Finite Group Databasebeta

← C23×#11564 #19159190

Subgroups — C23×#11564 #19159190

Order 5888.

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

normal subgroups only

This group has 444 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×#2547C211yesmaximal, characteristic
2C23×#1465C211yesmaximal, characteristic
3C23×#1459C211yesmaximal, characteristic
4C23×#1319C214yesmaximal, complemented
5C2×C23×C4⋊Q8C411yescharacteristic
6C2×C23×#124C412yes—
7C2×C4×D4×C23C2211yescharacteristic
8C23×#407—21no—
9C2×C23×C22⋊C8—21no—
10C23×#397—21no—
11C23×#397—21no—
12C23×#391C2212yes—
13C23×#380—21no—
14C23×#380—21no—
15C4×C4⋊C4×C23C411yescharacteristic
16C23×#324C2212yes—
17C23×#324C2212yes—
18C23×#323—21no—
19C23×#323—21no—
20C22×Q8×C23C2×C412yes—
21C22×D4×C23C2×C411yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C2×M16×C23—41no—
25C22×C8×C23—41no—
26C23×C4⋊Q8—41no—
27C23×C4⋊Q8—41no—
28C23×#124—44no—
29C4×D4×C23—41no—
30C4×D4×C23—41no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23C2311yescharacteristic
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23D411yescharacteristic
38C2×C22⋊C4×C23D411yescharacteristic
39C2×C42×C23C2×C411yescharacteristic
40C2×C42×C23C2×C412yes—
41C23×C22⋊C8—22no—
42C23×C22⋊C8—22no—
43C23×C22⋊C8—41no—
44C23×C22⋊C8—41no—
45C23×#95—21no—
46C23×#95—22no—
47C23×#95—22no—
48C24×C23C22⋊C411yescharacteristic
49C24×C23C22⋊C411yescharacteristic
50C2×Q8×C23—24no—
51C2×Q8×C23—42no—
52C2×Q8×C23—42no—
53C2×D4×C23—22no—
54C2×D4×C23—22no—
55C2×D4×C23—41no—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—22no—
61C22×C4×C23—22no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23C22⋊C411yescharacteristic
65C22×C4×C23C22⋊C411yescharacteristic
66C22×C4×C23C22×C411yescharacteristic
67C22×C4×C23C2×D411yescharacteristic
68C22×C4×C23C2×D411yescharacteristic
69M16×C23—82no—
70C2×C8×C23—42no—
71C2×C8×C23—42no—
72C2×C8×C23—82no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—42no—
75C4⋊C4×C23—42no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—41no—
78C22⋊C4×C23—41no—
79C22⋊C4×C23—41no—
80C22⋊C4×C23—44no—
81C22⋊C4×C23—44no—
82C42×C23—21no—
83C42×C23—21no—
84C42×C23—42no—
85#11564C2311yesmaximal, characteristic, direct product factor, Sylow
86C23×C23—21no—
87C23×C23—21no—
88C23×C23—21no—
89C23×C23—21no—
90C23×C23—21no—
91C23×C23—41no—
92C23×C23—41no—
93C23×C23—41no—
94C23×C23—41no—
95C23×C23#10012yes—
96C23×C23C2×C22⋊C411yescharacteristic
97Q8×C23—84no—
98D4×C23—44no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—22no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—41no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—81no—
117C2×C4×C23—81no—
118C2×C4×C23#9911yescharacteristic
119C2×C4×C23#9911yescharacteristic
120C2×C4×C23#13112yes—
121C8×C23—82no—
122C8×C23—82no—
123#2547C2×C2311yescharacteristic
124#1465C2×C2311yescharacteristic
125#1459C2×C2311yescharacteristic
126#1319C2×C2314yescomplemented
127C22×C23—21no—
128C22×C23—21no—
129C22×C23—22no—
130C22×C23—42no—
131C22×C23—42no—
132C22×C23—41no—
133C22×C23—41no—
134C22×C23—41no—
135C22×C23—41no—
136C22×C23—41no—
137C22×C23—81no—
138C22×C23—81no—
139C22×C23#40811yescharacteristic
140C22×C23C2×#9911yescharacteristic
141C22×C23C2×#10011yescharacteristic
142C4×C23—21no—
143C4×C23—22no—
144C4×C23—41no—
145C4×C23—41no—
146C4×C23—41no—
147C4×C23—82no—
148C4×C23—82no—
149C4×C23#35412yes—
150C2×C4⋊Q8C4×C2311yescharacteristic
151C2×#124C4×C2312yes—
152C2×C4×D4C22×C2311yescharacteristic
153#407—21no—
154C2×C22⋊C8—21no—
155#397—21no—
156#397—21no—
157#391C22×C2312yes—
158#380—21no—
159#380—21no—
160C4×C4⋊C4C4×C2311yescharacteristic
161#324C22×C2312yes—
162#324C22×C2312yes—
163#323—21no—
164#323—21no—
165C2×C23—21no—
166C2×C23—21no—
167C2×C23—42no—
168C2×C23—81no—
169C2×C23#147111yescharacteristic
170C2×C23C2×#35411yescharacteristic
171C2×C23#213511yescharacteristic
172C22×Q8C2×C4×C2312yes—
173C22×D4C2×C4×C2311yescharacteristic
174C23×C4D4×C2311yescharacteristic
175C23×C4D4×C2311yescharacteristic
176C2×M16—41no—
177C22×C8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180#124—44no—
181C4×D4—41no—
182C4×D4—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C23×C2311yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4D4×C2311yescharacteristic
190C2×C22⋊C4D4×C2311yescharacteristic
191C2×C42C2×C4×C2311yescharacteristic
192C2×C42C2×C4×C2312yes—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#95—21no—
198#95—22no—
199#95—22no—
200C23#1156411yescharacteristic, direct product factor, Sylow
201C24C22⋊C4×C2311yescharacteristic
202C24C22⋊C4×C2311yescharacteristic
203C2×Q8—24no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C22⋊C4×C2311yescharacteristic
218C22×C4C22⋊C4×C2311yescharacteristic
219C22×C4C22×C4×C2311yescharacteristic
220C22×C4C2×D4×C2311yescharacteristic
221C22×C4C2×D4×C2311yescharacteristic
222M16—82no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—82no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C42—21no—
236C42—21no—
237C42—42no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C23×#10012yes—
248C23C2×C22⋊C4×C2311yescharacteristic
249Q8—84no—
250D4—44no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4C23×#9911yescharacteristic
271C2×C4C23×#9911yescharacteristic
272C2×C4C23×#13112yes—
273C8—82no—
274C8—82no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22C23×#40811yescharacteristic
288C22C2×C23×#9911yescharacteristic
289C22C2×C23×#10011yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4C23×#35412yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2C23×#147111yescharacteristic
303C2C2×C23×#35411yescharacteristic
304C2C23×#213511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ 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×#1319C2semidirect
85. #11564C23direct (N × H)
126. #1319C2×C23semidirect
200. C23#11564direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25472
C23×#14652
C23×#14592
C23×#13192
#1156423