Finite Group Databasebeta

← C23×#12962 #19160588

Subgroups — C23×#12962 #19160588

Order 5888.

426 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2977C211yesmaximal, characteristic
2C23×#2120C211yesmaximal, characteristic
3C23×#1891C211yesmaximal, characteristic
4C23×#1868C211yesmaximal, characteristic, complemented
5C23×#1858C211yesmaximal, characteristic, complemented
6C23×#1395C211yesmaximal, characteristic, complemented
7C23×#1392C211yesmaximal, characteristic, complemented
8C23×#568C2211yescharacteristic
9C23×#525C2211yescharacteristic
10C23×#435—21no—
11C23×#435—21no—
12C8×D4×C23—21no—
13C8×D4×C23—21no—
14C23×#433—21no—
15C23×#430—21no—
16C23×#424—21no—
17C23×#413C2211yescharacteristic
18C23×#413C2211yescharacteristic
19C23×#410C2211yescharacteristic
20C23×#408C2211yescharacteristic
21C23×#408C2211yescharacteristic
22C23×#407—21no—
23C2×C23×C22⋊C8—21no—
24C23×#352C412yes—
25C23×AΣL(1,𝔽16)C412yes—
26C23×#344—21no—
27C23×#344—21no—
28C23×#341—21no—
29C2×#41×C23C2311yescharacteristic
30C23×C4×C23D411yescharacteristic
31C23×#131—42no—
32C2×M16×C23—21no—
33C2×M16×C23—21no—
34C2×M16×C23—21no—
35C2×M16×C23—21no—
36C2×M16×C23—21no—
37C2×M16×C23D411yescharacteristic
38C2×M16×C23D411yescharacteristic
39C22×C8×C23—21no—
40C22×C8×C23—21no—
41C22×C8×C23D411yescharacteristic
42C22×C8×C23D411yescharacteristic
43C23×#123C2×C411yescharacteristic
44C23×#123Q811yescharacteristic
45C23×C22⋊Q8—21no—
46C23×#121—21no—
47C23×#120C2×C411yescharacteristic
48C23×#120Q811yescharacteristic
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C23×#117—21no—
52C23×#117D411yescharacteristic
53C23×#108—42no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—42no—
57C23×#101—21no—
58C23×#101—21no—
59C23×#100—21no—
60C23×#100C2×C412yes—
61C23×#99—21no—
62C23×#99C2×C412yes—
63C23×C22⋊C8—21no—
64C23×C22⋊C8—21no—
65C23×C22⋊C8—21no—
66C23×C22⋊C8—21no—
67C23×C22⋊C8—21no—
68C23×C22⋊C8—21no—
69C23×C22⋊C8—42no—
70C23×#97—21no—
71C4×C8×C23—21no—
72C24×C23C4⋊C411yescharacteristic
73#41×C23—41no—
74#41×C23—41no—
75C2×Q8×C23#4111yescharacteristic
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23C22×C411yescharacteristic
79C2×D4×C23C2×Q811yescharacteristic
80C2×D4×C23#4111yescharacteristic
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—41no—
84C22×C4×C23—41no—
85C22×C4×C23C4⋊C411yescharacteristic
86C22×C4×C23C2×D411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23C2×D411yescharacteristic
89M16×C23—22no—
90M16×C23—22no—
91M16×C23—41no—
92M16×C23—41no—
93M16×C23—41no—
94M16×C23—41no—
95M16×C23—41no—
96M16×C23—41no—
97M16×C23—41no—
98M16×C23—41no—
99C2×C8×C23—22no—
100C2×C8×C23—22no—
101C2×C8×C23—21no—
102C2×C8×C23—21no—
103C2×C8×C23—21no—
104C2×C8×C23—21no—
105C2×C8×C23—41no—
106C2×C8×C23—41no—
107C2×C8×C23—41no—
108C2×C8×C23—41no—
109C2×C8×C23—41no—
110C2×C8×C23—81no—
111C2×C8×C23—81no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—41no—
116C22⋊C4×C23—21no—
117C22⋊C4×C23—21no—
118C22⋊C4×C23—21no—
119C22⋊C4×C23—21no—
120C22⋊C4×C23—22no—
121C22⋊C4×C23C4⋊C411yescharacteristic
122C22⋊C4×C23C4⋊C411yescharacteristic
123C42×C23—21no—
124C42×C23—41no—
125#12962C2311yesmaximal, characteristic, direct product factor, Sylow
126C23×C23—21no—
127C23×C23—41no—
128C23×C23—42no—
129C23×C23C2×C4⋊C411yescharacteristic
130C23×C23C4×D411yescharacteristic
131C23×C23C22⋊Q811yescharacteristic
132Q8×C23—41no—
133D4×C23—41no—
134D4×C23—41no—
135D4×C23—41no—
136D4×C23—42no—
137D4×C23—42no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—42no—
151C2×C4×C23—81no—
152C2×C4×C23C4×D411yescharacteristic
153C2×C4×C23#12011yescharacteristic
154C2×C4×C23C22⋊Q811yescharacteristic
155C2×C4×C23#12311yescharacteristic
156C8×C23—42no—
157C8×C23—42no—
158C8×C23—41no—
159C8×C23—41no—
160C8×C23—41no—
161C8×C23—41no—
162C8×C23—81no—
163#2977C2×C2311yescharacteristic
164#2120C2×C2311yescharacteristic
165#1891C2×C2311yescharacteristic
166#1868C2×C2311yescharacteristic, complemented
167#1858C2×C2311yescharacteristic, complemented
168#1395C2×C2311yescharacteristic, complemented
169#1392C2×C2311yescharacteristic, complemented
170C22×C23—22no—
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—41no—
175C22×C23—41no—
176C22×C23—41no—
177C22×C23—41no—
178C22×C23—42no—
179C22×C23—41no—
180C22×C23—81no—
181C22×C23#38511yescharacteristic
182C4×C23—21no—
183C4×C23—21no—
184C4×C23—21no—
185C4×C23—41no—
186C4×C23—41no—
187C4×C23—41no—
188C4×C23—82no—
189C4×C23B(4,𝔽2)11yescharacteristic
190C4×C23#45811yescharacteristic
191#568C22×C2311yescharacteristic
192#525C22×C2311yescharacteristic
193#435—21no—
194#435—21no—
195C8×D4—21no—
196C8×D4—21no—
197#433—21no—
198#430—21no—
199#424—21no—
200#413C22×C2311yescharacteristic
201#413C22×C2311yescharacteristic
202#410C22×C2311yescharacteristic
203#408C22×C2311yescharacteristic
204#408C22×C2311yescharacteristic
205#407—21no—
206C2×C22⋊C8—21no—
207#352C4×C2312yes—
208AΣL(1,𝔽16)C4×C2312yes—
209#344—21no—
210#344—21no—
211#341—21no—
212C2×C23—21no—
213C2×C23—41no—
214C2×C23—41no—
215C2×C23—42no—
216C2×C23—41no—
217C2×C23#186711yescharacteristic
218C2×#41C23×C2311yescharacteristic
219C23×C4D4×C2311yescharacteristic
220#131—42no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16D4×C2311yescharacteristic
227C2×M16D4×C2311yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230C22×C8D4×C2311yescharacteristic
231C22×C8D4×C2311yescharacteristic
232#123C2×C4×C2311yescharacteristic
233#123Q8×C2311yescharacteristic
234C22⋊Q8—21no—
235#121—21no—
236#120C2×C4×C2311yescharacteristic
237#120Q8×C2311yescharacteristic
238C4×D4—21no—
239C4×D4—21no—
240#117—21no—
241#117D4×C2311yescharacteristic
242#108—42no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246#101—21no—
247#101—21no—
248#100—21no—
249#100C2×C4×C2312yes—
250#99—21no—
251#99C2×C4×C2312yes—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—42no—
259#97—21no—
260C4×C8—21no—
261C23#1296211yescharacteristic, direct product factor, Sylow
262C24C4⋊C4×C2311yescharacteristic
263#41—41no—
264#41—41no—
265C2×Q8#41×C2311yescharacteristic
266C2×D4—21no—
267C2×D4—21no—
268C2×D4C22×C4×C2311yescharacteristic
269C2×D4C2×Q8×C2311yescharacteristic
270C2×D4#41×C2311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C4⋊C4×C2311yescharacteristic
276C22×C4C2×D4×C2311yescharacteristic
277C22×C4C2×D4×C2311yescharacteristic
278C22×C4C2×D4×C2311yescharacteristic
279M16—22no—
280M16—22no—
281M16—41no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—41no—
286M16—41no—
287M16—41no—
288M16—41no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—81no—
301C2×C8—81no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—22no—
311C22⋊C4C4⋊C4×C2311yescharacteristic
312C22⋊C4C4⋊C4×C2311yescharacteristic
313C42—21no—
314C42—41no—
315C23—21no—
316C23—41no—
317C23—42no—
318C23C2×C4⋊C4×C2311yescharacteristic
319C23C4×D4×C2311yescharacteristic
320C23C23×C22⋊Q811yescharacteristic
321Q8—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—42no—
326D4—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4C4×D4×C2311yescharacteristic
342C2×C4C23×#12011yescharacteristic
343C2×C4C23×C22⋊Q811yescharacteristic
344C2×C4C23×#12311yescharacteristic
345C8—42no—
346C8—42no—
347C8—41no—
348C8—41no—
349C8—41no—
350C8—41no—
351C8—81no—
352C22—22no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—81no—
363C22C23×#38511yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—82no—
371C4C23×B(4,𝔽2)11yescharacteristic
372C4C23×#45811yescharacteristic
373C2—21no—
374C2—41no—
375C2—42no—
376C2—41no—
377C2—41no—
378C2C23×#186711yescharacteristic
subgroupG/H
centerC4×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
socleC2×C23#1867
prime radicalC23×#525C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1868C2semidirect
5. C23×#1858C2semidirect
6. C23×#1395C2semidirect
7. C23×#1392C2semidirect
125. #12962C23direct (N × H)
166. #1868C2×C23semidirect
167. #1858C2×C23semidirect
168. #1395C2×C23semidirect
169. #1392C2×C23semidirect
261. C23#12962direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29772
C23×#21202
C23×#18912
C23×#18682
C23×#18582
C23×#13952
C23×#13922
#1296223