Finite Group Databasebeta

← C23×#9723 #19157349

Subgroups — C23×#9723 #19157349

Order 5888.

422 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3167C211yesmaximal, characteristic
2C4×C23×#108C211yesmaximal, characteristic, complemented
3C23×#1526C211yesmaximal, characteristic, complemented
4C23×#1525C211yesmaximal, characteristic
5C23×#1485C211yesmaximal, characteristic, complemented
6C23×#1484C211yesmaximal, characteristic
7C8×M16×C23C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C23×#498C411yescharacteristic
11C23×#492C411yescharacteristic
12C23×#487C411yescharacteristic
13C23×#486C411yescharacteristic
14C23×#465—21no—
15C23×#465—21no—
16C4×SD16×C23—21no—
17C4×SD16×C23—21no—
18C23×#431C2211yescharacteristic
19C2×C23×#108—21no—
20C4×M16×C23C2211yescharacteristic
21C4×M16×C23C2211yescharacteristic
22C4×M16×C23C2211yescharacteristic
23C2×C4×C8×C23C2211yescharacteristic
24C23×#341—21no—
25C23×#330—22no—
26C23×#329—22no—
27C23×Q8⋊C8—22no—
28C23×D4⋊C8—22no—
29C23×#322—21no—
30C82×C23—21no—
31C2×#41×C23—21no—
32C2×SD16×C23—21no—
33C2×SD16×C23—21no—
34C2×M16×C23—21no—
35C2×M16×C23—21no—
36C2×M16×C23—21no—
37C22×C8×C23D411yescharacteristic
38C22×C8×C23D411yescharacteristic
39C23×C4⋊Q8C2×C411yescharacteristic
40C23×C4⋊Q8C2×C411yescharacteristic
41C23×C4⋊Q8C2×C411yescharacteristic
42C23×#127C2×C411yescharacteristic
43C23×#125—21no—
44C23×#124—21no—
45C23×C22⋊Q8—21no—
46C23×C22⋊Q8—21no—
47C23×#121—21no—
48C23×#121—21no—
49C4×Q8×C23—21no—
50C4×Q8×C23—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C23×#117—21no—
54C2×C42×C23C2311yescharacteristic
55C23×#108—24no—
56C23×#106—21no—
57C23×#106—21no—
58C23×C4⋊C8D412yes—
59C23×#103—21no—
60C23×#103—21no—
61C23×#102—21no—
62C23×#102—21no—
63C23×C22⋊C8—21no—
64C23×#97—21no—
65C23×#97—21no—
66C23×#97D412yes—
67C4×C8×C23—21no—
68C4×C8×C23—21no—
69C4×C8×C23—21no—
70C4×C8×C23—21no—
71C4×C8×C23C2×C411yescharacteristic
72C4×C8×C23C2×C411yescharacteristic
73C4×C8×C23D411yescharacteristic
74C4×C8×C23D411yescharacteristic
75#41×C23—82no—
76C2×Q8×C23—21no—
77C2×Q8×C23—21no—
78C2×Q8×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23—41no—
83C22×C4×C23—41no—
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23#4111yescharacteristic
86C22×C4×C23#4111yescharacteristic
87SD16×C23—42no—
88SD16×C23—42no—
89M16×C23—24no—
90M16×C23—22no—
91M16×C23—22no—
92M16×C23—41no—
93M16×C23—41no—
94C2×C8×C23—22no—
95C2×C8×C23—22no—
96C2×C8×C23—22no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C2×C8×C23—41no—
100C2×C8×C23C22⋊C411yescharacteristic
101C2×C8×C23C22⋊C411yescharacteristic
102C2×C8×C23C22⋊C411yescharacteristic
103C2×C8×C23C22⋊C411yescharacteristic
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—41no—
115C42×C23—41no—
116C42×C23C22×C411yescharacteristic
117C42×C23C2×D411yescharacteristic
118C42×C23C2×D411yescharacteristic
119C42×C23C2×D411yescharacteristic
120#9723C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—41no—
122C23×C23#12411yescharacteristic
123Q8×C23—41no—
124Q8×C23—41no—
125Q8×C23—42no—
126Q8×C23—81no—
127D4×C23—41no—
128D4×C23—41no—
129D4×C23—42no—
130D4×C23—81no—
131D4×C23—82no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—22no—
135C2×C4×C23—22no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—82no—
140C2×C4×C23C2×C22⋊C411yescharacteristic
141C2×C4×C23C4×D411yescharacteristic
142C2×C4×C23C4×D411yescharacteristic
143C2×C4×C23#12111yescharacteristic
144C2×C4×C23#12111yescharacteristic
145C2×C4×C23#12711yescharacteristic
146C8×C23—24no—
147C8×C23—21no—
148C8×C23—21no—
149C8×C23—21no—
150C8×C23—42no—
151C8×C23—42no—
152C8×C23—42no—
153C8×C23#10412yes—
154#3167C2×C2311yescharacteristic
155C4×#108C2×C2311yescharacteristic, complemented
156#1526C2×C2311yescharacteristic, complemented
157#1525C2×C2311yescharacteristic
158#1485C2×C2311yescharacteristic, complemented
159#1484C2×C2311yescharacteristic
160C8×M16C2×C2311yescharacteristic, complemented
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—81no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23#39011yescharacteristic
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—81no—
172C4×C23—81no—
173C4×C23—81no—
174C4×C23C2×#10411yescharacteristic
175C4×C23#42111yescharacteristic
176C4×C23#44312yes—
177C4×C23#47112yes—
178#533C22×C2311yescharacteristic
179#532C22×C2311yescharacteristic
180#498C4×C2311yescharacteristic
181#492C4×C2311yescharacteristic
182#487C4×C2311yescharacteristic
183#486C4×C2311yescharacteristic
184#465—21no—
185#465—21no—
186C4×SD16—21no—
187C4×SD16—21no—
188#431C22×C2311yescharacteristic
189C2×#108—21no—
190C4×M16C22×C2311yescharacteristic
191C4×M16C22×C2311yescharacteristic
192C4×M16C22×C2311yescharacteristic
193C2×C4×C8C22×C2311yescharacteristic
194#341—21no—
195#330—22no—
196#329—22no—
197Q8⋊C8—22no—
198D4⋊C8—22no—
199#322—21no—
200C82—21no—
201C2×C23—41no—
202C2×C23—161no—
203C2×C23#197411yescharacteristic
204C2×C23#197711yescharacteristic
205C2×C23#198011yescharacteristic
206C2×#41—21no—
207C2×SD16—21no—
208C2×SD16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C22×C8D4×C2311yescharacteristic
213C22×C8D4×C2311yescharacteristic
214C4⋊Q8C2×C4×C2311yescharacteristic
215C4⋊Q8C2×C4×C2311yescharacteristic
216C4⋊Q8C2×C4×C2311yescharacteristic
217#127C2×C4×C2311yescharacteristic
218#125—21no—
219#124—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×D4—21no—
227C4×D4—21no—
228#117—21no—
229C2×C42C23×C2311yescharacteristic
230#108—24no—
231#106—21no—
232#106—21no—
233C4⋊C8D4×C2312yes—
234#103—21no—
235#103—21no—
236#102—21no—
237#102—21no—
238C22⋊C8—21no—
239#97—21no—
240#97—21no—
241#97D4×C2312yes—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8C2×C4×C2311yescharacteristic
247C4×C8C2×C4×C2311yescharacteristic
248C4×C8D4×C2311yescharacteristic
249C4×C8D4×C2311yescharacteristic
250C23#972311yescharacteristic, direct product factor, Sylow
251#41—82no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4C2×D4×C2311yescharacteristic
261C22×C4#41×C2311yescharacteristic
262C22×C4#41×C2311yescharacteristic
263SD16—42no—
264SD16—42no—
265M16—24no—
266M16—22no—
267M16—22no—
268M16—41no—
269M16—41no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8C22⋊C4×C2311yescharacteristic
277C2×C8C22⋊C4×C2311yescharacteristic
278C2×C8C22⋊C4×C2311yescharacteristic
279C2×C8C22⋊C4×C2311yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42—41no—
292C42C22×C4×C2311yescharacteristic
293C42C2×D4×C2311yescharacteristic
294C42C2×D4×C2311yescharacteristic
295C42C2×D4×C2311yescharacteristic
296C23—41no—
297C23C23×#12411yescharacteristic
298Q8—41no—
299Q8—41no—
300Q8—42no—
301Q8—81no—
302D4—41no—
303D4—41no—
304D4—42no—
305D4—81no—
306D4—82no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—82no—
315C2×C4C2×C22⋊C4×C2311yescharacteristic
316C2×C4C4×D4×C2311yescharacteristic
317C2×C4C4×D4×C2311yescharacteristic
318C2×C4C23×#12111yescharacteristic
319C2×C4C23×#12111yescharacteristic
320C2×C4C23×#12711yescharacteristic
321C8—24no—
322C8—21no—
323C8—21no—
324C8—21no—
325C8—42no—
326C8—42no—
327C8—42no—
328C8C23×#10412yes—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C23×#39011yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4C2×C23×#10411yescharacteristic
343C4C23×#42111yescharacteristic
344C4C23×#44312yes—
345C4C23×#47112yes—
346C2—41no—
347C2—161no—
348C2C23×#197411yescharacteristic
349C2C23×#197711yescharacteristic
350C2C23×#198011yescharacteristic
subgroupG/H
centerC2×C4×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#390
prime radicalC23×#532C22
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
2. C4×C23×#108C2semidirect
3. C23×#1526C2semidirect
5. C23×#1485C2semidirect
7. C8×M16×C23C2semidirect
120. #9723C23direct (N × H)
155. C4×#108C2×C23semidirect
156. #1526C2×C23semidirect
158. #1485C2×C23semidirect
160. C8×M16C2×C23semidirect
250. C23#9723direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31672
C4×C23×#1082
C23×#15262
C23×#15252
C23×#14852
C23×#14842
C8×M16×C232
#972323