130C2×C22⋊C8C22×C8913yes—
131C2×C22⋊C8C22×C8913yes—
132#386—26no—
133#380C22×C8916yes—
134#340C22×C8913yescomplemented
135#340C22×C8913yescomplemented
136C25D4×C8911yescharacteristic
137C22×D4—26no—
138C22×D4C23×C8912yes—
139C23×C4—43no—
140C23×C4—43no—
141C23×C4D4×C8913yes—
142C23×C4C23×C8913yes—
143C2×D8—43no—
144C22×C8C23×C8911yescharacteristic
145C22×C8C23×C8913yes—
146#121—46no—
147#121—46no—
148#120—48no—
149C2×C4⋊C4C23×C8916yes—
150C2×C22⋊C4—26no—
151C2×C22⋊C4—26no—
152#107—43no—
153#102—46no—
154C22⋊C8—43no—
155C22⋊C8—43no—
156#95—26no—
157C24—22no—
158C24—43no—
159C24—43no—
160C24D8×C8914yescomplemented
161C24C2×D4×C8913yes—
162C2×D4—26no—
163C2×D4—26no—
164C2×D4—412no—
165C2×D4—412no—
166C2×D4—412no—
167C22×C4—23no—
168C22×C4—23no—
169C22×C4—26no—
170C22×C4—26no—
171C22×C4—26no—
172C22×C4—43no—
173C22×C4—46no—
174C22×C4—412no—
175C22×C4—44no—
176C22×C4C2×D4×C8913yes—
177C22×C4C24×C8911yescharacteristic
178D8—44no—
179C2×C8—26no—
180C2×C8—23no—
181C2×C8—23no—
182C4⋊C4—26no—
183C4⋊C4—26no—
184C22⋊C4—412no—
185C22⋊C4—412no—
186C23—26no—
187C23—23no—
188C23—26no—
189C23—23no—
190C23—412no—
191C23—44no—
192C23—412no—
193C23—46no—
194C23—42no—
195C23—46no—
196C23C2×D8×C8916yes—
197C23C22×D4×C8911yescharacteristic
198D4—46no—
199D4—42no—
200D4—824no—
201C2×C4—26no—
202C2×C4—23no—
203C2×C4—412no—
204C2×C4—46no—
205C2×C4—43no—
206C2×C4—43no—
207C2×C4—46no—
208C2×C4—46no—
209C2×C4—46no—
210C2×C4H(5,𝔽2)×C8913yes—
211C2×C4H(5,𝔽2)×C8913yes—
212C8—43no—
213C8—41no—
214C22—212no—
215C22—26no—
216C22—416no—
217C22—43no—
218C22—46no—
219C22—43no—
220C22—82no—
221C22—86no—
222C22—86no—
223C22#534×C8913yes—
224C22C22×D8×C8911yescharacteristic
225C22#575×C8913yes—
226C4—43no—
227C4—86no—
228C4#560×C8913yes—
229C4H(3,𝔽4)×C8911yescharacteristic
230C2—46no—
231C2—162no—
232C2C89×#281211yescharacteristic
233C2C89×#318613yes—
234C2C89×#319713yes—
subgroupG/H
center, socleC23×C89C22×D4
commutator, Frattini, cosocleC22×C4C24×C89
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C89 ◁ C24×C4×C89 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C89×#2847C2semidirect
2. C89×#2075C2semidirect
3. C89×#2033C2semidirect
4. C89×#1999C2semidirect
5. C89×#1779C2semidirect
7. C22×D8×C89C22semidirect
10. C2×#107×C89C22semidirect
11. C2×#102×C89C22semidirect
16. #340×C89C22semidirect
17. #340×C89C22semidirect
42. C24×C89D8semidirect
112. #32706C89direct (N × H)
118. #2847C2×C89semidirect
119. #2075C2×C89semidirect
120. #2033C2×C89semidirect
121. #1999C2×C89semidirect
122. #1779C2×C89semidirect
123. C89#32706direct (N × H)
125. C22×D8C22×C89semidirect
128. C2×#107C22×C89semidirect
129. C2×#102C22×C89semidirect
134. #340C22×C89semidirect
135. #340C22×C89semidirect
160. C24D8×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#28472
C89×#20752
C89×#20332
C89×#19992
C89×#17792
#3270689