Finite Group Databasebeta

← C23×#11840 #19159466

Subgroups — C23×#11840 #19159466

Order 5888.

446 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2598C211yesmaximal, characteristic
2C23×#1820C211yesmaximal, characteristic, complemented
3C23×#1458C211yesmaximal, characteristic, complemented
4C23×#1352C212yesmaximal, complemented
5C23×#1343C212yesmaximal
6C2×C4×D4×C23C411yescharacteristic, complemented
7C2×C23×#117C2211yescharacteristic
8C23×#431—21no—
9C23×#400C411yescharacteristic, complemented
10C23×#399C2211yescharacteristic
11C23×#394C411yescharacteristic, complemented
12C23×#392C2211yescharacteristic
13C23×#388—21no—
14C23×#386—21no—
15C23×#386—21no—
16C23×#385—21no—
17C23×#383C411yescharacteristic, complemented
18C23×#381—21no—
19C23×#342—21no—
20C23×#342C2212yes—
21C23×#324—21no—
22C23×#323C2212yes—
23C22×D4×C23—21no—
24C22×D4×C23C2×C411yescharacteristic
25C23×C4×C23—21no—
26C23×C4×C23D411yescharacteristic
27C4×D4×C23—41no—
28C4×D4×C23—41no—
29C23×#117—41no—
30C23×#117—41no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C4⋊C4×C23C2×C411yescharacteristic
34C2×C4⋊C4×C23D411yescharacteristic
35C2×C22⋊C4×C23—21no—
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23D411yescharacteristic
42C2×C22⋊C4×C23C2311yescharacteristic
43C2×C42×C23C2×C411yescharacteristic
44C2×C42×C23D411yescharacteristic
45C23×C4⋊C8—41no—
46C23×C22⋊C8—22no—
47C4×C8×C23—41no—
48C23×#95—21no—
49C23×#95—21no—
50C23×#95—21no—
51C23×#95—21no—
52C23×#95—41no—
53C23×#95—41no—
54C23×#95C2×C411yescharacteristic
55C23×#95C2×C411yescharacteristic
56C24×C23—21no—
57C24×C23C2×D411yescharacteristic
58C2×D4×C23—41no—
59C2×D4×C23—42no—
60C2×D4×C23—42no—
61C2×D4×C23—42no—
62C2×D4×C23—81no—
63C22×C4×C23—22no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—41no—
70C22×C4×C23—42no—
71C22×C4×C23—42no—
72C22×C4×C23—81no—
73C22×C4×C23—81no—
74C22×C4×C23C22⋊C411yescharacteristic
75C22×C4×C23C22⋊C411yescharacteristic
76C22×C4×C23C22⋊C411yescharacteristic
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C22×C411yescharacteristic
79C22×C4×C23C2×D411yescharacteristic
80C2×C8×C23—42no—
81C4⋊C4×C23—41no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—41no—
84C4⋊C4×C23—81no—
85C22⋊C4×C23—41no—
86C22⋊C4×C23—41no—
87C22⋊C4×C23—42no—
88C22⋊C4×C23—42no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—41no—
91C22⋊C4×C23—42no—
92C22⋊C4×C23—42no—
93C22⋊C4×C23—81no—
94C22⋊C4×C23—81no—
95C42×C23—21no—
96C42×C23—21no—
97C42×C23—21no—
98C42×C23—21no—
99#11840C2311yesmaximal, characteristic, direct product factor, Sylow
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—22no—
103C23×C23—41no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23—41no—
110C23×C23—41no—
111C23×C23C2×C22⋊C411yescharacteristic
112C23×C23#13512yes—
113D4×C23—42no—
114D4×C23—42no—
115D4×C23—84no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
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—42no—
126C2×C4×C23—42no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
129C2×C4×C23—41no—
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—84no—
141C2×C4×C23—82no—
142C2×C4×C23—82no—
143C2×C4×C23—82no—
144C2×C4×C23#9911yescharacteristic
145C2×C4×C23#9911yescharacteristic
146C8×C23—82no—
147#2598C2×C2311yescharacteristic
148#1820C2×C2311yescharacteristic, complemented
149#1458C2×C2311yescharacteristic, complemented
150#1352C2×C2312yescomplemented
151#1343C2×C2312yes—
152C22×C23—21no—
153C22×C23—21no—
154C22×C23—22no—
155C22×C23—42no—
156C22×C23—42no—
157C22×C23—42no—
158C22×C23—41no—
159C22×C23—41no—
160C22×C23—42no—
161C22×C23—81no—
162C22×C23—81no—
163C22×C23—81no—
164C22×C23—81no—
165C22×C23—81no—
166C22×C23C2×#9911yescharacteristic
167C22×C23#41611yescharacteristic
168C22×C23#42111yescharacteristic
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23—41no—
175C4×C23—41no—
176C4×C23—81no—
177C4×C23—81no—
178C4×C23—81no—
179C4×C23—162no—
180C2×C4×D4C4×C2311yescharacteristic, complemented
181C2×#117C22×C2311yescharacteristic
182#431—21no—
183#400C4×C2311yescharacteristic, complemented
184#399C22×C2311yescharacteristic
185#394C4×C2311yescharacteristic, complemented
186#392C22×C2311yescharacteristic
187#388—21no—
188#386—21no—
189#386—21no—
190#385—21no—
191#383C4×C2311yescharacteristic, complemented
192#381—21no—
193#342—21no—
194#342C22×C2312yes—
195#324—21no—
196#323C22×C2312yes—
197C2×C23—21no—
198C2×C23—21no—
199C2×C23—42no—
200C2×C23—82no—
201C2×C23#151611yescharacteristic
202C2×C23#212011yescharacteristic
203C2×C23#212811yescharacteristic
204C22×D4—21no—
205C22×D4C2×C4×C2311yescharacteristic
206C23×C4—21no—
207C23×C4D4×C2311yescharacteristic
208C4×D4—41no—
209C4×D4—41no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×C4×C2311yescharacteristic
214C2×C4⋊C4C2×C4×C2311yescharacteristic
215C2×C4⋊C4D4×C2311yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4D4×C2311yescharacteristic
223C2×C22⋊C4C23×C2311yescharacteristic
224C2×C42C2×C4×C2311yescharacteristic
225C2×C42D4×C2311yescharacteristic
226C4⋊C8—41no—
227C22⋊C8—22no—
228C4×C8—41no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—41no—
234#95—41no—
235#95C2×C4×C2311yescharacteristic
236#95C2×C4×C2311yescharacteristic
237C23#1184011yescharacteristic, direct product factor, Sylow
238C24—21no—
239C24C2×D4×C2311yescharacteristic
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—81no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4C22⋊C4×C2311yescharacteristic
257C22×C4C22⋊C4×C2311yescharacteristic
258C22×C4C22⋊C4×C2311yescharacteristic
259C22×C4C22⋊C4×C2311yescharacteristic
260C22×C4C22×C4×C2311yescharacteristic
261C22×C4C2×D4×C2311yescharacteristic
262C2×C8—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C23—21no—
282C23—21no—
283C23—22no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊C4×C2311yescharacteristic
293C23C23×#13512yes—
294D4—42no—
295D4—42no—
296D4—84no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—84no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4C23×#9911yescharacteristic
326C2×C4C23×#9911yescharacteristic
327C8—82no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22C2×C23×#9911yescharacteristic
343C22C23×#41611yescharacteristic
344C22C23×#42111yescharacteristic
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—162no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—82no—
360C2C23×#151611yescharacteristic
361C2C23×#212011yescharacteristic
362C2C23×#212811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1820C2semidirect
3. C23×#1458C2semidirect
4. C23×#1352C2semidirect
6. C2×C4×D4×C23C4semidirect
9. C23×#400C4semidirect
11. C23×#394C4semidirect
17. C23×#383C4semidirect
99. #11840C23direct (N × H)
148. #1820C2×C23semidirect
149. #1458C2×C23semidirect
150. #1352C2×C23semidirect
180. C2×C4×D4C4×C23semidirect
183. #400C4×C23semidirect
185. #394C4×C23semidirect
191. #383C4×C23semidirect
237. C23#11840direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25982
C23×#18202
C23×#14582
C23×#13522
C23×#13432
#1184023