Finite Group Databasebeta

← C23×#9214 #19156840

Subgroups — C23×#9214 #19156840

Order 5888.

428 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3107C211yesmaximal, characteristic
2C23×#2922C211yesmaximal, characteristic
3C23×#1526C211yesmaximal, characteristic, complemented
4C23×#1493C211yesmaximal, characteristic, complemented
5C23×#1489C211yesmaximal, characteristic
6C23×#1299C211yesmaximal, characteristic, complemented
7C23×#1278C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C4×#41×C23C2211yescharacteristic
10C23×#485—21no—
11C23×#475C411yescharacteristic
12C23×#474C411yescharacteristic
13C23×#462C411yescharacteristic
14C23×#461C411yescharacteristic
15C23×#452—21no—
16C4×Q16×C23—21no—
17C4×SD16×C23—21no—
18C23×#433—22no—
19C23×#431—22no—
20C23×#431C2211yescharacteristic
21C23×#424—21no—
22C23×#423C2211yescharacteristic
23C23×#408—21no—
24C4×M16×C23C2211yescharacteristic
25C4×M16×C23C2211yescharacteristic
26C2×C4×C8×C23C2211yescharacteristic
27C23×#330—22no—
28C23×Q8⋊C8—22no—
29C23×Q8⋊C8—21no—
30C23×D4⋊C8—21no—
31C2×#41×C23D411yescharacteristic
32C2×Q16×C23—21no—
33C2×SD16×C23—21no—
34C2×M16×C23—21no—
35C2×M16×C23—21no—
36C2×M16×C23—21no—
37C22×C8×C23—21no—
38C23×C4⋊Q8C2×C411yescharacteristic
39C23×C4⋊Q8C2×C411yescharacteristic
40C23×#125—21no—
41C23×C22⋊Q8—21no—
42C23×C22⋊Q8—21no—
43C4×Q8×C23—21no—
44C4×Q8×C23—21no—
45C4×Q8×C23C2×C411yescharacteristic
46C4×Q8×C23D411yescharacteristic
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23C2×C411yescharacteristic
50C4×D4×C23D411yescharacteristic
51C23×#117—21no—
52C23×#117—21no—
53C23×#117D411yescharacteristic
54C2×C42×C23—21no—
55C2×C42×C23C2311yescharacteristic
56C23×#107—21no—
57C23×#106—21no—
58C23×C4⋊C8—24no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8C2×C411yescharacteristic
62C23×C4⋊C8C2×C411yescharacteristic
63C23×#103—21no—
64C23×#103—21no—
65C23×#103—21no—
66C23×#102—21no—
67C23×C22⋊C8—24no—
68C23×C22⋊C8—21no—
69C23×#97—21no—
70C23×#97D412yes—
71C4×C8×C23—21no—
72C4×C8×C23—21no—
73C4×C8×C23—21no—
74C4×C8×C23—21no—
75C4×C8×C23D412yes—
76#41×C23—41no—
77#41×C23—41no—
78C2×Q8×C23—21no—
79C2×Q8×C23—21no—
80C2×Q8×C23C22⋊C411yescharacteristic
81C2×D4×C23—21no—
82C2×D4×C23C22⋊C411yescharacteristic
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23C2×D411yescharacteristic
87C22×C4×C23#4111yescharacteristic
88C22×C4×C23#4111yescharacteristic
89Q16×C23—42no—
90SD16×C23—42no—
91M16×C23—22no—
92M16×C23—41no—
93M16×C23—41no—
94M16×C23—41no—
95M16×C23—81no—
96C2×C8×C23—21no—
97C2×C8×C23—21no—
98C2×C8×C23—22no—
99C2×C8×C23—22no—
100C2×C8×C23—41no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23C22⋊C411yescharacteristic
113C4⋊C4×C23C22⋊C411yescharacteristic
114C22⋊C4×C23—21no—
115C22⋊C4×C23—21no—
116C22⋊C4×C23—21no—
117C22⋊C4×C23—41no—
118C42×C23—21no—
119C42×C23—21no—
120C42×C23—21no—
121C42×C23—41no—
122C42×C23C22×C411yescharacteristic
123C42×C23C2×D411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23C2×D411yescharacteristic
126#9214C2311yesmaximal, characteristic, direct product factor, Sylow
127C23×C23—21no—
128C23×C23#12311yescharacteristic
129Q8×C23—21no—
130Q8×C23—21no—
131Q8×C23—41no—
132Q8×C23—41no—
133Q8×C23—41no—
134Q8×C23—41no—
135D4×C23—21no—
136D4×C23—21no—
137D4×C23—42no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—42no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23C2×C22⋊C411yescharacteristic
155C2×C4×C23C4×D411yescharacteristic
156C2×C4×C23C4×D411yescharacteristic
157C2×C4×C23#12011yescharacteristic
158C2×C4×C23#12111yescharacteristic
159C2×C4×C23#12111yescharacteristic
160C8×C23—42no—
161C8×C23—42no—
162C8×C23—42no—
163C8×C23—41no—
164C8×C23—41no—
165C8×C23—81no—
166#3107C2×C2311yescharacteristic
167#2922C2×C2311yescharacteristic
168#1526C2×C2311yescharacteristic, complemented
169#1493C2×C2311yescharacteristic, complemented
170#1489C2×C2311yescharacteristic
171#1299C2×C2311yescharacteristic, complemented
172#1278C2×C2311yescharacteristic, complemented
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—41no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23#38611yescharacteristic
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—41no—
184C4×C23—41no—
185C4×C23—41no—
186C4×C23—81no—
187C4×C23—81no—
188C4×C23#42111yescharacteristic
189C4×C23#42111yescharacteristic
190C4×C23#44311yescharacteristic
191C4×C23#44411yescharacteristic
192C4×C23#47111yescharacteristic
193C4×C23#47311yescharacteristic
194#533C22×C2311yescharacteristic
195C4×#41C22×C2311yescharacteristic
196#485—21no—
197#475C4×C2311yescharacteristic
198#474C4×C2311yescharacteristic
199#462C4×C2311yescharacteristic
200#461C4×C2311yescharacteristic
201#452—21no—
202C4×Q16—21no—
203C4×SD16—21no—
204#433—22no—
205#431—22no—
206#431C22×C2311yescharacteristic
207#424—21no—
208#423C22×C2311yescharacteristic
209#408—21no—
210C4×M16C22×C2311yescharacteristic
211C4×M16C22×C2311yescharacteristic
212C2×C4×C8C22×C2311yescharacteristic
213#330—22no—
214Q8⋊C8—21no—
215Q8⋊C8—22no—
216D4⋊C8—21no—
217C2×C23—41no—
218C2×C23—81no—
219C2×C23#187611yescharacteristic
220C2×C23#189711yescharacteristic
221C2×C23#190811yescharacteristic
222C2×#41D4×C2311yescharacteristic
223C2×Q16—21no—
224C2×SD16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C22×C8—21no—
229C4⋊Q8C2×C4×C2311yescharacteristic
230C4⋊Q8C2×C4×C2311yescharacteristic
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8C2×C4×C2311yescharacteristic
237C4×Q8D4×C2311yescharacteristic
238C4×D4—21no—
239C4×D4—21no—
240C4×D4C2×C4×C2311yescharacteristic
241C4×D4D4×C2311yescharacteristic
242#117—21no—
243#117—21no—
244#117D4×C2311yescharacteristic
245C2×C42—21no—
246C2×C42C23×C2311yescharacteristic
247#107—21no—
248#106—21no—
249C4⋊C8—24no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8C2×C4×C2311yescharacteristic
253C4⋊C8C2×C4×C2311yescharacteristic
254#103—21no—
255#103—21no—
256#103—21no—
257#102—21no—
258C22⋊C8—24no—
259C22⋊C8—21no—
260#97—21no—
261#97D4×C2312yes—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8D4×C2312yes—
267C23#921411yescharacteristic, direct product factor, Sylow
268#41—41no—
269#41—41no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8C22⋊C4×C2311yescharacteristic
273C2×D4—21no—
274C2×D4C22⋊C4×C2311yescharacteristic
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4C2×D4×C2311yescharacteristic
279C22×C4#41×C2311yescharacteristic
280C22×C4#41×C2311yescharacteristic
281Q16—42no—
282SD16—42no—
283M16—22no—
284M16—41no—
285M16—41no—
286M16—41no—
287M16—81no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4C22⋊C4×C2311yescharacteristic
305C4⋊C4C22⋊C4×C2311yescharacteristic
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—41no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—41no—
314C42C22×C4×C2311yescharacteristic
315C42C2×D4×C2311yescharacteristic
316C42C2×D4×C2311yescharacteristic
317C42C2×D4×C2311yescharacteristic
318C23—21no—
319C23C23×#12311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—41no—
323Q8—41no—
324Q8—41no—
325Q8—41no—
326D4—21no—
327D4—21no—
328D4—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4C2×C22⋊C4×C2311yescharacteristic
346C2×C4C4×D4×C2311yescharacteristic
347C2×C4C4×D4×C2311yescharacteristic
348C2×C4C23×#12011yescharacteristic
349C2×C4C23×#12111yescharacteristic
350C2×C4C23×#12111yescharacteristic
351C8—42no—
352C8—42no—
353C8—42no—
354C8—41no—
355C8—41no—
356C8—81no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22C23×#38611yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—81no—
371C4—81no—
372C4C23×#42111yescharacteristic
373C4C23×#42111yescharacteristic
374C4C23×#44311yescharacteristic
375C4C23×#44411yescharacteristic
376C4C23×#47111yescharacteristic
377C4C23×#47311yescharacteristic
378C2—41no—
379C2—81no—
380C2C23×#187611yescharacteristic
381C2C23×#189711yescharacteristic
382C2C23×#190811yescharacteristic
subgroupG/H
centerC2×C4×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC2×#41×C23D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1526C2semidirect
4. C23×#1493C2semidirect
6. C23×#1299C2semidirect
7. C23×#1278C2semidirect
126. #9214C23direct (N × H)
168. #1526C2×C23semidirect
169. #1493C2×C23semidirect
171. #1299C2×C23semidirect
172. #1278C2×C23semidirect
267. C23#9214direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31072
C23×#29222
C23×#15262
C23×#14932
C23×#14892
C23×#12992
C23×#12782
#921423