Finite Group Databasebeta

← C23×#7641 #19155267

Subgroups — C23×#7641 #19155267

Order 5888.

582 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#389C211yesmaximal, characteristic
2C23×#1797C211yesmaximal, characteristic
3C23×#1797C211yesmaximal, characteristic
4C23×#1271C214yesmaximal, complemented
5C22×C4⋊C4×C23—21no—
6C22×C4⋊C4×C23C411yescharacteristic
7C22×C4⋊C4×C23C411yescharacteristic
8C22×C42×C23C2211yescharacteristic
9C2×C23×C4⋊C8—21no—
10C2×C23×C4⋊C8—21no—
11C2×C23×C22⋊C8—21no—
12C2×C23×C22⋊C8—21no—
13C23×#389—21no—
14C23×#389—21no—
15C23×#389C412yes—
16C23×#389C2212yes—
17C2×C23×#95—21no—
18C23×#336C2212yes—
19C23×#336C2212yes—
20C23×C4×C23—21no—
21C23×C4×C23—21no—
22C23×C4×C23C2×C411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C22×C8×C23—22no—
26C22×C8×C23—22no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—24no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—42no—
33C2×C4⋊C4×C23—42no—
34C2×C4⋊C4×C23—42no—
35C2×C4⋊C4×C23C2×C412yes—
36C2×C4⋊C4×C23C2×C412yes—
37C2×C42×C23—42no—
38C2×C42×C23C2×C411yescharacteristic
39C2×C42×C23D411yescharacteristic
40C2×C42×C23D411yescharacteristic
41C2×C42×C23C2311yescharacteristic
42C23×C4⋊C8—42no—
43C23×C4⋊C8—42no—
44C23×C22⋊C8—42no—
45C23×C22⋊C8—42no—
46C23×#95—22no—
47C23×#95—22no—
48C24×C23C22⋊C411yescharacteristic
49C22×C4×C23—22no—
50C22×C4×C23—22no—
51C22×C4×C23—22no—
52C22×C4×C23—24no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—41no—
56C22×C4×C23—41no—
57C22×C4×C23—41no—
58C22×C4×C23—41no—
59C22×C4×C23—42no—
60C22×C4×C23—42no—
61C22×C4×C23—42no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23C22⋊C411yescharacteristic
65C22×C4×C23C22⋊C411yescharacteristic
66C22×C4×C23C22⋊C411yescharacteristic
67C22×C4×C23D812yes—
68C22×C4×C23SD1612yes—
69C22×C4×C23SD1612yes—
70C22×C4×C23Q1612yes—
71C22×C4×C23C22×C411yescharacteristic
72C22×C4×C23C2×D411yescharacteristic
73C22×C4×C23C2×D411yescharacteristic
74C2×C8×C23—42no—
75C2×C8×C23—42no—
76C2×C8×C23—42no—
77C2×C8×C23—42no—
78C2×C8×C23—42no—
79C2×C8×C23—42no—
80C4⋊C4×C23—24no—
81C4⋊C4×C23—22no—
82C4⋊C4×C23—42no—
83C4⋊C4×C23—42no—
84C4⋊C4×C23—44no—
85C4⋊C4×C23—42no—
86C4⋊C4×C23—42no—
87C4⋊C4×C23M1614yes—
88C42×C23—22no—
89C42×C23—22no—
90C42×C23—41no—
91C42×C23—41no—
92#7641C2311yesmaximal, characteristic, direct product factor, Sylow
93C23×C23—21no—
94C23×C23—21no—
95C23×C23—21no—
96C23×C23—21no—
97C23×C23#10112yes—
98C23×C23#10212yes—
99C23×C23#10312yes—
100C23×C23C2×C22⋊C411yescharacteristic
101C2×C4×C23—28no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—24no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—48no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—84no—
126C2×C4×C23#9912yes—
127C2×C4×C23#10212yes—
128C2×C4×C23#10312yes—
129C2×C4×C23C2×M1612yes—
130C2×C4×C23C2×D811yescharacteristic
131C2×C4×C23C2×SD1611yescharacteristic
132C2×C4×C23C2×SD1611yescharacteristic
133C2×C4×C23C2×Q1611yescharacteristic
134C8×C23—82no—
135C8×C23—82no—
136C2×#389C2×C2311yescharacteristic
137#1797C2×C2311yescharacteristic
138#1797C2×C2311yescharacteristic
139#1271C2×C2314yescomplemented
140C22×C23—22no—
141C22×C23—22no—
142C22×C23—22no—
143C22×C23—22no—
144C22×C23—22no—
145C22×C23—22no—
146C22×C23#33214yes—
147C22×C23#40711yescharacteristic
148C22×C23C2×#9911yescharacteristic
149C22×C23C2×#10111yescharacteristic
150C22×C23C2×#10211yescharacteristic
151C22×C23C2×#10311yescharacteristic
152C22×C23#42111yescharacteristic
153C22×C23#42111yescharacteristic
154C4×C23—22no—
155C4×C23—22no—
156C4×C23—21no—
157C4×C23—22no—
158C4×C23—21no—
159C4×C23—21no—
160C4×C23—22no—
161C4×C23—21no—
162C4×C23—44no—
163C4×C23—82no—
164C22×C4⋊C4—21no—
165C22×C4⋊C4C4×C2311yescharacteristic
166C22×C4⋊C4C4×C2311yescharacteristic
167C22×C42C22×C2311yescharacteristic
168C2×C4⋊C8—21no—
169C2×C4⋊C8—21no—
170C2×C22⋊C8—21no—
171C2×C22⋊C8—21no—
172#389—21no—
173#389—21no—
174#389C4×C2312yes—
175#389C22×C2312yes—
176C2×#95—21no—
177#336C22×C2312yes—
178#336C22×C2312yes—
179C2×C23—24no—
180C2×C23#146311yescharacteristic
181C2×C23#148611yescharacteristic
182C2×C23#148711yescharacteristic
183C2×C23#151911yescharacteristic
184C2×C23#152011yescharacteristic
185C2×C23#153411yescharacteristic
186C2×C23C2×#33211yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C2×C4×C2311yescharacteristic
190C23×C4D4×C2311yescharacteristic
191C23×C4D4×C2311yescharacteristic
192C22×C8—22no—
193C22×C8—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×C4×C2312yes—
203C2×C4⋊C4C2×C4×C2312yes—
204C2×C42—42no—
205C2×C42C2×C4×C2311yescharacteristic
206C2×C42D4×C2311yescharacteristic
207C2×C42D4×C2311yescharacteristic
208C2×C42C23×C2311yescharacteristic
209C4⋊C8—42no—
210C4⋊C8—42no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213#95—22no—
214#95—22no—
215C23#764111yescharacteristic, direct product factor, Sylow
216C24C22⋊C4×C2311yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4C22⋊C4×C2311yescharacteristic
233C22×C4C22⋊C4×C2311yescharacteristic
234C22×C4C22⋊C4×C2311yescharacteristic
235C22×C4D8×C2312yes—
236C22×C4SD16×C2312yes—
237C22×C4SD16×C2312yes—
238C22×C4Q16×C2312yes—
239C22×C4C22×C4×C2311yescharacteristic
240C22×C4C2×D4×C2311yescharacteristic
241C22×C4C2×D4×C2311yescharacteristic
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4M16×C2314yes—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23C23×#10112yes—
265C23C23×#10212yes—
266C23C23×#10312yes—
267C23C2×C22⋊C4×C2311yescharacteristic
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—84no—
293C2×C4C23×#9912yes—
294C2×C4C23×#10212yes—
295C2×C4C23×#10312yes—
296C2×C4C2×M16×C2312yes—
297C2×C4C2×D8×C2311yescharacteristic
298C2×C4C2×SD16×C2311yescharacteristic
299C2×C4C2×SD16×C2311yescharacteristic
300C2×C4C2×Q16×C2311yescharacteristic
301C8—82no—
302C8—82no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22C23×#33214yes—
310C22C23×#40711yescharacteristic
311C22C2×C23×#9911yescharacteristic
312C22C2×C23×#10111yescharacteristic
313C22C2×C23×#10211yescharacteristic
314C22C2×C23×#10311yescharacteristic
315C22C23×#42111yescharacteristic
316C22C23×#42111yescharacteristic
317C4—22no—
318C4—22no—
319C4—21no—
320C4—22no—
321C4—21no—
322C4—21no—
323C4—22no—
324C4—21no—
325C4—44no—
326C4—82no—
327C2—24no—
328C2C23×#146311yescharacteristic
329C2C23×#148611yescharacteristic
330C2C23×#148711yescharacteristic
331C2C23×#151911yescharacteristic
332C2C23×#152011yescharacteristic
333C2C23×#153411yescharacteristic
334C2C2×C23×#33211yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C22×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1271C2semidirect
92. #7641C23direct (N × H)
139. #1271C2×C23semidirect
215. C23#7641direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3892
C23×#17972
C23×#12712
#764123