Finite Group Databasebeta

← C23×#33955 #19181581

Subgroups — C23×#33955 #19181581

Order 5888.

458 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3383C211yesmaximal, characteristic, complemented
2C23×#3380C211yesmaximal, characteristic
3C23×#2255C211yesmaximal, characteristic, complemented
4C23×#2255C211yesmaximal, characteristic, complemented
5C23×#2253C211yesmaximal, characteristic, complemented
6C23×#2229C211yesmaximal, characteristic, complemented
7C23×#2229C211yesmaximal, characteristic, complemented
8C23×#2228C211yesmaximal, characteristic
9C23×#2228C211yesmaximal, characteristic
10C23×#2226C211yesmaximal, characteristic
11C23×#2225C211yesmaximal, characteristic, complemented
12C23×#2183C211yesmaximal, characteristic, complemented
13C23×#2175C211yesmaximal, characteristic
14C23×#2175C211yesmaximal, characteristic
15C4×C23×SD32C211yesmaximal, characteristic
16C23×#557—21no—
17C23×#554—21no—
18C2×C23×SD32C2211yescharacteristic
19C23×#500C2211yescharacteristic
20C23×#500C2211yescharacteristic
21C23×#499C2211yescharacteristic
22C23×#499C2211yescharacteristic
23C23×#499C2211yescharacteristic
24C23×#499C2211yescharacteristic
25C23×#479—21no—
26C23×#479—21no—
27C23×#478—21no—
28C23×#478—21no—
29C23×#476—21no—
30C23×#475—21no—
31C8×Q8×C23C2211yescharacteristic
32C4×Q16×C23C2211yescharacteristic
33C4×Q16×C23C2211yescharacteristic
34C4×Q16×C23C2211yescharacteristic
35C4×D8×C23C2211yescharacteristic
36C4×D8×C23C2211yescharacteristic
37C4×D8×C23C2211yescharacteristic
38C23×#367C2211yescharacteristic
39C23×#367C2211yescharacteristic
40C23×#367C2211yescharacteristic
41C23×#367C2211yescharacteristic
42C23×#367C2211yescharacteristic
43C23×#366C2211yescharacteristic
44C23×#366C2211yescharacteristic
45C23×#366C2211yescharacteristic
46C23×#366C2211yescharacteristic
47C23×C4⋊C16C2211yescharacteristic
48C23×C4⋊C16C2211yescharacteristic
49C23×C4⋊C16C2211yescharacteristic
50C23×#358C2211yescharacteristic
51C23×#358C2211yescharacteristic
52C23×#358C2211yescharacteristic
53C23×#357C2211yescharacteristic
54C23×#357C2211yescharacteristic
55C23×#357C2211yescharacteristic
56C23×#346C2211yescharacteristic
57C23×#346C2211yescharacteristic
58C4×C16×C23C2211yescharacteristic
59C2×Q16×C23C2311yescharacteristic
60C2×D8×C23C2311yescharacteristic
61C23×C4⋊Q8—21no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8—41no—
64C23×#125—21no—
65C23×#125—21no—
66C23×#125—21no—
67C23×#125—21no—
68C23×#125—41no—
69C23×#125—41no—
70C23×C22⋊Q8—41no—
71C23×C22⋊Q8—41no—
72C23×C22⋊Q8—41no—
73C4×Q8×C23—21no—
74C4×Q8×C23—21no—
75C4×Q8×C23—21no—
76C4×Q8×C23—41no—
77C4×Q8×C23D411yescharacteristic
78C4×D4×C23—21no—
79C4×D4×C23—21no—
80C4×D4×C23—21no—
81C2×C4⋊C4×C23—41no—
82C23×SD32Q814yescomplemented
83C2×C16×C23C2311yescharacteristic
84C2×C16×C23C2311yescharacteristic
85C2×C16×C23C2311yescharacteristic
86C2×C16×C23C2311yescharacteristic
87C23×#107—21no—
88C23×#107—21no—
89C23×#107C2311yescharacteristic
90C23×#107C2311yescharacteristic
91C23×#107C2311yescharacteristic
92C23×#107C2311yescharacteristic
93C23×#107C2311yescharacteristic
94C23×#107C2311yescharacteristic
95C23×#106—21no—
96C23×#106—21no—
97C23×#106—21no—
98C23×#106—21no—
99C23×C4⋊C8D411yescharacteristic
100C23×C4⋊C8D411yescharacteristic
101C23×C4⋊C8D411yescharacteristic
102C23×#103—21no—
103C23×#103—21no—
104C23×#103—21no—
105C23×#102—21no—
106C23×#102—21no—
107C23×#102—21no—
108C4×C8×C23C2311yescharacteristic
109C4×C8×C23C2311yescharacteristic
110C4×C8×C23C2311yescharacteristic
111C2×Q8×C23—21no—
112C2×Q8×C23—41no—
113C2×Q8×C23—41no—
114C2×Q8×C23D811yescharacteristic
115C2×D4×C23—21no—
116C22×C4×C23—41no—
117C22×C4×C23—41no—
118C22×C4×C23—41no—
119Q16×C23—21no—
120Q16×C23C2×Q812yes—
121D8×C23—21no—
122D8×C23C2×Q812yes—
123C2×C8×C23C2×D411yescharacteristic
124C2×C8×C23C2×D411yescharacteristic
125C2×C8×C23C2×D411yescharacteristic
126C2×C8×C23C2411yescharacteristic
127C4⋊C4×C23—21no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—21no—
130C4⋊C4×C23—21no—
131C4⋊C4×C23—21no—
132C4⋊C4×C23—21no—
133C4⋊C4×C23—41no—
134C4⋊C4×C23—41no—
135C4⋊C4×C23—41no—
136C4⋊C4×C23—41no—
137C4⋊C4×C23—41no—
138C4⋊C4×C23—41no—
139C4⋊C4×C23—41no—
140C4⋊C4×C23—41no—
141C4⋊C4×C23—41no—
142C4⋊C4×C23—41no—
143C4⋊C4×C23—41no—
144C4⋊C4×C23—41no—
145C4⋊C4×C23—41no—
146C4⋊C4×C23D811yescharacteristic
147C4⋊C4×C23D811yescharacteristic
148C4⋊C4×C23D811yescharacteristic
149C22⋊C4×C23—41no—
150C22⋊C4×C23—41no—
151C22⋊C4×C23—41no—
152C42×C23—41no—
153C42×C23—41no—
154C42×C23—41no—
155C42×C23C2×D411yescharacteristic
156C42×C23C2×D411yescharacteristic
157C42×C23C2×D411yescharacteristic
158C16×C23—21no—
159C16×C23—21no—
160C16×C23—21no—
161C16×C23C2×Q812yes—
162#33955C2311yesmaximal, characteristic, direct product factor, Sylow
163C23×C23—41no—
164Q8×C23—22no—
165Q8×C23—41no—
166Q8×C23—41no—
167Q8×C23—82no—
168Q8×C23—161no—
169D4×C23—22no—
170D4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—41no—
176C2×C4×C23—41no—
177C2×C4×C23—41no—
178C2×C4×C23—82no—
179C2×C4×C23—161no—
180C2×C4×C23—161no—
181C2×C4×C23C2×D811yescharacteristic
182C2×C4×C23C2×D811yescharacteristic
183C2×C4×C23C2×D811yescharacteristic
184C2×C4×C23C2×D811yescharacteristic
185C2×C4×C23C2×D811yescharacteristic
186C2×C4×C23C2×D811yescharacteristic
187C2×C4×C23C22×D411yescharacteristic
188C8×C23—21no—
189C8×C23—21no—
190C8×C23—21no—
191C8×C23C22×Q811yescharacteristic
192C8×C232−1+411yescharacteristic
193#3383C2×C2311yescharacteristic, complemented
194#3380C2×C2311yescharacteristic
195#2255C2×C2311yescharacteristic, complemented
196#2255C2×C2311yescharacteristic, complemented
197#2253C2×C2311yescharacteristic, complemented
198#2229C2×C2311yescharacteristic, complemented
199#2229C2×C2311yescharacteristic, complemented
200#2228C2×C2311yescharacteristic
201#2228C2×C2311yescharacteristic
202#2226C2×C2311yescharacteristic
203#2225C2×C2311yescharacteristic, complemented
204#2183C2×C2311yescharacteristic, complemented
205#2175C2×C2311yescharacteristic
206#2175C2×C2311yescharacteristic
207C4×SD32C2×C2311yescharacteristic
208C22×C23—42no—
209C22×C23—81no—
210C22×C23—81no—
211C22×C23C22×D811yescharacteristic
212C4×C23—21no—
213C4×C23—21no—
214C4×C23—21no—
215C4×C23—21no—
216C4×C23—21no—
217C4×C23—21no—
218C4×C23—42no—
219C4×C23—81no—
220C4×C23—81no—
221C4×C23—81no—
222C4×C23—81no—
223C4×C23—81no—
224C4×C23—81no—
225C4×C23D4×Q811yescharacteristic
226C4×C23#57811yescharacteristic
227#557—21no—
228#554—21no—
229C2×SD32C22×C2311yescharacteristic
230#500C22×C2311yescharacteristic
231#500C22×C2311yescharacteristic
232#499C22×C2311yescharacteristic
233#499C22×C2311yescharacteristic
234#499C22×C2311yescharacteristic
235#499C22×C2311yescharacteristic
236#479—21no—
237#479—21no—
238#478—21no—
239#478—21no—
240#476—21no—
241#475—21no—
242C8×Q8C22×C2311yescharacteristic
243C4×Q16C22×C2311yescharacteristic
244C4×Q16C22×C2311yescharacteristic
245C4×Q16C22×C2311yescharacteristic
246C4×D8C22×C2311yescharacteristic
247C4×D8C22×C2311yescharacteristic
248C4×D8C22×C2311yescharacteristic
249#367C22×C2311yescharacteristic
250#367C22×C2311yescharacteristic
251#367C22×C2311yescharacteristic
252#367C22×C2311yescharacteristic
253#367C22×C2311yescharacteristic
254#366C22×C2311yescharacteristic
255#366C22×C2311yescharacteristic
256#366C22×C2311yescharacteristic
257#366C22×C2311yescharacteristic
258C4⋊C16C22×C2311yescharacteristic
259C4⋊C16C22×C2311yescharacteristic
260C4⋊C16C22×C2311yescharacteristic
261#358C22×C2311yescharacteristic
262#358C22×C2311yescharacteristic
263#358C22×C2311yescharacteristic
264#357C22×C2311yescharacteristic
265#357C22×C2311yescharacteristic
266#357C22×C2311yescharacteristic
267#346C22×C2311yescharacteristic
268#346C22×C2311yescharacteristic
269C4×C16C22×C2311yescharacteristic
270C2×C23—82no—
271C2×C23Q8×D811yescharacteristic
272C2×C23#341411yescharacteristic
273C2×C23#341611yescharacteristic
274C2×Q16C23×C2311yescharacteristic
275C2×D8C23×C2311yescharacteristic
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—41no—
279#125—21no—
280#125—21no—
281#125—21no—
282#125—21no—
283#125—41no—
284#125—41no—
285C22⋊Q8—41no—
286C22⋊Q8—41no—
287C22⋊Q8—41no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—41no—
292C4×Q8D4×C2311yescharacteristic
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C2×C4⋊C4—41no—
297SD32Q8×C2314yescomplemented
298C2×C16C23×C2311yescharacteristic
299C2×C16C23×C2311yescharacteristic
300C2×C16C23×C2311yescharacteristic
301C2×C16C23×C2311yescharacteristic
302#107—21no—
303#107—21no—
304#107C23×C2311yescharacteristic
305#107C23×C2311yescharacteristic
306#107C23×C2311yescharacteristic
307#107C23×C2311yescharacteristic
308#107C23×C2311yescharacteristic
309#107C23×C2311yescharacteristic
310#106—21no—
311#106—21no—
312#106—21no—
313#106—21no—
314C4⋊C8D4×C2311yescharacteristic
315C4⋊C8D4×C2311yescharacteristic
316C4⋊C8D4×C2311yescharacteristic
317#103—21no—
318#103—21no—
319#103—21no—
320#102—21no—
321#102—21no—
322#102—21no—
323C4×C8C23×C2311yescharacteristic
324C4×C8C23×C2311yescharacteristic
325C4×C8C23×C2311yescharacteristic
326C23#3395511yescharacteristic, direct product factor, Sylow
327C2×Q8—21no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8D8×C2311yescharacteristic
331C2×D4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335Q16—21no—
336Q16C2×Q8×C2312yes—
337D8—21no—
338D8C2×Q8×C2312yes—
339C2×C8C2×D4×C2311yescharacteristic
340C2×C8C2×D4×C2311yescharacteristic
341C2×C8C2×D4×C2311yescharacteristic
342C2×C8C24×C2311yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4D8×C2311yescharacteristic
363C4⋊C4D8×C2311yescharacteristic
364C4⋊C4D8×C2311yescharacteristic
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42C2×D4×C2311yescharacteristic
372C42C2×D4×C2311yescharacteristic
373C42C2×D4×C2311yescharacteristic
374C16—21no—
375C16—21no—
376C16—21no—
377C16C2×Q8×C2312yes—
378C23—41no—
379Q8—22no—
380Q8—41no—
381Q8—41no—
382Q8—82no—
383Q8—161no—
384D4—22no—
385D4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—161no—
395C2×C4—161no—
396C2×C4C2×D8×C2311yescharacteristic
397C2×C4C2×D8×C2311yescharacteristic
398C2×C4C2×D8×C2311yescharacteristic
399C2×C4C2×D8×C2311yescharacteristic
400C2×C4C2×D8×C2311yescharacteristic
401C2×C4C2×D8×C2311yescharacteristic
402C2×C4C22×D4×C2311yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8C22×Q8×C2311yescharacteristic
407C8C23×2−1+411yescharacteristic
408C22—42no—
409C22—81no—
410C22—81no—
411C22C22×D8×C2311yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4D4×Q8×C2311yescharacteristic
426C4C23×#57811yescharacteristic
427C2—82no—
428C2Q8×D8×C2311yescharacteristic
429C2C23×#341411yescharacteristic
430C2C23×#341611yescharacteristic
subgroupG/H
center, socleC22×C23C22×D8
commutator, Frattini, cosocleC2×C8C24×C23
prime radicalC2×D8×C23C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C4×Q8×C23 ◁ C8×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3383C2semidirect
3. C23×#2255C2semidirect
4. C23×#2255C2semidirect
5. C23×#2253C2semidirect
6. C23×#2229C2semidirect
7. C23×#2229C2semidirect
11. C23×#2225C2semidirect
12. C23×#2183C2semidirect
82. C23×SD32Q8semidirect
162. #33955C23direct (N × H)
193. #3383C2×C23semidirect
195. #2255C2×C23semidirect
196. #2255C2×C23semidirect
197. #2253C2×C23semidirect
198. #2229C2×C23semidirect
199. #2229C2×C23semidirect
203. #2225C2×C23semidirect
204. #2183C2×C23semidirect
297. SD32Q8×C23semidirect
326. C23#33955direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#33832
C23×#33802
C23×#22552
C23×#22532
C23×#22292
C23×#22282
C23×#22262
C23×#22252
C23×#21832
C23×#21752
C4×C23×SD322
#3395523