Finite Group Databasebeta

← C23×#8628 #19156254

Subgroups — C23×#8628 #19156254

Order 5888.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2929C211yesmaximal, characteristic
2C23×#1810C211yesmaximal, characteristic, complemented
3C23×#1808C211yesmaximal, 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×#107—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×#107—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#8628C2311yesmaximal, characteristic, direct product factor, Sylow
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23#11711yescharacteristic
121C23×C23#12111yescharacteristic
122C23×C23C22⋊Q811yescharacteristic
123Q8×C23—41no—
124D4×C23—41no—
125D4×C23—42no—
126D4×C23—42no—
127D4×C23—41no—
128D4×C23—41no—
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—21no—
137C2×C4×C23—21no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—82no—
144C2×C4×C23—82no—
145C2×C4×C23—82no—
146C2×C4×C23C2×C22⋊C411yescharacteristic
147C2×C4×C23C2×C4⋊C411yescharacteristic
148C2×C4×C23#12111yescharacteristic
149C2×C4×C23C22⋊Q811yescharacteristic
150C8×C23—21no—
151C8×C23—21no—
152C8×C23—21no—
153C8×C23—41no—
154C8×C23—41no—
155C8×C23—82no—
156C8×C23#9912yes—
157#2929C2×C2311yescharacteristic
158#1810C2×C2311yescharacteristic, complemented
159#1808C2×C2311yescharacteristic, complemented
160#1513C2×C2312yes—
161#1305C2×C2312yescomplemented
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—21no—
166C22×C23—21no—
167C22×C23—21no—
168C22×C23—21no—
169C22×C23—21no—
170C22×C23—42no—
171C22×C23—42no—
172C22×C23—41no—
173C22×C23—41no—
174C22×C23—41no—
175C22×C23#38011yescharacteristic
176C22×C23#47211yescharacteristic
177C22×C23#47311yescharacteristic
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23—41no—
184C4×C23—162no—
185C4×C23C2×#9911yescharacteristic
186C4×C23#41011yescharacteristic
187C2×#131C4×C2311yescharacteristic, complemented
188C23×C8C22×C2311yescharacteristic
189#525C22×C2311yescharacteristic
190#435—21no—
191#435—21no—
192C8×D4—21no—
193C8×D4—21no—
194C2×#108—21no—
195C2×#107—21no—
196#424C4×C2311yescharacteristic, complemented
197#408C4×C2311yescharacteristic, complemented
198#407C22×C2312yes—
199#407C22×C2311yescharacteristic
200#405C4×C2311yescharacteristic, complemented
201#380C22×C2312yes—
202#341—21no—
203#340—21no—
204#328—22no—
205#327—22no—
206C2×C23—21no—
207C2×C23—21no—
208C2×C23—41no—
209C2×C23—41no—
210C2×C23—81no—
211C2×C23#179911yescharacteristic
212C2×C23#181411yescharacteristic
213C2×C23#181611yescharacteristic
214C2×#41C2×C4×C2311yescharacteristic
215C23×C4C23×C2311yescharacteristic
216#131—42no—
217C2×M16—21no—
218C2×M16—22no—
219C2×M16C2×C4×C2311yescharacteristic
220C2×M16C2×C4×C2311yescharacteristic
221C22×C8—21no—
222C22×C8—42no—
223C22×C8C2×C4×C2311yescharacteristic
224C22×C8C2×C4×C2311yescharacteristic
225C22×C8D4×C2311yescharacteristic
226C22×C8D4×C2311yescharacteristic
227C22×C8D4×C2311yescharacteristic
228C22×C8D4×C2311yescharacteristic
229#123—21no—
230C22⋊Q8D4×C2311yescharacteristic
231C22⋊Q8Q8×C2311yescharacteristic
232#121D4×C2311yescharacteristic
233#121Q8×C2311yescharacteristic
234#120—21no—
235C4×D4—21no—
236C4×D4—21no—
237#117C2×C4×C2311yescharacteristic
238C2×C4⋊C4—22no—
239C2×C22⋊C4—22no—
240#108—42no—
241#107—42no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250#97—21no—
251C4×C8—21no—
252#95—22no—
253C23#862811yescharacteristic, direct product factor, Sylow
254C24#41×C2311yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8C4⋊C4×C2311yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C4⋊C4×C2311yescharacteristic
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4C22×C4×C2311yescharacteristic
267C22×C4C2×D4×C2311yescharacteristic
268C22×C4C2×D4×C2311yescharacteristic
269C22×C4C2×D4×C2311yescharacteristic
270C22×C4C2×Q8×C2311yescharacteristic
271C22×C4#41×C2311yescharacteristic
272M16—42no—
273M16—41no—
274M16—41no—
275M16—42no—
276M16—42no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8C22⋊C4×C2311yescharacteristic
288C2×C8C22⋊C4×C2311yescharacteristic
289C2×C8C22⋊C4×C2311yescharacteristic
290C2×C8C22⋊C4×C2311yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4C4⋊C4×C2311yescharacteristic
296C4⋊C4C4⋊C4×C2311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—82no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23C23×#11711yescharacteristic
310C23C23×#12111yescharacteristic
311C23C23×C22⋊Q811yescharacteristic
312Q8—41no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C2×C22⋊C4×C2311yescharacteristic
336C2×C4C2×C4⋊C4×C2311yescharacteristic
337C2×C4C23×#12111yescharacteristic
338C2×C4C23×C22⋊Q811yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—41no—
343C8—41no—
344C8—82no—
345C8C23×#9912yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22C23×#38011yescharacteristic
360C22C23×#47211yescharacteristic
361C22C23×#47311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—162no—
369C4C2×C23×#9911yescharacteristic
370C4C23×#41011yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—81no—
376C2C23×#179911yescharacteristic
377C2C23×#181411yescharacteristic
378C2C23×#181611yescharacteristic
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×#1808C2semidirect
5. C23×#1305C2semidirect
6. C2×C23×#131C4semidirect
15. C23×#424C4semidirect
16. C23×#408C4semidirect
19. C23×#405C4semidirect
114. #8628C23direct (N × H)
158. #1810C2×C23semidirect
159. #1808C2×C23semidirect
161. #1305C2×C23semidirect
187. C2×#131C4×C23semidirect
196. #424C4×C23semidirect
197. #408C4×C23semidirect
200. #405C4×C23semidirect
253. C23#8628direct (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×#18082
C23×#15132
C23×#13052
#862823