Finite Group Databasebeta

← C23×#9212 #19156838

Subgroups — C23×#9212 #19156838

Order 5888.

514 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3107C211yesmaximal, characteristic
2C4×C23×#131C211yesmaximal, characteristic
3C23×#1526C211yesmaximal, characteristic, complemented
4C23×#1493C211yesmaximal, characteristic, complemented
5C23×#1489C211yesmaximal, characteristic
6C23×#1299C211yesmaximal, characteristic, complemented
7C23×#1278C211yesmaximal, characteristic, complemented
8C2×C23×#131—21no—
9C23×#533C2211yescharacteristic
10C4×#41×C23C2211yescharacteristic
11C23×#485—21no—
12C23×#475C411yescharacteristic
13C23×#474C411yescharacteristic
14C23×#462C411yescharacteristic
15C23×#461C411yescharacteristic
16C23×#452—21no—
17C4×Q16×C23—21no—
18C4×SD16×C23—21no—
19C23×#431C2211yescharacteristic
20C23×#423C2211yescharacteristic
21C23×#405—22no—
22C23×#405—21no—
23C4×M16×C23—21no—
24C4×M16×C23C2211yescharacteristic
25C4×M16×C23C2211yescharacteristic
26C2×C4×C8×C23—21no—
27C2×C4×C8×C23C2211yescharacteristic
28C23×#330—22no—
29C23×Q8⋊C8—21no—
30C23×Q8⋊C8—21no—
31C23×Q8⋊C8—21no—
32C23×D4⋊C8—21no—
33C2×#41×C23D411yescharacteristic
34C2×Q16×C23—21no—
35C2×SD16×C23—21no—
36C23×#131—24no—
37C23×#131—24no—
38C2×M16×C23—22no—
39C2×M16×C23—21no—
40C2×M16×C23—21no—
41C2×M16×C23—21no—
42C22×C8×C23—22no—
43C22×C8×C23—21no—
44C23×C4⋊Q8C2×C411yescharacteristic
45C23×C4⋊Q8C2×C411yescharacteristic
46C23×#125—21no—
47C23×C22⋊Q8—21no—
48C23×C22⋊Q8—21no—
49C4×Q8×C23—21no—
50C4×Q8×C23—21no—
51C4×Q8×C23C2×C411yescharacteristic
52C4×Q8×C23D411yescharacteristic
53C4×D4×C23—21no—
54C4×D4×C23—21no—
55C4×D4×C23C2×C411yescharacteristic
56C4×D4×C23D411yescharacteristic
57C23×#117—21no—
58C23×#117—21no—
59C23×#117D411yescharacteristic
60C2×C42×C23—21no—
61C2×C42×C23C2311yescharacteristic
62C23×#107—21no—
63C23×#106—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8C2×C411yescharacteristic
67C23×C4⋊C8C2×C411yescharacteristic
68C23×#103—21no—
69C23×#103—21no—
70C23×#103—21no—
71C23×#102—21no—
72C23×C22⋊C8—21no—
73C23×#97—22no—
74C23×#97—21no—
75C23×#97D412yes—
76C4×C8×C23—22no—
77C4×C8×C23—21no—
78C4×C8×C23—21no—
79C4×C8×C23—21no—
80C4×C8×C23—21no—
81C4×C8×C23D411yescharacteristic
82C4×C8×C23D411yescharacteristic
83#41×C23—22no—
84#41×C23—22no—
85C2×Q8×C23—21no—
86C2×Q8×C23—21no—
87C2×Q8×C23C22⋊C411yescharacteristic
88C2×D4×C23—21no—
89C2×D4×C23C22⋊C411yescharacteristic
90C22×C4×C23—21no—
91C22×C4×C23—21no—
92C22×C4×C23—21no—
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23#4111yescharacteristic
95C22×C4×C23#4111yescharacteristic
96Q16×C23—42no—
97SD16×C23—42no—
98M16×C23—28no—
99M16×C23—22no—
100M16×C23—22no—
101M16×C23—22no—
102M16×C23—41no—
103M16×C23—81no—
104C2×C8×C23—28no—
105C2×C8×C23—22no—
106C2×C8×C23—22no—
107C2×C8×C23—21no—
108C2×C8×C23—22no—
109C2×C8×C23—21no—
110C2×C8×C23—21no—
111C2×C8×C23—21no—
112C2×C8×C23—41no—
113C2×C8×C23—41no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23—21no—
120C4⋊C4×C23—41no—
121C4⋊C4×C23—41no—
122C4⋊C4×C23C22⋊C411yescharacteristic
123C4⋊C4×C23C22⋊C411yescharacteristic
124C22⋊C4×C23—21no—
125C22⋊C4×C23—21no—
126C22⋊C4×C23—21no—
127C22⋊C4×C23—41no—
128C42×C23—21no—
129C42×C23—21no—
130C42×C23—21no—
131C42×C23—41no—
132C42×C23C22×C411yescharacteristic
133C42×C23C2×D411yescharacteristic
134C42×C23C2×D411yescharacteristic
135C42×C23C2×D411yescharacteristic
136#9212C2311yesmaximal, characteristic, direct product factor, Sylow
137C23×C23—21no—
138C23×C23#12311yescharacteristic
139Q8×C23—21no—
140Q8×C23—41no—
141Q8×C23—41no—
142Q8×C23—41no—
143Q8×C23—41no—
144Q8×C23#10412yes—
145D4×C23—24no—
146D4×C23—21no—
147D4×C23#10412yes—
148C2×C4×C23—24no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—41no—
164C2×C4×C23C2×C22⋊C411yescharacteristic
165C2×C4×C23C4×D411yescharacteristic
166C2×C4×C23C4×D411yescharacteristic
167C2×C4×C23#12011yescharacteristic
168C2×C4×C23#12111yescharacteristic
169C2×C4×C23#12111yescharacteristic
170C8×C23—22no—
171C8×C23—24no—
172C8×C23—22no—
173C8×C23—42no—
174C8×C23—41no—
175C8×C23—41no—
176C8×C23—81no—
177#3107C2×C2311yescharacteristic
178C4×#131C2×C2311yescharacteristic
179#1526C2×C2311yescharacteristic, complemented
180#1493C2×C2311yescharacteristic, complemented
181#1489C2×C2311yescharacteristic
182#1299C2×C2311yescharacteristic, complemented
183#1278C2×C2311yescharacteristic, complemented
184C22×C23—21no—
185C22×C23—21no—
186C22×C23—22no—
187C22×C23—21no—
188C22×C23—41no—
189C22×C23—41no—
190C22×C23#38611yescharacteristic
191C4×C23—21no—
192C4×C23—21no—
193C4×C23—22no—
194C4×C23—21no—
195C4×C23—41no—
196C4×C23—41no—
197C4×C23—81no—
198C4×C23—81no—
199C4×C23C2×#10411yescharacteristic
200C4×C23C2×#10411yescharacteristic
201C4×C23#44311yescharacteristic
202C4×C23#44411yescharacteristic
203C4×C23#47111yescharacteristic
204C4×C23#47311yescharacteristic
205C2×#131—21no—
206#533C22×C2311yescharacteristic
207C4×#41C22×C2311yescharacteristic
208#485—21no—
209#475C4×C2311yescharacteristic
210#474C4×C2311yescharacteristic
211#462C4×C2311yescharacteristic
212#461C4×C2311yescharacteristic
213#452—21no—
214C4×Q16—21no—
215C4×SD16—21no—
216#431C22×C2311yescharacteristic
217#423C22×C2311yescharacteristic
218#405—21no—
219#405—22no—
220C4×M16—21no—
221C4×M16C22×C2311yescharacteristic
222C4×M16C22×C2311yescharacteristic
223C2×C4×C8—21no—
224C2×C4×C8C22×C2311yescharacteristic
225#330—22no—
226Q8⋊C8—21no—
227Q8⋊C8—21no—
228Q8⋊C8—21no—
229D4⋊C8—21no—
230C2×C23—41no—
231C2×C23—42no—
232C2×C23#187611yescharacteristic
233C2×C23#189611yescharacteristic
234C2×C23#190811yescharacteristic
235C2×#41D4×C2311yescharacteristic
236C2×Q16—21no—
237C2×SD16—21no—
238#131—24no—
239#131—24no—
240C2×M16—22no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C22×C8—22no—
245C22×C8—21no—
246C4⋊Q8C2×C4×C2311yescharacteristic
247C4⋊Q8C2×C4×C2311yescharacteristic
248#125—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C2×C4×C2311yescharacteristic
254C4×Q8D4×C2311yescharacteristic
255C4×D4—21no—
256C4×D4—21no—
257C4×D4C2×C4×C2311yescharacteristic
258C4×D4D4×C2311yescharacteristic
259#117—21no—
260#117—21no—
261#117D4×C2311yescharacteristic
262C2×C42—21no—
263C2×C42C23×C2311yescharacteristic
264#107—21no—
265#106—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8C2×C4×C2311yescharacteristic
269C4⋊C8C2×C4×C2311yescharacteristic
270#103—21no—
271#103—21no—
272#103—21no—
273#102—21no—
274C22⋊C8—21no—
275#97—22no—
276#97—21no—
277#97D4×C2312yes—
278C4×C8—22no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8D4×C2311yescharacteristic
284C4×C8D4×C2311yescharacteristic
285C23#921211yescharacteristic, direct product factor, Sylow
286#41—22no—
287#41—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8C22⋊C4×C2311yescharacteristic
291C2×D4—21no—
292C2×D4C22⋊C4×C2311yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4C2×D4×C2311yescharacteristic
297C22×C4#41×C2311yescharacteristic
298C22×C4#41×C2311yescharacteristic
299Q16—42no—
300SD16—42no—
301M16—28no—
302M16—22no—
303M16—22no—
304M16—22no—
305M16—41no—
306M16—81no—
307C2×C8—28no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4C22⋊C4×C2311yescharacteristic
326C4⋊C4C22⋊C4×C2311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—41no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—41no—
335C42C22×C4×C2311yescharacteristic
336C42C2×D4×C2311yescharacteristic
337C42C2×D4×C2311yescharacteristic
338C42C2×D4×C2311yescharacteristic
339C23—21no—
340C23C23×#12311yescharacteristic
341Q8—21no—
342Q8—41no—
343Q8—41no—
344Q8—41no—
345Q8—41no—
346Q8C23×#10412yes—
347D4—24no—
348D4—21no—
349D4C23×#10412yes—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4C2×C22⋊C4×C2311yescharacteristic
367C2×C4C4×D4×C2311yescharacteristic
368C2×C4C4×D4×C2311yescharacteristic
369C2×C4C23×#12011yescharacteristic
370C2×C4C23×#12111yescharacteristic
371C2×C4C23×#12111yescharacteristic
372C8—22no—
373C8—24no—
374C8—22no—
375C8—42no—
376C8—41no—
377C8—41no—
378C8—81no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22C23×#38611yescharacteristic
386C4—21no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4C2×C23×#10411yescharacteristic
395C4C2×C23×#10411yescharacteristic
396C4C23×#44311yescharacteristic
397C4C23×#44411yescharacteristic
398C4C23×#47111yescharacteristic
399C4C23×#47311yescharacteristic
400C2—41no—
401C2—42no—
402C2C23×#187611yescharacteristic
403C2C23×#189611yescharacteristic
404C2C23×#190811yescharacteristic
subgroupG/H
centerC2×C4×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC2×#41×C23D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1526C2semidirect
4. C23×#1493C2semidirect
6. C23×#1299C2semidirect
7. C23×#1278C2semidirect
136. #9212C23direct (N × H)
179. #1526C2×C23semidirect
180. #1493C2×C23semidirect
182. #1299C2×C23semidirect
183. #1278C2×C23semidirect
285. C23#9212direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31072
C4×C23×#1312
C23×#15262
C23×#14932
C23×#14892
C23×#12992
C23×#12782
#921223