Finite Group Databasebeta

← C23×#11576 #19159202

Subgroups — C23×#11576 #19159202

Order 5888.

426 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2487C211yesmaximal, characteristic
2C23×#1466C211yesmaximal, characteristic
3C23×#1459C211yesmaximal, characteristic
4C23×#1326C214yesmaximal, complemented
5C2×C4×D4×C23C2211yescharacteristic
6C22×C4⋊C4×C23C411yescharacteristic
7C23×#432—21no—
8C2×C23×C22⋊C8—21no—
9C23×#400—21no—
10C23×#399—21no—
11C23×#389—21no—
12C23×#386C412yes—
13C23×#385C2212yes—
14C23×#380—21no—
15C23×#380C411yescharacteristic
16C23×#324—21no—
17C23×#323—21no—
18C23×#323C2212yes—
19C23×#323C2212yes—
20C22×D4×C23C2×C411yescharacteristic
21C23×C4×C23C2×C411yescharacteristic
22C23×C4×C23C2×C412yes—
23C23×C4×C23D411yescharacteristic
24C22×C8×C23—41no—
25C4×D4×C23—41no—
26C4×D4×C23—41no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—41no—
32C2×C4⋊C4×C23—41no—
33C2×C4⋊C4×C23D411yescharacteristic
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23C2×C412yes—
36C2×C22⋊C4×C23D411yescharacteristic
37C2×C22⋊C4×C23C2311yescharacteristic
38C2×C42×C23D411yescharacteristic
39C23×C4⋊C8—41no—
40C23×C22⋊C8—22no—
41C23×C22⋊C8—22no—
42C23×C22⋊C8—41no—
43C23×#97—41no—
44C23×#95—21no—
45C23×#95—22no—
46C23×#95—22no—
47C24×C23C22⋊C411yescharacteristic
48C24×C23C22×C411yescharacteristic
49C2×D4×C23—22no—
50C2×D4×C23—22no—
51C2×D4×C23—41no—
52C22×C4×C23—21no—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23—21no—
56C22×C4×C23—22no—
57C22×C4×C23—22no—
58C22×C4×C23—42no—
59C22×C4×C23—42no—
60C22×C4×C23—42no—
61C22×C4×C23—82no—
62C22×C4×C23C22⋊C411yescharacteristic
63C22×C4×C23C22⋊C411yescharacteristic
64C22×C4×C23C22⋊C411yescharacteristic
65C22×C4×C23C2×D411yescharacteristic
66C22×C4×C23C2×D411yescharacteristic
67C2×C8×C23—42no—
68C2×C8×C23—42no—
69C2×C8×C23—82no—
70C4⋊C4×C23—41no—
71C4⋊C4×C23—42no—
72C4⋊C4×C23—42no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—82no—
76C22⋊C4×C23—24no—
77C22⋊C4×C23—41no—
78C22⋊C4×C23—41no—
79C22⋊C4×C23—82no—
80C42×C23—21no—
81C42×C23—21no—
82#11576C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—21no—
84C23×C23—21no—
85C23×C23—21no—
86C23×C23—21no—
87C23×C23—41no—
88C23×C23—41no—
89C23×C23—41no—
90C23×C23—41no—
91C23×C23#10112yes—
92C23×C23C2×C22⋊C411yescharacteristic
93C23×C23#13112yes—
94D4×C23—44no—
95C2×C4×C23—22no—
96C2×C4×C23—21no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—41no—
106C2×C4×C23—41no—
107C2×C4×C23—41no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—81no—
115C2×C4×C23—84no—
116C2×C4×C23#9911yescharacteristic
117C2×C4×C23#9911yescharacteristic
118C8×C23—82no—
119C8×C23—82no—
120#2487C2×C2311yescharacteristic
121#1466C2×C2311yescharacteristic
122#1459C2×C2311yescharacteristic
123#1326C2×C2314yescomplemented
124C22×C23—21no—
125C22×C23—21no—
126C22×C23—22no—
127C22×C23—22no—
128C22×C23—42no—
129C22×C23—42no—
130C22×C23—42no—
131C22×C23—41no—
132C22×C23—41no—
133C22×C23—41no—
134C22×C23—41no—
135C22×C23—81no—
136C22×C23#40811yescharacteristic
137C22×C23C2×#9911yescharacteristic
138C22×C23C2×#10111yescharacteristic
139C4×C23—21no—
140C4×C23—41no—
141C4×C23—41no—
142C4×C23—41no—
143C4×C23—41no—
144C4×C23—82no—
145C4×C23—82no—
146C4×C23#35212yes—
147C2×C4×D4C22×C2311yescharacteristic
148C22×C4⋊C4C4×C2311yescharacteristic
149#432—21no—
150C2×C22⋊C8—21no—
151#400—21no—
152#399—21no—
153#389—21no—
154#386C4×C2312yes—
155#385C22×C2312yes—
156#380—21no—
157#380C4×C2311yescharacteristic
158#324—21no—
159#323—21no—
160#323C22×C2312yes—
161#323C22×C2312yes—
162C2×C23—21no—
163C2×C23—21no—
164C2×C23—42no—
165C2×C23—42no—
166C2×C23#147211yescharacteristic
167C2×C23C2×#35211yescharacteristic
168C2×C23#212211yescharacteristic
169C22×D4C2×C4×C2311yescharacteristic
170C23×C4C2×C4×C2311yescharacteristic
171C23×C4C2×C4×C2312yes—
172C23×C4D4×C2311yescharacteristic
173C22×C8—41no—
174C4×D4—41no—
175C4×D4—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4D4×C2311yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4C2×C4×C2312yes—
185C2×C22⋊C4D4×C2311yescharacteristic
186C2×C22⋊C4C23×C2311yescharacteristic
187C2×C42D4×C2311yescharacteristic
188C4⋊C8—41no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192#97—41no—
193#95—21no—
194#95—22no—
195#95—22no—
196C23#1157611yescharacteristic, direct product factor, Sylow
197C24C22⋊C4×C2311yescharacteristic
198C24C22×C4×C2311yescharacteristic
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—82no—
212C22×C4C22⋊C4×C2311yescharacteristic
213C22×C4C22⋊C4×C2311yescharacteristic
214C22×C4C22⋊C4×C2311yescharacteristic
215C22×C4C2×D4×C2311yescharacteristic
216C22×C4C2×D4×C2311yescharacteristic
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—82no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—82no—
226C22⋊C4—24no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23C23×#10112yes—
241C23C2×C22⋊C4×C2311yescharacteristic
242C23C23×#13112yes—
243D4—44no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—84no—
265C2×C4C23×#9911yescharacteristic
266C2×C4C23×#9911yescharacteristic
267C8—82no—
268C8—82no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22C23×#40811yescharacteristic
282C22C2×C23×#9911yescharacteristic
283C22C2×C23×#10111yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—82no—
290C4—82no—
291C4C23×#35212yes—
292C2—21no—
293C2—21no—
294C2—42no—
295C2—42no—
296C2C23×#147211yescharacteristic
297C2C2×C23×#35211yescharacteristic
298C2C23×#212211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1326C2semidirect
82. #11576C23direct (N × H)
123. #1326C2×C23semidirect
196. C23#11576direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24872
C23×#14662
C23×#14592
C23×#13262
#1157623