Finite Group Databasebeta

← C23×#9294 #19156920

Subgroups — C23×#9294 #19156920

Order 5888.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3208C211yesmaximal, characteristic
2C23×#2961C211yesmaximal, characteristic, complemented
3C23×#1961C211yesmaximal, characteristic
4C23×#1545C211yesmaximal, characteristic
5C23×#1518C211yesmaximal, characteristic, complemented
6C23×#1286C211yesmaximal, characteristic, complemented
7C23×#1275C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C2×C23×#127C2211yescharacteristic
10C23×#492C412yescomplemented
11C23×#486C412yescomplemented
12C23×#460—22no—
13C23×#450—22no—
14C23×#446—22no—
15C23×#446—22no—
16C23×#431C2211yescharacteristic
17C23×#431C2211yescharacteristic
18C23×#424—21no—
19C23×#423C2211yescharacteristic
20C2×C23×#102—21no—
21C2×C23×#102—21no—
22C23×#405—21no—
23C2×C23×#97C2211yescharacteristic
24C23×#382C2211yescharacteristic
25C23×#331—22no—
26C23×#328—22no—
27C22×D4×C23—21no—
28C22×D4×C23—21no—
29C22×D4×C23—21no—
30C2×SD16×C23—24no—
31C2×M16×C23—21no—
32C22×C8×C23D411yescharacteristic
33C22×C8×C23D411yescharacteristic
34C23×C4⋊Q8C2×C411yescharacteristic
35C23×C4⋊Q8C2×C411yescharacteristic
36C23×#127—21no—
37C23×#127—21no—
38C23×#127—21no—
39C23×#127C2×C412yes—
40C23×#125—21no—
41C23×C22⋊Q8—21no—
42C23×C22⋊Q8D412yes—
43C4×Q8×C23—21no—
44C4×Q8×C23D412yes—
45C23×#117—21no—
46C2×C4⋊C4×C23—21no—
47C2×C42×C23C2311yescharacteristic
48C23×C4⋊C8—24no—
49C23×C4⋊C8—21no—
50C23×C4⋊C8—21no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8D412yes—
53C23×#102—24no—
54C23×#102—42no—
55C23×#102—42no—
56C23×C22⋊C8—21no—
57C23×C22⋊C8—21no—
58C23×#97—22no—
59C23×#97—21no—
60C23×#97—21no—
61C4×C8×C23—22no—
62C4×C8×C23—21no—
63C4×C8×C23C2×C412yes—
64C23×#95—21no—
65C23×#95—21no—
66C24×C23—41no—
67C2×Q8×C23—21no—
68C2×Q8×C23C22⋊C412yes—
69C2×D4×C23—22no—
70C2×D4×C23—22no—
71C2×D4×C23—22no—
72C2×D4×C23—41no—
73C2×D4×C23—41no—
74C2×D4×C23—41no—
75C2×D4×C23—41no—
76C2×D4×C23—41no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—82no—
80C2×D4×C23—82no—
81C22×C4×C23—41no—
82C22×C4×C23C2×D411yescharacteristic
83C22×C4×C23C2×D411yescharacteristic
84C22×C4×C23#4111yescharacteristic
85SD16×C23—84no—
86M16×C23—41no—
87M16×C23—41no—
88C2×C8×C23—21no—
89C2×C8×C23—21no—
90C2×C8×C23—22no—
91C2×C8×C23—21no—
92C2×C8×C23—22no—
93C2×C8×C23—21no—
94C2×C8×C23—21no—
95C2×C8×C23—41no—
96C2×C8×C23SD1612yescomplemented
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23C22⋊C412yes—
105C22⋊C4×C23—22no—
106C42×C23—22no—
107C42×C23C22×C411yescharacteristic
108C42×C23C2×D411yescharacteristic
109C42×C23C2×D411yescharacteristic
110C42×C23#4111yescharacteristic
111#9294C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—42no—
113C23×C23—82no—
114C23×C23—82no—
115C23×C23—81no—
116C23×C23—81no—
117C23×C23#12111yescharacteristic
118Q8×C23—42no—
119Q8×C23—81no—
120D4×C23—44no—
121D4×C23—42no—
122D4×C23—42no—
123D4×C23—82no—
124D4×C23—82no—
125D4×C23—82no—
126D4×C23—81no—
127D4×C23—81no—
128C2×C4×C23—22no—
129C2×C4×C23—22no—
130C2×C4×C23—41no—
131C2×C4×C23—41no—
132C2×C4×C23—81no—
133C2×C4×C23—81no—
134C2×C4×C23—81no—
135C2×C4×C23C2×C22⋊C411yescharacteristic
136C2×C4×C23C4×D411yescharacteristic
137C2×C4×C23C4×D411yescharacteristic
138C2×C4×C23#12011yescharacteristic
139C2×C4×C23#12111yescharacteristic
140C2×C4×C23#12311yescharacteristic
141C2×C4×C23C2×SD1611yescharacteristic
142C2×C4×C23#13511yescharacteristic
143C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C23#13711yescharacteristic
145C8×C23—22no—
146C8×C23—42no—
147C8×C23—41no—
148C8×C23—42no—
149C8×C23—81no—
150#3208C2×C2311yescharacteristic
151#2961C2×C2311yescharacteristic, complemented
152#1961C2×C2311yescharacteristic
153#1545C2×C2311yescharacteristic
154#1518C2×C2311yescharacteristic, complemented
155#1286C2×C2311yescharacteristic, complemented
156#1275C2×C2311yescharacteristic, complemented
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—82no—
160C22×C23—82no—
161C22×C23—82no—
162C22×C23—162no—
163C22×C23—162no—
164C22×C23#38611yescharacteristic
165C22×C23#46011yescharacteristic
166C22×C23#46411yescharacteristic
167C4×C23—21no—
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—42no—
172C4×C23—42no—
173C4×C23—161no—
174C4×C23C4×SD1611yescharacteristic
175C4×C23#44011yescharacteristic
176C4×C23#44811yescharacteristic
177C4×C23#44911yescharacteristic
178#533C22×C2311yescharacteristic
179C2×#127C22×C2311yescharacteristic
180#492C4×C2312yescomplemented
181#486C4×C2312yescomplemented
182#460—22no—
183#450—22no—
184#446—22no—
185#446—22no—
186#431C22×C2311yescharacteristic
187#431C22×C2311yescharacteristic
188#424—21no—
189#423C22×C2311yescharacteristic
190C2×#102—21no—
191C2×#102—21no—
192#405—21no—
193C2×#97C22×C2311yescharacteristic
194#382C22×C2311yescharacteristic
195#331—22no—
196#328—22no—
197C2×C23—21no—
198C2×C23—21no—
199C2×C23—162no—
200C2×C23#188011yescharacteristic
201C2×C23#190311yescharacteristic
202C2×C23#191011yescharacteristic
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—21no—
206C2×SD16—24no—
207C2×M16—21no—
208C22×C8D4×C2311yescharacteristic
209C22×C8D4×C2311yescharacteristic
210C4⋊Q8C2×C4×C2311yescharacteristic
211C4⋊Q8C2×C4×C2311yescharacteristic
212#127—21no—
213#127—21no—
214#127—21no—
215#127C2×C4×C2312yes—
216#125—21no—
217C22⋊Q8—21no—
218C22⋊Q8D4×C2312yes—
219C4×Q8—21no—
220C4×Q8D4×C2312yes—
221#117—21no—
222C2×C4⋊C4—21no—
223C2×C42C23×C2311yescharacteristic
224C4⋊C8—24no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8D4×C2312yes—
229#102—24no—
230#102—42no—
231#102—42no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234#97—22no—
235#97—21no—
236#97—21no—
237C4×C8—22no—
238C4×C8—21no—
239C4×C8C2×C4×C2312yes—
240#95—21no—
241#95—21no—
242C23#929411yescharacteristic, direct product factor, Sylow
243C24—41no—
244C2×Q8—21no—
245C2×Q8C22⋊C4×C2312yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—41no—
259C22×C4C2×D4×C2311yescharacteristic
260C22×C4C2×D4×C2311yescharacteristic
261C22×C4#41×C2311yescharacteristic
262SD16—84no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—41no—
273C2×C8SD16×C2312yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4C22⋊C4×C2312yes—
282C22⋊C4—22no—
283C42—22no—
284C42C22×C4×C2311yescharacteristic
285C42C2×D4×C2311yescharacteristic
286C42C2×D4×C2311yescharacteristic
287C42#41×C2311yescharacteristic
288C23—42no—
289C23—82no—
290C23—82no—
291C23—81no—
292C23—81no—
293C23C23×#12111yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—44no—
297D4—42no—
298D4—42no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—81no—
303D4—81no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4C2×C22⋊C4×C2311yescharacteristic
312C2×C4C4×D4×C2311yescharacteristic
313C2×C4C4×D4×C2311yescharacteristic
314C2×C4C23×#12011yescharacteristic
315C2×C4C23×#12111yescharacteristic
316C2×C4C23×#12311yescharacteristic
317C2×C4C2×SD16×C2311yescharacteristic
318C2×C4C23×#13511yescharacteristic
319C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
320C2×C4C23×#13711yescharacteristic
321C8—22no—
322C8—42no—
323C8—41no—
324C8—42no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22C23×#38611yescharacteristic
334C22C23×#46011yescharacteristic
335C22C23×#46411yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—161no—
343C4C4×SD16×C2311yescharacteristic
344C4C23×#44011yescharacteristic
345C4C23×#44811yescharacteristic
346C4C23×#44911yescharacteristic
347C2—21no—
348C2—21no—
349C2—162no—
350C2C23×#188011yescharacteristic
351C2C23×#190311yescharacteristic
352C2C23×#191011yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2961C2semidirect
5. C23×#1518C2semidirect
6. C23×#1286C2semidirect
7. C23×#1275C2semidirect
10. C23×#492C4semidirect
11. C23×#486C4semidirect
96. C2×C8×C23SD16semidirect
111. #9294C23direct (N × H)
151. #2961C2×C23semidirect
154. #1518C2×C23semidirect
155. #1286C2×C23semidirect
156. #1275C2×C23semidirect
180. #492C4×C23semidirect
181. #486C4×C23semidirect
242. C23#9294direct (N × H)
273. C2×C8SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32082
C23×#29612
C23×#19612
C23×#15452
C23×#15182
C23×#12862
C23×#12752
#929423