Finite Group Databasebeta

← C23×#11798 #19159424

Subgroups — C23×#11798 #19159424

Order 5888.

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

normal subgroups only

This group has 486 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×#2392C211yesmaximal, characteristic
2C23×#1820C211yesmaximal, characteristic, complemented
3C23×#1464C211yesmaximal, characteristic, complemented
4C23×#1344C212yesmaximal, complemented
5C23×#1343C212yesmaximal
6C2×C4×D4×C23—21no—
7C2×C4×D4×C23C411yescharacteristic, complemented
8C2×C23×#117C2211yescharacteristic
9C23×#432—21no—
10C23×#398C411yescharacteristic, complemented
11C23×#397C2211yescharacteristic
12C23×#396—21no—
13C23×#394—21no—
14C23×#394C411yescharacteristic, complemented
15C23×#392C2211yescharacteristic
16C23×#381—21no—
17C4×C4⋊C4×C23C411yescharacteristic, complemented
18C4×C22⋊C4×C23—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—
29C4×D4×C23—44no—
30C23×#117—41no—
31C23×#117—41no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23C2×C411yescharacteristic
35C2×C4⋊C4×C23C2×C411yescharacteristic
36C2×C4⋊C4×C23D411yescharacteristic
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—22no—
43C2×C22⋊C4×C23D411yescharacteristic
44C2×C22⋊C4×C23C2311yescharacteristic
45C2×C42×C23—21no—
46C2×C42×C23C2×C411yescharacteristic
47C2×C42×C23D411yescharacteristic
48C23×C4⋊C8—41no—
49C23×C22⋊C8—22no—
50C23×#97—41no—
51C23×#95—21no—
52C23×#95—21no—
53C23×#95—41no—
54C23×#95—41no—
55C23×#95C2×C411yescharacteristic
56C23×#95C2×C411yescharacteristic
57C24×C23—21no—
58C24×C23C2×D411yescharacteristic
59C2×D4×C23—41no—
60C2×D4×C23—42no—
61C2×D4×C23—42no—
62C2×D4×C23—42no—
63C2×D4×C23—42no—
64C22×C4×C23—21no—
65C22×C4×C23—22no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—41no—
71C22×C4×C23—42no—
72C22×C4×C23—42no—
73C22×C4×C23—41no—
74C22×C4×C23—41no—
75C22×C4×C23—42no—
76C22×C4×C23C22⋊C411yescharacteristic
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23C22×C411yescharacteristic
81C22×C4×C23C2×D411yescharacteristic
82C2×C8×C23—42no—
83C4⋊C4×C23—41no—
84C4⋊C4×C23—41no—
85C4⋊C4×C23—41no—
86C4⋊C4×C23—42no—
87C4⋊C4×C23—42no—
88C22⋊C4×C23—41no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—42no—
91C22⋊C4×C23—42no—
92C22⋊C4×C23—42no—
93C22⋊C4×C23—42no—
94C22⋊C4×C23—42no—
95C22⋊C4×C23—42no—
96C22⋊C4×C23—42no—
97C42×C23—21no—
98C42×C23—21no—
99C42×C23—21no—
100C42×C23—21no—
101C42×C23—42no—
102#11798C2311yesmaximal, characteristic, direct product factor, Sylow
103C23×C23—21no—
104C23×C23—21no—
105C23×C23—22no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23—41no—
110C23×C23—41no—
111C23×C23—41no—
112C23×C23—41no—
113C23×C23—41no—
114C23×C23C2×C22⋊C411yescharacteristic
115C23×C23AGL(1,ℤ/8)12yes—
116D4×C23—42no—
117D4×C23—48no—
118D4×C23—42no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—22no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—42no—
130C2×C4×C23—42no—
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—82no—
144C2×C4×C23—82no—
145C2×C4×C23—82no—
146C2×C4×C23—82no—
147C2×C4×C23—82no—
148C2×C4×C23#9911yescharacteristic
149C2×C4×C23#9911yescharacteristic
150C8×C23—82no—
151#2392C2×C2311yescharacteristic
152#1820C2×C2311yescharacteristic, complemented
153#1464C2×C2311yescharacteristic, complemented
154#1344C2×C2312yescomplemented
155#1343C2×C2312yes—
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—22no—
159C22×C23—42no—
160C22×C23—42no—
161C22×C23—42no—
162C22×C23—41no—
163C22×C23—41no—
164C22×C23—42no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23—81no—
169C22×C23—81no—
170C22×C23C2×#9911yescharacteristic
171C22×C23#41811yescharacteristic
172C22×C23#42111yescharacteristic
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—42no—
176C4×C23—41no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—41no—
180C4×C23—41no—
181C4×C23—81no—
182C4×C23—81no—
183C4×C23—162no—
184C2×C4×D4—21no—
185C2×C4×D4C4×C2311yescharacteristic, complemented
186C2×#117C22×C2311yescharacteristic
187#432—21no—
188#398C4×C2311yescharacteristic, complemented
189#397C22×C2311yescharacteristic
190#396—21no—
191#394—21no—
192#394C4×C2311yescharacteristic, complemented
193#392C22×C2311yescharacteristic
194#381—21no—
195C4×C4⋊C4C4×C2311yescharacteristic, complemented
196C4×C22⋊C4—21no—
197#342—21no—
198#342C22×C2312yes—
199#324—21no—
200#323C22×C2312yes—
201C2×C23—21no—
202C2×C23—21no—
203C2×C23—42no—
204C2×C23—82no—
205C2×C23#151011yescharacteristic
206C2×C23#212011yescharacteristic
207C2×C23#212611yescharacteristic
208C22×D4—21no—
209C22×D4C2×C4×C2311yescharacteristic
210C23×C4—21no—
211C23×C4D4×C2311yescharacteristic
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—44no—
215#117—41no—
216#117—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C2×C4×C2311yescharacteristic
220C2×C4⋊C4C2×C4×C2311yescharacteristic
221C2×C4⋊C4D4×C2311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4D4×C2311yescharacteristic
229C2×C22⋊C4C23×C2311yescharacteristic
230C2×C42—21no—
231C2×C42C2×C4×C2311yescharacteristic
232C2×C42D4×C2311yescharacteristic
233C4⋊C8—41no—
234C22⋊C8—22no—
235#97—41no—
236#95—21no—
237#95—21no—
238#95—41no—
239#95—41no—
240#95C2×C4×C2311yescharacteristic
241#95C2×C4×C2311yescharacteristic
242C23#1179811yescharacteristic, direct product factor, Sylow
243C24—21no—
244C24C2×D4×C2311yescharacteristic
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4C22⋊C4×C2311yescharacteristic
263C22×C4C22⋊C4×C2311yescharacteristic
264C22×C4C22⋊C4×C2311yescharacteristic
265C22×C4C22⋊C4×C2311yescharacteristic
266C22×C4C22×C4×C2311yescharacteristic
267C22×C4C2×D4×C2311yescharacteristic
268C2×C8—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—42no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊C4×C2311yescharacteristic
300C23C23×AGL(1,ℤ/8)12yes—
301D4—42no—
302D4—48no—
303D4—42no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
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—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4C23×#9911yescharacteristic
334C2×C4C23×#9911yescharacteristic
335C8—82no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C2×C23×#9911yescharacteristic
351C22C23×#41811yescharacteristic
352C22C23×#42111yescharacteristic
353C4—21no—
354C4—21no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—162no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—82no—
368C2C23×#151011yescharacteristic
369C2C23×#212011yescharacteristic
370C2C23×#212611yescharacteristic
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×#1464C2semidirect
4. C23×#1344C2semidirect
7. C2×C4×D4×C23C4semidirect
10. C23×#398C4semidirect
14. C23×#394C4semidirect
17. C4×C4⋊C4×C23C4semidirect
102. #11798C23direct (N × H)
152. #1820C2×C23semidirect
153. #1464C2×C23semidirect
154. #1344C2×C23semidirect
185. C2×C4×D4C4×C23semidirect
188. #398C4×C23semidirect
192. #394C4×C23semidirect
195. C4×C4⋊C4C4×C23semidirect
242. C23#11798direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23922
C23×#18202
C23×#14642
C23×#13442
C23×#13432
#1179823