Finite Group Databasebeta

← C23×#8842 #19156468

Subgroups — C23×#8842 #19156468

Order 5888.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2989C211yesmaximal, characteristic
2C23×#1831C211yesmaximal, characteristic, complemented
3C4×C23×#104C211yesmaximal, characteristic
4C23×#1565C211yesmaximal, characteristic, complemented
5C23×#1477C211yesmaximal, characteristic
6C23×#1283C211yesmaximal, characteristic, complemented
7C23×#1276C211yesmaximal, characteristic, complemented
8C4×#41×C23C2211yescharacteristic
9C23×#446—21no—
10C23×#436—21no—
11C23×#435—22no—
12C23×#432—21no—
13C23×#431C2211yescharacteristic
14C23×#431C2211yescharacteristic
15C23×#423—21no—
16C2×C23×C4⋊C8C2211yescharacteristic
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×#104C411yescharacteristic
19C2×C23×#104C411yescharacteristic
20C4×M16×C23C2211yescharacteristic
21C43×C23C2211yescharacteristic
22C23×#339C411yescharacteristic
23C23×#339C411yescharacteristic
24C23×#336—22no—
25C23×#335—21no—
26C23×#334—21no—
27C23×Q8⋊C8—21no—
28C23×D4⋊C8—21no—
29C2×#41×C23C2×C411yescharacteristic
30C2×M16×C23—22no—
31C2×M16×C23C2×C411yescharacteristic
32C2×M16×C23C2×C411yescharacteristic
33C22×C8×C23—21no—
34C22×C8×C23D412yes—
35C4×Q8×C23D411yescharacteristic
36C4×Q8×C23Q811yescharacteristic
37C4×D4×C23—21no—
38C4×D4×C23—21no—
39C4×D4×C23D411yescharacteristic
40C4×D4×C23Q811yescharacteristic
41C23×#117—21no—
42C23×#117C2×C411yescharacteristic
43C2×C42×C23—21no—
44C2×C42×C23—21no—
45C2×C42×C23—21no—
46C2×C42×C23C2×C411yescharacteristic
47C2×C42×C23C2×C411yescharacteristic
48C2×C42×C23C2311yescharacteristic
49C23×C4⋊C8—22no—
50C23×C4⋊C8—21no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8D411yescharacteristic
56C23×C4⋊C8D411yescharacteristic
57C23×#104—22no—
58C23×#104—22no—
59C23×C22⋊C8—22no—
60C23×C22⋊C8—21no—
61C23×C22⋊C8—21no—
62C23×#97—22no—
63C23×#97—21no—
64C4×C8×C23—21no—
65C4×C8×C23—21no—
66C4×C8×C23—21no—
67#41×C23—21no—
68#41×C23—21no—
69#41×C23—21no—
70#41×C23—21no—
71C2×Q8×C23C4⋊C411yescharacteristic
72C2×D4×C23—21no—
73C2×D4×C23C4⋊C411yescharacteristic
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23C22×C411yescharacteristic
80C22×C4×C23C2×D411yescharacteristic
81C22×C4×C23#4111yescharacteristic
82M16×C23—21no—
83M16×C23—21no—
84M16×C23—21no—
85M16×C23—82no—
86M16×C23M1612yescomplemented
87C2×C8×C23—21no—
88C2×C8×C23—21no—
89C2×C8×C23—22no—
90C2×C8×C23—21no—
91C2×C8×C23—21no—
92C2×C8×C23—42no—
93C2×C8×C23—42no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C4⋊C4×C23—21no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23C4⋊C411yescharacteristic
99C4⋊C4×C23C4⋊C411yescharacteristic
100C22⋊C4×C23—21no—
101C22⋊C4×C23—21no—
102C22⋊C4×C23—21no—
103C42×C23—21no—
104C42×C23—21no—
105C42×C23—21no—
106C42×C23—21no—
107C42×C23—21no—
108C42×C23—21no—
109C42×C23—41no—
110C42×C23—41no—
111C42×C23—42no—
112C42×C23M1612yescomplemented
113C42×C23C2×D411yescharacteristic
114C42×C23C2×D411yescharacteristic
115C42×C23C2×Q811yescharacteristic
116C42×C23#4111yescharacteristic
117#8842C2311yesmaximal, characteristic, direct product factor, Sylow
118C23×C23—21no—
119C23×C23C4×D411yescharacteristic
120Q8×C23—21no—
121Q8×C23—21no—
122D4×C23—21no—
123D4×C23—21no—
124D4×C23—41no—
125D4×C23—41no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—22no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—22no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—81no—
143C2×C4×C23C2×C4⋊C411yescharacteristic
144C2×C4×C23C4×D411yescharacteristic
145C2×C4×C23#12011yescharacteristic
146C2×C4×C23C22⋊Q811yescharacteristic
147C2×C4×C23C22⋊Q811yescharacteristic
148C2×C4×C23#12311yescharacteristic
149C2×C4×C23C2×M1611yescharacteristic
150C2×C4×C23C2×M1611yescharacteristic
151C2×C4×C23#13111yescharacteristic
152C2×C4×C23#13111yescharacteristic
153C8×C23—22no—
154C8×C23—41no—
155C8×C23—41no—
156C8×C23—42no—
157C8×C23—41no—
158C8×C23—81no—
159C8×C23—81no—
160#2989C2×C2311yescharacteristic
161#1831C2×C2311yescharacteristic, complemented
162C4×#104C2×C2311yescharacteristic
163#1565C2×C2311yescharacteristic, complemented
164#1477C2×C2311yescharacteristic
165#1283C2×C2311yescharacteristic, complemented
166#1276C2×C2311yescharacteristic, complemented
167C22×C23—21no—
168C22×C23—21no—
169C22×C23—41no—
170C22×C23—41no—
171C22×C23—41no—
172C22×C23#38511yescharacteristic
173C22×C23#43511yescharacteristic
174C22×C23#43611yescharacteristic
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—21no—
178C4×C23—42no—
179C4×C23—42no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23#42311yescharacteristic
184C4×C23#42411yescharacteristic
185C4×C23#43511yescharacteristic
186C4×C23#43611yescharacteristic
187C4×C23#45411yescharacteristic
188C4×C23#45511yescharacteristic
189C4×#41C22×C2311yescharacteristic
190#446—21no—
191#436—21no—
192#435—22no—
193#432—21no—
194#431C22×C2311yescharacteristic
195#431C22×C2311yescharacteristic
196#423—21no—
197C2×C4⋊C8C22×C2311yescharacteristic
198C2×C4⋊C8C22×C2311yescharacteristic
199C2×#104C4×C2311yescharacteristic
200C2×#104C4×C2311yescharacteristic
201C4×M16C22×C2311yescharacteristic
202C43C22×C2311yescharacteristic
203#339C4×C2311yescharacteristic
204#339C4×C2311yescharacteristic
205#336—22no—
206#335—21no—
207#334—21no—
208Q8⋊C8—21no—
209D4⋊C8—21no—
210C2×C23—21no—
211C2×C23—21no—
212C2×C23—81no—
213C2×C23#185111yescharacteristic
214C2×C23#186011yescharacteristic
215C2×C23#186111yescharacteristic
216C2×#41C2×C4×C2311yescharacteristic
217C2×M16—22no—
218C2×M16C2×C4×C2311yescharacteristic
219C2×M16C2×C4×C2311yescharacteristic
220C22×C8—21no—
221C22×C8D4×C2312yes—
222C4×Q8D4×C2311yescharacteristic
223C4×Q8Q8×C2311yescharacteristic
224C4×D4—21no—
225C4×D4—21no—
226C4×D4D4×C2311yescharacteristic
227C4×D4Q8×C2311yescharacteristic
228#117—21no—
229#117C2×C4×C2311yescharacteristic
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C2×C4×C2311yescharacteristic
234C2×C42C2×C4×C2311yescharacteristic
235C2×C42C23×C2311yescharacteristic
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8D4×C2311yescharacteristic
243C4⋊C8D4×C2311yescharacteristic
244#104—22no—
245#104—22no—
246C22⋊C8—22no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249#97—22no—
250#97—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C23#884211yescharacteristic, direct product factor, Sylow
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259C2×Q8C4⋊C4×C2311yescharacteristic
260C2×D4—21no—
261C2×D4C4⋊C4×C2311yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4C22×C4×C2311yescharacteristic
268C22×C4C2×D4×C2311yescharacteristic
269C22×C4#41×C2311yescharacteristic
270M16—21no—
271M16—21no—
272M16—21no—
273M16—82no—
274M16M16×C2312yescomplemented
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4C4⋊C4×C2311yescharacteristic
287C4⋊C4C4⋊C4×C2311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—41no—
298C42—41no—
299C42—42no—
300C42M16×C2312yescomplemented
301C42C2×D4×C2311yescharacteristic
302C42C2×D4×C2311yescharacteristic
303C42C2×Q8×C2311yescharacteristic
304C42#41×C2311yescharacteristic
305C23—21no—
306C23C4×D4×C2311yescharacteristic
307Q8—21no—
308Q8—21no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—41no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4C2×C4⋊C4×C2311yescharacteristic
331C2×C4C4×D4×C2311yescharacteristic
332C2×C4C23×#12011yescharacteristic
333C2×C4C23×C22⋊Q811yescharacteristic
334C2×C4C23×C22⋊Q811yescharacteristic
335C2×C4C23×#12311yescharacteristic
336C2×C4C2×M16×C2311yescharacteristic
337C2×C4C2×M16×C2311yescharacteristic
338C2×C4C23×#13111yescharacteristic
339C2×C4C23×#13111yescharacteristic
340C8—22no—
341C8—41no—
342C8—41no—
343C8—42no—
344C8—41no—
345C8—81no—
346C8—81no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22C23×#38511yescharacteristic
353C22C23×#43511yescharacteristic
354C22C23×#43611yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4C23×#42311yescharacteristic
364C4C23×#42411yescharacteristic
365C4C23×#43511yescharacteristic
366C4C23×#43611yescharacteristic
367C4C23×#45411yescharacteristic
368C4C23×#45511yescharacteristic
369C2—21no—
370C2—21no—
371C2—81no—
372C2C23×#185111yescharacteristic
373C2C23×#186011yescharacteristic
374C2C23×#186111yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#385
prime radicalC2×#41×C23C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1831C2semidirect
4. C23×#1565C2semidirect
6. C23×#1283C2semidirect
7. C23×#1276C2semidirect
86. M16×C23M16semidirect
112. C42×C23M16semidirect
117. #8842C23direct (N × H)
161. #1831C2×C23semidirect
163. #1565C2×C23semidirect
165. #1283C2×C23semidirect
166. #1276C2×C23semidirect
254. C23#8842direct (N × H)
274. M16M16×C23semidirect
300. C42M16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#29892
C23×#18312
C4×C23×#1042
C23×#15652
C23×#14772
C23×#12832
C23×#12762
#884223