Finite Group Databasebeta

← C23×#9371 #19156997

Subgroups — C23×#9371 #19156997

Order 5888.

442 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×#3214C211yesmaximal, characteristic
2C23×#2981C211yesmaximal, characteristic, complemented
3C23×#1551C211yesmaximal, characteristic, complemented
4C23×#1536C211yesmaximal, characteristic, complemented
5C23×#1480C211yesmaximal, characteristic, complemented
6C23×#1290C211yesmaximal, characteristic, complemented
7C23×#1287C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C23×#496C412yes—
11C23×#489C412yes—
12C23×#461—21no—
13C23×#459—21no—
14C23×#449—22no—
15C23×#436—21no—
16C23×#435—22no—
17C8×D4×C23—21no—
18C23×#432C2211yescharacteristic
19C23×#432C2211yescharacteristic
20C23×#432C2211yescharacteristic
21C23×#431C2211yescharacteristic
22C2×C23×C4⋊C8C2211yescharacteristic
23C23×#408—22no—
24C23×#333—21no—
25C23×#331—21no—
26C23×#329—22no—
27C23×D4⋊C8—21no—
28C23×D4⋊C8—21no—
29C2×#41×C23—21no—
30C2×#41×C23D412yes—
31C2×SD16×C23—22no—
32C2×D8×C23—22no—
33C2×M16×C23—22no—
34C22×C8×C23—22no—
35C22×C8×C23—21no—
36C23×C4⋊Q8C2×C411yescharacteristic
37C23×C4⋊Q8D411yescharacteristic
38C23×#127C2×C411yescharacteristic
39C23×#127D411yescharacteristic
40C23×#124—21no—
41C23×#124C2×C412yes—
42C23×#121—21no—
43C23×#121—21no—
44C23×#121—21no—
45C23×#121—21no—
46C4×D4×C23—21no—
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C2×C42×C23C2311yescharacteristic
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×#103—22no—
58C23×#102—22no—
59C23×C22⋊C8—22no—
60C23×C22⋊C8—22no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8—21no—
63C23×C22⋊C8—21no—
64C23×C22⋊C8—21no—
65C23×#97—21no—
66C23×#97C2×C412yes—
67C23×#97D412yes—
68C4×C8×C23D411yescharacteristic
69C4×C8×C23D411yescharacteristic
70#41×C23—81no—
71#41×C23—81no—
72#41×C23—82no—
73C2×Q8×C23—21no—
74C2×Q8×C23C22⋊C412yes—
75C2×D4×C23—22no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—41no—
82C2×D4×C23C22⋊C412yes—
83C22×C4×C23—22no—
84C22×C4×C23—41no—
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23#4111yescharacteristic
87C22×C4×C23#4111yescharacteristic
88SD16×C23—82no—
89D8×C23—82no—
90M16×C23—82no—
91C2×C8×C23—22no—
92C2×C8×C23—21no—
93C2×C8×C23—22no—
94C2×C8×C23—21no—
95C2×C8×C23—41no—
96C2×C8×C23—41no—
97C2×C8×C23—82no—
98C2×C8×C23—81no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—21no—
103C22⋊C4×C23—22no—
104C22⋊C4×C23—22no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C42×C23C22×C411yescharacteristic
108C42×C23C2×D411yescharacteristic
109C42×C23C2×D411yescharacteristic
110C42×C23C2×D411yescharacteristic
111#9371C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—22no—
113C23×C23—41no—
114C23×C23#12311yescharacteristic
115Q8×C23—42no—
116Q8×C23—81no—
117D4×C23—42no—
118D4×C23—41no—
119D4×C23—41no—
120D4×C23—42no—
121D4×C23—41no—
122D4×C23—41no—
123D4×C23—81no—
124D4×C23—81no—
125D4×C23—82no—
126D4×C23—82no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—22no—
130C2×C4×C23—41no—
131C2×C4×C23—41no—
132C2×C4×C23—42no—
133C2×C4×C23—42no—
134C2×C4×C23—41no—
135C2×C4×C23—81no—
136C2×C4×C23—81no—
137C2×C4×C23C2×C22⋊C411yescharacteristic
138C2×C4×C23C4×D411yescharacteristic
139C2×C4×C23C4×D411yescharacteristic
140C2×C4×C23#12011yescharacteristic
141C2×C4×C23#12111yescharacteristic
142C2×C4×C23#12111yescharacteristic
143C8×C23—41no—
144C8×C23—42no—
145C8×C23—41no—
146C8×C23—42no—
147C8×C23—81no—
148C8×C23—81no—
149#3214C2×C2311yescharacteristic
150#2981C2×C2311yescharacteristic, complemented
151#1551C2×C2311yescharacteristic, complemented
152#1536C2×C2311yescharacteristic, complemented
153#1480C2×C2311yescharacteristic, complemented
154#1290C2×C2311yescharacteristic, complemented
155#1287C2×C2311yescharacteristic, complemented
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—42no—
160C22×C23—42no—
161C22×C23—42no—
162C22×C23—81no—
163C22×C23—81no—
164C22×C23—81no—
165C22×C23#38611yescharacteristic
166C4×C23—21no—
167C4×C23—21no—
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—42no—
172C4×C23—81no—
173C4×C23#47111yescharacteristic
174C4×C23#47211yescharacteristic
175#532C22×C2311yescharacteristic
176#532C22×C2311yescharacteristic
177#496C4×C2312yes—
178#489C4×C2312yes—
179#461—21no—
180#459—21no—
181#449—22no—
182#436—21no—
183#435—22no—
184C8×D4—21no—
185#432C22×C2311yescharacteristic
186#432C22×C2311yescharacteristic
187#432C22×C2311yescharacteristic
188#431C22×C2311yescharacteristic
189C2×C4⋊C8C22×C2311yescharacteristic
190#408—22no—
191#333—21no—
192#331—21no—
193#329—22no—
194D4⋊C8—21no—
195D4⋊C8—21no—
196C2×C23—41no—
197C2×C23—82no—
198C2×C23—161no—
199C2×C23#188111yescharacteristic
200C2×C23#190711yescharacteristic
201C2×C23#191111yescharacteristic
202C2×#41—21no—
203C2×#41D4×C2312yes—
204C2×SD16—22no—
205C2×D8—22no—
206C2×M16—22no—
207C22×C8—22no—
208C22×C8—21no—
209C4⋊Q8C2×C4×C2311yescharacteristic
210C4⋊Q8D4×C2311yescharacteristic
211#127C2×C4×C2311yescharacteristic
212#127D4×C2311yescharacteristic
213#124—21no—
214#124C2×C4×C2312yes—
215#121—21no—
216#121—21no—
217#121—21no—
218#121—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C2×C42C23×C2311yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230#103—22no—
231#102—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238#97—21no—
239#97C2×C4×C2312yes—
240#97D4×C2312yes—
241C4×C8D4×C2311yescharacteristic
242C4×C8D4×C2311yescharacteristic
243C23#937111yescharacteristic, direct product factor, Sylow
244#41—81no—
245#41—81no—
246#41—82no—
247C2×Q8—21no—
248C2×Q8C22⋊C4×C2312yes—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4C22⋊C4×C2312yes—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4C2×D4×C2311yescharacteristic
260C22×C4#41×C2311yescharacteristic
261C22×C4#41×C2311yescharacteristic
262SD16—82no—
263D8—82no—
264M16—82no—
265C2×C8—22no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—82no—
272C2×C8—81no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42C22×C4×C2311yescharacteristic
282C42C2×D4×C2311yescharacteristic
283C42C2×D4×C2311yescharacteristic
284C42C2×D4×C2311yescharacteristic
285C23—22no—
286C23—41no—
287C23C23×#12311yescharacteristic
288Q8—42no—
289Q8—81no—
290D4—42no—
291D4—41no—
292D4—41no—
293D4—42no—
294D4—41no—
295D4—41no—
296D4—81no—
297D4—81no—
298D4—82no—
299D4—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C2×C22⋊C4×C2311yescharacteristic
311C2×C4C4×D4×C2311yescharacteristic
312C2×C4C4×D4×C2311yescharacteristic
313C2×C4C23×#12011yescharacteristic
314C2×C4C23×#12111yescharacteristic
315C2×C4C23×#12111yescharacteristic
316C8—41no—
317C8—42no—
318C8—41no—
319C8—42no—
320C8—81no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22C23×#38611yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—42no—
338C4—81no—
339C4C23×#47111yescharacteristic
340C4C23×#47211yescharacteristic
341C2—41no—
342C2—82no—
343C2—161no—
344C2C23×#188111yescharacteristic
345C2C23×#190711yescharacteristic
346C2C23×#191111yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3214C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2981C2semidirect
3. C23×#1551C2semidirect
4. C23×#1536C2semidirect
5. C23×#1480C2semidirect
6. C23×#1290C2semidirect
7. C23×#1287C2semidirect
111. #9371C23direct (N × H)
150. #2981C2×C23semidirect
151. #1551C2×C23semidirect
152. #1536C2×C23semidirect
153. #1480C2×C23semidirect
154. #1290C2×C23semidirect
155. #1287C2×C23semidirect
243. C23#9371direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32142
C23×#29812
C23×#15512
C23×#15362
C23×#14802
C23×#12902
C23×#12872
#937123