Finite Group Databasebeta

← C23×#13299 #19160925

Subgroups — C23×#13299 #19160925

Order 5888.

500 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2563C211yesmaximal, characteristic
2C23×#1799C211yesmaximal, characteristic
3C4×C23×#99C211yesmaximal, characteristic
4C23×#1393C214yesmaximal
5C2×C23×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×D4×C23C2211yescharacteristic
7C2×C23×#99C2212yes—
8C2×C23×#99C2212yes—
9C23×#393—21no—
10C23×#393—21no—
11C23×#390C2212yes—
12C23×#382C412yescomplemented
13C23×#380—21no—
14C23×#380—21no—
15C23×#380—21no—
16C4×C4⋊C4×C23C411yescharacteristic, complemented
17C4×C22⋊C4×C23—21no—
18C23×#342—21no—
19C23×#342—21no—
20C22×Q8×C23—21no—
21C22×D4×C23C2311yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4⋊Q8—41no—
25C23×C4⋊Q8—41no—
26C4×D4×C23—41no—
27C4×D4×C23—41no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—41no—
31C2×C4⋊C4×C23C2×C412yes—
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23D411yescharacteristic
38C2×C22⋊C4×C23D411yescharacteristic
39C2×C42×C23—41no—
40C2×C42×C23C2×C411yescharacteristic
41C2×C42×C23C2×C412yes—
42C23×#99—22no—
43C23×#99—24no—
44C23×#99Q814yescomplemented
45C23×#95—21no—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—41no—
49C23×#95—41no—
50C24×C23C2×D411yescharacteristic
51C24×C23C2×D411yescharacteristic
52C2×Q8×C23—42no—
53C2×Q8×C23—82no—
54C2×D4×C23—41no—
55C2×D4×C23—41no—
56C2×D4×C23C2×Q812yes—
57C2×D4×C23#4112yes—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—22no—
63C22×C4×C23—22no—
64C22×C4×C23—42no—
65C22×C4×C23—42no—
66C22×C4×C23—42no—
67C22×C4×C23—42no—
68C22×C4×C23C22⋊C411yescharacteristic
69C22×C4×C23C22⋊C411yescharacteristic
70C22×C4×C23C22⋊C411yescharacteristic
71C22×C4×C23C22⋊C411yescharacteristic
72C22×C4×C23C22×C411yescharacteristic
73C4⋊C4×C23—41no—
74C4⋊C4×C23—42no—
75C4⋊C4×C23—42no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—82no—
79C22⋊C4×C23—24no—
80C22⋊C4×C23—41no—
81C22⋊C4×C23—41no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—42no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—82no—
86C22⋊C4×C23—82no—
87C42×C23—21no—
88C42×C23—21no—
89C42×C23—42no—
90C42×C23—82no—
91#13299C2311yesmaximal, characteristic, direct product factor, Sylow
92C23×C23—21no—
93C23×C23—21no—
94C23×C23—21no—
95C23×C23—21no—
96C23×C23—41no—
97C23×C23—41no—
98C23×C23—41no—
99C23×C23—41no—
100C23×C23C2×C22⋊C411yescharacteristic
101C23×C23C22⋊Q812yes—
102C23×C23#12411yescharacteristic
103C23×C23C4⋊Q811yescharacteristic
104Q8×C23—84no—
105D4×C23—42no—
106D4×C23—42no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—22no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—41no—
116C2×C4×C23—41no—
117C2×C4×C23—41no—
118C2×C4×C23—44no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—81no—
126C2×C4×C23—81no—
127C2×C4×C23—82no—
128C2×C4×C23—82no—
129C2×C4×C23—82no—
130C2×C4×C23—82no—
131C2×C4×C23—82no—
132C2×C4×C23#9911yescharacteristic
133C2×C4×C23#9911yescharacteristic
134C2×C4×C23C4×Q812yes—
135#2563C2×C2311yescharacteristic
136#1799C2×C2311yescharacteristic
137C4×#99C2×C2311yescharacteristic
138#1393C2×C2314yes—
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—22no—
142C22×C23—21no—
143C22×C23—21no—
144C22×C23—42no—
145C22×C23—41no—
146C22×C23—41no—
147C22×C23—41no—
148C22×C23—41no—
149C22×C23—41no—
150C22×C23—41no—
151C22×C23—81no—
152C22×C23—81no—
153C22×C23#39111yescharacteristic
154C22×C23C2×#9911yescharacteristic
155C22×C23#41011yescharacteristic
156C4×C23—21no—
157C4×C23—22no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—82no—
162C4×C23—84no—
163C4×C23—82no—
164C4×C23—162no—
165C4×C23#35412yes—
166C2×C4⋊Q8C4×C2311yescharacteristic, complemented
167C2×C4×D4C22×C2311yescharacteristic
168C2×#99C22×C2312yes—
169C2×#99C22×C2312yes—
170#393—21no—
171#393—21no—
172#390C22×C2312yes—
173#382C4×C2312yescomplemented
174#380—21no—
175#380—21no—
176#380—21no—
177C4×C4⋊C4C4×C2311yescharacteristic, complemented
178C4×C22⋊C4—21no—
179#342—21no—
180#342—21no—
181C2×C23—21no—
182C2×C23—21no—
183C2×C23—42no—
184C2×C23—41no—
185C2×C23—41no—
186C2×C23#199611yescharacteristic
187C2×C23C2×#35411yescharacteristic
188C2×C23#212811yescharacteristic
189C22×Q8—21no—
190C22×D4C23×C2311yescharacteristic
191C23×C4D4×C2311yescharacteristic
192C23×C4D4×C2311yescharacteristic
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4×D4—41no—
196C4×D4—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×C4×C2312yes—
201C2×C4⋊C4C2×C4×C2311yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4D4×C2311yescharacteristic
207C2×C22⋊C4D4×C2311yescharacteristic
208C2×C42—41no—
209C2×C42C2×C4×C2311yescharacteristic
210C2×C42C2×C4×C2312yes—
211#99—22no—
212#99—24no—
213#99Q8×C2314yescomplemented
214#95—21no—
215#95—22no—
216#95—22no—
217#95—41no—
218#95—41no—
219C23#1329911yescharacteristic, direct product factor, Sylow
220C24C2×D4×C2311yescharacteristic
221C24C2×D4×C2311yescharacteristic
222C2×Q8—42no—
223C2×Q8—82no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4C2×Q8×C2312yes—
227C2×D4#41×C2312yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C22⋊C4×C2311yescharacteristic
239C22×C4C22⋊C4×C2311yescharacteristic
240C22×C4C22⋊C4×C2311yescharacteristic
241C22×C4C22⋊C4×C2311yescharacteristic
242C22×C4C22×C4×C2311yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—82no—
249C22⋊C4—24no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C42—21no—
258C42—21no—
259C42—42no—
260C42—82no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23C2×C22⋊C4×C2311yescharacteristic
270C23C23×C22⋊Q812yes—
271C23C23×#12411yescharacteristic
272C23C23×C4⋊Q811yescharacteristic
273Q8—84no—
274D4—42no—
275D4—42no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4C23×#9911yescharacteristic
302C2×C4C23×#9911yescharacteristic
303C2×C4C4×Q8×C2312yes—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—81no—
318C22C23×#39111yescharacteristic
319C22C2×C23×#9911yescharacteristic
320C22C23×#41011yescharacteristic
321C4—22no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—84no—
328C4—82no—
329C4—162no—
330C4C23×#35412yes—
331C2—21no—
332C2—21no—
333C2—42no—
334C2—41no—
335C2—41no—
336C2C23×#199611yescharacteristic
337C2C2×C23×#35411yescharacteristic
338C2C23×#212811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC22×D4×C23C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×C23×C4⋊Q8C4semidirect
12. C23×#382C4semidirect
16. C4×C4⋊C4×C23C4semidirect
44. C23×#99Q8semidirect
91. #13299C23direct (N × H)
166. C2×C4⋊Q8C4×C23semidirect
173. #382C4×C23semidirect
177. C4×C4⋊C4C4×C23semidirect
213. #99Q8×C23semidirect
219. C23#13299direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25632
C23×#17992
C4×C23×#992
C23×#13932
#1329923