Finite Group Databasebeta

← C23×#10550 #19158176

Subgroups — C23×#10550 #19158176

Order 5888.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3152C211yesmaximal, characteristic
2C23×#1950C211yesmaximal, characteristic, complemented
3C23×#1547C211yesmaximal, characteristic
4C23×#1546C211yesmaximal, characteristic, complemented
5C2×C23×#332C211yesmaximal, characteristic, complemented
6C23×#1480C211yesmaximal, characteristic
7C23×#1283C211yesmaximal, characteristic, complemented
8C2×C23×#137—21no—
9C2×C23×#137—21no—
10C23×#532C2211yescharacteristic
11C2×C23×C4⋊Q8C2211yescharacteristic
12C23×#497—22no—
13C23×#494—21no—
14C23×#492—21no—
15C23×#431C2211yescharacteristic
16C2×C23×#108—21no—
17C23×#423C2211yescharacteristic
18C2×C23×C4⋊C8C2211yescharacteristic
19C2×C23×C4⋊C8C2211yescharacteristic
20C4×M16×C23C2211yescharacteristic
21C23×#341—21no—
22C23×#332—22no—
23C23×#332—21no—
24C23×#332—22no—
25C23×#331—21no—
26C23×D4⋊C8—22no—
27C2×#41×C23—21no—
28C22×Q8×C23—21no—
29C23×#137—42no—
30C23×#137—42no—
31C23×#137—42no—
32C23×#137—42no—
33C2×Q16×C23—22no—
34C2×Q16×C23—22no—
35C2×SD16×C23—22no—
36C2×SD16×C23—22no—
37C2×M16×C23—21no—
38C2×M16×C23D411yescharacteristic
39C2×M16×C23D411yescharacteristic
40C22×C8×C23—21no—
41C22×C8×C23—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8—21no—
44C23×C4⋊Q8—21no—
45C23×C4⋊Q8D411yescharacteristic
46C23×C4⋊Q8D412yes—
47C23×#127D411yescharacteristic
48C23×#124—21no—
49C23×#121—21no—
50C23×#121—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C42×C23C2311yescharacteristic
56C23×#108—42no—
57C23×C4⋊C8—21no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8D412yes—
64C23×C22⋊C8—21no—
65C23×#97—21no—
66C4×C8×C23—21no—
67C4×C8×C23—21no—
68#41×C23—41no—
69#41×C23—41no—
70#41×C23—41no—
71#41×C23—41no—
72C2×Q8×C23—21no—
73C2×Q8×C23—22no—
74C2×Q8×C23—41no—
75C2×Q8×C23—41no—
76C2×Q8×C23—41no—
77C2×Q8×C23—41no—
78C2×Q8×C23—41no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23—41no—
83C22×C4×C23—41no—
84C22×C4×C23—41no—
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23#4111yescharacteristic
87C22×C4×C23#4111yescharacteristic
88Q16×C23—44no—
89Q16×C23—44no—
90SD16×C23—44no—
91SD16×C23—44no—
92M16×C23—21no—
93M16×C23—21no—
94M16×C23—41no—
95M16×C23—41no—
96M16×C23D812yes—
97M16×C23SD1612yes—
98C2×C8×C23—21no—
99C2×C8×C23—21no—
100C2×C8×C23—22no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C2×C8×C23—41no—
105C2×C8×C23—41no—
106C2×C8×C23—81no—
107C4⋊C4×C23—22no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—22no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—41no—
115C42×C23C2×D411yescharacteristic
116C42×C23C2×D411yescharacteristic
117C42×C23C2×D411yescharacteristic
118C42×C23#4111yescharacteristic
119#10550C2311yesmaximal, characteristic, direct product factor, Sylow
120C23×C23—41no—
121C23×C23#12411yescharacteristic
122Q8×C23—42no—
123Q8×C23—41no—
124Q8×C23—42no—
125Q8×C23—41no—
126Q8×C23—81no—
127Q8×C23—81no—
128D4×C23—41no—
129D4×C23—41no—
130D4×C23—41no—
131D4×C23—41no—
132D4×C23—81no—
133D4×C23—81no—
134D4×C23—81no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—41no—
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#12011yescharacteristic
145C2×C4×C23#12111yescharacteristic
146C2×C4×C23#12111yescharacteristic
147C2×C4×C23#12111yescharacteristic
148C2×C4×C23#12311yescharacteristic
149C2×C4×C23#12311yescharacteristic
150C2×C4×C23C2×D811yescharacteristic
151C2×C4×C23C2×SD1611yescharacteristic
152C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
154C8×C23—22no—
155C8×C23—22no—
156C8×C23—42no—
157C8×C23—81no—
158C8×C23—81no—
159C8×C23—81no—
160#3152C2×C2311yescharacteristic
161#1950C2×C2311yescharacteristic, complemented
162#1547C2×C2311yescharacteristic
163#1546C2×C2311yescharacteristic, complemented
164C2×#332C2×C2311yescharacteristic, complemented
165#1480C2×C2311yescharacteristic
166#1283C2×C2311yescharacteristic, complemented
167C22×C23—21no—
168C22×C23—21no—
169C22×C23—81no—
170C22×C23—81no—
171C22×C23—81no—
172C22×C23#39411yescharacteristic
173C22×C23#48611yescharacteristic
174C22×C23#49011yescharacteristic
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—21no—
178C4×C23—81no—
179C4×C23—82no—
180C4×C23#44711yescharacteristic
181C4×C23#45011yescharacteristic
182C4×C23#45911yescharacteristic
183C4×C23#46011yescharacteristic
184C4×C23#47111yescharacteristic
185C4×C23#47311yescharacteristic
186C2×#137—21no—
187C2×#137—21no—
188#532C22×C2311yescharacteristic
189C2×C4⋊Q8C22×C2311yescharacteristic
190#497—22no—
191#494—21no—
192#492—21no—
193#431C22×C2311yescharacteristic
194C2×#108—21no—
195#423C22×C2311yescharacteristic
196C2×C4⋊C8C22×C2311yescharacteristic
197C2×C4⋊C8C22×C2311yescharacteristic
198C4×M16C22×C2311yescharacteristic
199#341—21no—
200#332—22no—
201#332—21no—
202#332—22no—
203#331—21no—
204D4⋊C8—22no—
205C2×C23—21no—
206C2×C23—21no—
207C2×C23—161no—
208C2×C23#204111yescharacteristic
209C2×C23#204911yescharacteristic
210C2×C23#205211yescharacteristic
211C2×#41—21no—
212C22×Q8—21no—
213#137—42no—
214#137—42no—
215#137—42no—
216#137—42no—
217C2×Q16—22no—
218C2×Q16—22no—
219C2×SD16—22no—
220C2×SD16—22no—
221C2×M16—21no—
222C2×M16D4×C2311yescharacteristic
223C2×M16D4×C2311yescharacteristic
224C22×C8—21no—
225C22×C8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8D4×C2312yes—
230C4⋊Q8D4×C2311yescharacteristic
231#127D4×C2311yescharacteristic
232#124—21no—
233#121—21no—
234#121—21no—
235C4×D4—21no—
236C4×D4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C42C23×C2311yescharacteristic
240#108—42no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8D4×C2312yes—
248C22⋊C8—21no—
249#97—21no—
250C4×C8—21no—
251C4×C8—21no—
252C23#1055011yescharacteristic, direct product factor, Sylow
253#41—41no—
254#41—41no—
255#41—41no—
256#41—41no—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C2×D4×C2311yescharacteristic
271C22×C4#41×C2311yescharacteristic
272C22×C4#41×C2311yescharacteristic
273Q16—44no—
274Q16—44no—
275SD16—44no—
276SD16—44no—
277M16—21no—
278M16—21no—
279M16—41no—
280M16—41no—
281M16D8×C2312yes—
282M16SD16×C2312yes—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—81no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42C2×D4×C2311yescharacteristic
301C42C2×D4×C2311yescharacteristic
302C42C2×D4×C2311yescharacteristic
303C42#41×C2311yescharacteristic
304C23—41no—
305C23C23×#12411yescharacteristic
306Q8—42no—
307Q8—41no—
308Q8—42no—
309Q8—41no—
310Q8—81no—
311Q8—81no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—81no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4C23×#12011yescharacteristic
329C2×C4C23×#12111yescharacteristic
330C2×C4C23×#12111yescharacteristic
331C2×C4C23×#12111yescharacteristic
332C2×C4C23×#12311yescharacteristic
333C2×C4C23×#12311yescharacteristic
334C2×C4C2×D8×C2311yescharacteristic
335C2×C4C2×SD16×C2311yescharacteristic
336C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
337C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
338C8—22no—
339C8—22no—
340C8—42no—
341C8—81no—
342C8—81no—
343C8—81no—
344C22—21no—
345C22—21no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C23×#39411yescharacteristic
350C22C23×#48611yescharacteristic
351C22C23×#49011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—82no—
357C4C23×#44711yescharacteristic
358C4C23×#45011yescharacteristic
359C4C23×#45911yescharacteristic
360C4C23×#46011yescharacteristic
361C4C23×#47111yescharacteristic
362C4C23×#47311yescharacteristic
363C2—21no—
364C2—21no—
365C2—161no—
366C2C23×#204111yescharacteristic
367C2C23×#204911yescharacteristic
368C2C23×#205211yescharacteristic
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×#1950C2semidirect
4. C23×#1546C2semidirect
5. C2×C23×#332C2semidirect
7. C23×#1283C2semidirect
119. #10550C23direct (N × H)
161. #1950C2×C23semidirect
163. #1546C2×C23semidirect
164. C2×#332C2×C23semidirect
166. #1283C2×C23semidirect
252. C23#10550direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31522
C23×#19502
C23×#15472
C23×#15462
C2×C23×#3322
C23×#14802
C23×#12832
#1055023