Finite Group Databasebeta

← #1513×C181 #37529456

Subgroups — #1513×C181 #37529456

Order 23168.

286 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 92 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
1#525×C181C211yesmaximal, characteristic
2#407×C181C211yesmaximal, characteristic, complemented
3#380×C181C211yesmaximal, characteristic, complemented
4#328×C181C212yesmaximal, complemented
5#327×C181C212yesmaximal, complemented
6C2×#41×C181C411yescharacteristic, complemented
7C23×C4×C181C2211yescharacteristic
8C2×M16×C181—21no—
9#123×C181—21no—
10C22⋊Q8×C181C411yescharacteristic, complemented
11C22⋊Q8×C181C2211yescharacteristic
12#121×C181C411yescharacteristic, complemented
13#121×C181C2211yescharacteristic
14#120×C181—21no—
15C4×D4×C181—21no—
16C4×D4×C181—21no—
17#117×C181C411yescharacteristic, complemented
18C2×C4⋊C4×C181—21no—
19C2×C22⋊C4×C181—21no—
20C22⋊C8×C181—21no—
21C22⋊C8×C181C2212yes—
22#95×C181C2212yes—
23C24×C181D411yescharacteristic
24#41×C181—42no—
25C2×Q8×C181C2×C411yescharacteristic
26C2×D4×C181—21no—
27C2×D4×C181—21no—
28C2×D4×C181—21no—
29C2×D4×C181C2×C411yescharacteristic
30C22×C4×C181—21no—
31C22×C4×C181—22no—
32C22×C4×C181—42no—
33C22×C4×C181C2×C411yescharacteristic
34C22×C4×C181C2×C411yescharacteristic
35C22×C4×C181D411yescharacteristic
36C22×C4×C181D411yescharacteristic
37C22×C4×C181D411yescharacteristic
38C22×C4×C181C2311yescharacteristic
39M16×C181—42no—
40C2×C8×C181—22no—
41C4⋊C4×C181—21no—
42C4⋊C4×C181—21no—
43C4⋊C4×C181—42no—
44C4⋊C4×C181C2×C411yescharacteristic
45C4⋊C4×C181C2×C411yescharacteristic
46C22⋊C4×C181—21no—
47C22⋊C4×C181—21no—
48C22⋊C4×C181—21no—
49C22⋊C4×C181—21no—
50C22⋊C4×C181—21no—
51C22⋊C4×C181—42no—
52C42×C181—21no—
53C23×C181—21no—
54C23×C181—21no—
55C23×C181—21no—
56C23×C181—41no—
57C23×C181—41no—
58C23×C181C22⋊C411yescharacteristic
59C23×C181C22⋊C411yescharacteristic
60C23×C181C2×D411yescharacteristic
61Q8×C181—41no—
62D4×C181—41no—
63D4×C181—42no—
64D4×C181—42no—
65D4×C181—41no—
66D4×C181—41no—
67C2×C4×C181—22no—
68C2×C4×C181—22no—
69C2×C4×C181—22no—
70C2×C4×C181—21no—
71C2×C4×C181—21no—
72C2×C4×C181—21no—
73C2×C4×C181—41no—
74C2×C4×C181—41no—
75C2×C4×C181—41no—
76C2×C4×C181—42no—
77C2×C4×C181—42no—
78C2×C4×C181—42no—
79C2×C4×C181—42no—
80C2×C4×C181C22⋊C411yescharacteristic
81C2×C4×C181C22⋊C411yescharacteristic
82C2×C4×C181C22×C411yescharacteristic
83C2×C4×C181C2×D411yescharacteristic
84C8×C181—42no—
85C22×C181—21no—
86C22×C181—21no—
87C22×C181—21no—
88C22×C181—21no—
89C22×C181—21no—
90C22×C181—21no—
91C22×C181—21no—
92C22×C181—21no—
93C22×C181—42no—
94C22×C181—42no—
95C22×C181—41no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181C2×C22⋊C411yescharacteristic
99C22×C181AGL(1,ℤ/8)11yescharacteristic
100C22×C181#13711yescharacteristic
101C4×C181—21no—
102C4×C181—21no—
103C4×C181—21no—
104C4×C181—41no—
105C4×C181—41no—
106C4×C181—41no—
107C4×C181—82no—
108C4×C181#9912yes—
109C2×C181—21no—
110C2×C181—21no—
111C2×C181—41no—
112C2×C181—41no—
113C2×C181—81no—
114C2×C181C2×#9911yescharacteristic
115C2×C181#41711yescharacteristic
116C2×C181#42111yescharacteristic
117C181#151311yescharacteristic, direct product factor, Sylow
118#1513C18111yesmaximal, characteristic, direct product factor, Sylow
119#525C2×C18111yescharacteristic
120#407C2×C18111yescharacteristic, complemented
121#380C2×C18111yescharacteristic, complemented
122#328C2×C18112yescomplemented
123#327C2×C18112yescomplemented
124C2×#41C4×C18111yescharacteristic, complemented
125C23×C4C22×C18111yescharacteristic
126C2×M16—21no—
127#123—21no—
128C22⋊Q8C4×C18111yescharacteristic, complemented
129C22⋊Q8C22×C18111yescharacteristic
130#121C4×C18111yescharacteristic, complemented
131#121C22×C18111yescharacteristic
132#120—21no—
133C4×D4—21no—
134C4×D4—21no—
135#117C4×C18111yescharacteristic, complemented
136C2×C4⋊C4—21no—
137C2×C22⋊C4—21no—
138C22⋊C8—21no—
139C22⋊C8C22×C18112yes—
140#95C22×C18112yes—
141C24D4×C18111yescharacteristic
142#41—42no—
143C2×Q8C2×C4×C18111yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C2×C4×C18111yescharacteristic
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—42no—
151C22×C4C2×C4×C18111yescharacteristic
152C22×C4C2×C4×C18111yescharacteristic
153C22×C4D4×C18111yescharacteristic
154C22×C4D4×C18111yescharacteristic
155C22×C4D4×C18111yescharacteristic
156C22×C4C23×C18111yescharacteristic
157M16—42no—
158C2×C8—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—42no—
162C4⋊C4C2×C4×C18111yescharacteristic
163C4⋊C4C2×C4×C18111yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—42no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23C22⋊C4×C18111yescharacteristic
177C23C22⋊C4×C18111yescharacteristic
178C23C2×D4×C18111yescharacteristic
179Q8—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183D4—41no—
184D4—41no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4C22⋊C4×C18111yescharacteristic
199C2×C4C22⋊C4×C18111yescharacteristic
200C2×C4C22×C4×C18111yescharacteristic
201C2×C4C2×D4×C18111yescharacteristic
202C8—42no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22C2×C22⋊C4×C18111yescharacteristic
217C22AGL(1,ℤ/8)×C18111yescharacteristic
218C22#137×C18111yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—82no—
226C4#99×C18112yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—81no—
232C2C2×#99×C18111yescharacteristic
233C2#417×C18111yescharacteristic
234C2#421×C18111yescharacteristic
subgroupG/H
center, socleC22×C181C2×C22⋊C4
commutatorC2×C4C22×C4×C181
Frattini, cosocleC22×C4C23×C181
prime radical#525×C181C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C181 ◁ C23×C4×C181 ◁ G

Complements

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

NH ≅ G/Ntype
2. #407×C181C2semidirect
3. #380×C181C2semidirect
4. #328×C181C2semidirect
5. #327×C181C2semidirect
6. C2×#41×C181C4semidirect
10. C22⋊Q8×C181C4semidirect
12. #121×C181C4semidirect
17. #117×C181C4semidirect
117. C181#1513direct (N × H)
118. #1513C181direct (N × H)
120. #407C2×C181semidirect
121. #380C2×C181semidirect
122. #328C2×C181semidirect
123. #327C2×C181semidirect
124. C2×#41C4×C181semidirect
128. C22⋊Q8C4×C181semidirect
130. #121C4×C181semidirect
135. #117C4×C181semidirect

Maximal subgroups

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

subgroupindex
#525×C1812
#407×C1812
#380×C1812
#328×C1812
#327×C1812
#1513181