Finite Group Databasebeta

← C23×#9375 #19157001

Subgroups — C23×#9375 #19157001

Order 5888.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3238C211yesmaximal, characteristic
2C23×#2992C211yesmaximal, characteristic
3C23×#1988C211yesmaximal, characteristic, complemented
4C23×#1558C211yesmaximal, characteristic, complemented
5C23×#1509C211yesmaximal, characteristic
6C23×#1290C211yesmaximal, characteristic, complemented
7C23×#1285C211yesmaximal, characteristic, complemented
8C2×C23×C4⋊Q8C2211yescharacteristic
9C23×#529C2211yescharacteristic
10C23×#501C412yescomplemented
11C23×#489C412yescomplemented
12C23×#483—21no—
13C23×#480—21no—
14C23×#477—21no—
15C23×#476—21no—
16C23×#446—21no—
17C23×#435—21no—
18C23×#435—21no—
19C8×D4×C23—21no—
20C23×#433—21no—
21C23×#432C2211yescharacteristic
22C23×#432C2211yescharacteristic
23C23×#432C2211yescharacteristic
24C23×#424—21no—
25C2×C23×C4⋊C8C2211yescharacteristic
26C4×C4⋊C4×C23C2211yescharacteristic
27C23×#340—21no—
28C23×#340—21no—
29C23×#333—21no—
30C23×#333—21no—
31C23×#328—21no—
32C23×#327—21no—
33C22×Q8×C23—21no—
34C2×M16×C23—21no—
35C22×C8×C23—21no—
36C22×C8×C23D411yescharacteristic
37C22×C8×C23D411yescharacteristic
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—21no—
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8C2×C412yes—
42C23×#126—21no—
43C23×#125C2×C411yescharacteristic
44C23×#124C2×C411yescharacteristic
45C23×#123—21no—
46C23×C22⋊Q8D411yescharacteristic
47C23×#121D411yescharacteristic
48C4×Q8×C23D411yescharacteristic
49C4×D4×C23—21no—
50C4×D4×C23D411yescharacteristic
51C23×#117—21no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C42×C23—21no—
56C2×C42×C23C2311yescharacteristic
57C23×#107—22no—
58C23×#106—22no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8D411yescharacteristic
66C23×C4⋊C8D411yescharacteristic
67C23×#103—22no—
68C23×#102—22no—
69C23×C22⋊C8—21no—
70C23×C22⋊C8—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×#97—21no—
75C23×#97—21no—
76C23×#97—21no—
77C23×#97C2×C412yes—
78C4×C8×C23—21no—
79C23×#95—21no—
80C2×Q8×C23—22no—
81C2×Q8×C23—41no—
82C2×Q8×C23—81no—
83C2×Q8×C23—81no—
84C2×Q8×C23C22⋊C411yescharacteristic
85C2×D4×C23—21no—
86C2×D4×C23C22⋊C411yescharacteristic
87C22×C4×C23—21no—
88C22×C4×C23—41no—
89C22×C4×C23—41no—
90C22×C4×C23C2×D411yescharacteristic
91C22×C4×C23C2×D411yescharacteristic
92C22×C4×C23#4111yescharacteristic
93M16×C23—81no—
94C2×C8×C23—22no—
95C2×C8×C23—21no—
96C2×C8×C23—21no—
97C2×C8×C23—21no—
98C2×C8×C23—21no—
99C2×C8×C23—41no—
100C2×C8×C23—41no—
101C2×C8×C23—41no—
102C2×C8×C23—81no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—22no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—22no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23C22⋊C411yescharacteristic
114C4⋊C4×C23C22⋊C411yescharacteristic
115C22⋊C4×C23—21no—
116C22⋊C4×C23—21no—
117C22⋊C4×C23—21no—
118C22⋊C4×C23—21no—
119C22⋊C4×C23—21no—
120C42×C23—21no—
121C42×C23—81no—
122C42×C23C22×C411yescharacteristic
123C42×C23C2×D411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23#4111yescharacteristic
126#9375C2311yesmaximal, characteristic, direct product factor, Sylow
127C23×C23—21no—
128C23×C23#12111yescharacteristic
129Q8×C23—41no—
130Q8×C23—82no—
131Q8×C23—82no—
132D4×C23—41no—
133D4×C23—41no—
134D4×C23—41no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—81no—
148C2×C4×C23—81no—
149C2×C4×C23—81no—
150C2×C4×C23—81no—
151C2×C4×C23C2×C22⋊C411yescharacteristic
152C2×C4×C23C4×D411yescharacteristic
153C2×C4×C23C4×D411yescharacteristic
154C2×C4×C23#12011yescharacteristic
155C2×C4×C23#12111yescharacteristic
156C2×C4×C23#12311yescharacteristic
157C8×C23—42no—
158C8×C23—41no—
159C8×C23—41no—
160C8×C23—41no—
161C8×C23—41no—
162C8×C23—81no—
163#3238C2×C2311yescharacteristic
164#2992C2×C2311yescharacteristic
165#1988C2×C2311yescharacteristic, complemented
166#1558C2×C2311yescharacteristic, complemented
167#1509C2×C2311yescharacteristic
168#1290C2×C2311yescharacteristic, complemented
169#1285C2×C2311yescharacteristic, complemented
170C22×C23—21no—
171C22×C23—21no—
172C22×C23—41no—
173C22×C23—41no—
174C22×C23—41no—
175C22×C23#38611yescharacteristic
176C22×C23#47111yescharacteristic
177C22×C23#47211yescharacteristic
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—41no—
184C4×C23—41no—
185C4×C23—41no—
186C4×C23—82no—
187C4×C23—82no—
188C4×C23#45511yescharacteristic
189C4×C23#45611yescharacteristic
190C2×C4⋊Q8C22×C2311yescharacteristic
191#529C22×C2311yescharacteristic
192#501C4×C2312yescomplemented
193#489C4×C2312yescomplemented
194#483—21no—
195#480—21no—
196#477—21no—
197#476—21no—
198#446—21no—
199#435—21no—
200#435—21no—
201C8×D4—21no—
202#433—21no—
203#432C22×C2311yescharacteristic
204#432C22×C2311yescharacteristic
205#432C22×C2311yescharacteristic
206#424—21no—
207C2×C4⋊C8C22×C2311yescharacteristic
208C4×C4⋊C4C22×C2311yescharacteristic
209#340—21no—
210#340—21no—
211#333—21no—
212#333—21no—
213#328—21no—
214#327—21no—
215C2×C23—21no—
216C2×C23—21no—
217C2×C23—81no—
218C2×C23#188211yescharacteristic
219C2×C23#190711yescharacteristic
220C2×C23#190911yescharacteristic
221C22×Q8—21no—
222C2×M16—21no—
223C22×C8—21no—
224C22×C8D4×C2311yescharacteristic
225C22×C8D4×C2311yescharacteristic
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8C2×C4×C2312yes—
230#126—21no—
231#125C2×C4×C2311yescharacteristic
232#124C2×C4×C2311yescharacteristic
233#123—21no—
234C22⋊Q8D4×C2311yescharacteristic
235#121D4×C2311yescharacteristic
236C4×Q8D4×C2311yescharacteristic
237C4×D4—21no—
238C4×D4D4×C2311yescharacteristic
239#117—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C42—21no—
244C2×C42C23×C2311yescharacteristic
245#107—22no—
246#106—22no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8D4×C2311yescharacteristic
254C4⋊C8D4×C2311yescharacteristic
255#103—22no—
256#102—22no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262#97—21no—
263#97—21no—
264#97—21no—
265#97C2×C4×C2312yes—
266C4×C8—21no—
267#95—21no—
268C23#937511yescharacteristic, direct product factor, Sylow
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—81no—
272C2×Q8—81no—
273C2×Q8C22⋊C4×C2311yescharacteristic
274C2×D4—21no—
275C2×D4C22⋊C4×C2311yescharacteristic
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C2×D4×C2311yescharacteristic
280C22×C4C2×D4×C2311yescharacteristic
281C22×C4#41×C2311yescharacteristic
282M16—81no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—81no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4C22⋊C4×C2311yescharacteristic
303C4⋊C4C22⋊C4×C2311yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C42—21no—
310C42—81no—
311C42C22×C4×C2311yescharacteristic
312C42C2×D4×C2311yescharacteristic
313C42C2×D4×C2311yescharacteristic
314C42#41×C2311yescharacteristic
315C23—21no—
316C23C23×#12111yescharacteristic
317Q8—41no—
318Q8—82no—
319Q8—82no—
320D4—41no—
321D4—41no—
322D4—41no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4C2×C22⋊C4×C2311yescharacteristic
340C2×C4C4×D4×C2311yescharacteristic
341C2×C4C4×D4×C2311yescharacteristic
342C2×C4C23×#12011yescharacteristic
343C2×C4C23×#12111yescharacteristic
344C2×C4C23×#12311yescharacteristic
345C8—42no—
346C8—41no—
347C8—41no—
348C8—41no—
349C8—41no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22C23×#38611yescharacteristic
357C22C23×#47111yescharacteristic
358C22C23×#47211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—82no—
368C4—82no—
369C4C23×#45511yescharacteristic
370C4C23×#45611yescharacteristic
371C2—21no—
372C2—21no—
373C2—81no—
374C2C23×#188211yescharacteristic
375C2C23×#190711yescharacteristic
376C2C23×#190911yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#121D4
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
3. C23×#1988C2semidirect
4. C23×#1558C2semidirect
6. C23×#1290C2semidirect
7. C23×#1285C2semidirect
10. C23×#501C4semidirect
11. C23×#489C4semidirect
126. #9375C23direct (N × H)
165. #1988C2×C23semidirect
166. #1558C2×C23semidirect
168. #1290C2×C23semidirect
169. #1285C2×C23semidirect
192. #501C4×C23semidirect
193. #489C4×C23semidirect
268. C23#9375direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32382
C23×#29922
C23×#19882
C23×#15582
C23×#15092
C23×#12902
C23×#12852
#937523