Finite Group Databasebeta

← C23×#9766 #19157392

Subgroups — C23×#9766 #19157392

Order 5888.

396 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3167C211yesmaximal, characteristic
2C23×#1746C211yesmaximal, characteristic, complemented
3C23×#1568C211yesmaximal, characteristic, complemented
4C23×#1531C211yesmaximal, characteristic, complemented
5C23×#1530C211yesmaximal, characteristic
6C23×#1506C211yesmaximal, characteristic, complemented
7C23×#1505C211yesmaximal, characteristic
8C23×#533C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C23×#498C411yescharacteristic, complemented
11C23×#492C411yescharacteristic, complemented
12C23×#487C411yescharacteristic, complemented
13C23×#486C411yescharacteristic, complemented
14C23×#465—21no—
15C23×#465—21no—
16C4×SD16×C23—21no—
17C4×SD16×C23—21no—
18C23×#432C2211yescharacteristic
19C23×#432C2211yescharacteristic
20C2×C23×#97C2211yescharacteristic
21C2×C4×C8×C23C2211yescharacteristic
22C23×#376C2211yescharacteristic
23C23×#336—21no—
24C23×#336—21no—
25C23×#335—21no—
26C23×#334—21no—
27C23×#330—22no—
28C23×#329—22no—
29C23×#328—22no—
30C23×#327—22no—
31C2×#41×C23—21no—
32C2×SD16×C23—21no—
33C2×SD16×C23—21no—
34C22×C8×C23—21no—
35C22×C8×C23D411yescharacteristic
36C22×C8×C23D411yescharacteristic
37C23×C4⋊Q8C2×C411yescharacteristic
38C23×C4⋊Q8C2×C411yescharacteristic
39C23×C4⋊Q8C2×C411yescharacteristic
40C23×#127C2×C411yescharacteristic
41C23×#125—21no—
42C23×#124—21no—
43C23×C22⋊Q8—21no—
44C23×C22⋊Q8—21no—
45C23×#121—21no—
46C23×#121—21no—
47C4×Q8×C23—21no—
48C4×Q8×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C23×#117—21no—
52C2×C42×C23—21no—
53C2×C42×C23C2311yescharacteristic
54C23×#106—21no—
55C23×#106—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×#103—21no—
59C23×#103—21no—
60C23×#102—21no—
61C23×#102—21no—
62C23×C22⋊C8—21no—
63C23×C22⋊C8—21no—
64C23×#97—21no—
65C23×#97—21no—
66C23×#97—21no—
67C23×#97D412yes—
68C4×C8×C23C2×C411yescharacteristic
69C4×C8×C23C2×C411yescharacteristic
70C4×C8×C23D411yescharacteristic
71C4×C8×C23D411yescharacteristic
72C23×#95—21no—
73C23×#95D412yes—
74#41×C23—82no—
75C2×Q8×C23—21no—
76C2×Q8×C23—21no—
77C2×Q8×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—41no—
82C22×C4×C23—22no—
83C22×C4×C23—41no—
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23#4111yescharacteristic
87SD16×C23—42no—
88SD16×C23—42no—
89C2×C8×C23—22no—
90C2×C8×C23—41no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C2×C8×C23C22⋊C411yescharacteristic
97C2×C8×C23C22⋊C411yescharacteristic
98C2×C8×C23C22⋊C411yescharacteristic
99C2×C8×C23C22⋊C411yescharacteristic
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—41no—
107C4⋊C4×C23—41no—
108C22⋊C4×C23—41no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C42×C23—41no—
112C42×C23—41no—
113C42×C23—41no—
114C42×C23C22×C411yescharacteristic
115C42×C23C2×D411yescharacteristic
116C42×C23C2×D411yescharacteristic
117C42×C23#4111yescharacteristic
118#9766C2311yesmaximal, characteristic, direct product factor, Sylow
119C23×C23—41no—
120C23×C23#12111yescharacteristic
121Q8×C23—41no—
122Q8×C23—41no—
123Q8×C23—81no—
124Q8×C23—81no—
125D4×C23—41no—
126D4×C23—41no—
127D4×C23—81no—
128D4×C23—81no—
129D4×C23—82no—
130C2×C4×C23—22no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—42no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—42no—
140C2×C4×C23—82no—
141C2×C4×C23C2×C22⋊C411yescharacteristic
142C2×C4×C23C4×D411yescharacteristic
143C2×C4×C23C4×D411yescharacteristic
144C2×C4×C23#12111yescharacteristic
145C2×C4×C23#12411yescharacteristic
146C2×C4×C23#12711yescharacteristic
147C8×C23—21no—
148C8×C23—21no—
149C8×C23—21no—
150C8×C23—21no—
151C8×C23—42no—
152C8×C23—81no—
153C8×C23—81no—
154#3167C2×C2311yescharacteristic
155#1746C2×C2311yescharacteristic, complemented
156#1568C2×C2311yescharacteristic, complemented
157#1531C2×C2311yescharacteristic, complemented
158#1530C2×C2311yescharacteristic
159#1506C2×C2311yescharacteristic, complemented
160#1505C2×C2311yescharacteristic
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—81no—
164C22×C23—81no—
165C22×C23—81no—
166C22×C23#39011yescharacteristic
167C22×C23#47112yes—
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—82no—
174C4×C23—81no—
175C4×C23—81no—
176C4×C23—81no—
177C4×C23#42111yescharacteristic
178C4×C23#42111yescharacteristic
179#533C22×C2311yescharacteristic
180#532C22×C2311yescharacteristic
181#498C4×C2311yescharacteristic, complemented
182#492C4×C2311yescharacteristic, complemented
183#487C4×C2311yescharacteristic, complemented
184#486C4×C2311yescharacteristic, complemented
185#465—21no—
186#465—21no—
187C4×SD16—21no—
188C4×SD16—21no—
189#432C22×C2311yescharacteristic
190#432C22×C2311yescharacteristic
191C2×#97C22×C2311yescharacteristic
192C2×C4×C8C22×C2311yescharacteristic
193#376C22×C2311yescharacteristic
194#336—21no—
195#336—21no—
196#335—21no—
197#334—21no—
198#330—22no—
199#329—22no—
200#328—22no—
201#327—22no—
202C2×C23—22no—
203C2×C23—161no—
204C2×C23#196311yescharacteristic
205C2×C23#197611yescharacteristic
206C2×C23#197711yescharacteristic
207C2×#41—21no—
208C2×SD16—21no—
209C2×SD16—21no—
210C22×C8—21no—
211C22×C8D4×C2311yescharacteristic
212C22×C8D4×C2311yescharacteristic
213C4⋊Q8C2×C4×C2311yescharacteristic
214C4⋊Q8C2×C4×C2311yescharacteristic
215C4⋊Q8C2×C4×C2311yescharacteristic
216#127C2×C4×C2311yescharacteristic
217#125—21no—
218#124—21no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221#121—21no—
222#121—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×D4—21no—
226C4×D4—21no—
227#117—21no—
228C2×C42—21no—
229C2×C42C23×C2311yescharacteristic
230#106—21no—
231#106—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234#103—21no—
235#103—21no—
236#102—21no—
237#102—21no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240#97—21no—
241#97—21no—
242#97—21no—
243#97D4×C2312yes—
244C4×C8C2×C4×C2311yescharacteristic
245C4×C8C2×C4×C2311yescharacteristic
246C4×C8D4×C2311yescharacteristic
247C4×C8D4×C2311yescharacteristic
248#95—21no—
249#95D4×C2312yes—
250C23#976611yescharacteristic, direct product factor, Sylow
251#41—82no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4C2×D4×C2311yescharacteristic
262C22×C4C2×D4×C2311yescharacteristic
263C22×C4#41×C2311yescharacteristic
264SD16—42no—
265SD16—42no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8C22⋊C4×C2311yescharacteristic
274C2×C8C22⋊C4×C2311yescharacteristic
275C2×C8C22⋊C4×C2311yescharacteristic
276C2×C8C22⋊C4×C2311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42C22×C4×C2311yescharacteristic
292C42C2×D4×C2311yescharacteristic
293C42C2×D4×C2311yescharacteristic
294C42#41×C2311yescharacteristic
295C23—41no—
296C23C23×#12111yescharacteristic
297Q8—41no—
298Q8—41no—
299Q8—81no—
300Q8—81no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—82no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—82no—
317C2×C4C2×C22⋊C4×C2311yescharacteristic
318C2×C4C4×D4×C2311yescharacteristic
319C2×C4C4×D4×C2311yescharacteristic
320C2×C4C23×#12111yescharacteristic
321C2×C4C23×#12411yescharacteristic
322C2×C4C23×#12711yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—21no—
327C8—42no—
328C8—81no—
329C8—81no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C23×#39011yescharacteristic
336C22C23×#47112yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4C23×#42111yescharacteristic
347C4C23×#42111yescharacteristic
348C2—22no—
349C2—161no—
350C2C23×#196311yescharacteristic
351C2C23×#197611yescharacteristic
352C2C23×#197711yescharacteristic
subgroupG/H
center, socleC22×C23#390
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1746C2semidirect
3. C23×#1568C2semidirect
4. C23×#1531C2semidirect
6. C23×#1506C2semidirect
10. C23×#498C4semidirect
11. C23×#492C4semidirect
12. C23×#487C4semidirect
13. C23×#486C4semidirect
118. #9766C23direct (N × H)
155. #1746C2×C23semidirect
156. #1568C2×C23semidirect
157. #1531C2×C23semidirect
159. #1506C2×C23semidirect
181. #498C4×C23semidirect
182. #492C4×C23semidirect
183. #487C4×C23semidirect
184. #486C4×C23semidirect
250. C23#9766direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31672
C23×#17462
C23×#15682
C23×#15312
C23×#15302
C23×#15062
C23×#15052
#976623