Finite Group Databasebeta

← C23×#11900 #19159526

Subgroups — C23×#11900 #19159526

Order 5888.

452 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 118 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
3C23×#1459C211yesmaximal, characteristic
4C23×#1351C214yesmaximal, complemented
5C2×C23×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×D4×C23C2211yescharacteristic
7C2×C23×#99—21no—
8C2×C23×C22⋊C8—21no—
9C23×#393C412yescomplemented
10C23×#393C2212yes—
11C23×#390—21no—
12C23×#382—21no—
13C23×#380—21no—
14C23×#380—21no—
15C23×#380—21no—
16C4×C4⋊C4×C23C411yescharacteristic, complemented
17C23×#342C2212yes—
18C23×#324—21no—
19C23×#323C2212yes—
20C22×Q8×C23C2×C412yes—
21C22×D4×C23D411yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C22×C8×C23—41no—
25C23×C4⋊Q8—41no—
26C23×C4⋊Q8—41no—
27C4×D4×C23—41no—
28C4×D4×C23—41no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—41no—
33C2×C4⋊C4×C23D411yescharacteristic
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23C2×C411yescharacteristic
38C2×C22⋊C4×C23C2311yescharacteristic
39C2×C42×C23—21no—
40C2×C42×C23C2×C411yescharacteristic
41C23×#99—42no—
42C23×C22⋊C8—22no—
43C23×C22⋊C8—41no—
44C23×#95—22no—
45C23×#95—22no—
46C23×#95—41no—
47C23×#95C2×C412yes—
48C24×C23C22⋊C411yescharacteristic
49C24×C23C2×D411yescharacteristic
50C2×Q8×C23—42no—
51C2×Q8×C23—82no—
52C2×D4×C23—41no—
53C2×D4×C23—41no—
54C2×D4×C23SD1612yes—
55C2×D4×C23Q1612yes—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—22no—
61C22×C4×C23—22no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23—42no—
65C22×C4×C23C22⋊C411yescharacteristic
66C22×C4×C23C22⋊C411yescharacteristic
67C22×C4×C23C22⋊C411yescharacteristic
68C22×C4×C23C22×C411yescharacteristic
69C22×C4×C23C2×D411yescharacteristic
70C2×C8×C23—42no—
71C2×C8×C23—82no—
72C4⋊C4×C23—41no—
73C4⋊C4×C23—42no—
74C4⋊C4×C23—42no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—82no—
78C22⋊C4×C23—41no—
79C22⋊C4×C23—41no—
80C22⋊C4×C23—42no—
81C22⋊C4×C23—42no—
82C22⋊C4×C23—82no—
83C22⋊C4×C23—82no—
84C42×C23—21no—
85C42×C23—21no—
86C42×C23—42no—
87#11900C2311yesmaximal, characteristic, direct product factor, Sylow
88C23×C23—21no—
89C23×C23—21no—
90C23×C23—21no—
91C23×C23—21no—
92C23×C23—41no—
93C23×C23—41no—
94C23×C23—41no—
95C23×C23—41no—
96C23×C23#10312yes—
97C23×C23C2×C22⋊C411yescharacteristic
98C23×C23C2×SD1611yescharacteristic
99C23×C23C2×Q1611yescharacteristic
100Q8×C23—84no—
101D4×C23—42no—
102D4×C23—42no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—22no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—41no—
112C2×C4×C23—41no—
113C2×C4×C23—41no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—81no—
121C2×C4×C23—81no—
122C2×C4×C23—82no—
123C2×C4×C23—82no—
124C2×C4×C23—82no—
125C2×C4×C23#9911yescharacteristic
126C2×C4×C23#9911yescharacteristic
127C2×C4×C23#10312yes—
128C8×C23—82no—
129#2563C2×C2311yescharacteristic
130#1799C2×C2311yescharacteristic
131#1459C2×C2311yescharacteristic
132#1351C2×C2314yescomplemented
133C22×C23—21no—
134C22×C23—21no—
135C22×C23—21no—
136C22×C23—22no—
137C22×C23—21no—
138C22×C23—42no—
139C22×C23—41no—
140C22×C23—41no—
141C22×C23—41no—
142C22×C23—41no—
143C22×C23—41no—
144C22×C23—41no—
145C22×C23—81no—
146C22×C23—81no—
147C22×C23C2×#9911yescharacteristic
148C22×C23C2×#10311yescharacteristic
149C22×C23#42111yescharacteristic
150C4×C23—21no—
151C4×C23—22no—
152C4×C23—41no—
153C4×C23—41no—
154C4×C23—41no—
155C4×C23—82no—
156C4×C23—82no—
157C4×C23—162no—
158C4×C23#35412yes—
159C2×C4⋊Q8C4×C2311yescharacteristic, complemented
160C2×C4×D4C22×C2311yescharacteristic
161C2×#99—21no—
162C2×C22⋊C8—21no—
163#393C4×C2312yescomplemented
164#393C22×C2312yes—
165#390—21no—
166#382—21no—
167#380—21no—
168#380—21no—
169#380—21no—
170C4×C4⋊C4C4×C2311yescharacteristic, complemented
171#342C22×C2312yes—
172#324—21no—
173#323C22×C2312yes—
174C2×C23—21no—
175C2×C23—21no—
176C2×C23—41no—
177C2×C23—42no—
178C2×C23—41no—
179C2×C23#152011yescharacteristic
180C2×C23#212211yescharacteristic
181C2×C23C2×#35411yescharacteristic
182C22×Q8C2×C4×C2312yes—
183C22×D4D4×C2311yescharacteristic
184C23×C4D4×C2311yescharacteristic
185C23×C4D4×C2311yescharacteristic
186C22×C8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4D4×C2311yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×C4×C2311yescharacteristic
200C2×C22⋊C4C23×C2311yescharacteristic
201C2×C42—21no—
202C2×C42C2×C4×C2311yescharacteristic
203#99—42no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206#95—22no—
207#95—22no—
208#95—41no—
209#95C2×C4×C2312yes—
210C23#1190011yescharacteristic, direct product factor, Sylow
211C24C22⋊C4×C2311yescharacteristic
212C24C2×D4×C2311yescharacteristic
213C2×Q8—42no—
214C2×Q8—82no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4SD16×C2312yes—
218C2×D4Q16×C2312yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊C4×C2311yescharacteristic
229C22×C4C22⋊C4×C2311yescharacteristic
230C22×C4C22⋊C4×C2311yescharacteristic
231C22×C4C22×C4×C2311yescharacteristic
232C22×C4C2×D4×C2311yescharacteristic
233C2×C8—42no—
234C2×C8—82no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—82no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C42—21no—
248C42—21no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C23×#10312yes—
259C23C2×C22⋊C4×C2311yescharacteristic
260C23C2×SD16×C2311yescharacteristic
261C23C2×Q16×C2311yescharacteristic
262Q8—84no—
263D4—42no—
264D4—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4C23×#9911yescharacteristic
288C2×C4C23×#9911yescharacteristic
289C2×C4C23×#10312yes—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×C23×#9911yescharacteristic
306C22C2×C23×#10311yescharacteristic
307C22C23×#42111yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C4—82no—
315C4—162no—
316C4C23×#35412yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2C23×#152011yescharacteristic
323C2C23×#212211yescharacteristic
324C2C2×C23×#35411yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
4. C23×#1351C2semidirect
5. C2×C23×C4⋊Q8C4semidirect
9. C23×#393C4semidirect
16. C4×C4⋊C4×C23C4semidirect
87. #11900C23direct (N × H)
132. #1351C2×C23semidirect
159. C2×C4⋊Q8C4×C23semidirect
163. #393C4×C23semidirect
170. C4×C4⋊C4C4×C23semidirect
210. C23#11900direct (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
C23×#14592
C23×#13512
#1190023