Finite Group Databasebeta

← C23×#11577 #19159203

Subgroups — C23×#11577 #19159203

Order 5888.

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

normal subgroups only

This group has 424 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×#2579C211yesmaximal, characteristic
2C23×#1465C211yesmaximal, characteristic
3C23×#1460C211yesmaximal, characteristic
4C23×#1326C212yesmaximal, complemented
5C23×#1326C212yesmaximal, complemented
6C2×C4×D4×C23C2211yescharacteristic
7C22×C4⋊C4×C23C411yescharacteristic
8C23×#431—21no—
9C23×#407—21no—
10C23×#398—21no—
11C23×#396—21no—
12C23×#387—21no—
13C23×#386C411yescharacteristic
14C23×#386C411yescharacteristic
15C23×#385—21no—
16C23×#385C2211yescharacteristic
17C23×#385C2211yescharacteristic
18C23×#380C411yescharacteristic
19C23×#324—21no—
20C23×#323—21no—
21C23×#323C2212yes—
22C23×#323C2212yes—
23C22×D4×C23C2×C411yescharacteristic
24C23×C4×C23C2×C411yescharacteristic
25C23×C4×C23C2×C411yescharacteristic
26C23×C4×C23C2×C411yescharacteristic
27C23×C4×C23D411yescharacteristic
28C2×M16×C23—41no—
29C4×D4×C23—41no—
30C4×D4×C23—41no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—41no—
37C2×C4⋊C4×C23—41no—
38C2×C4⋊C4×C23D411yescharacteristic
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23C2×C411yescharacteristic
42C2×C22⋊C4×C23C2×C411yescharacteristic
43C2×C22⋊C4×C23D411yescharacteristic
44C2×C22⋊C4×C23C2311yescharacteristic
45C2×C42×C23D411yescharacteristic
46C23×C4⋊C8—41no—
47C23×C22⋊C8—22no—
48C23×C22⋊C8—22no—
49C23×C22⋊C8—41no—
50C4×C8×C23—41no—
51C23×#95—21no—
52C23×#95—21no—
53C23×#95—21no—
54C23×#95—21no—
55C23×#95—21no—
56C24×C23C22⋊C411yescharacteristic
57C24×C23C22×C411yescharacteristic
58C2×D4×C23—22no—
59C2×D4×C23—22no—
60C2×D4×C23—41no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—42no—
70C22×C4×C23—41no—
71C22×C4×C23—41no—
72C22×C4×C23—81no—
73C22×C4×C23—81no—
74C22×C4×C23—81no—
75C22×C4×C23C22⋊C411yescharacteristic
76C22×C4×C23C22⋊C411yescharacteristic
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C2×D411yescharacteristic
79C22×C4×C23C2×D411yescharacteristic
80M16×C23—82no—
81C2×C8×C23—42no—
82C2×C8×C23—42no—
83C4⋊C4×C23—41no—
84C4⋊C4×C23—42no—
85C4⋊C4×C23—42no—
86C4⋊C4×C23—41no—
87C4⋊C4×C23—41no—
88C4⋊C4×C23—81no—
89C4⋊C4×C23—81no—
90C22⋊C4×C23—22no—
91C22⋊C4×C23—22no—
92C22⋊C4×C23—41no—
93C22⋊C4×C23—41no—
94C22⋊C4×C23—42no—
95C22⋊C4×C23—81no—
96C42×C23—21no—
97C42×C23—21no—
98#11577C2311yesmaximal, characteristic, direct product factor, Sylow
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—21no—
103C23×C23—21no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23C2×C22⋊C411yescharacteristic
109C23×C23#13112yes—
110D4×C23—44no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—22no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—41no—
118C2×C4×C23—42no—
119C2×C4×C23—41no—
120C2×C4×C23—42no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—41no—
124C2×C4×C23—41no—
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—81no—
136C2×C4×C23—81no—
137C2×C4×C23—81no—
138C2×C4×C23—82no—
139C2×C4×C23#9911yescharacteristic
140C2×C4×C23#9911yescharacteristic
141C2×C4×C23#10112yes—
142C8×C23—82no—
143C8×C23—82no—
144#2579C2×C2311yescharacteristic
145#1465C2×C2311yescharacteristic
146#1460C2×C2311yescharacteristic
147#1326C2×C2312yescomplemented
148#1326C2×C2312yescomplemented
149C22×C23—21no—
150C22×C23—21no—
151C22×C23—22no—
152C22×C23—42no—
153C22×C23—42no—
154C22×C23—42no—
155C22×C23—41no—
156C22×C23—41no—
157C22×C23—41no—
158C22×C23—41no—
159C22×C23—41no—
160C22×C23—81no—
161C22×C23#40811yescharacteristic
162C22×C23C2×#9911yescharacteristic
163C22×C23C2×#10111yescharacteristic
164C4×C23—21no—
165C4×C23—21no—
166C4×C23—22no—
167C4×C23—41no—
168C4×C23—42no—
169C4×C23—41no—
170C4×C23—41no—
171C4×C23—81no—
172C4×C23—81no—
173C4×C23—81no—
174C2×C4×D4C22×C2311yescharacteristic
175C22×C4⋊C4C4×C2311yescharacteristic
176#431—21no—
177#407—21no—
178#398—21no—
179#396—21no—
180#387—21no—
181#386C4×C2311yescharacteristic
182#386C4×C2311yescharacteristic
183#385—21no—
184#385C22×C2311yescharacteristic
185#385C22×C2311yescharacteristic
186#380C4×C2311yescharacteristic
187#324—21no—
188#323—21no—
189#323C22×C2312yes—
190#323C22×C2312yes—
191C2×C23—21no—
192C2×C23—21no—
193C2×C23—42no—
194C2×C23—81no—
195C2×C23#147211yescharacteristic
196C2×C23#212011yescharacteristic
197C2×C23#212111yescharacteristic
198C22×D4C2×C4×C2311yescharacteristic
199C23×C4C2×C4×C2311yescharacteristic
200C23×C4C2×C4×C2311yescharacteristic
201C23×C4C2×C4×C2311yescharacteristic
202C23×C4D4×C2311yescharacteristic
203C2×M16—41no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4D4×C2311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4C2×C4×C2311yescharacteristic
217C2×C22⋊C4C2×C4×C2311yescharacteristic
218C2×C22⋊C4D4×C2311yescharacteristic
219C2×C22⋊C4C23×C2311yescharacteristic
220C2×C42D4×C2311yescharacteristic
221C4⋊C8—41no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C4×C8—41no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C23#1157711yescharacteristic, direct product factor, Sylow
232C24C22⋊C4×C2311yescharacteristic
233C24C22×C4×C2311yescharacteristic
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C22⋊C4×C2311yescharacteristic
252C22×C4C22⋊C4×C2311yescharacteristic
253C22×C4C22⋊C4×C2311yescharacteristic
254C22×C4C2×D4×C2311yescharacteristic
255C22×C4C2×D4×C2311yescharacteristic
256M16—82no—
257C2×C8—42no—
258C2×C8—42no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—81no—
272C42—21no—
273C42—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×C22⋊C4×C2311yescharacteristic
284C23C23×#13112yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
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—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4C23×#9911yescharacteristic
315C2×C4C23×#9911yescharacteristic
316C2×C4C23×#10112yes—
317C8—82no—
318C8—82no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22C23×#40811yescharacteristic
332C22C2×C23×#9911yescharacteristic
333C22C2×C23×#10111yescharacteristic
334C4—22no—
335C4—21no—
336C4—21no—
337C4—41no—
338C4—42no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2C23×#147211yescharacteristic
349C2C23×#212011yescharacteristic
350C2C23×#212111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1326C2semidirect
5. C23×#1326C2semidirect
98. #11577C23direct (N × H)
147. #1326C2×C23semidirect
148. #1326C2×C23semidirect
231. C23#11577direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25792
C23×#14652
C23×#14602
C23×#13262
#1157723