Finite Group Databasebeta

← C23×#11846 #19159472

Subgroups — C23×#11846 #19159472

Order 5888.

382 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2611C211yesmaximal, characteristic
2C23×#1820C211yesmaximal, characteristic
3C23×#1458C211yesmaximal, characteristic
4C23×#1352C212yesmaximal
5C23×#1344C212yesmaximal
6C2×C23×#125C411yescharacteristic, complemented
7C2×C23×#117C2211yescharacteristic
8C23×#431—21no—
9C23×#400C411yescharacteristic, complemented
10C23×#399C2211yescharacteristic
11C23×#398C411yescharacteristic, complemented
12C23×#397C2211yescharacteristic
13C23×#389—21no—
14C23×#388—21no—
15C23×#387—21no—
16C23×#385—21no—
17C23×#381—21no—
18C23×#376C411yescharacteristic, complemented
19C23×#342—21no—
20C23×#342C2212yes—
21C23×#324—21no—
22C23×#323C2212yes—
23C23×C4×C23D411yescharacteristic
24C23×#125—41no—
25C23×#125—41no—
26C23×#117—41no—
27C23×#117—41no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C4⋊C4×C23C2×C411yescharacteristic
34C2×C4⋊C4×C23C2×C411yescharacteristic
35C2×C4⋊C4×C23D411yescharacteristic
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23D411yescharacteristic
40C2×C22⋊C4×C23C2311yescharacteristic
41C2×C42×C23—21no—
42C2×C42×C23C2×C411yescharacteristic
43C2×C42×C23D411yescharacteristic
44C23×C4⋊C8—41no—
45C23×C22⋊C8—22no—
46C4×C8×C23—41no—
47C23×#95—21no—
48C23×#95—21no—
49C23×#95—21no—
50C23×#95—21no—
51C23×#95—21no—
52C23×#95—41no—
53C23×#95—41no—
54C23×#95C2×C411yescharacteristic
55C23×#95C2×C411yescharacteristic
56C24×C23C2×D411yescharacteristic
57C22×C4×C23—21no—
58C22×C4×C23—22no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—41no—
65C22×C4×C23—42no—
66C22×C4×C23C22⋊C411yescharacteristic
67C22×C4×C23C22⋊C411yescharacteristic
68C22×C4×C23C22⋊C411yescharacteristic
69C22×C4×C23C22⋊C411yescharacteristic
70C22×C4×C23C22×C411yescharacteristic
71C22×C4×C23C2×D411yescharacteristic
72C2×C8×C23—42no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—42no—
78C4⋊C4×C23—42no—
79C4⋊C4×C23—42no—
80C4⋊C4×C23—41no—
81C4⋊C4×C23—41no—
82C22⋊C4×C23—41no—
83C22⋊C4×C23—41no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—42no—
87C22⋊C4×C23—42no—
88C42×C23—21no—
89C42×C23—21no—
90C42×C23—21no—
91C42×C23—21no—
92C42×C23—42no—
93#11846C2311yesmaximal, characteristic, direct product factor, Sylow
94C23×C23—21no—
95C23×C23—21no—
96C23×C23—41no—
97C23×C23—41no—
98C23×C23C2×C22⋊C411yescharacteristic
99C23×C23#13512yes—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—22no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—41no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—41no—
117C2×C4×C23—41no—
118C2×C4×C23—41no—
119C2×C4×C23—41no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—41no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—41no—
127C2×C4×C23—82no—
128C2×C4×C23—82no—
129C2×C4×C23—82no—
130C2×C4×C23#9911yescharacteristic
131C2×C4×C23#9911yescharacteristic
132C8×C23—82no—
133#2611C2×C2311yescharacteristic
134#1820C2×C2311yescharacteristic
135#1458C2×C2311yescharacteristic
136#1352C2×C2312yes—
137#1344C2×C2312yes—
138C22×C23—21no—
139C22×C23—21no—
140C22×C23—22no—
141C22×C23—42no—
142C22×C23—41no—
143C22×C23—41no—
144C22×C23—81no—
145C22×C23C2×#9911yescharacteristic
146C22×C23#41611yescharacteristic
147C22×C23#42111yescharacteristic
148C4×C23—21no—
149C4×C23—21no—
150C4×C23—42no—
151C4×C23—41no—
152C4×C23—41no—
153C4×C23—41no—
154C4×C23—41no—
155C4×C23—41no—
156C4×C23—81no—
157C4×C23—81no—
158C4×C23—81no—
159C4×C23—162no—
160C2×#125C4×C2311yescharacteristic, complemented
161C2×#117C22×C2311yescharacteristic
162#431—21no—
163#400C4×C2311yescharacteristic, complemented
164#399C22×C2311yescharacteristic
165#398C4×C2311yescharacteristic, complemented
166#397C22×C2311yescharacteristic
167#389—21no—
168#388—21no—
169#387—21no—
170#385—21no—
171#381—21no—
172#376C4×C2311yescharacteristic, complemented
173#342—21no—
174#342C22×C2312yes—
175#324—21no—
176#323C22×C2312yes—
177C2×C23—21no—
178C2×C23—21no—
179C2×C23—42no—
180C2×C23#151611yescharacteristic
181C2×C23#212311yescharacteristic
182C2×C23#212611yescharacteristic
183C23×C4D4×C2311yescharacteristic
184#125—41no—
185#125—41no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×C4×C2311yescharacteristic
193C2×C4⋊C4C2×C4×C2311yescharacteristic
194C2×C4⋊C4C2×C4×C2311yescharacteristic
195C2×C4⋊C4D4×C2311yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4D4×C2311yescharacteristic
200C2×C22⋊C4C23×C2311yescharacteristic
201C2×C42—21no—
202C2×C42C2×C4×C2311yescharacteristic
203C2×C42D4×C2311yescharacteristic
204C4⋊C8—41no—
205C22⋊C8—22no—
206C4×C8—41no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—41no—
213#95—41no—
214#95C2×C4×C2311yescharacteristic
215#95C2×C4×C2311yescharacteristic
216C23#1184611yescharacteristic, direct product factor, Sylow
217C24C2×D4×C2311yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4C22⋊C4×C2311yescharacteristic
228C22×C4C22⋊C4×C2311yescharacteristic
229C22×C4C22⋊C4×C2311yescharacteristic
230C22×C4C22⋊C4×C2311yescharacteristic
231C22×C4C22×C4×C2311yescharacteristic
232C22×C4C2×D4×C2311yescharacteristic
233C2×C8—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊C4×C2311yescharacteristic
259C23C23×#13512yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4C23×#9911yescharacteristic
291C2×C4C23×#9911yescharacteristic
292C8—82no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×C23×#9911yescharacteristic
301C22C23×#41611yescharacteristic
302C22C23×#42111yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—162no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2C23×#151611yescharacteristic
319C2C23×#212311yescharacteristic
320C2C23×#212611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C23×#125C4semidirect
9. C23×#400C4semidirect
11. C23×#398C4semidirect
18. C23×#376C4semidirect
93. #11846C23direct (N × H)
160. C2×#125C4×C23semidirect
163. #400C4×C23semidirect
165. #398C4×C23semidirect
172. #376C4×C23semidirect
216. C23#11846direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26112
C23×#18202
C23×#14582
C23×#13522
C23×#13442
#1184623