Finite Group Databasebeta

← C23×#11572 #19159198

Subgroups — C23×#11572 #19159198

Order 5888.

416 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2566C211yesmaximal, characteristic
2C23×#1465C211yesmaximal, characteristic
3C23×#1459C211yesmaximal, characteristic
4C23×#1325C212yesmaximal, complemented
5C23×#1324C212yesmaximal, complemented
6C2×C23×#125C411yescharacteristic
7C2×C4×D4×C23C2211yescharacteristic
8C23×#407—21no—
9C2×C23×C22⋊C8—21no—
10C23×#396—21no—
11C23×#396—21no—
12C23×#390C411yescharacteristic
13C23×#390C411yescharacteristic
14C23×#385—21no—
15C23×#385—21no—
16C23×#384C2211yescharacteristic
17C23×#382C2211yescharacteristic
18C4×C4⋊C4×C23C411yescharacteristic
19C23×#324C2212yes—
20C23×#324C2212yes—
21C23×#323—21no—
22C23×#323—21no—
23C22×D4×C23C2×C411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C23×C4×C23D411yescharacteristic
26C2×M16×C23—41no—
27C22×C8×C23—41no—
28C23×#125—41no—
29C23×#125—41no—
30C4×D4×C23—41no—
31C4×D4×C23—41no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23C2×C411yescharacteristic
38C2×C4⋊C4×C23C2×C411yescharacteristic
39C2×C4⋊C4×C23C2311yescharacteristic
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23D411yescharacteristic
45C2×C22⋊C4×C23D411yescharacteristic
46C2×C42×C23C2×C411yescharacteristic
47C2×C42×C23C2×C411yescharacteristic
48C2×C42×C23C2×C411yescharacteristic
49C23×C22⋊C8—22no—
50C23×C22⋊C8—22no—
51C23×C22⋊C8—41no—
52C23×C22⋊C8—41no—
53C23×#95—21no—
54C23×#95—21no—
55C23×#95—21no—
56C24×C23C22⋊C411yescharacteristic
57C24×C23C22⋊C411yescharacteristic
58C2×D4×C23—22no—
59C2×D4×C23—22no—
60C2×D4×C23—41no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—81no—
70C22×C4×C23—81no—
71C22×C4×C23C22⋊C411yescharacteristic
72C22×C4×C23C22⋊C411yescharacteristic
73C22×C4×C23C22×C411yescharacteristic
74C22×C4×C23C2×D411yescharacteristic
75C22×C4×C23C2×D411yescharacteristic
76M16×C23—82no—
77C2×C8×C23—42no—
78C2×C8×C23—42no—
79C2×C8×C23—82no—
80C4⋊C4×C23—41no—
81C4⋊C4×C23—41no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—42no—
84C4⋊C4×C23—42no—
85C4⋊C4×C23—41no—
86C4⋊C4×C23—41no—
87C4⋊C4×C23—81no—
88C4⋊C4×C23—81no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—41no—
91C22⋊C4×C23—42no—
92C22⋊C4×C23—42no—
93C22⋊C4×C23—81no—
94C22⋊C4×C23—81no—
95C42×C23—21no—
96C42×C23—21no—
97C42×C23—22no—
98C42×C23—22no—
99#11572C2311yesmaximal, characteristic, direct product factor, Sylow
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—21no—
103C23×C23—21no—
104C23×C23—21no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23#10112yes—
110C23×C23C2×C22⋊C411yescharacteristic
111D4×C23—44no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—22no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—41no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
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—41no—
128C2×C4×C23—41no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—81no—
136C2×C4×C23—81no—
137C2×C4×C23#9911yescharacteristic
138C2×C4×C23#9911yescharacteristic
139C2×C4×C23#13112yes—
140C8×C23—82no—
141C8×C23—82no—
142#2566C2×C2311yescharacteristic
143#1465C2×C2311yescharacteristic
144#1459C2×C2311yescharacteristic
145#1325C2×C2312yescomplemented
146#1324C2×C2312yescomplemented
147C22×C23—21no—
148C22×C23—21no—
149C22×C23—22no—
150C22×C23—42no—
151C22×C23—42no—
152C22×C23—41no—
153C22×C23—41no—
154C22×C23—41no—
155C22×C23—41no—
156C22×C23—41no—
157C22×C23—81no—
158C22×C23—81no—
159C22×C23#40811yescharacteristic
160C22×C23C2×#9911yescharacteristic
161C22×C23C2×#10111yescharacteristic
162C4×C23—21no—
163C4×C23—21no—
164C4×C23—22no—
165C4×C23—42no—
166C4×C23—41no—
167C4×C23—41no—
168C4×C23—41no—
169C4×C23—81no—
170C4×C23—81no—
171C4×C23—81no—
172C2×#125C4×C2311yescharacteristic
173C2×C4×D4C22×C2311yescharacteristic
174#407—21no—
175C2×C22⋊C8—21no—
176#396—21no—
177#396—21no—
178#390C4×C2311yescharacteristic
179#390C4×C2311yescharacteristic
180#385—21no—
181#385—21no—
182#384C22×C2311yescharacteristic
183#382C22×C2311yescharacteristic
184C4×C4⋊C4C4×C2311yescharacteristic
185#324C22×C2312yes—
186#324C22×C2312yes—
187#323—21no—
188#323—21no—
189C2×C23—21no—
190C2×C23—21no—
191C2×C23—42no—
192C2×C23—81no—
193C2×C23#147211yescharacteristic
194C2×C23#212711yescharacteristic
195C2×C23#213211yescharacteristic
196C22×D4C2×C4×C2311yescharacteristic
197C23×C4D4×C2311yescharacteristic
198C23×C4D4×C2311yescharacteristic
199C2×M16—41no—
200C22×C8—41no—
201#125—41no—
202#125—41no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C2×C4×C2311yescharacteristic
211C2×C4⋊C4C2×C4×C2311yescharacteristic
212C2×C4⋊C4C23×C2311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4D4×C2311yescharacteristic
218C2×C22⋊C4D4×C2311yescharacteristic
219C2×C42C2×C4×C2311yescharacteristic
220C2×C42C2×C4×C2311yescharacteristic
221C2×C42C2×C4×C2311yescharacteristic
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226#95—21no—
227#95—21no—
228#95—21no—
229C23#1157211yescharacteristic, direct product factor, Sylow
230C24C22⋊C4×C2311yescharacteristic
231C24C22⋊C4×C2311yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4C22⋊C4×C2311yescharacteristic
246C22×C4C22⋊C4×C2311yescharacteristic
247C22×C4C22×C4×C2311yescharacteristic
248C22×C4C2×D4×C2311yescharacteristic
249C22×C4C2×D4×C2311yescharacteristic
250M16—82no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—82no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C42—21no—
270C42—21no—
271C42—22no—
272C42—22no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23C23×#10112yes—
283C23C2×C22⋊C4×C2311yescharacteristic
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C23×#9911yescharacteristic
311C2×C4C23×#9911yescharacteristic
312C2×C4C23×#13112yes—
313C8—82no—
314C8—82no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22C23×#40811yescharacteristic
328C22C2×C23×#9911yescharacteristic
329C22C2×C23×#10111yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2C23×#147211yescharacteristic
345C2C23×#212711yescharacteristic
346C2C23×#213211yescharacteristic
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×#1325C2semidirect
5. C23×#1324C2semidirect
99. #11572C23direct (N × H)
145. #1325C2×C23semidirect
146. #1324C2×C23semidirect
229. C23#11572direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25662
C23×#14652
C23×#14592
C23×#13252
C23×#13242
#1157223