Finite Group Databasebeta

← C23×#11786 #19159412

Subgroups — C23×#11786 #19159412

Order 5888.

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

normal subgroups only

This group has 462 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
2C4×C23×#99C211yesmaximal, characteristic
3C23×#1466C211yesmaximal, characteristic
4C23×#1352C214yesmaximal, complemented
5C2×C4×D4×C23C2211yescharacteristic
6C22×C4⋊C4×C23C411yescharacteristic, complemented
7C23×#432—21no—
8C2×C23×#99—21no—
9C23×#400C412yescomplemented
10C23×#399C2212yes—
11C23×#389—21no—
12C23×#386—21no—
13C23×#385—21no—
14C23×#380—21no—
15C23×#380C411yescharacteristic, complemented
16C4×C22⋊C4×C23—21no—
17C23×#342C2212yes—
18C23×#323—21no—
19C23×#323C2212yes—
20C22×D4×C23D411yescharacteristic
21C23×C4×C23—21no—
22C23×C4×C23C2×C411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C4×D4×C23—41no—
25C4×D4×C23—41no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—41no—
30C2×C4⋊C4×C23—41no—
31C2×C4⋊C4×C23C2×C412yes—
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23—21no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23D411yescharacteristic
37C2×C22⋊C4×C23C2311yescharacteristic
38C2×C42×C23—41no—
39C2×C42×C23D411yescharacteristic
40C23×C4⋊C8—41no—
41C23×#99—24no—
42C23×C22⋊C8—22no—
43C23×#97—41no—
44C23×#95—22no—
45C23×#95—22no—
46C23×#95—41no—
47C23×#95C2×C412yes—
48C24×C23C22⋊C411yescharacteristic
49C24×C23C2×D411yescharacteristic
50C2×D4×C23—22no—
51C2×D4×C23—22no—
52C2×D4×C23—41no—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23—21no—
56C22×C4×C23—21no—
57C22×C4×C23—22no—
58C22×C4×C23—22no—
59C22×C4×C23—42no—
60C22×C4×C23—42no—
61C22×C4×C23—42no—
62C22×C4×C23—42no—
63C22×C4×C23—82no—
64C22×C4×C23C22⋊C411yescharacteristic
65C22×C4×C23C22⋊C411yescharacteristic
66C22×C4×C23C22⋊C411yescharacteristic
67C22×C4×C23C22×C411yescharacteristic
68C22×C4×C23C2×D411yescharacteristic
69C2×C8×C23—42no—
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—42no—
80C22⋊C4×C23—42no—
81C22⋊C4×C23—82no—
82C42×C23—21no—
83C42×C23—21no—
84C42×C23—82no—
85#11786C2311yesmaximal, characteristic, direct product factor, Sylow
86C23×C23—21no—
87C23×C23—21no—
88C23×C23—21no—
89C23×C23—21no—
90C23×C23—41no—
91C23×C23—41no—
92C23×C23—41no—
93C23×C23—41no—
94C23×C23#10412yes—
95C23×C23C2×C22⋊C411yescharacteristic
96C23×C23#13712yes—
97D4×C23—44no—
98C2×C4×C23—22no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—41no—
111C2×C4×C23—44no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—81no—
119C2×C4×C23—82no—
120C2×C4×C23—82no—
121C2×C4×C23—84no—
122C2×C4×C23#9911yescharacteristic
123C2×C4×C23#9911yescharacteristic
124C8×C23—82no—
125#2487C2×C2311yescharacteristic
126C4×#99C2×C2311yescharacteristic
127#1466C2×C2311yescharacteristic
128#1352C2×C2314yescomplemented
129C22×C23—21no—
130C22×C23—21no—
131C22×C23—22no—
132C22×C23—22no—
133C22×C23—42no—
134C22×C23—42no—
135C22×C23—42no—
136C22×C23—41no—
137C22×C23—41no—
138C22×C23—41no—
139C22×C23—41no—
140C22×C23—81no—
141C22×C23C2×#9911yescharacteristic
142C22×C23#41911yescharacteristic
143C22×C23C2×#10411yescharacteristic
144C4×C23—21no—
145C4×C23—41no—
146C4×C23—41no—
147C4×C23—41no—
148C4×C23—41no—
149C4×C23—82no—
150C4×C23—84no—
151C4×C23—82no—
152C4×C23#35212yes—
153C2×C4×D4C22×C2311yescharacteristic
154C22×C4⋊C4C4×C2311yescharacteristic, complemented
155#432—21no—
156C2×#99—21no—
157#400C4×C2312yescomplemented
158#399C22×C2312yes—
159#389—21no—
160#386—21no—
161#385—21no—
162#380—21no—
163#380C4×C2311yescharacteristic, complemented
164C4×C22⋊C4—21no—
165#342C22×C2312yes—
166#323—21no—
167#323C22×C2312yes—
168C2×C23—21no—
169C2×C23—21no—
170C2×C23—42no—
171C2×C23—42no—
172C2×C23#150811yescharacteristic
173C2×C23C2×#35211yescharacteristic
174C2×C23#212811yescharacteristic
175C22×D4D4×C2311yescharacteristic
176C23×C4—21no—
177C23×C4C2×C4×C2311yescharacteristic
178C23×C4D4×C2311yescharacteristic
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4C2×C4×C2312yes—
187C2×C4⋊C4C2×C4×C2311yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4D4×C2311yescharacteristic
192C2×C22⋊C4C23×C2311yescharacteristic
193C2×C42—41no—
194C2×C42D4×C2311yescharacteristic
195C4⋊C8—41no—
196#99—24no—
197C22⋊C8—22no—
198#97—41no—
199#95—22no—
200#95—22no—
201#95—41no—
202#95C2×C4×C2312yes—
203C23#1178611yescharacteristic, direct product factor, Sylow
204C24C22⋊C4×C2311yescharacteristic
205C24C2×D4×C2311yescharacteristic
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×C4—42no—
218C22×C4—42no—
219C22×C4—82no—
220C22×C4C22⋊C4×C2311yescharacteristic
221C22×C4C22⋊C4×C2311yescharacteristic
222C22×C4C22⋊C4×C2311yescharacteristic
223C22×C4C22×C4×C2311yescharacteristic
224C22×C4C2×D4×C2311yescharacteristic
225C2×C8—42no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—82no—
232C22⋊C4—24no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—82no—
238C42—21no—
239C42—21no—
240C42—82no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23C23×#10412yes—
250C23C2×C22⋊C4×C2311yescharacteristic
251C23C23×#13712yes—
252D4—44no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—84no—
276C2×C4—82no—
277C2×C4C23×#9911yescharacteristic
278C2×C4C23×#9911yescharacteristic
279C8—82no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×C23×#9911yescharacteristic
293C22C23×#41911yescharacteristic
294C22C2×C23×#10411yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—84no—
302C4—82no—
303C4C23×#35212yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2C23×#150811yescharacteristic
309C2C2×C23×#35211yescharacteristic
310C2C23×#212811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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×#1352C2semidirect
6. C22×C4⋊C4×C23C4semidirect
9. C23×#400C4semidirect
15. C23×#380C4semidirect
85. #11786C23direct (N × H)
128. #1352C2×C23semidirect
154. C22×C4⋊C4C4×C23semidirect
157. #400C4×C23semidirect
163. #380C4×C23semidirect
203. C23#11786direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24872
C4×C23×#992
C23×#14662
C23×#13522
#1178623