Finite Group Databasebeta

← C23×#9202 #19156828

Subgroups — C23×#9202 #19156828

Order 5888.

516 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C23×#2627C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic
3C23×#1520C212yesmaximal
4C2×C23×#324C211yesmaximal, characteristic
5C23×#1289C212yesmaximal
6C2×C23×C22⋊Q8C2211yescharacteristic
7C22×C4⋊C4×C23C2211yescharacteristic
8C2×C23×C22⋊C8C2212yes—
9C23×#400C412yescomplemented
10C23×#398—22no—
11C23×#395C412yescomplemented
12C23×#387—24no—
13C23×#385—21no—
14C23×#381—21no—
15C23×#381—21no—
16C23×#380C2211yescharacteristic
17C23×#380C2212yes—
18C4×C22⋊C4×C23—21no—
19C23×#328—24no—
20C23×#324—22no—
21C22×Q8×C23C2×C411yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23C2311yescharacteristic
25C22×C8×C23—22no—
26C23×C22⋊Q8—22no—
27C23×C22⋊Q8—22no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23C2×C411yescharacteristic
35C2×C4⋊C4×C23C2×C412yes—
36C2×C4⋊C4×C23D412yes—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C42×C23—22no—
41C23×C22⋊C8—22no—
42C23×C22⋊C8—22no—
43C23×#95—22no—
44C23×#95—24no—
45C23×#95—22no—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95C2×C412yes—
49C23×#95D414yes—
50C24×C23C2×D411yescharacteristic
51C2×Q8×C23—22no—
52C2×Q8×C23—81no—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23—21no—
56C22×C4×C23—22no—
57C22×C4×C23—22no—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—22no—
61C22×C4×C23—21no—
62C22×C4×C23—24no—
63C22×C4×C23—81no—
64C22×C4×C23—81no—
65C22×C4×C23C22⋊C412yes—
66C22×C4×C23C22⋊C412yes—
67C22×C4×C23C22×C411yescharacteristic
68C22×C4×C23C2×D412yes—
69C22×C4×C23C2×D411yescharacteristic
70C22×C4×C23#4112yes—
71C2×C8×C23—42no—
72C2×C8×C23—82no—
73C4⋊C4×C23—24no—
74C4⋊C4×C23—24no—
75C4⋊C4×C23—22no—
76C4⋊C4×C23—42no—
77C4⋊C4×C23—82no—
78C4⋊C4×C23—81no—
79C4⋊C4×C23—82no—
80C22⋊C4×C23—41no—
81C22⋊C4×C23—41no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—42no—
84C22⋊C4×C23—42no—
85C42×C23—82no—
86#9202C2311yesmaximal, characteristic, direct product factor, Sylow
87C23×C23—41no—
88C23×C23—41no—
89C23×C23C2×C22⋊C411yescharacteristic
90C23×C23#12011yescharacteristic
91C23×C23#12311yescharacteristic
92C23×C23#13512yes—
93C23×C23#13712yes—
94Q8×C23—82no—
95C2×C4×C23—22no—
96C2×C4×C23—22no—
97C2×C4×C23—22no—
98C2×C4×C23—22no—
99C2×C4×C23—41no—
100C2×C4×C23—42no—
101C2×C4×C23—42no—
102C2×C4×C23—41no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—44no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—44no—
112C2×C4×C23—42no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23—42no—
116C2×C4×C23—44no—
117C2×C4×C23—81no—
118C2×C4×C23C4×D412yes—
119C2×C4×C23#12112yes—
120C8×C23—82no—
121#2627C2×C2311yescharacteristic
122#2313C2×C2311yescharacteristic
123#1520C2×C2312yes—
124C2×#324C2×C2311yescharacteristic
125#1289C2×C2312yes—
126C22×C23—21no—
127C22×C23—21no—
128C22×C23—21no—
129C22×C23—21no—
130C22×C23—21no—
131C22×C23—21no—
132C22×C23—42no—
133C22×C23—42no—
134C22×C23#38611yescharacteristic
135C22×C23#41611yescharacteristic
136C22×C23#41911yescharacteristic
137C22×C23#45212yes—
138C22×C23#48512yes—
139C4×C23—22no—
140C4×C23—22no—
141C4×C23—42no—
142C4×C23—42no—
143C4×C23—82no—
144C4×C23—84no—
145C4×C23—81no—
146C4×C23—81no—
147C2×C22⋊Q8C22×C2311yescharacteristic
148C22×C4⋊C4C22×C2311yescharacteristic
149C2×C22⋊C8C22×C2312yes—
150#400C4×C2312yescomplemented
151#398—22no—
152#395C4×C2312yescomplemented
153#387—24no—
154#385—21no—
155#381—21no—
156#381—21no—
157#380C22×C2311yescharacteristic
158#380C22×C2312yes—
159C4×C22⋊C4—21no—
160#328—24no—
161#324—22no—
162C2×C23—21no—
163C2×C23—22no—
164C2×C23—21no—
165C2×C23—22no—
166C2×C23#187311yescharacteristic
167C2×C23#191211yescharacteristic
168C2×C23#191311yescharacteristic
169C22×Q8C2×C4×C2311yescharacteristic
170C23×C4D4×C2311yescharacteristic
171C23×C4D4×C2311yescharacteristic
172C23×C4C23×C2311yescharacteristic
173C22×C8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×C4×C2311yescharacteristic
183C2×C4⋊C4C2×C4×C2312yes—
184C2×C4⋊C4D4×C2312yes—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C42—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191#95—22no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95C2×C4×C2312yes—
197#95D4×C2314yes—
198C23#920211yescharacteristic, direct product factor, Sylow
199C24C2×D4×C2311yescharacteristic
200C2×Q8—22no—
201C2×Q8—81no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4C22⋊C4×C2312yes—
215C22×C4C22⋊C4×C2312yes—
216C22×C4C22×C4×C2311yescharacteristic
217C22×C4C2×D4×C2312yes—
218C22×C4C2×D4×C2311yescharacteristic
219C22×C4#41×C2312yes—
220C2×C8—42no—
221C2×C8—82no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C4⋊C4—81no—
228C4⋊C4—82no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C42—82no—
235C23—41no—
236C23—41no—
237C23C2×C22⋊C4×C2311yescharacteristic
238C23C23×#12011yescharacteristic
239C23C23×#12311yescharacteristic
240C23C23×#13512yes—
241C23C23×#13712yes—
242Q8—82no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—81no—
266C2×C4C4×D4×C2312yes—
267C2×C4C23×#12112yes—
268C8—82no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22C23×#38611yescharacteristic
278C22C23×#41611yescharacteristic
279C22C23×#41911yescharacteristic
280C22C23×#45212yes—
281C22C23×#48512yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—82no—
287C4—84no—
288C4—81no—
289C4—81no—
290C2—21no—
291C2—22no—
292C2—21no—
293C2—22no—
294C2C23×#187311yescharacteristic
295C2C23×#191211yescharacteristic
296C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
9. C23×#400C4semidirect
11. C23×#395C4semidirect
86. #9202C23direct (N × H)
150. #400C4×C23semidirect
152. #395C4×C23semidirect
198. C23#9202direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26272
C23×#23132
C23×#15202
C2×C23×#3242
C23×#12892
#920223