Finite Group Databasebeta

← C23×#10013 #19157639

Subgroups — C23×#10013 #19157639

Order 5888.

598 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3103C211yesmaximal, characteristic, complemented
2C23×#3100C211yesmaximal, characteristic
3C23×#2398C211yesmaximal, characteristic, complemented
4C23×#1963C211yesmaximal, characteristic, complemented
5C23×#1487C211yesmaximal, characteristic, complemented
6C23×#1486C211yesmaximal, characteristic, complemented
7C23×#1274C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C4×Q8×C23C2211yescharacteristic
10C2×C4×D4×C23C2211yescharacteristic
11C23×#481—21no—
12C23×#480—21no—
13C23×#475—21no—
14C23×#474—21no—
15C23×#460—21no—
16C23×#459—21no—
17C23×#448—21no—
18C23×#447—21no—
19C4×SD16×C23—22no—
20C4×D8×C23—22no—
21C23×#431C2211yescharacteristic
22C23×#431C2211yescharacteristic
23C23×#423C2211yescharacteristic
24C2×C23×#104—22no—
25C23×#400—21no—
26C23×#397—21no—
27C23×#394—21no—
28C23×#393—21no—
29C4×C22⋊C4×C23—22no—
30C23×#376C2211yescharacteristic
31C23×Q8⋊C8—22no—
32C23×D4⋊C8—22no—
33C2×#41×C23—21no—
34C22×Q8×C23D411yescharacteristic
35C22×D4×C23D411yescharacteristic
36C23×C4×C23—21no—
37C2×SD16×C23—22no—
38C2×D8×C23—22no—
39C2×M16×C23—21no—
40C23×C4⋊Q8D411yescharacteristic
41C23×#127D411yescharacteristic
42C23×#124—21no—
43C23×#121—21no—
44C23×#121—21no—
45C4×Q8×C23—21no—
46C4×Q8×C23—21no—
47C4×Q8×C23—21no—
48C4×Q8×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C4×D4×C23—21no—
54C4×D4×C23—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23D411yescharacteristic
57C2×C4⋊C4×C23D411yescharacteristic
58C2×C22⋊C4×C23—22no—
59C2×C22⋊C4×C23—22no—
60C2×C22⋊C4×C23—21no—
61C2×C42×C23—21no—
62C2×C42×C23D412yes—
63C2×C42×C23C2311yescharacteristic
64C23×#107—22no—
65C23×#106—22no—
66C23×C4⋊C8—21no—
67C23×C4⋊C8—21no—
68C23×C4⋊C8—21no—
69C23×C4⋊C8—21no—
70C23×#104—44no—
71C23×#103—22no—
72C23×#102—22no—
73C23×#102—22no—
74C23×#102—22no—
75C23×C22⋊C8—21no—
76C23×C22⋊C8—21no—
77C4×C8×C23D412yes—
78C4×C8×C23D412yes—
79C23×#95—22no—
80C23×#95—22no—
81C23×#95—21no—
82C23×#95—21no—
83C24×C23—21no—
84#41×C23—42no—
85#41×C23—42no—
86C2×Q8×C23—21no—
87C2×Q8×C23—21no—
88C2×Q8×C23—21no—
89C2×Q8×C23—42no—
90C2×D4×C23—21no—
91C2×D4×C23—21no—
92C2×D4×C23—21no—
93C2×D4×C23—21no—
94C2×D4×C23—21no—
95C2×D4×C23—42no—
96C2×D4×C23—41no—
97C22×C4×C23—21no—
98C22×C4×C23—21no—
99C22×C4×C23—22no—
100C22×C4×C23—21no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23—42no—
105C22×C4×C23—42no—
106C22×C4×C23C2×D411yescharacteristic
107C22×C4×C23C2×D411yescharacteristic
108C22×C4×C23C2×D411yescharacteristic
109SD16×C23—44no—
110D8×C23—44no—
111M16×C23—42no—
112C2×C8×C23—22no—
113C2×C8×C23—22no—
114C2×C8×C23—41no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23—21no—
120C4⋊C4×C23—21no—
121C4⋊C4×C23—41no—
122C4⋊C4×C23—41no—
123C22⋊C4×C23—44no—
124C22⋊C4×C23—44no—
125C22⋊C4×C23—41no—
126C22⋊C4×C23—41no—
127C22⋊C4×C23—41no—
128C22⋊C4×C23—41no—
129C42×C23—22no—
130C42×C23—22no—
131C42×C23—41no—
132C42×C23—41no—
133C42×C23C2×D411yescharacteristic
134C42×C23C2×D411yescharacteristic
135C42×C23C2×D411yescharacteristic
136C42×C23#4111yescharacteristic
137#10013C2311yesmaximal, characteristic, direct product factor, Sylow
138C23×C23—22no—
139C23×C23—42no—
140C23×C23—41no—
141C23×C23—41no—
142C23×C23—42no—
143C23×C23—41no—
144C23×C23#12011yescharacteristic
145Q8×C23—22no—
146Q8×C23—22no—
147Q8×C23—41no—
148Q8×C23—41no—
149Q8×C23—42no—
150D4×C23—22no—
151D4×C23—22no—
152D4×C23—41no—
153D4×C23—41no—
154D4×C23—42no—
155D4×C23—42no—
156D4×C23—42no—
157D4×C23—82no—
158C2×C4×C23—22no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—42no—
164C2×C4×C23—42no—
165C2×C4×C23—42no—
166C2×C4×C23—41no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—42no—
170C2×C4×C23—42no—
171C2×C4×C23—82no—
172C2×C4×C23—81no—
173C2×C4×C23—81no—
174C2×C4×C23—81no—
175C2×C4×C23—81no—
176C2×C4×C23#12011yescharacteristic
177C2×C4×C23#12011yescharacteristic
178C2×C4×C23#12111yescharacteristic
179C2×C4×C23#12111yescharacteristic
180C2×C4×C23#12111yescharacteristic
181C2×C4×C23#12711yescharacteristic
182C2×C4×C23#13512yes—
183C2×C4×C23AGL(1,ℤ/8)12yes—
184C8×C23—42no—
185C8×C23—42no—
186C8×C23—81no—
187#3103C2×C2311yescharacteristic, complemented
188#3100C2×C2311yescharacteristic
189#2398C2×C2311yescharacteristic, complemented
190#1963C2×C2311yescharacteristic, complemented
191#1487C2×C2311yescharacteristic, complemented
192#1486C2×C2311yescharacteristic, complemented
193#1274C2×C2311yescharacteristic, complemented
194C22×C23—21no—
195C22×C23—21no—
196C22×C23—44no—
197C22×C23—42no—
198C22×C23—82no—
199C22×C23—81no—
200C22×C23—81no—
201C22×C23—81no—
202C22×C23—81no—
203C22×C23—81no—
204C22×C23#39211yescharacteristic
205C22×C23#44912yes—
206C4×C23—21no—
207C4×C23—21no—
208C4×C23—21no—
209C4×C23—21no—
210C4×C23—42no—
211C4×C23—82no—
212C4×C23—81no—
213C4×C23—81no—
214C4×C23—81no—
215C4×C23#46312yes—
216C4×C23#49511yescharacteristic
217C4×C23#49611yescharacteristic
218#532C22×C2311yescharacteristic
219C2×C4×Q8C22×C2311yescharacteristic
220C2×C4×D4C22×C2311yescharacteristic
221#481—21no—
222#480—21no—
223#475—21no—
224#474—21no—
225#460—21no—
226#459—21no—
227#448—21no—
228#447—21no—
229C4×SD16—22no—
230C4×D8—22no—
231#431C22×C2311yescharacteristic
232#431C22×C2311yescharacteristic
233#423C22×C2311yescharacteristic
234C2×#104—22no—
235#400—21no—
236#397—21no—
237#394—21no—
238#393—21no—
239C4×C22⋊C4—22no—
240#376C22×C2311yescharacteristic
241Q8⋊C8—22no—
242D4⋊C8—22no—
243C2×C23—22no—
244C2×C23—82no—
245C2×C23—161no—
246C2×C23#201011yescharacteristic
247C2×C23#201011yescharacteristic
248C2×C23#201211yescharacteristic
249C2×#41—21no—
250C22×Q8D4×C2311yescharacteristic
251C22×D4D4×C2311yescharacteristic
252C23×C4—21no—
253C2×SD16—22no—
254C2×D8—22no—
255C2×M16—21no—
256C4⋊Q8D4×C2311yescharacteristic
257#127D4×C2311yescharacteristic
258#124—21no—
259#121—21no—
260#121—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4D4×C2311yescharacteristic
273C2×C4⋊C4D4×C2311yescharacteristic
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C42—21no—
278C2×C42D4×C2312yes—
279C2×C42C23×C2311yescharacteristic
280#107—22no—
281#106—22no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286#104—44no—
287#103—22no—
288#102—22no—
289#102—22no—
290#102—22no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C4×C8D4×C2312yes—
294C4×C8D4×C2312yes—
295#95—22no—
296#95—22no—
297#95—21no—
298#95—21no—
299C23#1001311yescharacteristic, direct product factor, Sylow
300C24—21no—
301#41—42no—
302#41—42no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—42no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—41no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4C2×D4×C2311yescharacteristic
324C22×C4C2×D4×C2311yescharacteristic
325C22×C4C2×D4×C2311yescharacteristic
326SD16—44no—
327D8—44no—
328M16—42no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C42C2×D4×C2311yescharacteristic
351C42C2×D4×C2311yescharacteristic
352C42C2×D4×C2311yescharacteristic
353C42#41×C2311yescharacteristic
354C23—22no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—41no—
360C23C23×#12011yescharacteristic
361Q8—22no—
362Q8—22no—
363Q8—41no—
364Q8—41no—
365Q8—42no—
366D4—22no—
367D4—22no—
368D4—41no—
369D4—41no—
370D4—42no—
371D4—42no—
372D4—42no—
373D4—82no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4C23×#12011yescharacteristic
393C2×C4C23×#12011yescharacteristic
394C2×C4C23×#12111yescharacteristic
395C2×C4C23×#12111yescharacteristic
396C2×C4C23×#12111yescharacteristic
397C2×C4C23×#12711yescharacteristic
398C2×C4C23×#13512yes—
399C2×C4C23×AGL(1,ℤ/8)12yes—
400C8—42no—
401C8—42no—
402C8—81no—
403C22—21no—
404C22—21no—
405C22—44no—
406C22—42no—
407C22—82no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22C23×#39211yescharacteristic
414C22C23×#44912yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—82no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4C23×#46312yes—
425C4C23×#49511yescharacteristic
426C4C23×#49611yescharacteristic
427C2—22no—
428C2—82no—
429C2—161no—
430C2C23×#201011yescharacteristic
431C2C23×#201011yescharacteristic
432C2C23×#201211yescharacteristic
subgroupG/H
centerC2×C4×C23#127
commutator, Frattini, cosocleC2×C42C23×C23
socleC22×C23#392
prime radicalC23×#3100C2
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
1. C23×#3103C2semidirect
3. C23×#2398C2semidirect
4. C23×#1963C2semidirect
5. C23×#1487C2semidirect
6. C23×#1486C2semidirect
7. C23×#1274C2semidirect
137. #10013C23direct (N × H)
187. #3103C2×C23semidirect
189. #2398C2×C23semidirect
190. #1963C2×C23semidirect
191. #1487C2×C23semidirect
192. #1486C2×C23semidirect
193. #1274C2×C23semidirect
299. C23#10013direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31032
C23×#31002
C23×#23982
C23×#19632
C23×#14872
C23×#14862
C23×#12742
#1001323