Finite Group Databasebeta

← C23×#10547 #19158173

Subgroups — C23×#10547 #19158173

Order 5888.

386 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×#3135C211yesmaximal, characteristic
2C23×#1940C211yesmaximal, characteristic, complemented
3C23×#1542C211yesmaximal, characteristic, complemented
4C23×#1540C211yesmaximal, characteristic
5C23×#1503C211yesmaximal, characteristic, complemented
6C23×#1502C211yesmaximal, characteristic
7C23×#1275C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C23×#491—21no—
11C23×#490—21no—
12C23×#489—21no—
13C23×#489—21no—
14C23×#431C2211yescharacteristic
15C23×#431C2211yescharacteristic
16C2×C23×C4⋊C8—21no—
17C2×C23×C4⋊C8C2211yescharacteristic
18C23×#416—21no—
19C23×#416—21no—
20C2×C23×#97C2211yescharacteristic
21C4×C4⋊C4×C23C2211yescharacteristic
22C23×#336—21no—
23C23×#333—21no—
24C23×#332—21no—
25C23×#332—21no—
26C23×#331—21no—
27C23×#328—21no—
28C23×#328—21no—
29C23×#327—21no—
30C23×#327—21no—
31C2×#41×C23—21no—
32C22×C8×C23—21no—
33C22×C8×C23D411yescharacteristic
34C22×C8×C23D411yescharacteristic
35C23×C4⋊Q8D411yescharacteristic
36C23×C4⋊Q8D411yescharacteristic
37C23×C4⋊Q8D411yescharacteristic
38C23×#127D411yescharacteristic
39C23×#125—21no—
40C23×#124—21no—
41C23×C22⋊Q8—21no—
42C23×C22⋊Q8—21no—
43C23×#121—21no—
44C23×#121—21no—
45C4×Q8×C23—21no—
46C4×Q8×C23—21no—
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C23×#117—21no—
50C2×C4⋊C4×C23—21no—
51C2×C42×C23—21no—
52C2×C42×C23C2311yescharacteristic
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—42no—
58C23×#103—21no—
59C23×#103—21no—
60C23×#103—21no—
61C23×#103—21no—
62C23×#102—21no—
63C23×#102—21no—
64C23×#102—21no—
65C23×#102—21no—
66C23×C22⋊C8—21no—
67C23×C22⋊C8—21no—
68C23×#97—21no—
69C23×#97—21no—
70C4×C8×C23—21no—
71C4×C8×C23—21no—
72C23×#95D411yescharacteristic
73C23×#95D411yescharacteristic
74#41×C23—41no—
75#41×C23—41no—
76#41×C23—41no—
77#41×C23—41no—
78C2×Q8×C23—21no—
79C2×Q8×C23—21no—
80C2×Q8×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—41no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23#4111yescharacteristic
91C2×C8×C23—21no—
92C2×C8×C23—21no—
93C2×C8×C23—21no—
94C2×C8×C23—21no—
95C2×C8×C23—21no—
96C2×C8×C23—21no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C2×C8×C23—41no—
100C2×C8×C23—41no—
101C2×C8×C23—41no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—42no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C42×C23—41no—
115C42×C23—41no—
116C42×C23—41no—
117C42×C23C2×D411yescharacteristic
118C42×C23C2×D411yescharacteristic
119C42×C23#4111yescharacteristic
120C42×C23#4111yescharacteristic
121#10547C2311yesmaximal, characteristic, direct product factor, Sylow
122C23×C23—41no—
123C23×C23#12111yescharacteristic
124Q8×C23—41no—
125Q8×C23—41no—
126Q8×C23—81no—
127Q8×C23—81no—
128D4×C23—41no—
129D4×C23—41no—
130D4×C23—81no—
131D4×C23—81no—
132D4×C23—81no—
133D4×C23—81no—
134C2×C4×C23—22no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—81no—
146C2×C4×C23—81no—
147C2×C4×C23#12011yescharacteristic
148C2×C4×C23#12111yescharacteristic
149C2×C4×C23#12111yescharacteristic
150C2×C4×C23#12311yescharacteristic
151C2×C4×C23#12311yescharacteristic
152C2×C4×C23#12411yescharacteristic
153C2×C4×C23#13511yescharacteristic
154C2×C4×C23#13511yescharacteristic
155C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C23#13711yescharacteristic
157C8×C23—41no—
158C8×C23—41no—
159C8×C23—81no—
160C8×C23—81no—
161C8×C23—81no—
162#3135C2×C2311yescharacteristic
163#1940C2×C2311yescharacteristic, complemented
164#1542C2×C2311yescharacteristic, complemented
165#1540C2×C2311yescharacteristic
166#1503C2×C2311yescharacteristic, complemented
167#1502C2×C2311yescharacteristic
168#1275C2×C2311yescharacteristic, complemented
169C22×C23—21no—
170C22×C23—21no—
171C22×C23—81no—
172C22×C23—81no—
173C22×C23—81no—
174C22×C23#39411yescharacteristic
175C22×C23#46311yescharacteristic
176C22×C23#46411yescharacteristic
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—42no—
182C4×C23—81no—
183C4×C23—81no—
184C4×C23—81no—
185C4×C23—81no—
186C4×C23#44911yescharacteristic
187C4×C23#45211yescharacteristic
188C4×C23#48811yescharacteristic
189C4×C23#48911yescharacteristic
190#533C22×C2311yescharacteristic
191#532C22×C2311yescharacteristic
192#491—21no—
193#490—21no—
194#489—21no—
195#489—21no—
196#431C22×C2311yescharacteristic
197#431C22×C2311yescharacteristic
198C2×C4⋊C8—21no—
199C2×C4⋊C8C22×C2311yescharacteristic
200#416—21no—
201#416—21no—
202C2×#97C22×C2311yescharacteristic
203C4×C4⋊C4C22×C2311yescharacteristic
204#336—21no—
205#333—21no—
206#332—21no—
207#332—21no—
208#331—21no—
209#328—21no—
210#328—21no—
211#327—21no—
212#327—21no—
213C2×C23—21no—
214C2×C23—21no—
215C2×C23—161no—
216C2×C23#204211yescharacteristic
217C2×C23#205011yescharacteristic
218C2×C23#205311yescharacteristic
219C2×#41—21no—
220C22×C8—21no—
221C22×C8D4×C2311yescharacteristic
222C22×C8D4×C2311yescharacteristic
223C4⋊Q8D4×C2311yescharacteristic
224C4⋊Q8D4×C2311yescharacteristic
225C4⋊Q8D4×C2311yescharacteristic
226#127D4×C2311yescharacteristic
227#125—21no—
228#124—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—21no—
232#121—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—21no—
236C4×D4—21no—
237#117—21no—
238C2×C4⋊C4—21no—
239C2×C42—21no—
240C2×C42C23×C2311yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246#103—21no—
247#103—21no—
248#103—21no—
249#103—21no—
250#102—21no—
251#102—21no—
252#102—21no—
253#102—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256#97—21no—
257#97—21no—
258C4×C8—21no—
259C4×C8—21no—
260#95D4×C2311yescharacteristic
261#95D4×C2311yescharacteristic
262C23#1054711yescharacteristic, direct product factor, Sylow
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—41no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4C2×D4×C2311yescharacteristic
278C22×C4C2×D4×C2311yescharacteristic
279C22×C4#41×C2311yescharacteristic
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42C2×D4×C2311yescharacteristic
307C42C2×D4×C2311yescharacteristic
308C42#41×C2311yescharacteristic
309C42#41×C2311yescharacteristic
310C23—41no—
311C23C23×#12111yescharacteristic
312Q8—41no—
313Q8—41no—
314Q8—81no—
315Q8—81no—
316D4—41no—
317D4—41no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4C23×#12011yescharacteristic
336C2×C4C23×#12111yescharacteristic
337C2×C4C23×#12111yescharacteristic
338C2×C4C23×#12311yescharacteristic
339C2×C4C23×#12311yescharacteristic
340C2×C4C23×#12411yescharacteristic
341C2×C4C23×#13511yescharacteristic
342C2×C4C23×#13511yescharacteristic
343C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
344C2×C4C23×#13711yescharacteristic
345C8—41no—
346C8—41no—
347C8—81no—
348C8—81no—
349C8—81no—
350C22—21no—
351C22—21no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22C23×#39411yescharacteristic
356C22C23×#46311yescharacteristic
357C22C23×#46411yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—42no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4C23×#44911yescharacteristic
368C4C23×#45211yescharacteristic
369C4C23×#48811yescharacteristic
370C4C23×#48911yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2C23×#204211yescharacteristic
375C2C23×#205011yescharacteristic
376C2C23×#205311yescharacteristic
subgroupG/H
centerC2×C4×C23#124
commutator, Frattini, cosocleC2×C42C23×C23
socleC22×C23#394
prime radicalC23×#532C22
LCSG ▷ C2×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
2. C23×#1940C2semidirect
3. C23×#1542C2semidirect
5. C23×#1503C2semidirect
7. C23×#1275C2semidirect
121. #10547C23direct (N × H)
163. #1940C2×C23semidirect
164. #1542C2×C23semidirect
166. #1503C2×C23semidirect
168. #1275C2×C23semidirect
262. C23#10547direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31352
C23×#19402
C23×#15422
C23×#15402
C23×#15032
C23×#15022
C23×#12752
#1054723