Finite Group Databasebeta

← C23×#9167 #19156793

Subgroups — C23×#9167 #19156793

Order 5888.

478 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3079C211yesmaximal, characteristic
2C23×#2906C211yesmaximal, characteristic
3C23×#1548C211yesmaximal, characteristic, complemented
4C2×C23×#332C211yesmaximal, characteristic, complemented
5C23×#1477C211yesmaximal, characteristic
6C23×#1301C211yesmaximal, characteristic, complemented
7C23×#1292C211yesmaximal, characteristic, complemented
8C2×C23×#131—21no—
9C2×C23×C4⋊Q8C2211yescharacteristic
10C4×#41×C23C2211yescharacteristic
11C23×#477C412yes—
12C23×#476C412yes—
13C23×#475—21no—
14C23×#474—21no—
15C23×#431C2211yescharacteristic
16C23×#428—21no—
17C23×#424—21no—
18C23×#424—22no—
19C23×#423—21no—
20C23×#423C2211yescharacteristic
21C23×#423C2211yescharacteristic
22C2×C23×C4⋊C8—21no—
23C2×C23×C4⋊C8C2211yescharacteristic
24C2×C23×C4⋊C8C2211yescharacteristic
25C23×#417—21no—
26C23×#333—22no—
27C23×#332—21no—
28C23×#332—21no—
29C23×Q8⋊C8—21no—
30C23×D4⋊C8—21no—
31C2×#41×C23D411yescharacteristic
32C22×Q8×C23—21no—
33C23×#131—42no—
34C23×#131—42no—
35C2×M16×C23—22no—
36C2×M16×C23—21no—
37C2×M16×C23—21no—
38C22×C8×C23—22no—
39C22×C8×C23—21no—
40C22×C8×C23—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8—21no—
44C23×C4⋊Q8C2×C412yes—
45C4×Q8×C23C2×C411yescharacteristic
46C4×Q8×C23D411yescharacteristic
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23C2×C411yescharacteristic
50C4×D4×C23D411yescharacteristic
51C23×#117—21no—
52C23×#117D411yescharacteristic
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—22no—
60C23×C4⋊C8—22no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8—21no—
67C23×C4⋊C8C2×C412yes—
68C23×C4⋊C8D411yescharacteristic
69C23×C4⋊C8D411yescharacteristic
70C23×C4⋊C8D412yes—
71C23×#103—22no—
72C23×#102—22no—
73C23×C22⋊C8—21no—
74C4×C8×C23—21no—
75#41×C23—21no—
76#41×C23—21no—
77#41×C23SD1612yes—
78#41×C23Q1612yes—
79C2×Q8×C23—22no—
80C2×Q8×C23—41no—
81C2×Q8×C23—81no—
82C2×Q8×C23—81no—
83C2×Q8×C23C22⋊C411yescharacteristic
84C2×D4×C23—21no—
85C2×D4×C23C22⋊C411yescharacteristic
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—21no—
89C22×C4×C23—41no—
90C22×C4×C23C2×D411yescharacteristic
91C22×C4×C23#4111yescharacteristic
92C22×C4×C23#4111yescharacteristic
93M16×C23—44no—
94M16×C23—41no—
95M16×C23—41no—
96M16×C23—41no—
97M16×C23—41no—
98C2×C8×C23—22no—
99C2×C8×C23—21no—
100C2×C8×C23—22no—
101C2×C8×C23—21no—
102C2×C8×C23—44no—
103C2×C8×C23—41no—
104C2×C8×C23—41no—
105C2×C8×C23—41no—
106C2×C8×C23—41no—
107C2×C8×C23—81no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—22no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23—41no—
114C4⋊C4×C23C22⋊C411yescharacteristic
115C4⋊C4×C23C22⋊C411yescharacteristic
116C22⋊C4×C23—21no—
117C22⋊C4×C23—21no—
118C22⋊C4×C23—21no—
119C42×C23—21no—
120C42×C23—21no—
121C42×C23—21no—
122C42×C23C22×C411yescharacteristic
123C42×C23C2×D411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23C2×D411yescharacteristic
126#9167C2311yesmaximal, characteristic, direct product factor, Sylow
127C23×C23—21no—
128C23×C23#12311yescharacteristic
129Q8×C23—21no—
130Q8×C23—82no—
131Q8×C23—81no—
132Q8×C23—81no—
133Q8×C23#10312yes—
134D4×C23—22no—
135D4×C23—22no—
136D4×C23—21no—
137D4×C23#10312yes—
138C2×C4×C23—21no—
139C2×C4×C23—22no—
140C2×C4×C23—22no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—42no—
149C2×C4×C23—81no—
150C2×C4×C23—82no—
151C2×C4×C23C2×C22⋊C411yescharacteristic
152C2×C4×C23C4×D411yescharacteristic
153C2×C4×C23C4×D411yescharacteristic
154C2×C4×C23#12011yescharacteristic
155C2×C4×C23#12111yescharacteristic
156C2×C4×C23#12111yescharacteristic
157C2×C4×C23C2×SD1611yescharacteristic
158C2×C4×C23C2×Q1611yescharacteristic
159C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C23#13711yescharacteristic
161C8×C23—42no—
162C8×C23—41no—
163C8×C23—42no—
164C8×C23—41no—
165C8×C23—81no—
166C8×C23—81no—
167#3079C2×C2311yescharacteristic
168#2906C2×C2311yescharacteristic
169#1548C2×C2311yescharacteristic, complemented
170C2×#332C2×C2311yescharacteristic, complemented
171#1477C2×C2311yescharacteristic
172#1301C2×C2311yescharacteristic, complemented
173#1292C2×C2311yescharacteristic, complemented
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—22no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23#38611yescharacteristic
180C22×C23#48311yescharacteristic
181C22×C23#48411yescharacteristic
182C4×C23—22no—
183C4×C23—21no—
184C4×C23—21no—
185C4×C23—21no—
186C4×C23—41no—
187C4×C23—41no—
188C4×C23—82no—
189C4×C23C2×#10311yescharacteristic
190C4×C23#41711yescharacteristic
191C4×C23#45011yescharacteristic
192C4×C23#45111yescharacteristic
193C4×C23#47111yescharacteristic
194C4×C23#47211yescharacteristic
195C2×#131—21no—
196C2×C4⋊Q8C22×C2311yescharacteristic
197C4×#41C22×C2311yescharacteristic
198#477C4×C2312yes—
199#476C4×C2312yes—
200#475—21no—
201#474—21no—
202#431C22×C2311yescharacteristic
203#428—21no—
204#424—21no—
205#424—22no—
206#423—21no—
207#423C22×C2311yescharacteristic
208#423C22×C2311yescharacteristic
209C2×C4⋊C8—21no—
210C2×C4⋊C8C22×C2311yescharacteristic
211C2×C4⋊C8C22×C2311yescharacteristic
212#417—21no—
213#333—22no—
214#332—21no—
215#332—21no—
216Q8⋊C8—21no—
217D4⋊C8—21no—
218C2×C23—21no—
219C2×C23—21no—
220C2×C23—42no—
221C2×C23#187211yescharacteristic
222C2×C23#190211yescharacteristic
223C2×C23#190711yescharacteristic
224C2×#41D4×C2311yescharacteristic
225C22×Q8—21no—
226#131—42no—
227#131—42no—
228C2×M16—22no—
229C2×M16—21no—
230C2×M16—21no—
231C22×C8—22no—
232C22×C8—21no—
233C22×C8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8C2×C4×C2312yes—
238C4×Q8C2×C4×C2311yescharacteristic
239C4×Q8D4×C2311yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C2×C4×C2311yescharacteristic
243C4×D4D4×C2311yescharacteristic
244#117—21no—
245#117D4×C2311yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C42—21no—
249C2×C42C23×C2311yescharacteristic
250#107—22no—
251#106—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8C2×C4×C2312yes—
261C4⋊C8D4×C2311yescharacteristic
262C4⋊C8D4×C2311yescharacteristic
263C4⋊C8D4×C2312yes—
264#103—22no—
265#102—22no—
266C22⋊C8—21no—
267C4×C8—21no—
268C23#916711yescharacteristic, direct product factor, Sylow
269#41—21no—
270#41—21no—
271#41SD16×C2312yes—
272#41Q16×C2312yes—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×Q8C22⋊C4×C2311yescharacteristic
278C2×D4—21no—
279C2×D4C22⋊C4×C2311yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4C2×D4×C2311yescharacteristic
285C22×C4#41×C2311yescharacteristic
286C22×C4#41×C2311yescharacteristic
287M16—44no—
288M16—41no—
289M16—41no—
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—44no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—81no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4C22⋊C4×C2311yescharacteristic
309C4⋊C4C22⋊C4×C2311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42C22×C4×C2311yescharacteristic
317C42C2×D4×C2311yescharacteristic
318C42C2×D4×C2311yescharacteristic
319C42C2×D4×C2311yescharacteristic
320C23—21no—
321C23C23×#12311yescharacteristic
322Q8—21no—
323Q8—82no—
324Q8—81no—
325Q8—81no—
326Q8C23×#10312yes—
327D4—22no—
328D4—22no—
329D4—21no—
330D4C23×#10312yes—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4C2×C22⋊C4×C2311yescharacteristic
345C2×C4C4×D4×C2311yescharacteristic
346C2×C4C4×D4×C2311yescharacteristic
347C2×C4C23×#12011yescharacteristic
348C2×C4C23×#12111yescharacteristic
349C2×C4C23×#12111yescharacteristic
350C2×C4C2×SD16×C2311yescharacteristic
351C2×C4C2×Q16×C2311yescharacteristic
352C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
353C2×C4C23×#13711yescharacteristic
354C8—42no—
355C8—41no—
356C8—42no—
357C8—41no—
358C8—81no—
359C8—81no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22C23×#38611yescharacteristic
366C22C23×#48311yescharacteristic
367C22C23×#48411yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—82no—
375C4C2×C23×#10311yescharacteristic
376C4C23×#41711yescharacteristic
377C4C23×#45011yescharacteristic
378C4C23×#45111yescharacteristic
379C4C23×#47111yescharacteristic
380C4C23×#47211yescharacteristic
381C2—21no—
382C2—21no—
383C2—42no—
384C2C23×#187211yescharacteristic
385C2C23×#190211yescharacteristic
386C2C23×#190711yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×#41×C23D4
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×#1548C2semidirect
4. C2×C23×#332C2semidirect
6. C23×#1301C2semidirect
7. C23×#1292C2semidirect
126. #9167C23direct (N × H)
169. #1548C2×C23semidirect
170. C2×#332C2×C23semidirect
172. #1301C2×C23semidirect
173. #1292C2×C23semidirect
268. C23#9167direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30792
C23×#29062
C23×#15482
C2×C23×#3322
C23×#14772
C23×#13012
C23×#12922
#916723