Finite Group Databasebeta

← C23×#9531 #19157157

Subgroups — C23×#9531 #19157157

Order 5888.

412 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3117C211yesmaximal, characteristic
2C23×#2967C211yesmaximal, characteristic
3C23×#1535C211yesmaximal, characteristic
4C23×#1491C211yesmaximal, characteristic, complemented
5C23×#1467C211yesmaximal, characteristic, complemented
6C23×#1308C211yesmaximal, characteristic, complemented
7C23×#1291C211yesmaximal, characteristic, complemented
8C23×#529C2211yescharacteristic
9C2×C4×Q8×C23C2211yescharacteristic
10C23×#484C411yescharacteristic
11C23×#481C411yescharacteristic
12C23×#479—21no—
13C23×#464—21no—
14C23×#451C411yescharacteristic
15C23×#448C411yescharacteristic
16C23×#446—21no—
17C23×#441—21no—
18C23×#440—21no—
19C23×#436—21no—
20C23×#435—21no—
21C23×#435—21no—
22C23×#433—21no—
23C23×#432C2211yescharacteristic
24C23×#432C2211yescharacteristic
25C23×#432C2211yescharacteristic
26C23×#432C2211yescharacteristic
27C23×#405—21no—
28C4×M16×C23C2211yescharacteristic
29C23×#330—21no—
30C23×#329—21no—
31C23×Q8⋊C8—21no—
32C23×Q8⋊C8—21no—
33C23×#324—21no—
34C23×#324—21no—
35C22×Q8×C23C2×C411yescharacteristic
36C2×Q16×C23—21no—
37C2×SD16×C23—21no—
38C2×M16×C23—21no—
39C2×M16×C23—21no—
40C22×C8×C23—21no—
41C23×#126—21no—
42C23×#125D411yescharacteristic
43C23×#124D411yescharacteristic
44C23×#123—21no—
45C23×C22⋊Q8C2×C411yescharacteristic
46C23×#121C2×C411yescharacteristic
47C4×Q8×C23—21no—
48C4×Q8×C23—21no—
49C4×Q8×C23—21no—
50C4×Q8×C23—21no—
51C4×Q8×C23D411yescharacteristic
52C4×D4×C23—21no—
53C4×D4×C23D411yescharacteristic
54C23×#117—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23C2×C411yescharacteristic
57C2×C42×C23—21no—
58C2×C42×C23C2311yescharacteristic
59C23×#107—21no—
60C23×#106—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8—21no—
67C23×#103—21no—
68C23×#103—21no—
69C23×#103—21no—
70C23×#102—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8—21no—
75C23×C22⋊C8C2×C411yescharacteristic
76C23×C22⋊C8C2×C411yescharacteristic
77C23×C22⋊C8D411yescharacteristic
78C23×C22⋊C8D411yescharacteristic
79C23×#97—21no—
80C23×#97—21no—
81C23×#97—21no—
82C23×#97—21no—
83C23×#97—21no—
84C23×#97—21no—
85C4×C8×C23—21no—
86C4×C8×C23D411yescharacteristic
87C4×C8×C23D411yescharacteristic
88C2×Q8×C23—21no—
89C2×Q8×C23—21no—
90C2×Q8×C23—42no—
91C2×Q8×C23C22⋊C411yescharacteristic
92C2×D4×C23—21no—
93C2×D4×C23C22⋊C411yescharacteristic
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23C22×C411yescharacteristic
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23#4111yescharacteristic
100Q16×C23—42no—
101SD16×C23—42no—
102M16×C23—42no—
103M16×C23—42no—
104C2×C8×C23—21no—
105C2×C8×C23—21no—
106C2×C8×C23—21no—
107C2×C8×C23—21no—
108C2×C8×C23—21no—
109C2×C8×C23—21no—
110C2×C8×C23—42no—
111C2×C8×C23—41no—
112C2×C8×C23—41no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23—41no—
120C4⋊C4×C23—41no—
121C4⋊C4×C23C22⋊C411yescharacteristic
122C4⋊C4×C23C22⋊C411yescharacteristic
123C22⋊C4×C23—21no—
124C22⋊C4×C23—21no—
125C22⋊C4×C23—21no—
126C22⋊C4×C23—21no—
127C22⋊C4×C23—21no—
128C42×C23—21no—
129C42×C23—41no—
130C42×C23—41no—
131C42×C23C2×D411yescharacteristic
132C42×C23C2×D411yescharacteristic
133C42×C23C2×D411yescharacteristic
134C42×C23#4111yescharacteristic
135#9531C2311yesmaximal, characteristic, direct product factor, Sylow
136C23×C23—21no—
137C23×C23C4×D411yescharacteristic
138Q8×C23—22no—
139Q8×C23—22no—
140Q8×C23—41no—
141Q8×C23—41no—
142Q8×C23—41no—
143D4×C23—41no—
144D4×C23—81no—
145C2×C4×C23—22no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—81no—
158C2×C4×C23C2×C22⋊C411yescharacteristic
159C2×C4×C23C4×D411yescharacteristic
160C2×C4×C23#12011yescharacteristic
161C2×C4×C23#12111yescharacteristic
162C2×C4×C23#12111yescharacteristic
163C2×C4×C23#12311yescharacteristic
164C8×C23—22no—
165C8×C23—41no—
166C8×C23—41no—
167C8×C23—41no—
168C8×C23—41no—
169C8×C23—41no—
170C8×C23—81no—
171C8×C23—81no—
172#3117C2×C2311yescharacteristic
173#2967C2×C2311yescharacteristic
174#1535C2×C2311yescharacteristic
175#1491C2×C2311yescharacteristic, complemented
176#1467C2×C2311yescharacteristic, complemented
177#1308C2×C2311yescharacteristic, complemented
178#1291C2×C2311yescharacteristic, complemented
179C22×C23—21no—
180C22×C23—21no—
181C22×C23—21no—
182C22×C23—41no—
183C22×C23—41no—
184C22×C23—41no—
185C22×C23#38611yescharacteristic
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—21no—
189C4×C23—21no—
190C4×C23—21no—
191C4×C23—21no—
192C4×C23—42no—
193C4×C23—41no—
194C4×C23—41no—
195C4×C23—41no—
196C4×C23—81no—
197#529C22×C2311yescharacteristic
198C2×C4×Q8C22×C2311yescharacteristic
199#484C4×C2311yescharacteristic
200#481C4×C2311yescharacteristic
201#479—21no—
202#464—21no—
203#451C4×C2311yescharacteristic
204#448C4×C2311yescharacteristic
205#446—21no—
206#441—21no—
207#440—21no—
208#436—21no—
209#435—21no—
210#435—21no—
211#433—21no—
212#432C22×C2311yescharacteristic
213#432C22×C2311yescharacteristic
214#432C22×C2311yescharacteristic
215#432C22×C2311yescharacteristic
216#405—21no—
217C4×M16C22×C2311yescharacteristic
218#330—21no—
219#329—21no—
220Q8⋊C8—21no—
221Q8⋊C8—21no—
222#324—21no—
223#324—21no—
224C2×C23—41no—
225C2×C23—81no—
226C2×C23#189111yescharacteristic
227C2×C23#190211yescharacteristic
228C2×C23#190611yescharacteristic
229C22×Q8C2×C4×C2311yescharacteristic
230C2×Q16—21no—
231C2×SD16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C22×C8—21no—
235#126—21no—
236#125D4×C2311yescharacteristic
237#124D4×C2311yescharacteristic
238#123—21no—
239C22⋊Q8C2×C4×C2311yescharacteristic
240#121C2×C4×C2311yescharacteristic
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8D4×C2311yescharacteristic
246C4×D4—21no—
247C4×D4D4×C2311yescharacteristic
248#117—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C2×C4×C2311yescharacteristic
251C2×C42—21no—
252C2×C42C23×C2311yescharacteristic
253#107—21no—
254#106—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261#103—21no—
262#103—21no—
263#103—21no—
264#102—21no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8C2×C4×C2311yescharacteristic
270C22⋊C8C2×C4×C2311yescharacteristic
271C22⋊C8D4×C2311yescharacteristic
272C22⋊C8D4×C2311yescharacteristic
273#97—21no—
274#97—21no—
275#97—21no—
276#97—21no—
277#97—21no—
278#97—21no—
279C4×C8—21no—
280C4×C8D4×C2311yescharacteristic
281C4×C8D4×C2311yescharacteristic
282C23#953111yescharacteristic, direct product factor, Sylow
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8C22⋊C4×C2311yescharacteristic
287C2×D4—21no—
288C2×D4C22⋊C4×C2311yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4C22×C4×C2311yescharacteristic
293C22×C4C2×D4×C2311yescharacteristic
294C22×C4#41×C2311yescharacteristic
295Q16—42no—
296SD16—42no—
297M16—42no—
298M16—42no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4C22⋊C4×C2311yescharacteristic
317C4⋊C4C22⋊C4×C2311yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C42—21no—
324C42—41no—
325C42—41no—
326C42C2×D4×C2311yescharacteristic
327C42C2×D4×C2311yescharacteristic
328C42C2×D4×C2311yescharacteristic
329C42#41×C2311yescharacteristic
330C23—21no—
331C23C4×D4×C2311yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337D4—41no—
338D4—81no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4C2×C22⋊C4×C2311yescharacteristic
353C2×C4C4×D4×C2311yescharacteristic
354C2×C4C23×#12011yescharacteristic
355C2×C4C23×#12111yescharacteristic
356C2×C4C23×#12111yescharacteristic
357C2×C4C23×#12311yescharacteristic
358C8—22no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—41no—
363C8—41no—
364C8—81no—
365C8—81no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22C23×#38611yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—81no—
384C2—41no—
385C2—81no—
386C2C23×#189111yescharacteristic
387C2C23×#190211yescharacteristic
388C2C23×#190611yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1491C2semidirect
5. C23×#1467C2semidirect
6. C23×#1308C2semidirect
7. C23×#1291C2semidirect
135. #9531C23direct (N × H)
175. #1491C2×C23semidirect
176. #1467C2×C23semidirect
177. #1308C2×C23semidirect
178. #1291C2×C23semidirect
282. C23#9531direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31172
C23×#29672
C23×#15352
C23×#14912
C23×#14672
C23×#13082
C23×#12912
#953123