Finite Group Databasebeta

← C23×#10501 #19158127

Subgroups — C23×#10501 #19158127

Order 5888.

440 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3206C211yesmaximal, characteristic
2C23×#1997C211yesmaximal, characteristic, complemented
3C23×#1981C211yesmaximal, characteristic, complemented
4C23×#1980C211yesmaximal, characteristic
5C23×#1565C211yesmaximal, characteristic, complemented
6C23×#1547C211yesmaximal, characteristic
7C23×#1303C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C23×#492—21no—
11C23×#487—21no—
12C23×#461—21no—
13C23×#461—21no—
14C23×#448—22no—
15C23×#431C2211yescharacteristic
16C23×#423C2211yescharacteristic
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×#104—21no—
19C2×C23×#104—21no—
20C2×C23×#103—22no—
21C4×M16×C23C2211yescharacteristic
22C4×C4⋊C4×C23C2211yescharacteristic
23C23×#343—21no—
24C23×#343—21no—
25C23×#335—21no—
26C23×#334—21no—
27C23×#332—21no—
28C23×#331—21no—
29C2×#41×C23—21no—
30C22×Q8×C23D412yes—
31C2×SD16×C23—22no—
32C2×SD16×C23—22no—
33C2×M16×C23—21no—
34C2×M16×C23D411yescharacteristic
35C2×M16×C23D411yescharacteristic
36C22×C8×C23—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×C23—21no—
51C2×C42×C23—21no—
52C2×C42×C23C2311yescharacteristic
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8D411yescharacteristic
58C23×C4⋊C8D411yescharacteristic
59C23×#104—42no—
60C23×#104—42no—
61C23×#103—22no—
62C23×#103—22no—
63C23×#103—44no—
64C23×C22⋊C8—21no—
65C23×#97—21no—
66C4×C8×C23—21no—
67C4×C8×C23—21no—
68C23×#95—21no—
69#41×C23—41no—
70#41×C23—41no—
71#41×C23—41no—
72#41×C23—41no—
73C2×Q8×C23—22no—
74C2×Q8×C23—21no—
75C2×Q8×C23—22no—
76C2×Q8×C23—42no—
77C2×Q8×C23—42no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—41no—
82C22×C4×C23—22no—
83C22×C4×C23—41no—
84C22×C4×C23—41no—
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23C2×D411yescharacteristic
87C22×C4×C23#4111yescharacteristic
88SD16×C23—82no—
89SD16×C23—82no—
90M16×C23—21no—
91M16×C23—21no—
92M16×C23—21no—
93M16×C23—41no—
94M16×C23—41no—
95M16×C23SD1612yescomplemented
96C2×C8×C23—21no—
97C2×C8×C23—21no—
98C2×C8×C23—21no—
99C2×C8×C23—21no—
100C2×C8×C23—42no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C4⋊C4×C23—22no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—41no—
107C4⋊C4×C23—41no—
108C4⋊C4×C23—41no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C42×C23—41no—
114C42×C23—41no—
115C42×C23C2×D411yescharacteristic
116C42×C23C2×D411yescharacteristic
117C42×C23#4111yescharacteristic
118C42×C23#4111yescharacteristic
119#10501C2311yesmaximal, characteristic, direct product factor, Sylow
120C23×C23—41no—
121C23×C23#12111yescharacteristic
122Q8×C23—44no—
123Q8×C23—42no—
124Q8×C23—42no—
125Q8×C23—41no—
126Q8×C23—41no—
127D4×C23—41no—
128D4×C23—41no—
129D4×C23—41no—
130D4×C23—41no—
131D4×C23—81no—
132D4×C23—81no—
133D4×C23—81no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—41no—
137C2×C4×C23—42no—
138C2×C4×C23—41no—
139C2×C4×C23—42no—
140C2×C4×C23—81no—
141C2×C4×C23—81no—
142C2×C4×C23—81no—
143C2×C4×C23—81no—
144C2×C4×C23—82no—
145C2×C4×C23#12011yescharacteristic
146C2×C4×C23#12111yescharacteristic
147C2×C4×C23#12111yescharacteristic
148C2×C4×C23#12311yescharacteristic
149C2×C4×C23#12311yescharacteristic
150C2×C4×C23#12411yescharacteristic
151C2×C4×C23C2×SD1611yescharacteristic
152C2×C4×C23#13511yescharacteristic
153C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C23#13711yescharacteristic
155C8×C23—22no—
156C8×C23—41no—
157C8×C23—41no—
158C8×C23—41no—
159C8×C23—81no—
160C8×C23—81no—
161#3206C2×C2311yescharacteristic
162#1997C2×C2311yescharacteristic, complemented
163#1981C2×C2311yescharacteristic, complemented
164#1980C2×C2311yescharacteristic
165#1565C2×C2311yescharacteristic, complemented
166#1547C2×C2311yescharacteristic
167#1303C2×C2311yescharacteristic, complemented
168C22×C23—21no—
169C22×C23—21no—
170C22×C23—81no—
171C22×C23—81no—
172C22×C23—81no—
173C22×C23#39411yescharacteristic
174C22×C23#46011yescharacteristic
175C22×C23#46411yescharacteristic
176C4×C23—21no—
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—82no—
180C4×C23—81no—
181C4×C23—82no—
182C4×C23#45311yescharacteristic
183C4×C23#45511yescharacteristic
184C4×C23#46111yescharacteristic
185C4×C23#46311yescharacteristic
186C4×C23#46511yescharacteristic
187C4×C23#46811yescharacteristic
188#532C22×C2311yescharacteristic
189C2×C4⋊Q8C22×C2311yescharacteristic
190#492—21no—
191#487—21no—
192#461—21no—
193#461—21no—
194#448—22no—
195#431C22×C2311yescharacteristic
196#423C22×C2311yescharacteristic
197C2×C4⋊C8C22×C2311yescharacteristic
198C2×#104—21no—
199C2×#104—21no—
200C2×#103—22no—
201C4×M16C22×C2311yescharacteristic
202C4×C4⋊C4C22×C2311yescharacteristic
203#343—21no—
204#343—21no—
205#335—21no—
206#334—21no—
207#332—21no—
208#331—21no—
209C2×C23—21no—
210C2×C23—21no—
211C2×C23—161no—
212C2×C23#203711yescharacteristic
213C2×C23#204911yescharacteristic
214C2×C23#205511yescharacteristic
215C2×#41—21no—
216C22×Q8D4×C2312yes—
217C2×SD16—22no—
218C2×SD16—22no—
219C2×M16—21no—
220C2×M16D4×C2311yescharacteristic
221C2×M16D4×C2311yescharacteristic
222C22×C8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8D4×C2311yescharacteristic
228#127D4×C2311yescharacteristic
229#124—21no—
230#121—21no—
231#121—21no—
232C4×D4—21no—
233C4×D4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C42—21no—
238C2×C42C23×C2311yescharacteristic
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8D4×C2311yescharacteristic
244C4⋊C8D4×C2311yescharacteristic
245#104—42no—
246#104—42no—
247#103—22no—
248#103—22no—
249#103—44no—
250C22⋊C8—21no—
251#97—21no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255C23#1050111yescharacteristic, direct product factor, Sylow
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260C2×Q8—22no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4C2×D4×C2311yescharacteristic
273C22×C4C2×D4×C2311yescharacteristic
274C22×C4#41×C2311yescharacteristic
275SD16—82no—
276SD16—82no—
277M16—21no—
278M16—21no—
279M16—21no—
280M16—41no—
281M16—41no—
282M16SD16×C2312yescomplemented
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—22no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—41no—
301C42—41no—
302C42C2×D4×C2311yescharacteristic
303C42C2×D4×C2311yescharacteristic
304C42#41×C2311yescharacteristic
305C42#41×C2311yescharacteristic
306C23—41no—
307C23C23×#12111yescharacteristic
308Q8—44no—
309Q8—42no—
310Q8—42no—
311Q8—41no—
312Q8—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4—81no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4C23×#12011yescharacteristic
332C2×C4C23×#12111yescharacteristic
333C2×C4C23×#12111yescharacteristic
334C2×C4C23×#12311yescharacteristic
335C2×C4C23×#12311yescharacteristic
336C2×C4C23×#12411yescharacteristic
337C2×C4C2×SD16×C2311yescharacteristic
338C2×C4C23×#13511yescharacteristic
339C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
340C2×C4C23×#13711yescharacteristic
341C8—22no—
342C8—41no—
343C8—41no—
344C8—41no—
345C8—81no—
346C8—81no—
347C22—21no—
348C22—21no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22C23×#39411yescharacteristic
353C22C23×#46011yescharacteristic
354C22C23×#46411yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4—81no—
361C4C23×#45311yescharacteristic
362C4C23×#45511yescharacteristic
363C4C23×#46111yescharacteristic
364C4C23×#46311yescharacteristic
365C4C23×#46511yescharacteristic
366C4C23×#46811yescharacteristic
367C2—21no—
368C2—21no—
369C2—161no—
370C2C23×#203711yescharacteristic
371C2C23×#204911yescharacteristic
372C2C23×#205511yescharacteristic
subgroupG/H
center, socleC22×C23#394
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
2. C23×#1997C2semidirect
3. C23×#1981C2semidirect
5. C23×#1565C2semidirect
7. C23×#1303C2semidirect
95. M16×C23SD16semidirect
119. #10501C23direct (N × H)
162. #1997C2×C23semidirect
163. #1981C2×C23semidirect
165. #1565C2×C23semidirect
167. #1303C2×C23semidirect
255. C23#10501direct (N × H)
282. M16SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32062
C23×#19972
C23×#19812
C23×#19802
C23×#15652
C23×#15472
C23×#13032
#1050123