Finite Group Databasebeta

← C23×#8626 #19156252

Subgroups — C23×#8626 #19156252

Order 5888.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2929C211yesmaximal, characteristic
2C23×#1810C211yesmaximal, characteristic, complemented
3C23×#1807C211yesmaximal, characteristic, complemented
4C23×#1513C212yesmaximal
5C23×#1305C212yesmaximal, complemented
6C2×C23×#131C411yescharacteristic, complemented
7C23×C8×C23C2211yescharacteristic
8C23×#525C2211yescharacteristic
9C23×#435—21no—
10C23×#435—21no—
11C8×D4×C23—21no—
12C8×D4×C23—21no—
13C2×C23×#108—21no—
14C2×C23×#106—21no—
15C23×#424C411yescharacteristic, complemented
16C23×#408C411yescharacteristic, complemented
17C23×#407C2212yes—
18C23×#407C2211yescharacteristic
19C23×#405C411yescharacteristic, complemented
20C23×#380C2212yes—
21C23×#341—21no—
22C23×#340—21no—
23C23×#328—22no—
24C23×#327—22no—
25C2×#41×C23C2×C411yescharacteristic
26C23×C4×C23C2311yescharacteristic
27C23×#131—42no—
28C2×M16×C23—21no—
29C2×M16×C23—22no—
30C2×M16×C23C2×C411yescharacteristic
31C2×M16×C23C2×C411yescharacteristic
32C22×C8×C23—21no—
33C22×C8×C23—42no—
34C22×C8×C23C2×C411yescharacteristic
35C22×C8×C23C2×C411yescharacteristic
36C22×C8×C23D411yescharacteristic
37C22×C8×C23D411yescharacteristic
38C22×C8×C23D411yescharacteristic
39C22×C8×C23D411yescharacteristic
40C23×#123—21no—
41C23×C22⋊Q8D411yescharacteristic
42C23×C22⋊Q8Q811yescharacteristic
43C23×#121D411yescharacteristic
44C23×#121Q811yescharacteristic
45C23×#120—21no—
46C4×D4×C23—21no—
47C4×D4×C23—21no—
48C23×#117C2×C411yescharacteristic
49C2×C4⋊C4×C23—22no—
50C2×C22⋊C4×C23—22no—
51C23×#108—42no—
52C23×#106—42no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C22⋊C8—21no—
56C23×C22⋊C8—21no—
57C23×C22⋊C8—21no—
58C23×C22⋊C8—21no—
59C23×C22⋊C8—22no—
60C23×C22⋊C8—22no—
61C23×#97—21no—
62C4×C8×C23—21no—
63C23×#95—22no—
64C24×C23#4111yescharacteristic
65#41×C23—41no—
66#41×C23—41no—
67C2×Q8×C23C4⋊C411yescharacteristic
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23—21no—
71C2×D4×C23C4⋊C411yescharacteristic
72C22×C4×C23—21no—
73C22×C4×C23—41no—
74C22×C4×C23—41no—
75C22×C4×C23—42no—
76C22×C4×C23C22×C411yescharacteristic
77C22×C4×C23C2×D411yescharacteristic
78C22×C4×C23C2×D411yescharacteristic
79C22×C4×C23C2×D411yescharacteristic
80C22×C4×C23C2×Q811yescharacteristic
81C22×C4×C23#4111yescharacteristic
82M16×C23—42no—
83M16×C23—41no—
84M16×C23—41no—
85M16×C23—42no—
86M16×C23—42no—
87C2×C8×C23—22no—
88C2×C8×C23—22no—
89C2×C8×C23—22no—
90C2×C8×C23—21no—
91C2×C8×C23—21no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23—42no—
96C2×C8×C23—42no—
97C2×C8×C23C22⋊C411yescharacteristic
98C2×C8×C23C22⋊C411yescharacteristic
99C2×C8×C23C22⋊C411yescharacteristic
100C2×C8×C23C22⋊C411yescharacteristic
101C4⋊C4×C23—21no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—42no—
104C4⋊C4×C23—42no—
105C4⋊C4×C23C4⋊C411yescharacteristic
106C4⋊C4×C23C4⋊C411yescharacteristic
107C22⋊C4×C23—21no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23—21no—
110C22⋊C4×C23—21no—
111C22⋊C4×C23—21no—
112C22⋊C4×C23—82no—
113C42×C23—21no—
114#8626C2311yesmaximal, characteristic, direct product factor, Sylow
115C23×C23—22no—
116C23×C23—21no—
117C23×C23—41no—
118C23×C23—41no—
119C23×C23#11711yescharacteristic
120C23×C23#12111yescharacteristic
121C23×C23C22⋊Q811yescharacteristic
122Q8×C23—41no—
123D4×C23—41no—
124D4×C23—42no—
125D4×C23—42no—
126D4×C23—42no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—82no—
142C2×C4×C23—82no—
143C2×C4×C23—82no—
144C2×C4×C23C2×C22⋊C411yescharacteristic
145C2×C4×C23C2×C4⋊C411yescharacteristic
146C2×C4×C23#12111yescharacteristic
147C2×C4×C23C22⋊Q811yescharacteristic
148C8×C23—21no—
149C8×C23—21no—
150C8×C23—21no—
151C8×C23—41no—
152C8×C23—41no—
153C8×C23—82no—
154C8×C23#9912yes—
155#2929C2×C2311yescharacteristic
156#1810C2×C2311yescharacteristic, complemented
157#1807C2×C2311yescharacteristic, complemented
158#1513C2×C2312yes—
159#1305C2×C2312yescomplemented
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—21no—
166C22×C23—21no—
167C22×C23—21no—
168C22×C23—44no—
169C22×C23—41no—
170C22×C23—41no—
171C22×C23—41no—
172C22×C23#38011yescharacteristic
173C22×C23#47112yes—
174C4×C23—21no—
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—41no—
180C4×C23—162no—
181C4×C23C2×#9911yescharacteristic
182C4×C23#41011yescharacteristic
183C2×#131C4×C2311yescharacteristic, complemented
184C23×C8C22×C2311yescharacteristic
185#525C22×C2311yescharacteristic
186#435—21no—
187#435—21no—
188C8×D4—21no—
189C8×D4—21no—
190C2×#108—21no—
191C2×#106—21no—
192#424C4×C2311yescharacteristic, complemented
193#408C4×C2311yescharacteristic, complemented
194#407C22×C2312yes—
195#407C22×C2311yescharacteristic
196#405C4×C2311yescharacteristic, complemented
197#380C22×C2312yes—
198#341—21no—
199#340—21no—
200#328—22no—
201#327—22no—
202C2×C23—22no—
203C2×C23—41no—
204C2×C23—41no—
205C2×C23—81no—
206C2×C23#179911yescharacteristic
207C2×C23#181411yescharacteristic
208C2×C23#181511yescharacteristic
209C2×#41C2×C4×C2311yescharacteristic
210C23×C4C23×C2311yescharacteristic
211#131—42no—
212C2×M16—21no—
213C2×M16—22no—
214C2×M16C2×C4×C2311yescharacteristic
215C2×M16C2×C4×C2311yescharacteristic
216C22×C8—21no—
217C22×C8—42no—
218C22×C8C2×C4×C2311yescharacteristic
219C22×C8C2×C4×C2311yescharacteristic
220C22×C8D4×C2311yescharacteristic
221C22×C8D4×C2311yescharacteristic
222C22×C8D4×C2311yescharacteristic
223C22×C8D4×C2311yescharacteristic
224#123—21no—
225C22⋊Q8D4×C2311yescharacteristic
226C22⋊Q8Q8×C2311yescharacteristic
227#121D4×C2311yescharacteristic
228#121Q8×C2311yescharacteristic
229#120—21no—
230C4×D4—21no—
231C4×D4—21no—
232#117C2×C4×C2311yescharacteristic
233C2×C4⋊C4—22no—
234C2×C22⋊C4—22no—
235#108—42no—
236#106—42no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245#97—21no—
246C4×C8—21no—
247#95—22no—
248C23#862611yescharacteristic, direct product factor, Sylow
249C24#41×C2311yescharacteristic
250#41—41no—
251#41—41no—
252C2×Q8C4⋊C4×C2311yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C4⋊C4×C2311yescharacteristic
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4C22×C4×C2311yescharacteristic
262C22×C4C2×D4×C2311yescharacteristic
263C22×C4C2×D4×C2311yescharacteristic
264C22×C4C2×D4×C2311yescharacteristic
265C22×C4C2×Q8×C2311yescharacteristic
266C22×C4#41×C2311yescharacteristic
267M16—42no—
268M16—41no—
269M16—41no—
270M16—42no—
271M16—42no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8C22⋊C4×C2311yescharacteristic
283C2×C8C22⋊C4×C2311yescharacteristic
284C2×C8C22⋊C4×C2311yescharacteristic
285C2×C8C22⋊C4×C2311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4C4⋊C4×C2311yescharacteristic
291C4⋊C4C4⋊C4×C2311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—82no—
298C42—21no—
299C23—22no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23C23×#11711yescharacteristic
304C23C23×#12111yescharacteristic
305C23C23×C22⋊Q811yescharacteristic
306Q8—41no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4C2×C22⋊C4×C2311yescharacteristic
329C2×C4C2×C4⋊C4×C2311yescharacteristic
330C2×C4C23×#12111yescharacteristic
331C2×C4C23×C22⋊Q811yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—41no—
336C8—41no—
337C8—82no—
338C8C23×#9912yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22C23×#38011yescharacteristic
352C22C23×#47112yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—162no—
360C4C2×C23×#9911yescharacteristic
361C4C23×#41011yescharacteristic
362C2—22no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2C23×#179911yescharacteristic
367C2C23×#181411yescharacteristic
368C2C23×#181511yescharacteristic
subgroupG/H
center, socleC22×C23#380
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1810C2semidirect
3. C23×#1807C2semidirect
5. C23×#1305C2semidirect
6. C2×C23×#131C4semidirect
15. C23×#424C4semidirect
16. C23×#408C4semidirect
19. C23×#405C4semidirect
114. #8626C23direct (N × H)
156. #1810C2×C23semidirect
157. #1807C2×C23semidirect
159. #1305C2×C23semidirect
183. C2×#131C4×C23semidirect
192. #424C4×C23semidirect
193. #408C4×C23semidirect
196. #405C4×C23semidirect
248. C23#8626direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29292
C23×#18102
C23×#18072
C23×#15132
C23×#13052
#862623