Finite Group Databasebeta

← C23×#11550 #19159176

Subgroups — C23×#11550 #19159176

Order 5888.

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

normal subgroups only

This group has 408 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×#2500C211yesmaximal, characteristic
2C23×#1464C211yesmaximal, characteristic, complemented
3C23×#1458C211yesmaximal, characteristic, complemented
4C23×#1324C212yesmaximal, complemented
5C23×#1318C212yesmaximal, complemented
6C2×C23×#121C411yescharacteristic
7C2×C23×#117C2211yescharacteristic
8C23×#432—21no—
9C23×#431—21no—
10C23×#394—21no—
11C23×#394—21no—
12C23×#390C411yescharacteristic
13C23×#390C2211yescharacteristic
14C23×#388—21no—
15C23×#388—21no—
16C23×#384C411yescharacteristic
17C23×#382C2211yescharacteristic
18C4×C22⋊C4×C23C411yescharacteristic
19C23×#324C2212yes—
20C23×#324C2212yes—
21C23×#323—21no—
22C23×#323—21no—
23C22×D4×C23—21no—
24C22×D4×C23C2×C411yescharacteristic
25C23×C4×C23C2×C411yescharacteristic
26C23×#121—41no—
27C23×#121—41no—
28C23×#117—41no—
29C23×#117—41no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C4⋊C4×C23C2×C411yescharacteristic
34C2×C4⋊C4×C23C2311yescharacteristic
35C2×C22⋊C4×C23—21no—
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23D411yescharacteristic
42C2×C22⋊C4×C23D411yescharacteristic
43C2×C42×C23C2×C411yescharacteristic
44C2×C42×C23C2×C411yescharacteristic
45C2×C42×C23D411yescharacteristic
46C2×C42×C23D411yescharacteristic
47C23×C4⋊C8—41no—
48C23×C4⋊C8—41no—
49C23×C22⋊C8—22no—
50C23×C22⋊C8—22no—
51C23×#97—41no—
52C4×C8×C23—41no—
53C23×#95—21no—
54C23×#95—21no—
55C23×#95—21no—
56C24×C23—21no—
57C24×C23C22⋊C411yescharacteristic
58C2×D4×C23—22no—
59C2×D4×C23—41no—
60C2×D4×C23—41no—
61C2×D4×C23—42no—
62C2×D4×C23—81no—
63C2×D4×C23—81no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—22no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—41no—
71C22×C4×C23C22⋊C411yescharacteristic
72C22×C4×C23C22⋊C411yescharacteristic
73C22×C4×C23C22⋊C411yescharacteristic
74C22×C4×C23C22×C411yescharacteristic
75C22×C4×C23C2×D411yescharacteristic
76C22×C4×C23C2×D411yescharacteristic
77C2×C8×C23—42no—
78C2×C8×C23—42no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—41no—
81C4⋊C4×C23—41no—
82C4⋊C4×C23—81no—
83C4⋊C4×C23—81no—
84C22⋊C4×C23—41no—
85C22⋊C4×C23—41no—
86C22⋊C4×C23—42no—
87C22⋊C4×C23—42no—
88C22⋊C4×C23—42no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—41no—
91C22⋊C4×C23—81no—
92C22⋊C4×C23—81no—
93C42×C23—21no—
94C42×C23—21no—
95C42×C23—21no—
96C42×C23—21no—
97C42×C23—22no—
98C42×C23—41no—
99#11550C2311yesmaximal, characteristic, direct product factor, Sylow
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—21no—
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×C23—41no—
112C23×C23C2×C22⋊C411yescharacteristic
113D4×C23—82no—
114D4×C23—84no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—22no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—42no—
125C2×C4×C23—42no—
126C2×C4×C23—41no—
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#9911yescharacteristic
139C2×C4×C23#9911yescharacteristic
140C2×C4×C23#13112yes—
141C8×C23—82no—
142C8×C23—82no—
143#2500C2×C2311yescharacteristic
144#1464C2×C2311yescharacteristic, complemented
145#1458C2×C2311yescharacteristic, complemented
146#1324C2×C2312yescomplemented
147#1318C2×C2312yescomplemented
148C22×C23—21no—
149C22×C23—21no—
150C22×C23—41no—
151C22×C23—41no—
152C22×C23—41no—
153C22×C23—81no—
154C22×C23—81no—
155C22×C23—81no—
156C22×C23—81no—
157C22×C23—81no—
158C22×C23—81no—
159C22×C23—81no—
160C22×C23—81no—
161C22×C23—81no—
162C22×C23#40811yescharacteristic
163C22×C23C2×#9911yescharacteristic
164C22×C23#41311yescharacteristic
165C4×C23—21no—
166C4×C23—21no—
167C4×C23—22no—
168C4×C23—42no—
169C4×C23—41no—
170C4×C23—41no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23—81no—
174C4×C23—81no—
175C2×#121C4×C2311yescharacteristic
176C2×#117C22×C2311yescharacteristic
177#432—21no—
178#431—21no—
179#394—21no—
180#394—21no—
181#390C4×C2311yescharacteristic
182#390C22×C2311yescharacteristic
183#388—21no—
184#388—21no—
185#384C4×C2311yescharacteristic
186#382C22×C2311yescharacteristic
187C4×C22⋊C4C4×C2311yescharacteristic
188#324C22×C2312yes—
189#324C22×C2312yes—
190#323—21no—
191#323—21no—
192C2×C23—21no—
193C2×C23—21no—
194C2×C23—81no—
195C2×C23—161no—
196C2×C23#146811yescharacteristic
197C2×C23#212611yescharacteristic
198C2×C23#213411yescharacteristic
199C22×D4—21no—
200C22×D4C2×C4×C2311yescharacteristic
201C23×C4C2×C4×C2311yescharacteristic
202#121—41no—
203#121—41no—
204#117—41no—
205#117—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×C4×C2311yescharacteristic
209C2×C4⋊C4C2×C4×C2311yescharacteristic
210C2×C4⋊C4C23×C2311yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4D4×C2311yescharacteristic
218C2×C22⋊C4D4×C2311yescharacteristic
219C2×C42C2×C4×C2311yescharacteristic
220C2×C42C2×C4×C2311yescharacteristic
221C2×C42D4×C2311yescharacteristic
222C2×C42D4×C2311yescharacteristic
223C4⋊C8—41no—
224C4⋊C8—41no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227#97—41no—
228C4×C8—41no—
229#95—21no—
230#95—21no—
231#95—21no—
232C23#1155011yescharacteristic, direct product factor, Sylow
233C24—21no—
234C24C22⋊C4×C2311yescharacteristic
235C2×D4—22no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—81no—
240C2×D4—81no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4C22⋊C4×C2311yescharacteristic
249C22×C4C22⋊C4×C2311yescharacteristic
250C22×C4C22⋊C4×C2311yescharacteristic
251C22×C4C22×C4×C2311yescharacteristic
252C22×C4C2×D4×C2311yescharacteristic
253C22×C4C2×D4×C2311yescharacteristic
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×C22⋊C4×C2311yescharacteristic
289D4—82no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4C23×#9911yescharacteristic
315C2×C4C23×#9911yescharacteristic
316C2×C4C23×#13112yes—
317C8—82no—
318C8—82no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C23×#40811yescharacteristic
334C22C2×C23×#9911yescharacteristic
335C22C23×#41311yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—161no—
350C2C23×#146811yescharacteristic
351C2C23×#212611yescharacteristic
352C2C23×#213411yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC2×C23×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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×#1464C2semidirect
3. C23×#1458C2semidirect
4. C23×#1324C2semidirect
5. C23×#1318C2semidirect
99. #11550C23direct (N × H)
144. #1464C2×C23semidirect
145. #1458C2×C23semidirect
146. #1324C2×C23semidirect
147. #1318C2×C23semidirect
232. C23#11550direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25002
C23×#14642
C23×#14582
C23×#13242
C23×#13182
#1155023