Finite Group Databasebeta

← C23×#7644 #19155270

Subgroups — C23×#7644 #19155270

Order 5888.

402 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2295C211yesmaximal, characteristic
2C23×#1839C211yesmaximal, characteristic
3C23×#1831C211yesmaximal, characteristic
4C23×#1272C212yesmaximal
5C23×#1271C212yesmaximal
6C2×C4×Q8×C23C411yescharacteristic
7C2×C23×C4⋊C8—21no—
8C23×#391C411yescharacteristic
9C23×#391C2211yescharacteristic
10C23×#389C411yescharacteristic
11C23×#389C2211yescharacteristic
12C23×#387—21no—
13C23×#387—21no—
14C4×C4⋊C4×C23—21no—
15C4×C4⋊C4×C23C411yescharacteristic
16C23×#376—21no—
17C43×C23C2211yescharacteristic
18C23×#336—21no—
19C23×#336—21no—
20C23×#336—21no—
21C23×#336C2212yes—
22C23×#336C2212yes—
23C22×Q8×C23C2×C411yescharacteristic
24C22×C8×C23—22no—
25C22×C8×C23—22no—
26C4×Q8×C23—22no—
27C4×Q8×C23—22no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23C2×C411yescharacteristic
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C4⋊C4×C23C2×C411yescharacteristic
34C2×C42×C23—21no—
35C2×C42×C23—21no—
36C2×C42×C23C2×C411yescharacteristic
37C2×C42×C23C2×C411yescharacteristic
38C2×C42×C23D411yescharacteristic
39C2×C42×C23D411yescharacteristic
40C2×C42×C23D411yescharacteristic
41C2×C42×C23D411yescharacteristic
42C2×C42×C23C2311yescharacteristic
43C23×C4⋊C8—42no—
44C23×#95—21no—
45C23×#95—21no—
46C23×#95—21no—
47C23×#95—21no—
48C23×#95—21no—
49C23×#95—21no—
50C23×#95—21no—
51C2×Q8×C23—21no—
52C2×Q8×C23—41no—
53C2×Q8×C23—41no—
54C2×Q8×C23M1612yes—
55C22×C4×C23—21no—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23C22⋊C411yescharacteristic
60C22×C4×C23C22⋊C411yescharacteristic
61C22×C4×C23C22⋊C411yescharacteristic
62C22×C4×C23C22⋊C411yescharacteristic
63C22×C4×C23C22×C411yescharacteristic
64C22×C4×C23C2×D411yescharacteristic
65C22×C4×C23C2×D411yescharacteristic
66C2×C8×C23—42no—
67C2×C8×C23—42no—
68C2×C8×C23—42no—
69C2×C8×C23—42no—
70C2×C8×C23—42no—
71C2×C8×C23—42no—
72C4⋊C4×C23—24no—
73C4⋊C4×C23—21no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—42no—
77C4⋊C4×C23—42no—
78C4⋊C4×C23M1612yes—
79C42×C23—24no—
80C42×C23—21no—
81C42×C23—21no—
82C42×C23—22no—
83C42×C23—41no—
84C42×C23—41no—
85C42×C23—41no—
86C42×C23—41no—
87C42×C23—42no—
88C42×C23SD1612yes—
89C42×C23Q1612yes—
90#7644C2311yesmaximal, characteristic, direct product factor, Sylow
91C23×C23C2×C22⋊C411yescharacteristic
92Q8×C23—42no—
93Q8×C23—42no—
94C2×C4×C23—21no—
95C2×C4×C23—21no—
96C2×C4×C23—21no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—22no—
106C2×C4×C23—22no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—41no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—41no—
117C2×C4×C23—41no—
118C2×C4×C23—41no—
119C2×C4×C23—41no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23#10312yes—
123C2×C4×C23#10312yes—
124C2×C4×C23C2×M1611yescharacteristic
125C2×C4×C23C2×M1611yescharacteristic
126C2×C4×C23C2×SD1611yescharacteristic
127C2×C4×C23C2×Q1611yescharacteristic
128C2×C4×C23#13512yes—
129C8×C23—82no—
130C8×C23—82no—
131#2295C2×C2311yescharacteristic
132#1839C2×C2311yescharacteristic
133#1831C2×C2311yescharacteristic
134#1272C2×C2312yes—
135#1271C2×C2312yes—
136C22×C23#40711yescharacteristic
137C22×C23#41011yescharacteristic
138C22×C23#41311yescharacteristic
139C22×C23C2×#10311yescharacteristic
140C22×C23#41611yescharacteristic
141C22×C23#42111yescharacteristic
142C22×C23#42111yescharacteristic
143C4×C23—22no—
144C4×C23—21no—
145C4×C23—21no—
146C4×C23—22no—
147C4×C23—21no—
148C4×C23—21no—
149C4×C23—22no—
150C4×C23—42no—
151C4×C23—41no—
152C4×C23—42no—
153C4×C23—41no—
154C4×C23—81no—
155C4×C23—81no—
156C2×C4×Q8C4×C2311yescharacteristic
157C2×C4⋊C8—21no—
158#391C4×C2311yescharacteristic
159#391C22×C2311yescharacteristic
160#389C4×C2311yescharacteristic
161#389C22×C2311yescharacteristic
162#387—21no—
163#387—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4C4×C2311yescharacteristic
166#376—21no—
167C43C22×C2311yescharacteristic
168#336—21no—
169#336—21no—
170#336—21no—
171#336C22×C2312yes—
172#336C22×C2312yes—
173C2×C23#146411yescharacteristic
174C2×C23#148711yescharacteristic
175C2×C23#148811yescharacteristic
176C2×C23#151811yescharacteristic
177C2×C23#152111yescharacteristic
178C2×C23#153511yescharacteristic
179C2×C23#154211yescharacteristic
180C22×Q8C2×C4×C2311yescharacteristic
181C22×C8—22no—
182C22×C8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C2×C4×C2311yescharacteristic
189C2×C4⋊C4C2×C4×C2311yescharacteristic
190C2×C4⋊C4C2×C4×C2311yescharacteristic
191C2×C42—21no—
192C2×C42—21no—
193C2×C42C2×C4×C2311yescharacteristic
194C2×C42C2×C4×C2311yescharacteristic
195C2×C42D4×C2311yescharacteristic
196C2×C42D4×C2311yescharacteristic
197C2×C42D4×C2311yescharacteristic
198C2×C42D4×C2311yescharacteristic
199C2×C42C23×C2311yescharacteristic
200C4⋊C8—42no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208C23#764411yescharacteristic, direct product factor, Sylow
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8M16×C2312yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C22⋊C4×C2311yescharacteristic
218C22×C4C22⋊C4×C2311yescharacteristic
219C22×C4C22⋊C4×C2311yescharacteristic
220C22×C4C22⋊C4×C2311yescharacteristic
221C22×C4C22×C4×C2311yescharacteristic
222C22×C4C2×D4×C2311yescharacteristic
223C22×C4C2×D4×C2311yescharacteristic
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C4⋊C4—24no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4M16×C2312yes—
237C42—24no—
238C42—21no—
239C42—21no—
240C42—22no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—42no—
246C42SD16×C2312yes—
247C42Q16×C2312yes—
248C23C2×C22⋊C4×C2311yescharacteristic
249Q8—42no—
250Q8—42no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4C23×#10312yes—
280C2×C4C23×#10312yes—
281C2×C4C2×M16×C2311yescharacteristic
282C2×C4C2×M16×C2311yescharacteristic
283C2×C4C2×SD16×C2311yescharacteristic
284C2×C4C2×Q16×C2311yescharacteristic
285C2×C4C23×#13512yes—
286C8—82no—
287C8—82no—
288C22C23×#40711yescharacteristic
289C22C23×#41011yescharacteristic
290C22C23×#41311yescharacteristic
291C22C2×C23×#10311yescharacteristic
292C22C23×#41611yescharacteristic
293C22C23×#42111yescharacteristic
294C22C23×#42111yescharacteristic
295C4—21no—
296C4—21no—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—42no—
303C4—41no—
304C4—42no—
305C4—41no—
306C4—81no—
307C4—81no—
308C2C23×#146411yescharacteristic
309C2C23×#148711yescharacteristic
310C2C23×#148811yescharacteristic
311C2C23×#151811yescharacteristic
312C2C23×#152111yescharacteristic
313C2C23×#153511yescharacteristic
314C2C23×#154211yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C43×C23 ◁ G

Complements

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

NH ≅ G/Ntype
90. #7644C23direct (N × H)
208. C23#7644direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#22952
C23×#18392
C23×#18312
C23×#12722
C23×#12712
#764423