Finite Group Databasebeta

← C23×#11874 #19159500

Subgroups — C23×#11874 #19159500

Order 5888.

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

normal subgroups only

This group has 472 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×#2486C211yesmaximal, characteristic
2C23×#1799C211yesmaximal, characteristic
3C23×#1460C211yesmaximal, characteristic
4C23×#1344C214yesmaximal, complemented
5C2×C23×#123—21no—
6C2×C4×D4×C23C2211yescharacteristic
7C22×C4⋊C4×C23C411yescharacteristic, complemented
8C23×#431—21no—
9C2×C23×#99—21no—
10C23×#398C412yescomplemented
11C23×#397C2212yes—
12C23×#389—21no—
13C23×#381—21no—
14C23×#380—21no—
15C23×#380—21no—
16C23×#380C411yescharacteristic, complemented
17C23×#342C2212yes—
18C23×#323—21no—
19C23×#323C2212yes—
20C22×D4×C23D411yescharacteristic
21C23×C4×C23—21no—
22C23×C4×C23C2×C411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×#123—42no—
25C23×#123—42no—
26C4×D4×C23—41no—
27C4×D4×C23—41no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—41no—
32C2×C4⋊C4×C23—41no—
33C2×C4⋊C4×C23—41no—
34C2×C4⋊C4×C23C2×C412yes—
35C2×C4⋊C4×C23C2×C411yescharacteristic
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23D411yescharacteristic
40C2×C22⋊C4×C23C2311yescharacteristic
41C2×C42×C23D411yescharacteristic
42C23×C4⋊C8—41no—
43C23×#99—42no—
44C23×C22⋊C8—22no—
45C4×C8×C23—41no—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—41no—
49C23×#95C2×C412yes—
50C24×C23C22⋊C411yescharacteristic
51C24×C23C2×D411yescharacteristic
52C2×D4×C23—41no—
53C2×D4×C23—41no—
54C2×D4×C23D812yes—
55C2×D4×C23SD1612yes—
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×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×C23C2×D411yescharacteristic
73C2×C8×C23—42no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—42no—
76C4⋊C4×C23—42no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—44no—
80C4⋊C4×C23—82no—
81C22⋊C4×C23—41no—
82C22⋊C4×C23—41no—
83C22⋊C4×C23—42no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—42no—
87C22⋊C4×C23—42no—
88C42×C23—21no—
89C42×C23—21no—
90#11874C2311yesmaximal, characteristic, direct product factor, Sylow
91C23×C23—21no—
92C23×C23—21no—
93C23×C23—21no—
94C23×C23—21no—
95C23×C23—41no—
96C23×C23—41no—
97C23×C23—41no—
98C23×C23—41no—
99C23×C23#10212yes—
100C23×C23C2×C22⋊C411yescharacteristic
101C23×C23C2×D811yescharacteristic
102C23×C23C2×SD1611yescharacteristic
103D4×C23—42no—
104D4×C23—42no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—22no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—41no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—42no—
123C2×C4×C23—81no—
124C2×C4×C23—84no—
125C2×C4×C23—82no—
126C2×C4×C23—82no—
127C2×C4×C23—82no—
128C2×C4×C23#9911yescharacteristic
129C2×C4×C23#9911yescharacteristic
130C2×C4×C23#10212yes—
131C8×C23—82no—
132#2486C2×C2311yescharacteristic
133#1799C2×C2311yescharacteristic
134#1460C2×C2311yescharacteristic
135#1344C2×C2314yescomplemented
136C22×C23—21no—
137C22×C23—21no—
138C22×C23—22no—
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—42no—
142C22×C23—41no—
143C22×C23—41no—
144C22×C23—42no—
145C22×C23—41no—
146C22×C23—41no—
147C22×C23—41no—
148C22×C23—41no—
149C22×C23—81no—
150C22×C23C2×#9911yescharacteristic
151C22×C23C2×#10211yescharacteristic
152C22×C23#42111yescharacteristic
153C4×C23—21no—
154C4×C23—22no—
155C4×C23—41no—
156C4×C23—41no—
157C4×C23—41no—
158C4×C23—82no—
159C4×C23—82no—
160C4×C23—162no—
161C4×C23#35212yes—
162C2×#123—21no—
163C2×C4×D4C22×C2311yescharacteristic
164C22×C4⋊C4C4×C2311yescharacteristic, complemented
165#431—21no—
166C2×#99—21no—
167#398C4×C2312yescomplemented
168#397C22×C2312yes—
169#389—21no—
170#381—21no—
171#380—21no—
172#380—21no—
173#380C4×C2311yescharacteristic, complemented
174#342C22×C2312yes—
175#323—21no—
176#323C22×C2312yes—
177C2×C23—21no—
178C2×C23—21no—
179C2×C23—41no—
180C2×C23—42no—
181C2×C23—41no—
182C2×C23#151911yescharacteristic
183C2×C23C2×#35211yescharacteristic
184C2×C23#212911yescharacteristic
185C22×D4D4×C2311yescharacteristic
186C23×C4—21no—
187C23×C4C2×C4×C2311yescharacteristic
188C23×C4D4×C2311yescharacteristic
189#123—42no—
190#123—42no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C2×C4×C2312yes—
200C2×C4⋊C4C2×C4×C2311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4D4×C2311yescharacteristic
205C2×C22⋊C4C23×C2311yescharacteristic
206C2×C42D4×C2311yescharacteristic
207C4⋊C8—41no—
208#99—42no—
209C22⋊C8—22no—
210C4×C8—41no—
211#95—22no—
212#95—22no—
213#95—41no—
214#95C2×C4×C2312yes—
215C23#1187411yescharacteristic, direct product factor, Sylow
216C24C22⋊C4×C2311yescharacteristic
217C24C2×D4×C2311yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4D8×C2312yes—
221C2×D4SD16×C2312yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4C22⋊C4×C2311yescharacteristic
235C22×C4C22⋊C4×C2311yescharacteristic
236C22×C4C22⋊C4×C2311yescharacteristic
237C22×C4C22×C4×C2311yescharacteristic
238C22×C4C2×D4×C2311yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—44no—
246C4⋊C4—82no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C23×#10212yes—
265C23C2×C22⋊C4×C2311yescharacteristic
266C23C2×D8×C2311yescharacteristic
267C23C2×SD16×C2311yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—84no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4C23×#9911yescharacteristic
294C2×C4C23×#9911yescharacteristic
295C2×C4C23×#10212yes—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×C23×#9911yescharacteristic
312C22C2×C23×#10211yescharacteristic
313C22C23×#42111yescharacteristic
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—162no—
322C4C23×#35212yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2C23×#151911yescharacteristic
329C2C2×C23×#35211yescharacteristic
330C2C23×#212911yescharacteristic
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×#1344C2semidirect
7. C22×C4⋊C4×C23C4semidirect
10. C23×#398C4semidirect
16. C23×#380C4semidirect
90. #11874C23direct (N × H)
135. #1344C2×C23semidirect
164. C22×C4⋊C4C4×C23semidirect
167. #398C4×C23semidirect
173. #380C4×C23semidirect
215. C23#11874direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24862
C23×#17992
C23×#14602
C23×#13442
#1187423