Finite Group Databasebeta

← C23×#13381 #19161007

Subgroups — C23×#13381 #19161007

Order 5888.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#466C211yesmaximal, characteristic
2C2×C23×#367C211yesmaximal, characteristic, complemented
3C2×C23×#357C211yesmaximal, characteristic, complemented
4C2×C23×#357C211yesmaximal, characteristic, complemented
5C2×C23×C22⋊C16C211yesmaximal, characteristic, complemented
6C23×#1808C211yesmaximal, characteristic, complemented
7C23×#1380C211yesmaximal, characteristic, complemented
8C22×D8×C23C2211yescharacteristic
9C23×C8×C23C2211yescharacteristic
10C2×C23×#121—21no—
11C22×C16×C23C2211yescharacteristic
12C22×C16×C23C2211yescharacteristic
13C23×#466—22no—
14C23×#466—22no—
15C2×C23×#107C2211yescharacteristic
16C2×C23×#107C2211yescharacteristic
17C2×C23×#107C2211yescharacteristic
18C2×C23×#102—21no—
19C23×#380—21no—
20C23×#367—22no—
21C23×#357—22no—
22C23×#357—22no—
23C23×C22⋊C16—22no—
24C23×#340—21no—
25C22×D4×C23—21no—
26C22×D4×C23—41no—
27C23×C4×C23D411yescharacteristic
28C2×D8×C23—21no—
29C2×D8×C23—21no—
30C2×D8×C23—21no—
31C2×D8×C23—21no—
32C2×D8×C23—21no—
33C2×D8×C23D412yes—
34C22×C8×C23—21no—
35C22×C8×C23—21no—
36C22×C8×C23—21no—
37C22×C8×C23—21no—
38C22×C8×C23—21no—
39C22×C8×C23—21no—
40C22×C8×C23D411yescharacteristic
41C22×C8×C23C2311yescharacteristic
42C23×#121—24no—
43C23×#121—42no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23—21no—
47C2×C22⋊C4×C23—41no—
48C2×C22⋊C4×C23—41no—
49C2×C16×C23—21no—
50C2×C16×C23—21no—
51C2×C16×C23—21no—
52C2×C16×C23—21no—
53C2×C16×C23—21no—
54C2×C16×C23D412yescomplemented
55C23×#107—21no—
56C23×#107—21no—
57C23×#107—21no—
58C23×#107—21no—
59C23×#107—21no—
60C23×#107D412yes—
61C23×#102—24no—
62C23×#95—41no—
63C24×C23—41no—
64C24×C23D811yescharacteristic
65C2×D4×C23—22no—
66C2×D4×C23—44no—
67C2×D4×C23—41no—
68C2×D4×C23—41no—
69C2×D4×C23—41no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23—82no—
73C2×D4×C23—82no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—41no—
81C22×C4×C23—41no—
82C22×C4×C23—41no—
83C22×C4×C23D811yescharacteristic
84C22×C4×C23C2×D411yescharacteristic
85D8×C23—22no—
86D8×C23—22no—
87D8×C23—41no—
88D8×C23—41no—
89C2×C8×C23—21no—
90C2×C8×C23—21no—
91C2×C8×C23—21no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C2×C8×C23C2×D411yescharacteristic
97C2×C8×C23C2×D411yescharacteristic
98C2×C8×C23C2×D411yescharacteristic
99C2×C8×C23#4111yescharacteristic
100C2×C8×C23#4111yescharacteristic
101C2×C8×C23#4111yescharacteristic
102C4⋊C4×C23—22no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—41no—
105C4⋊C4×C23—41no—
106C4⋊C4×C23—41no—
107C4⋊C4×C23—41no—
108C22⋊C4×C23—44no—
109C22⋊C4×C23—82no—
110C16×C23—22no—
111C16×C23—41no—
112#13381C2311yesmaximal, characteristic, direct product factor, Sylow
113C23×C23—21no—
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—42no—
120C23×C23—81no—
121C23×C23—81no—
122C23×C23—81no—
123C23×C23—81no—
124C23×C23—81no—
125C23×C23—81no—
126C23×C23SD3212yes—
127C23×C23C2×D811yescharacteristic
128D4×C23—42no—
129D4×C23—42no—
130D4×C23—88no—
131D4×C23—81no—
132D4×C23—81no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—42no—
141C2×C4×C23—81no—
142C2×C4×C23—81no—
143C2×C4×C23—81no—
144C2×C4×C23—81no—
145C2×C4×C23—81no—
146C2×C4×C23—81no—
147C2×C4×C23—81no—
148C2×C4×C23—81no—
149C2×C4×C23#12011yescharacteristic
150C2×C4×C23#12111yescharacteristic
151C2×C4×C23#12311yescharacteristic
152C2×C4×C23#13511yescharacteristic
153C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C23#13711yescharacteristic
155C8×C23—41no—
156C8×C23#12111yescharacteristic
157C8×C23#12111yescharacteristic
158C8×C23#12311yescharacteristic
159C8×C23#12411yescharacteristic
160C2×#466C2×C2311yescharacteristic
161C2×#367C2×C2311yescharacteristic, complemented
162C2×#357C2×C2311yescharacteristic, complemented
163C2×#357C2×C2311yescharacteristic, complemented
164C2×C22⋊C16C2×C2311yescharacteristic, complemented
165#1808C2×C2311yescharacteristic, complemented
166#1380C2×C2311yescharacteristic, complemented
167C22×C23—22no—
168C22×C23—22no—
169C22×C23—22no—
170C22×C23—41no—
171C22×C23—41no—
172C22×C23—41no—
173C22×C23—41no—
174C22×C23—82no—
175C22×C23—82no—
176C22×C23—82no—
177C22×C23—161no—
178C22×C23—161no—
179C22×C23—161no—
180C22×C23—161no—
181C22×C23#44711yescharacteristic
182C22×C23#46611yescharacteristic
183C22×C23#48011yescharacteristic
184C22×C23C2×SD3211yescharacteristic
185C22×C23#50811yescharacteristic
186C22×C23#50911yescharacteristic
187C22×C23#50911yescharacteristic
188C4×C23—41no—
189C4×C23—82no—
190C4×C23—161no—
191C4×C23—161no—
192C4×C23#39411yescharacteristic
193C4×C23#45211yescharacteristic
194C4×C23#46811yescharacteristic
195C4×C23#48211yescharacteristic
196C22×D8C22×C2311yescharacteristic
197C23×C8C22×C2311yescharacteristic
198C2×#121—21no—
199C22×C16C22×C2311yescharacteristic
200C22×C16C22×C2311yescharacteristic
201#466—22no—
202#466—22no—
203C2×#107C22×C2311yescharacteristic
204C2×#107C22×C2311yescharacteristic
205C2×#107C22×C2311yescharacteristic
206C2×#102—21no—
207#380—21no—
208#367—22no—
209#357—22no—
210#357—22no—
211C22⋊C16—22no—
212#340—21no—
213C2×C23—42no—
214C2×C23—162no—
215C2×C23#203311yescharacteristic
216C2×C23#218611yescharacteristic
217C2×C23#218711yescharacteristic
218C2×C23#221711yescharacteristic
219C2×C23#221811yescharacteristic
220C2×C23#223311yescharacteristic
221C2×C23#223411yescharacteristic
222C22×D4—21no—
223C22×D4—41no—
224C23×C4D4×C2311yescharacteristic
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8D4×C2312yes—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8D4×C2311yescharacteristic
238C22×C8C23×C2311yescharacteristic
239#121—24no—
240#121—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16D4×C2312yescomplemented
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107D4×C2312yes—
258#102—24no—
259#95—41no—
260C23#1338111yescharacteristic, direct product factor, Sylow
261C24—41no—
262C24D8×C2311yescharacteristic
263C2×D4—22no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—82no—
271C2×D4—82no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4D8×C2311yescharacteristic
282C22×C4C2×D4×C2311yescharacteristic
283D8—22no—
284D8—22no—
285D8—41no—
286D8—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8C2×D4×C2311yescharacteristic
295C2×C8C2×D4×C2311yescharacteristic
296C2×C8C2×D4×C2311yescharacteristic
297C2×C8#41×C2311yescharacteristic
298C2×C8#41×C2311yescharacteristic
299C2×C8#41×C2311yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C22⋊C4—44no—
307C22⋊C4—82no—
308C16—22no—
309C16—41no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—42no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23C23×SD3212yes—
324C23C2×D8×C2311yescharacteristic
325D4—42no—
326D4—42no—
327D4—88no—
328D4—81no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4C23×#12011yescharacteristic
347C2×C4C23×#12111yescharacteristic
348C2×C4C23×#12311yescharacteristic
349C2×C4C23×#13511yescharacteristic
350C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
351C2×C4C23×#13711yescharacteristic
352C8—41no—
353C8C23×#12111yescharacteristic
354C8C23×#12111yescharacteristic
355C8C23×#12311yescharacteristic
356C8C23×#12411yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—161no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22C23×#44711yescharacteristic
372C22C23×#46611yescharacteristic
373C22C23×#48011yescharacteristic
374C22C2×C23×SD3211yescharacteristic
375C22C23×#50811yescharacteristic
376C22C23×#50911yescharacteristic
377C22C23×#50911yescharacteristic
378C4—41no—
379C4—82no—
380C4—161no—
381C4—161no—
382C4C23×#39411yescharacteristic
383C4C23×#45211yescharacteristic
384C4C23×#46811yescharacteristic
385C4C23×#48211yescharacteristic
386C2—42no—
387C2—162no—
388C2C23×#203311yescharacteristic
389C2C23×#218611yescharacteristic
390C2C23×#218711yescharacteristic
391C2C23×#221711yescharacteristic
392C2C23×#221811yescharacteristic
393C2C23×#223311yescharacteristic
394C2C23×#223411yescharacteristic
subgroupG/H
center, socleC23×C23C2×D8
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC2×C23×#466C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ C23×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#367C2semidirect
3. C2×C23×#357C2semidirect
4. C2×C23×#357C2semidirect
5. C2×C23×C22⋊C16C2semidirect
6. C23×#1808C2semidirect
7. C23×#1380C2semidirect
54. C2×C16×C23D4semidirect
112. #13381C23direct (N × H)
161. C2×#367C2×C23semidirect
162. C2×#357C2×C23semidirect
163. C2×#357C2×C23semidirect
164. C2×C22⋊C16C2×C23semidirect
165. #1808C2×C23semidirect
166. #1380C2×C23semidirect
251. C2×C16D4×C23semidirect
260. C23#13381direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4662
C2×C23×#3672
C2×C23×#3572
C2×C23×C22⋊C162
C23×#18082
C23×#13802
#1338123