Finite Group Databasebeta

← C23×#11852 #19159478

Subgroups — C23×#11852 #19159478

Order 5888.

438 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2591C211yesmaximal, characteristic
2C23×#1821C211yesmaximal, characteristic, complemented
3C2×C23×#324C211yesmaximal, characteristic, complemented
4C23×#1353C212yesmaximal, complemented
5C23×#1345C212yesmaximal, complemented
6C2×C23×#121C411yescharacteristic
7C22×C4⋊C4×C23C2211yescharacteristic
8C23×#407—21no—
9C2×C23×C22⋊C8—21no—
10C23×#400C411yescharacteristic
11C23×#400C2211yescharacteristic
12C23×#394C411yescharacteristic
13C23×#394C2211yescharacteristic
14C23×#388—21no—
15C23×#388—21no—
16C23×#386—21no—
17C23×#386—21no—
18C4×C22⋊C4×C23C411yescharacteristic
19C23×#343—21no—
20C23×#343C2212yes—
21C23×#324—21no—
22C23×#324C2212yes—
23C22×D4×C23—21no—
24C22×D4×C23C2×C411yescharacteristic
25C23×C4×C23C2×C411yescharacteristic
26C23×C4×C23D411yescharacteristic
27C23×C4×C23D411yescharacteristic
28C2×M16×C23—22no—
29C22×C8×C23—41no—
30C23×#121—41no—
31C23×#121—41no—
32C2×C4⋊C4×C23—41no—
33C2×C4⋊C4×C23—41no—
34C2×C4⋊C4×C23C2×C411yescharacteristic
35C2×C4⋊C4×C23C2×C411yescharacteristic
36C2×C4⋊C4×C23D411yescharacteristic
37C2×C4⋊C4×C23D411yescharacteristic
38C2×C4⋊C4×C23C2311yescharacteristic
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C42×C23—21no—
46C23×C22⋊C8—22no—
47C23×C22⋊C8—41no—
48C23×C22⋊C8—41no—
49C23×C22⋊C8—41no—
50C23×#95—21no—
51C23×#95—21no—
52C23×#95—21no—
53C23×#95—21no—
54C23×#95C2×C411yescharacteristic
55C23×#95C2×C411yescharacteristic
56C24×C23—21no—
57C24×C23C22⋊C411yescharacteristic
58C2×D4×C23—41no—
59C2×D4×C23—42no—
60C2×D4×C23—42no—
61C2×D4×C23—41no—
62C2×D4×C23—41no—
63C2×D4×C23—81no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—22no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—41no—
75C22×C4×C23—81no—
76C22×C4×C23—81no—
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23C22×C411yescharacteristic
81C22×C4×C23C2×D411yescharacteristic
82C22×C4×C23C2×D411yescharacteristic
83M16×C23—42no—
84M16×C23—42no—
85C2×C8×C23—42no—
86C2×C8×C23—42no—
87C2×C8×C23—82no—
88C4⋊C4×C23—41no—
89C4⋊C4×C23—41no—
90C4⋊C4×C23—41no—
91C4⋊C4×C23—41no—
92C4⋊C4×C23—41no—
93C22⋊C4×C23—42no—
94C22⋊C4×C23—42no—
95C22⋊C4×C23—42no—
96C22⋊C4×C23—42no—
97C22⋊C4×C23—81no—
98C22⋊C4×C23—81no—
99C42×C23—81no—
100#11852C2311yesmaximal, characteristic, direct product factor, Sylow
101C23×C23—22no—
102C23×C23—21no—
103C23×C23—21no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23—41no—
110C23×C23—41no—
111C23×C23—41no—
112C23×C23#9911yescharacteristic
113C23×C23#9911yescharacteristic
114C23×C23C2×C22⋊C411yescharacteristic
115D4×C23—82no—
116D4×C23—82no—
117D4×C23—82no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—42no—
128C2×C4×C23—42no—
129C2×C4×C23—42no—
130C2×C4×C23—41no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—81no—
144C2×C4×C23—81no—
145C2×C4×C23—81no—
146C2×C4×C23#13512yes—
147C8×C23—82no—
148C8×C23—82no—
149#2591C2×C2311yescharacteristic
150#1821C2×C2311yescharacteristic, complemented
151C2×#324C2×C2311yescharacteristic, complemented
152#1353C2×C2312yescomplemented
153#1345C2×C2312yescomplemented
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—21no—
160C22×C23—21no—
161C22×C23—42no—
162C22×C23—42no—
163C22×C23—81no—
164C22×C23—81no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23—81no—
169C22×C23—81no—
170C22×C23#35512yes—
171C22×C23C2×#9911yescharacteristic
172C22×C23#41611yescharacteristic
173C22×C23#42111yescharacteristic
174C4×C23—22no—
175C4×C23—41no—
176C4×C23—41no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—41no—
180C4×C23—81no—
181C4×C23—81no—
182C4×C23—81no—
183C2×#121C4×C2311yescharacteristic
184C22×C4⋊C4C22×C2311yescharacteristic
185#407—21no—
186C2×C22⋊C8—21no—
187#400C4×C2311yescharacteristic
188#400C22×C2311yescharacteristic
189#394C4×C2311yescharacteristic
190#394C22×C2311yescharacteristic
191#388—21no—
192#388—21no—
193#386—21no—
194#386—21no—
195C4×C22⋊C4C4×C2311yescharacteristic
196#343—21no—
197#343C22×C2312yes—
198#324—21no—
199#324C22×C2312yes—
200C2×C23—21no—
201C2×C23—21no—
202C2×C23—22no—
203C2×C23—41no—
204C2×C23—161no—
205C2×C23#151611yescharacteristic
206C2×C23#212111yescharacteristic
207C2×C23C2×#35511yescharacteristic
208C22×D4—21no—
209C22×D4C2×C4×C2311yescharacteristic
210C23×C4C2×C4×C2311yescharacteristic
211C23×C4D4×C2311yescharacteristic
212C23×C4D4×C2311yescharacteristic
213C2×M16—22no—
214C22×C8—41no—
215#121—41no—
216#121—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C2×C4×C2311yescharacteristic
220C2×C4⋊C4C2×C4×C2311yescharacteristic
221C2×C4⋊C4D4×C2311yescharacteristic
222C2×C4⋊C4D4×C2311yescharacteristic
223C2×C4⋊C4C23×C2311yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C22⋊C8—22no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234C22⋊C8—41no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95C2×C4×C2311yescharacteristic
240#95C2×C4×C2311yescharacteristic
241C23#1185211yescharacteristic, direct product factor, Sylow
242C24—21no—
243C24C22⋊C4×C2311yescharacteristic
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—81no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C22⋊C4×C2311yescharacteristic
264C22×C4C22⋊C4×C2311yescharacteristic
265C22×C4C22⋊C4×C2311yescharacteristic
266C22×C4C22×C4×C2311yescharacteristic
267C22×C4C2×D4×C2311yescharacteristic
268C22×C4C2×D4×C2311yescharacteristic
269M16—42no—
270M16—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—82no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C42—81no—
286C23—22no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C23×#9911yescharacteristic
298C23C23×#9911yescharacteristic
299C23C2×C22⋊C4×C2311yescharacteristic
300D4—82no—
301D4—82no—
302D4—82no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4C23×#13512yes—
332C8—82no—
333C8—82no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C23×#35512yes—
351C22C2×C23×#9911yescharacteristic
352C22C23×#41611yescharacteristic
353C22C23×#42111yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—161no—
368C2C23×#151611yescharacteristic
369C2C23×#212111yescharacteristic
370C2C2×C23×#35511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC2×C23×#121C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C22×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1821C2semidirect
3. C2×C23×#324C2semidirect
4. C23×#1353C2semidirect
5. C23×#1345C2semidirect
100. #11852C23direct (N × H)
150. #1821C2×C23semidirect
151. C2×#324C2×C23semidirect
152. #1353C2×C23semidirect
153. #1345C2×C23semidirect
241. C23#11852direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25912
C23×#18212
C2×C23×#3242
C23×#13532
C23×#13452
#1185223