Finite Group Databasebeta

← C23×#17632 #19165258

Subgroups — C23×#17632 #19165258

Order 5888.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3502C211yesmaximal, characteristic
2C23×#2983C211yesmaximal, characteristic
3C23×#2356C211yesmaximal, characteristic
4C23×#1518C212yesmaximal
5C23×#1515C214yesmaximal
6C23×#1506C214yesmaximal
7C23×#1503C212yesmaximal
8Q82×C23C414yescomplemented
9C23×#557—24no—
10C23×#557C414yescomplemented
11C23×#544—21no—
12C23×#544—22no—
13C23×#533C2212yes—
14C23×#533C2214yes—
15C23×#519—21no—
16C2×C4×Q8×C23C2211yescharacteristic
17C23×#446—24no—
18C23×#432C2214yes—
19C23×#431C2212yes—
20C23×#391—21no—
21C23×#387—22no—
22C23×#383C2212yes—
23C23×#382C2211yescharacteristic
24C4×C4⋊C4×C23—21no—
25C4×C4⋊C4×C23C2211yescharacteristic
26C23×#376C2212yes—
27C23×#328C2214yes—
28C23×#328C2214yes—
29C23×#328C2218yes—
30C22×Q8×C23D411yescharacteristic
31C23×C4⋊Q8—28no—
32C23×C4⋊Q8—24no—
33C23×C4⋊Q8C2×C412yes—
34C23×C4⋊Q8C2×C414yes—
35C23×C4⋊Q8C2×C412yes—
36C23×C4⋊Q8C2×C414yes—
37C23×#125—28no—
38C23×#125—24no—
39C23×#125—22no—
40C23×#125—24no—
41C23×C22⋊Q8—24no—
42C23×C22⋊Q8—22no—
43C23×C22⋊Q8C2314yes—
44C4×Q8×C23—24no—
45C4×Q8×C23—24no—
46C4×Q8×C23—22no—
47C4×Q8×C23—24no—
48C4×Q8×C23C2×C414yes—
49C4×Q8×C23C2×C418yes—
50C4×Q8×C23C2×C414yes—
51C4×Q8×C23D414yes—
52C23×#117—24no—
53C23×#117—22no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23D411yescharacteristic
57C2×C4⋊C4×C23D412yes—
58C2×C42×C23—22no—
59C2×C42×C23—21no—
60C2×C42×C23C2312yes—
61C2×C42×C23C2311yescharacteristic
62C23×C4⋊C8—22no—
63C23×C4⋊C8—24no—
64C23×C22⋊C8C2314yes—
65C23×#97—24no—
66C4×C8×C23—22no—
67C23×#95—22no—
68C23×#95—22no—
69C23×#95—21no—
70C23×#95—22no—
71C23×#95C2311yescharacteristic
72C23×#95C2312yes—
73C23×#95C2311yescharacteristic
74C2×Q8×C23—22no—
75C2×Q8×C23—24no—
76C2×Q8×C23—22no—
77C2×Q8×C23—22no—
78C2×Q8×C23C22⋊C414yes—
79C2×Q8×C23C22×C414yes—
80C22×C4×C23—21no—
81C22×C4×C23—22no—
82C22×C4×C23—21no—
83C22×C4×C23C2×D412yes—
84C22×C4×C23C2×D412yes—
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23C2×D411yescharacteristic
87C22×C4×C23C2411yescharacteristic
88C2×C8×C23—24no—
89C4⋊C4×C23—24no—
90C4⋊C4×C23—22no—
91C4⋊C4×C23—24no—
92C4⋊C4×C23—22no—
93C4⋊C4×C23—22no—
94C4⋊C4×C23—24no—
95C4⋊C4×C23—24no—
96C4⋊C4×C23—28no—
97C4⋊C4×C23—24no—
98C4⋊C4×C23—42no—
99C4⋊C4×C23—42no—
100C4⋊C4×C23C22⋊C418yes—
101C4⋊C4×C23C22⋊C414yes—
102C4⋊C4×C23C22×C414yes—
103C22⋊C4×C23—24no—
104C42×C23—28no—
105C42×C23—24no—
106C42×C23—44no—
107C42×C23—42no—
108C42×C23C22×C414yes—
109C42×C23C22×C412yes—
110C42×C23C2×D412yes—
111C42×C23C2×D414yes—
112#17632C2311yesmaximal, characteristic, direct product factor, Sylow
113C23×C23C22×D411yescharacteristic
114Q8×C23—28no—
115Q8×C23—44no—
116Q8×C23—84no—
117Q8×C23—82no—
118C2×C4×C23—22no—
119C2×C4×C23—22no—
120C2×C4×C23—22no—
121C2×C4×C23—24no—
122C2×C4×C23—24no—
123C2×C4×C23—42no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—41no—
127C2×C4×C23—41no—
128C2×C4×C23—42no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
131C2×C4×C23—42no—
132C2×C4×C23—41no—
133C2×C4×C23—42no—
134C2×C4×C23C2×C22⋊C414yes—
135C2×C4×C23C2×C22⋊C412yes—
136C2×C4×C23C2×C22⋊C414yes—
137C2×C4×C23C2×C22⋊C412yes—
138C2×C4×C23#13712yes—
139C2×C4×C23#13712yes—
140C2×C4×C23C23×C411yescharacteristic
141C2×C4×C23C22×D411yescharacteristic
142C8×C23—44no—
143#3502C2×C2311yescharacteristic
144#2983C2×C2311yescharacteristic
145#2356C2×C2311yescharacteristic
146#1518C2×C2312yes—
147#1515C2×C2314yes—
148#1506C2×C2314yes—
149#1503C2×C2312yes—
150C22×C23—21no—
151C22×C23—21no—
152C22×C23C22×C22⋊C411yescharacteristic
153C22×C23C2×#13711yescharacteristic
154C22×C23C2×#13711yescharacteristic
155C4×C23—21no—
156C4×C23—21no—
157C4×C23—24no—
158C4×C23—24no—
159C4×C23—22no—
160C4×C23—42no—
161C4×C23—44no—
162C4×C23—44no—
163C4×C23—81no—
164C4×C23—82no—
165C4×C23#41914yes—
166Q82C4×C2314yescomplemented
167#557—24no—
168#557C4×C2314yescomplemented
169#544—21no—
170#544—22no—
171#533C22×C2312yes—
172#533C22×C2314yes—
173#519—21no—
174C2×C4×Q8C22×C2311yescharacteristic
175#446—24no—
176#432C22×C2314yes—
177#431C22×C2312yes—
178#391—21no—
179#387—22no—
180#383C22×C2312yes—
181#382C22×C2311yescharacteristic
182C4×C4⋊C4—21no—
183C4×C4⋊C4C22×C2311yescharacteristic
184#376C22×C2312yes—
185#328C22×C2314yes—
186#328C22×C2314yes—
187#328C22×C2318yes—
188C2×C23—21no—
189C2×C23—21no—
190C2×C23#288411yescharacteristic
191C2×C23C2×#41911yescharacteristic
192C2×C23#290111yescharacteristic
193C22×Q8D4×C2311yescharacteristic
194C4⋊Q8—28no—
195C4⋊Q8—24no—
196C4⋊Q8C2×C4×C2312yes—
197C4⋊Q8C2×C4×C2314yes—
198C4⋊Q8C2×C4×C2312yes—
199C4⋊Q8C2×C4×C2314yes—
200#125—28no—
201#125—24no—
202#125—22no—
203#125—24no—
204C22⋊Q8—24no—
205C22⋊Q8—22no—
206C22⋊Q8C23×C2314yes—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—22no—
210C4×Q8—24no—
211C4×Q8C2×C4×C2314yes—
212C4×Q8C2×C4×C2318yes—
213C4×Q8C2×C4×C2314yes—
214C4×Q8D4×C2314yes—
215#117—24no—
216#117—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4D4×C2311yescharacteristic
220C2×C4⋊C4D4×C2312yes—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42C23×C2312yes—
224C2×C42C23×C2311yescharacteristic
225C4⋊C8—22no—
226C4⋊C8—24no—
227C22⋊C8C23×C2314yes—
228#97—24no—
229C4×C8—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95C23×C2311yescharacteristic
235#95C23×C2312yes—
236#95C23×C2311yescharacteristic
237C23#1763211yescharacteristic, direct product factor, Sylow
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8C22⋊C4×C2314yes—
243C2×Q8C22×C4×C2314yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4C2×D4×C2312yes—
248C22×C4C2×D4×C2312yes—
249C22×C4C2×D4×C2311yescharacteristic
250C22×C4C2×D4×C2311yescharacteristic
251C22×C4C24×C2311yescharacteristic
252C2×C8—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4C22⋊C4×C2318yes—
265C4⋊C4C22⋊C4×C2314yes—
266C4⋊C4C22×C4×C2314yes—
267C22⋊C4—24no—
268C42—28no—
269C42—24no—
270C42—44no—
271C42—42no—
272C42C22×C4×C2314yes—
273C42C22×C4×C2312yes—
274C42C2×D4×C2312yes—
275C42C2×D4×C2314yes—
276C23C22×D4×C2311yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—84no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4C2×C22⋊C4×C2314yes—
298C2×C4C2×C22⋊C4×C2312yes—
299C2×C4C2×C22⋊C4×C2314yes—
300C2×C4C2×C22⋊C4×C2312yes—
301C2×C4C23×#13712yes—
302C2×C4C23×#13712yes—
303C2×C4C23×C4×C2311yescharacteristic
304C2×C4C22×D4×C2311yescharacteristic
305C8—44no—
306C22—21no—
307C22—21no—
308C22C22×C22⋊C4×C2311yescharacteristic
309C22C2×C23×#13711yescharacteristic
310C22C2×C23×#13711yescharacteristic
311C4—21no—
312C4—21no—
313C4—24no—
314C4—24no—
315C4—22no—
316C4—42no—
317C4—44no—
318C4—44no—
319C4—81no—
320C4—82no—
321C4C23×#41914yes—
322C2—21no—
323C2—21no—
324C2C23×#288411yescharacteristic
325C2C2×C23×#41911yescharacteristic
326C2C23×#290111yescharacteristic
subgroupG/H
center, socleC22×C23C22×C22⋊C4
commutatorC2×C4C23×C4×C23
Frattini, cosocleC22×C4C24×C23
prime radicalC23×C23C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C4×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
8. Q82×C23C4semidirect
10. C23×#557C4semidirect
112. #17632C23direct (N × H)
166. Q82C4×C23semidirect
168. #557C4×C23semidirect
237. C23#17632direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35022
C23×#29832
C23×#23562
C23×#15182
C23×#15152
C23×#15062
C23×#15032
#1763223