Finite Group Databasebeta

← C23×#9274 #19156900

Subgroups — C23×#9274 #19156900

Order 5888.

492 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×#3208C211yesmaximal, characteristic
2C23×#2962C211yesmaximal, characteristic, complemented
3C23×#1965C211yesmaximal, characteristic
4C23×#1550C211yesmaximal, characteristic, complemented
5C2×C23×#331C211yesmaximal, characteristic
6C23×#1301C211yesmaximal, characteristic, complemented
7C23×#1283C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C2×C23×#127C2211yescharacteristic
10C23×#492—21no—
11C23×#486—21no—
12C23×#460—22no—
13C23×#450C414yes—
14C23×#446—22no—
15C23×#446—21no—
16C8×Q8×C23—21no—
17C23×#431C2211yescharacteristic
18C23×#424—21no—
19C23×#423C2211yescharacteristic
20C2×C23×C4⋊C8C2211yescharacteristic
21C2×C23×C4⋊C8C2211yescharacteristic
22C2×C23×#100—21no—
23C2×C23×#100—21no—
24C23×#405—21no—
25C4×M16×C23C2211yescharacteristic
26C23×#333—21no—
27C23×#332—21no—
28C23×#331—21no—
29C23×#331—21no—
30C22×D4×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23C2×C412yes—
33C2×SD16×C23—24no—
34C2×M16×C23—21no—
35C2×M16×C23D411yescharacteristic
36C2×M16×C23D411yescharacteristic
37C22×C8×C23—21no—
38C22×C8×C23—21no—
39C23×C4⋊Q8D411yescharacteristic
40C23×C4⋊Q8D411yescharacteristic
41C23×#127—21no—
42C23×#127—21no—
43C23×#127—21no—
44C23×#127—21no—
45C23×#125—21no—
46C23×C22⋊Q8—21no—
47C23×C22⋊Q8C2×C412yes—
48C4×Q8×C23—21no—
49C4×Q8×C23D412yes—
50C23×#117—21no—
51C2×C42×C23C2311yescharacteristic
52C23×C4⋊C8—22no—
53C23×C4⋊C8—22no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8D411yescharacteristic
61C23×C4⋊C8D411yescharacteristic
62C23×#102—24no—
63C23×#100—42no—
64C23×#100—42no—
65C23×C22⋊C8C2×C412yes—
66C23×#97—22no—
67C23×#97—21no—
68C4×C8×C23—22no—
69C4×C8×C23—21no—
70C4×C8×C23—21no—
71C24×C23—22no—
72C2×Q8×C23—21no—
73C2×Q8×C23C22⋊C412yes—
74C2×D4×C23—22no—
75C2×D4×C23—22no—
76C2×D4×C23—41no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C2×D4×C23—81no—
85C2×D4×C23—81no—
86C2×D4×C23—82no—
87C22×C4×C23C22×C411yescharacteristic
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23#4111yescharacteristic
90SD16×C23—84no—
91M16×C23—21no—
92M16×C23—21no—
93M16×C23—21no—
94M16×C23—41no—
95M16×C23—41no—
96M16×C23SD1612yescomplemented
97C2×C8×C23—21no—
98C2×C8×C23—22no—
99C2×C8×C23—21no—
100C2×C8×C23—21no—
101C2×C8×C23—21no—
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—21no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—22no—
112C4⋊C4×C23C22⋊C412yes—
113C22⋊C4×C23—22no—
114C42×C23—22no—
115C42×C23C2×D411yescharacteristic
116C42×C23C2×D411yescharacteristic
117C42×C23C2×D411yescharacteristic
118C42×C23#4111yescharacteristic
119#9274C2311yesmaximal, characteristic, direct product factor, Sylow
120C23×C23—42no—
121C23×C23—42no—
122C23×C23—42no—
123C23×C23—82no—
124C23×C23—82no—
125C23×C23C4×D411yescharacteristic
126Q8×C23—42no—
127Q8×C23—41no—
128Q8×C23—41no—
129D4×C23—44no—
130D4×C23—82no—
131D4×C23—82no—
132D4×C23—81no—
133D4×C23—81no—
134D4×C23—81no—
135D4×C23—81no—
136D4×C23—81no—
137D4×C23—81no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—22no—
141C2×C4×C23—22no—
142C2×C4×C23—41no—
143C2×C4×C23C2×C22⋊C411yescharacteristic
144C2×C4×C23C4×D411yescharacteristic
145C2×C4×C23#12011yescharacteristic
146C2×C4×C23#12111yescharacteristic
147C2×C4×C23#12111yescharacteristic
148C2×C4×C23#12311yescharacteristic
149C2×C4×C23C2×SD1611yescharacteristic
150C2×C4×C23#13511yescharacteristic
151C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C23#13711yescharacteristic
153C8×C23—22no—
154C8×C23—42no—
155C8×C23—41no—
156C8×C23—41no—
157C8×C23—41no—
158C8×C23—81no—
159C8×C23—81no—
160#3208C2×C2311yescharacteristic
161#2962C2×C2311yescharacteristic, complemented
162#1965C2×C2311yescharacteristic
163#1550C2×C2311yescharacteristic, complemented
164C2×#331C2×C2311yescharacteristic
165#1301C2×C2311yescharacteristic, complemented
166#1283C2×C2311yescharacteristic, complemented
167C22×C23—21no—
168C22×C23—21no—
169C22×C23—82no—
170C22×C23—82no—
171C22×C23—82no—
172C22×C23—162no—
173C22×C23—162no—
174C22×C23#38611yescharacteristic
175C22×C23C4×SD1611yescharacteristic
176C22×C23#44011yescharacteristic
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—42no—
181C4×C23—42no—
182C4×C23#44811yescharacteristic
183C4×C23#44911yescharacteristic
184C4×C23#46011yescharacteristic
185C4×C23#46411yescharacteristic
186C4×C23#47111yescharacteristic
187C4×C23#47211yescharacteristic
188#533C22×C2311yescharacteristic
189C2×#127C22×C2311yescharacteristic
190#492—21no—
191#486—21no—
192#460—22no—
193#450C4×C2314yes—
194#446—22no—
195#446—21no—
196C8×Q8—21no—
197#431C22×C2311yescharacteristic
198#424—21no—
199#423C22×C2311yescharacteristic
200C2×C4⋊C8C22×C2311yescharacteristic
201C2×C4⋊C8C22×C2311yescharacteristic
202C2×#100—21no—
203C2×#100—21no—
204#405—21no—
205C4×M16C22×C2311yescharacteristic
206#333—21no—
207#332—21no—
208#331—21no—
209#331—21no—
210C2×C23—21no—
211C2×C23—21no—
212C2×C23—162no—
213C2×C23#188011yescharacteristic
214C2×C23#190711yescharacteristic
215C2×C23#191411yescharacteristic
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C2×C4×C2312yes—
219C2×SD16—24no—
220C2×M16—21no—
221C2×M16D4×C2311yescharacteristic
222C2×M16D4×C2311yescharacteristic
223C22×C8—21no—
224C22×C8—21no—
225C4⋊Q8D4×C2311yescharacteristic
226C4⋊Q8D4×C2311yescharacteristic
227#127—21no—
228#127—21no—
229#127—21no—
230#127—21no—
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8C2×C4×C2312yes—
234C4×Q8—21no—
235C4×Q8D4×C2312yes—
236#117—21no—
237C2×C42C23×C2311yescharacteristic
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8D4×C2311yescharacteristic
247C4⋊C8D4×C2311yescharacteristic
248#102—24no—
249#100—42no—
250#100—42no—
251C22⋊C8C2×C4×C2312yes—
252#97—22no—
253#97—21no—
254C4×C8—22no—
255C4×C8—21no—
256C4×C8—21no—
257C23#927411yescharacteristic, direct product factor, Sylow
258C24—22no—
259C2×Q8—21no—
260C2×Q8C22⋊C4×C2312yes—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—81no—
273C2×D4—82no—
274C22×C4C22×C4×C2311yescharacteristic
275C22×C4C2×D4×C2311yescharacteristic
276C22×C4#41×C2311yescharacteristic
277SD16—84no—
278M16—21no—
279M16—21no—
280M16—21no—
281M16—41no—
282M16—41no—
283M16SD16×C2312yescomplemented
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—81no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4C22⋊C4×C2312yes—
300C22⋊C4—22no—
301C42—22no—
302C42C2×D4×C2311yescharacteristic
303C42C2×D4×C2311yescharacteristic
304C42C2×D4×C2311yescharacteristic
305C42#41×C2311yescharacteristic
306C23—42no—
307C23—42no—
308C23—42no—
309C23—82no—
310C23—82no—
311C23C4×D4×C2311yescharacteristic
312Q8—42no—
313Q8—41no—
314Q8—41no—
315D4—44no—
316D4—82no—
317D4—82no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4C2×C22⋊C4×C2311yescharacteristic
330C2×C4C4×D4×C2311yescharacteristic
331C2×C4C23×#12011yescharacteristic
332C2×C4C23×#12111yescharacteristic
333C2×C4C23×#12111yescharacteristic
334C2×C4C23×#12311yescharacteristic
335C2×C4C2×SD16×C2311yescharacteristic
336C2×C4C23×#13511yescharacteristic
337C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
338C2×C4C23×#13711yescharacteristic
339C8—22no—
340C8—42no—
341C8—41no—
342C8—41no—
343C8—41no—
344C8—81no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—162no—
352C22—162no—
353C22C23×#38611yescharacteristic
354C22C4×SD16×C2311yescharacteristic
355C22C23×#44011yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4C23×#44811yescharacteristic
362C4C23×#44911yescharacteristic
363C4C23×#46011yescharacteristic
364C4C23×#46411yescharacteristic
365C4C23×#47111yescharacteristic
366C4C23×#47211yescharacteristic
367C2—21no—
368C2—21no—
369C2—162no—
370C2C23×#188011yescharacteristic
371C2C23×#190711yescharacteristic
372C2C23×#191411yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#127C22
LCSG ▷ C22×C4 ▷ 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×#2962C2semidirect
4. C23×#1550C2semidirect
6. C23×#1301C2semidirect
7. C23×#1283C2semidirect
96. M16×C23SD16semidirect
119. #9274C23direct (N × H)
161. #2962C2×C23semidirect
163. #1550C2×C23semidirect
165. #1301C2×C23semidirect
166. #1283C2×C23semidirect
257. C23#9274direct (N × H)
283. M16SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32082
C23×#29622
C23×#19652
C23×#15502
C2×C23×#3312
C23×#13012
C23×#12832
#927423