Finite Group Databasebeta

← C23×#10286 #19157912

Subgroups — C23×#10286 #19157912

Order 5888.

542 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3227C211yesmaximal, characteristic
2C23×#3226C211yesmaximal, characteristic
3C23×#2862C211yesmaximal, characteristic, complemented
4C23×#1962C211yesmaximal, characteristic
5C23×#1549C211yesmaximal, characteristic, complemented
6C23×#1546C211yesmaximal, characteristic, complemented
7C23×#1276C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C2×C23×C4⋊Q8C2211yescharacteristic
11C23×#500—21no—
12C23×#498—21no—
13C23×#486—21no—
14C23×#486—21no—
15C23×#481—22no—
16C23×#480—22no—
17C23×#476—22no—
18C23×#474—22no—
19C23×#431C2211yescharacteristic
20C23×#431C2211yescharacteristic
21C23×#423C2211yescharacteristic
22C2×C23×#104—22no—
23C23×#391—22no—
24C23×#391—22no—
25C23×#382—22no—
26C43×C23C2211yescharacteristic
27C23×#332—22no—
28C23×#332—22no—
29C2×#41×C23—21no—
30C22×Q8×C23—21no—
31C22×Q8×C23—21no—
32C2×M16×C23—21no—
33C23×C4⋊Q8—21no—
34C23×C4⋊Q8—21no—
35C23×C4⋊Q8—21no—
36C23×C4⋊Q8—21no—
37C23×C4⋊Q8—21no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—21no—
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8D411yescharacteristic
42C23×#127D411yescharacteristic
43C23×#124—21no—
44C23×#121—21no—
45C23×#121—21no—
46C4×D4×C23—21no—
47C4×D4×C23—21no—
48C2×C4⋊C4×C23—21no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23D412yes—
51C2×C4⋊C4×C23D412yes—
52C2×C42×C23—21no—
53C2×C42×C23—21no—
54C2×C42×C23D412yes—
55C2×C42×C23C2311yescharacteristic
56C23×#107—24no—
57C23×#106—24no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8D412yes—
61C23×C4⋊C8D412yes—
62C23×#104—44no—
63C23×#102—24no—
64C23×#102—24no—
65C23×C22⋊C8—21no—
66C23×C22⋊C8—21no—
67C4×C8×C23—21no—
68C4×C8×C23—21no—
69C23×#95—28no—
70#41×C23—42no—
71#41×C23—42no—
72C2×Q8×C23—21no—
73C2×Q8×C23—44no—
74C2×Q8×C23—41no—
75C2×Q8×C23—41no—
76C2×Q8×C23—41no—
77C2×Q8×C23—41no—
78C2×Q8×C23—42no—
79C2×Q8×C23—42no—
80C2×D4×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—41no—
84C22×C4×C23—22no—
85C22×C4×C23—22no—
86C22×C4×C23—22no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23C2×D411yescharacteristic
91M16×C23—42no—
92C2×C8×C23—22no—
93C2×C8×C23—22no—
94C2×C8×C23—41no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—22no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—41no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C42×C23—24no—
108C42×C23—22no—
109C42×C23—22no—
110C42×C23—42no—
111C42×C23C2×D411yescharacteristic
112C42×C23C2×D411yescharacteristic
113C42×C23C2×D411yescharacteristic
114C42×C23#4111yescharacteristic
115#10286C2311yesmaximal, characteristic, direct product factor, Sylow
116C23×C23—41no—
117C23×C23#12011yescharacteristic
118Q8×C23—42no—
119Q8×C23—82no—
120Q8×C23—82no—
121Q8×C23—82no—
122Q8×C23—81no—
123Q8×C23—81no—
124D4×C23—42no—
125D4×C23—42no—
126D4×C23—42no—
127D4×C23—82no—
128C2×C4×C23—22no—
129C2×C4×C23—22no—
130C2×C4×C23—24no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—44no—
134C2×C4×C23—41no—
135C2×C4×C23—42no—
136C2×C4×C23—42no—
137C2×C4×C23—44no—
138C2×C4×C23—81no—
139C2×C4×C23—81no—
140C2×C4×C23#12011yescharacteristic
141C2×C4×C23#12011yescharacteristic
142C2×C4×C23#12111yescharacteristic
143C2×C4×C23#12111yescharacteristic
144C2×C4×C23#12111yescharacteristic
145C2×C4×C23#12711yescharacteristic
146C2×C4×C23#13512yes—
147C2×C4×C23AGL(1,ℤ/8)12yes—
148C8×C23—42no—
149C8×C23—42no—
150C8×C23—81no—
151#3227C2×C2311yescharacteristic
152#3226C2×C2311yescharacteristic
153#2862C2×C2311yescharacteristic, complemented
154#1962C2×C2311yescharacteristic
155#1549C2×C2311yescharacteristic, complemented
156#1546C2×C2311yescharacteristic, complemented
157#1276C2×C2311yescharacteristic, complemented
158C22×C23—21no—
159C22×C23—21no—
160C22×C23—81no—
161C22×C23—81no—
162C22×C23—81no—
163C22×C23#39211yescharacteristic
164C22×C23#44912yes—
165C4×C23—21no—
166C4×C23—21no—
167C4×C23—44no—
168C4×C23—82no—
169C4×C23—82no—
170C4×C23—81no—
171C4×C23#45412yes—
172C4×C23#45612yes—
173C4×C23#46312yes—
174C4×C23#49511yescharacteristic
175C4×C23#49611yescharacteristic
176#532C22×C2311yescharacteristic
177C2×C4⋊Q8C22×C2311yescharacteristic
178C2×C4⋊Q8C22×C2311yescharacteristic
179#500—21no—
180#498—21no—
181#486—21no—
182#486—21no—
183#481—22no—
184#480—22no—
185#476—22no—
186#474—22no—
187#431C22×C2311yescharacteristic
188#431C22×C2311yescharacteristic
189#423C22×C2311yescharacteristic
190C2×#104—22no—
191#391—22no—
192#391—22no—
193#382—22no—
194C43C22×C2311yescharacteristic
195#332—22no—
196#332—22no—
197C2×C23—22no—
198C2×C23—161no—
199C2×C23#200311yescharacteristic
200C2×C23#200511yescharacteristic
201C2×C23#201211yescharacteristic
202C2×#41—21no—
203C22×Q8—21no—
204C22×Q8—21no—
205C2×M16—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8D4×C2311yescharacteristic
215#127D4×C2311yescharacteristic
216#124—21no—
217#121—21no—
218#121—21no—
219C4×D4—21no—
220C4×D4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4D4×C2312yes—
224C2×C4⋊C4D4×C2312yes—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42D4×C2312yes—
228C2×C42C23×C2311yescharacteristic
229#107—24no—
230#106—24no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8D4×C2312yes—
234C4⋊C8D4×C2312yes—
235#104—44no—
236#102—24no—
237#102—24no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242#95—28no—
243C23#1028611yescharacteristic, direct product factor, Sylow
244#41—42no—
245#41—42no—
246C2×Q8—21no—
247C2×Q8—44no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4C2×D4×C2311yescharacteristic
263C22×C4C2×D4×C2311yescharacteristic
264C22×C4C2×D4×C2311yescharacteristic
265M16—42no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—41no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42—24no—
282C42—22no—
283C42—22no—
284C42—42no—
285C42C2×D4×C2311yescharacteristic
286C42C2×D4×C2311yescharacteristic
287C42C2×D4×C2311yescharacteristic
288C42#41×C2311yescharacteristic
289C23—41no—
290C23C23×#12011yescharacteristic
291Q8—42no—
292Q8—82no—
293Q8—82no—
294Q8—82no—
295Q8—81no—
296Q8—81no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4C23×#12011yescharacteristic
314C2×C4C23×#12011yescharacteristic
315C2×C4C23×#12111yescharacteristic
316C2×C4C23×#12111yescharacteristic
317C2×C4C23×#12111yescharacteristic
318C2×C4C23×#12711yescharacteristic
319C2×C4C23×#13512yes—
320C2×C4C23×AGL(1,ℤ/8)12yes—
321C8—42no—
322C8—42no—
323C8—81no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22C23×#39211yescharacteristic
330C22C23×#44912yes—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—82no—
335C4—82no—
336C4—81no—
337C4C23×#45412yes—
338C4C23×#45612yes—
339C4C23×#46312yes—
340C4C23×#49511yescharacteristic
341C4C23×#49611yescharacteristic
342C2—22no—
343C2—161no—
344C2C23×#200311yescharacteristic
345C2C23×#200511yescharacteristic
346C2C23×#201211yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#2862C2semidirect
5. C23×#1549C2semidirect
6. C23×#1546C2semidirect
7. C23×#1276C2semidirect
115. #10286C23direct (N × H)
153. #2862C2×C23semidirect
155. #1549C2×C23semidirect
156. #1546C2×C23semidirect
157. #1276C2×C23semidirect
243. C23#10286direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32272
C23×#32262
C23×#28622
C23×#19622
C23×#15492
C23×#15462
C23×#12762
#1028623