Finite Group Databasebeta

← C23×#9330 #19156956

Subgroups — C23×#9330 #19156956

Order 5888.

482 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3112C211yesmaximal, characteristic
2C23×#2320C211yesmaximal, characteristic, complemented
3C23×#1980C211yesmaximal, characteristic
4C23×#1505C211yesmaximal, characteristic
5C23×#1491C211yesmaximal, characteristic, complemented
6C23×#1300C211yesmaximal, characteristic, complemented
7C23×#1285C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C4×Q8×C23C2211yescharacteristic
10C23×#480—21no—
11C23×#477C412yescomplemented
12C23×#460C412yescomplemented
13C23×#448—21no—
14C23×#440—22no—
15C23×#432C2211yescharacteristic
16C23×#432C2211yescharacteristic
17C2×C23×#104—22no—
18C4×M16×C23C2211yescharacteristic
19C23×#391—21no—
20C23×#389—21no—
21C23×#387—22no—
22C23×#383—21no—
23C23×#382—21no—
24C4×C4⋊C4×C23C2211yescharacteristic
25C23×#376C2211yescharacteristic
26C23×#327—21no—
27C23×#327—21no—
28C23×Q8⋊C8—22no—
29C2×#41×C23—21no—
30C22×Q8×C23D411yescharacteristic
31C2×SD16×C23—22no—
32C2×M16×C23—21no—
33C23×C4⋊Q8C2×C411yescharacteristic
34C23×#127C2×C411yescharacteristic
35C23×#124—21no—
36C23×#121—21no—
37C23×#121—21no—
38C4×Q8×C23—21no—
39C4×Q8×C23—21no—
40C4×Q8×C23C2×C412yes—
41C4×Q8×C23D412yes—
42C4×D4×C23—21no—
43C4×D4×C23—21no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23—22no—
47C2×C4⋊C4×C23D411yescharacteristic
48C2×C42×C23—21no—
49C2×C42×C23—21no—
50C2×C42×C23D412yes—
51C2×C42×C23C2311yescharacteristic
52C23×#107—22no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8C2×C412yes—
55C23×#104—44no—
56C23×#103—22no—
57C23×#102—22no—
58C23×C22⋊C8—21no—
59C23×C22⋊C8—21no—
60C23×#97—21no—
61C23×#97—21no—
62C23×#97—21no—
63C4×C8×C23—21no—
64C23×#95—22no—
65C23×#95—22no—
66C23×#95—22no—
67C23×#95—21no—
68C23×#95—21no—
69C23×#95D411yescharacteristic
70C23×#95D411yescharacteristic
71#41×C23—42no—
72#41×C23—42no—
73C2×Q8×C23—21no—
74C2×Q8×C23—21no—
75C2×Q8×C23—42no—
76C2×Q8×C23C22⋊C412yes—
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—41no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—22no—
86C22×C4×C23—41no—
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90SD16×C23—44no—
91M16×C23—22no—
92M16×C23—41no—
93C2×C8×C23—22no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C4⋊C4×C23—22no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—42no—
102C4⋊C4×C23—82no—
103C4⋊C4×C23C22⋊C412yes—
104C22⋊C4×C23—41no—
105C22⋊C4×C23—41no—
106C42×C23—22no—
107C42×C23—22no—
108C42×C23—42no—
109C42×C23—42no—
110C42×C23C22×C411yescharacteristic
111C42×C23C2×D411yescharacteristic
112C42×C23#4111yescharacteristic
113C42×C23#4111yescharacteristic
114#9330C2311yesmaximal, characteristic, direct product factor, Sylow
115C23×C23—41no—
116C23×C23#12011yescharacteristic
117Q8×C23—22no—
118Q8×C23—22no—
119Q8×C23—42no—
120Q8×C23—42no—
121D4×C23—42no—
122D4×C23—42no—
123D4×C23—81no—
124D4×C23—81no—
125D4×C23—81no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—22no—
129C2×C4×C23—22no—
130C2×C4×C23—22no—
131C2×C4×C23—41no—
132C2×C4×C23—42no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—42no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—42no—
141C2×C4×C23—41no—
142C2×C4×C23—42no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—82no—
146C2×C4×C23C2×C22⋊C411yescharacteristic
147C2×C4×C23C4×D411yescharacteristic
148C2×C4×C23C4×D411yescharacteristic
149C2×C4×C23#12111yescharacteristic
150C2×C4×C23#12111yescharacteristic
151C2×C4×C23#12311yescharacteristic
152C8×C23—42no—
153C8×C23—42no—
154C8×C23—81no—
155#3112C2×C2311yescharacteristic
156#2320C2×C2311yescharacteristic, complemented
157#1980C2×C2311yescharacteristic
158#1505C2×C2311yescharacteristic
159#1491C2×C2311yescharacteristic, complemented
160#1300C2×C2311yescharacteristic, complemented
161#1285C2×C2311yescharacteristic, complemented
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—81no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23#38611yescharacteristic
168C22×C23#45311yescharacteristic
169C22×C23#45411yescharacteristic
170C4×C23—21no—
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—42no—
176C4×C23—42no—
177C4×C23—81no—
178C4×C23—82no—
179C4×C23—81no—
180C4×C23—81no—
181C4×C23#42112yes—
182#532C22×C2311yescharacteristic
183C2×C4×Q8C22×C2311yescharacteristic
184#480—21no—
185#477C4×C2312yescomplemented
186#460C4×C2312yescomplemented
187#448—21no—
188#440—22no—
189#432C22×C2311yescharacteristic
190#432C22×C2311yescharacteristic
191C2×#104—22no—
192C4×M16C22×C2311yescharacteristic
193#391—21no—
194#389—21no—
195#387—22no—
196#383—21no—
197#382—21no—
198C4×C4⋊C4C22×C2311yescharacteristic
199#376C22×C2311yescharacteristic
200#327—21no—
201#327—21no—
202Q8⋊C8—22no—
203C2×C23—21no—
204C2×C23—21no—
205C2×C23—161no—
206C2×C23#188311yescharacteristic
207C2×C23#189711yescharacteristic
208C2×C23#190511yescharacteristic
209C2×#41—21no—
210C22×Q8D4×C2311yescharacteristic
211C2×SD16—22no—
212C2×M16—21no—
213C4⋊Q8C2×C4×C2311yescharacteristic
214#127C2×C4×C2311yescharacteristic
215#124—21no—
216#121—21no—
217#121—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8C2×C4×C2312yes—
221C4×Q8D4×C2312yes—
222C4×D4—21no—
223C4×D4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4D4×C2311yescharacteristic
228C2×C42—21no—
229C2×C42—21no—
230C2×C42D4×C2312yes—
231C2×C42C23×C2311yescharacteristic
232#107—22no—
233C4⋊C8—21no—
234C4⋊C8C2×C4×C2312yes—
235#104—44no—
236#103—22no—
237#102—22no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240#97—21no—
241#97—21no—
242#97—21no—
243C4×C8—21no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95D4×C2311yescharacteristic
250#95D4×C2311yescharacteristic
251C23#933011yescharacteristic, direct product factor, Sylow
252#41—42no—
253#41—42no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—42no—
257C2×Q8C22⋊C4×C2312yes—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4C2×D4×C2311yescharacteristic
269C22×C4C2×D4×C2311yescharacteristic
270C22×C4C2×D4×C2311yescharacteristic
271SD16—44no—
272M16—22no—
273M16—41no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—42no—
283C4⋊C4—82no—
284C4⋊C4C22⋊C4×C2312yes—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—22no—
288C42—22no—
289C42—42no—
290C42—42no—
291C42C22×C4×C2311yescharacteristic
292C42C2×D4×C2311yescharacteristic
293C42#41×C2311yescharacteristic
294C42#41×C2311yescharacteristic
295C23—41no—
296C23C23×#12011yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—42no—
300Q8—42no—
301D4—42no—
302D4—42no—
303D4—81no—
304D4—81no—
305D4—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4C2×C22⋊C4×C2311yescharacteristic
327C2×C4C4×D4×C2311yescharacteristic
328C2×C4C4×D4×C2311yescharacteristic
329C2×C4C23×#12111yescharacteristic
330C2×C4C23×#12111yescharacteristic
331C2×C4C23×#12311yescharacteristic
332C8—42no—
333C8—42no—
334C8—81no—
335C22—21no—
336C22—21no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C23×#38611yescharacteristic
341C22C23×#45311yescharacteristic
342C22C23×#45411yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—81no—
351C4—82no—
352C4—81no—
353C4—81no—
354C4C23×#42112yes—
355C2—21no—
356C2—21no—
357C2—161no—
358C2C23×#188311yescharacteristic
359C2C23×#189711yescharacteristic
360C2C23×#190511yescharacteristic
subgroupG/H
center, socleC22×C23#386
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×#2320C2semidirect
5. C23×#1491C2semidirect
6. C23×#1300C2semidirect
7. C23×#1285C2semidirect
11. C23×#477C4semidirect
12. C23×#460C4semidirect
114. #9330C23direct (N × H)
156. #2320C2×C23semidirect
159. #1491C2×C23semidirect
160. #1300C2×C23semidirect
161. #1285C2×C23semidirect
185. #477C4×C23semidirect
186. #460C4×C23semidirect
251. C23#9330direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31122
C23×#23202
C23×#19802
C23×#15052
C23×#14912
C23×#13002
C23×#12852
#933023