Finite Group Databasebeta

← C23×#23896 #19171522

Subgroups — C23×#23896 #19171522

Order 5888.

1764 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3485C212yesmaximal, complemented
2C23×#3026C214yesmaximal, complemented
3C23×#2022C214yesmaximal, complemented
4C23×#1913C214yesmaximal, complemented
5C23×#1782C211yesmaximal, characteristic, complemented
6C2×C23×H(5,𝔽2)C2212yescomplemented
7C22×#41×C23C2211yescharacteristic
8C23×#547—28no—
9C23×#546—28no—
10C23×#536—22no—
11C23×#534—22no—
12C23×#534C2214yes—
13C23×#534C2214yes—
14C23×#525—22no—
15C2×C23×#123—22no—
16C2×C23×#123C2214yescomplemented
17C23×#518C2212yescomplemented
18C23×#458—28no—
19C23×B(4,𝔽2)—28no—
20C2×C23×#99C2218yescomplemented
21C2×C23×#99C2214yescomplemented
22C23×#381C2212yescomplemented
23C23×#342C2214yescomplemented
24C23×H(5,𝔽2)—28no—
25C23×H(5,𝔽2)—44no—
26C2×#41×C23—28no—
27C2×#41×C23—24no—
28C2×#41×C23—22no—
29C2×#41×C23—42no—
30C22×Q8×C23D411yescharacteristic, complemented
31C22×D4×C23—22no—
32C22×D4×C23—24no—
33C22×D4×C23D411yescharacteristic, complemented
34C22×D4×C23C2314yes—
35C22×D4×C23C2312yes—
36C23×C4×C23D412yescomplemented
37C23×C4×C23C2311yescharacteristic
38C23×C4⋊Q8—24no—
39C23×#124—24no—
40C23×#123—24no—
41C23×#123—28no—
42C23×#123—24no—
43C23×#123—24no—
44C23×#123—24no—
45C23×#123—28no—
46C23×#123—28no—
47C23×#123D418yescomplemented
48C23×C22⋊Q8—24no—
49C23×C22⋊Q8—28no—
50C23×#121—28no—
51C23×#121—24no—
52C23×#121—28no—
53C23×#121—44no—
54C23×#120—24no—
55C23×#120—24no—
56C23×#120—28no—
57C23×#120D418yescomplemented
58C4×D4×C23—28no—
59C4×D4×C23—28no—
60C23×#117—22no—
61C2×C4⋊C4×C23—24no—
62C2×C4⋊C4×C23—24no—
63C2×C22⋊C4×C23—24no—
64C2×C22⋊C4×C23—22no—
65C2×C22⋊C4×C23—24no—
66C2×C22⋊C4×C23C2314yes—
67C2×C22⋊C4×C23C2314yes—
68C23×#99—28no—
69C23×#99—216no—
70C23×#95—42no—
71C24×C23—22no—
72C24×C23C2×D414yes—
73C24×C23C2×D412yes—
74C24×C23C2411yescharacteristic
75#41×C23—416no—
76#41×C23—44no—
77#41×C23—416no—
78#41×C23—44no—
79C2×Q8×C23—22no—
80C2×Q8×C23—24no—
81C2×Q8×C23—41no—
82C2×D4×C23—24no—
83C2×D4×C23—24no—
84C2×D4×C23—24no—
85C2×D4×C23—22no—
86C2×D4×C23—24no—
87C2×D4×C23—28no—
88C2×D4×C23—24no—
89C2×D4×C23—24no—
90C2×D4×C23—28no—
91C2×D4×C23—28no—
92C2×D4×C23—41no—
93C2×D4×C23—416no—
94C2×D4×C23—44no—
95C2×D4×C23—44no—
96C2×D4×C23—44no—
97C2×D4×C23C2×D418yes—
98C22×C4×C23—22no—
99C22×C4×C23—24no—
100C22×C4×C23—24no—
101C22×C4×C23—24no—
102C22×C4×C23—22no—
103C22×C4×C23—24no—
104C22×C4×C23—24no—
105C22×C4×C23—24no—
106C22×C4×C23—24no—
107C22×C4×C23—42no—
108C22×C4×C23C2×D414yes—
109C22×C4×C23C2×D411yescharacteristic
110C22×C4×C23C2×D411yescharacteristic
111C22×C4×C23C2×D412yes—
112C4⋊C4×C23—28no—
113C4⋊C4×C23—28no—
114C4⋊C4×C23—24no—
115C4⋊C4×C23—28no—
116C4⋊C4×C23—44no—
117C22⋊C4×C23—28no—
118C22⋊C4×C23—24no—
119C22⋊C4×C23—28no—
120C22⋊C4×C23—28no—
121C22⋊C4×C23—24no—
122C22⋊C4×C23—28no—
123C22⋊C4×C23—28no—
124C22⋊C4×C23—48no—
125C22⋊C4×C23C2×D418yes—
126C42×C23—24no—
127#23896C2311yesmaximal, characteristic, direct product factor, Sylow
128C23×C23—22no—
129C23×C23—22no—
130C23×C23—24no—
131C23×C23—28no—
132C23×C23—44no—
133C23×C23—81no—
134C23×C23—82no—
135C23×C23—84no—
136C23×C23—84no—
137C23×C23#12014yes—
138C23×C23C22×D411yescharacteristic
139C23×C23C22×D414yes—
140C23×C23H(5,𝔽2)12yes—
141Q8×C23—44no—
142Q8×C23—44no—
143D4×C23—216no—
144D4×C23—44no—
145D4×C23—44no—
146D4×C23—48no—
147D4×C23—44no—
148D4×C23—44no—
149D4×C23—48no—
150D4×C23—88no—
151D4×C23—164no—
152C2×C4×C23—22no—
153C2×C4×C23—22no—
154C2×C4×C23—28no—
155C2×C4×C23—22no—
156C2×C4×C23—22no—
157C2×C4×C23—24no—
158C2×C4×C23—21no—
159C2×C4×C23—21no—
160C2×C4×C23—28no—
161C2×C4×C23—48no—
162C2×C4×C23—48no—
163C2×C4×C23—44no—
164C2×C4×C23—44no—
165C2×C4×C23—44no—
166C2×C4×C23—84no—
167C2×C4×C23—84no—
168C2×C4×C23—84no—
169C2×C4×C23#12014yes—
170#3485C2×C2312yescomplemented
171#3026C2×C2314yescomplemented
172#2022C2×C2314yescomplemented
173#1913C2×C2314yescomplemented
174#1782C2×C2311yescharacteristic, complemented
175C22×C23—24no—
176C22×C23—42no—
177C22×C23—44no—
178C22×C23—41no—
179C22×C23—44no—
180C22×C23—44no—
181C22×C23—48no—
182C22×C23—44no—
183C22×C23—82no—
184C22×C23—88no—
185C22×C23—164no—
186C22×C23C2×#12012yes—
187C22×C23#53411yescharacteristic
188C22×C23D4214yes—
189C4×C23—24no—
190C4×C23—48no—
191C4×C23—41no—
192C4×C23—44no—
193C4×C23—41no—
194C4×C23—84no—
195C2×H(5,𝔽2)C22×C2312yescomplemented
196C22×#41C22×C2311yescharacteristic
197#547—28no—
198#546—28no—
199#536—22no—
200#534—22no—
201#534C22×C2314yes—
202#534C22×C2314yes—
203#525—22no—
204C2×#123—22no—
205C2×#123C22×C2314yescomplemented
206#518C22×C2312yescomplemented
207#458—28no—
208B(4,𝔽2)—28no—
209C2×#99C22×C2318yescomplemented
210C2×#99C22×C2314yescomplemented
211#381C22×C2312yescomplemented
212#342C22×C2314yescomplemented
213C2×C23—22no—
214C2×C23—24no—
215C2×C23—21no—
216C2×C23—44no—
217C2×C23—84no—
218C2×C23#240311yescharacteristic
219H(5,𝔽2)—28no—
220H(5,𝔽2)—44no—
221C2×#41—28no—
222C2×#41—22no—
223C2×#41—24no—
224C2×#41—42no—
225C22×Q8D4×C2311yescharacteristic, complemented
226C22×D4—22no—
227C22×D4—24no—
228C22×D4D4×C2311yescharacteristic, complemented
229C22×D4C23×C2314yes—
230C22×D4C23×C2312yes—
231C23×C4D4×C2312yescomplemented
232C23×C4C23×C2311yescharacteristic
233C4⋊Q8—24no—
234#124—24no—
235#123—28no—
236#123—24no—
237#123—24no—
238#123—24no—
239#123—24no—
240#123—28no—
241#123—28no—
242#123D4×C2318yescomplemented
243C22⋊Q8—24no—
244C22⋊Q8—28no—
245#121—24no—
246#121—28no—
247#121—28no—
248#121—44no—
249#120—24no—
250#120—24no—
251#120—28no—
252#120D4×C2318yescomplemented
253C4×D4—28no—
254C4×D4—28no—
255#117—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4C23×C2314yes—
262C2×C22⋊C4C23×C2314yes—
263#99—28no—
264#99—216no—
265#95—42no—
266C23#2389611yescharacteristic, direct product factor, Sylow
267C24—22no—
268C24C2×D4×C2314yes—
269C24C2×D4×C2312yes—
270C24C24×C2311yescharacteristic
271#41—416no—
272#41—44no—
273#41—416no—
274#41—44no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—41no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—28no—
287C2×D4—28no—
288C2×D4—41no—
289C2×D4—416no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4C2×D4×C2318yes—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—42no—
304C22×C4C2×D4×C2314yes—
305C22×C4C2×D4×C2311yescharacteristic
306C22×C4C2×D4×C2311yescharacteristic
307C22×C4C2×D4×C2312yes—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—44no—
313C22⋊C4—28no—
314C22⋊C4—24no—
315C22⋊C4—28no—
316C22⋊C4—28no—
317C22⋊C4—24no—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—48no—
321C22⋊C4C2×D4×C2318yes—
322C42—24no—
323C23—22no—
324C23—22no—
325C23—24no—
326C23—28no—
327C23—44no—
328C23—81no—
329C23—82no—
330C23—84no—
331C23—84no—
332C23C23×#12014yes—
333C23C22×D4×C2311yescharacteristic
334C23C22×D4×C2314yes—
335C23C23×H(5,𝔽2)12yes—
336Q8—44no—
337Q8—44no—
338D4—216no—
339D4—44no—
340D4—44no—
341D4—48no—
342D4—44no—
343D4—44no—
344D4—48no—
345D4—88no—
346D4—164no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—28no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4C23×#12014yes—
365C22—24no—
366C22—42no—
367C22—44no—
368C22—41no—
369C22—44no—
370C22—44no—
371C22—48no—
372C22—44no—
373C22—82no—
374C22—88no—
375C22—164no—
376C22C2×C23×#12012yes—
377C22C23×#53411yescharacteristic
378C22D42×C2314yes—
379C4—24no—
380C4—48no—
381C4—41no—
382C4—44no—
383C4—41no—
384C4—84no—
385C2—22no—
386C2—24no—
387C2—21no—
388C2—44no—
389C2—84no—
390C2C23×#240311yescharacteristic
subgroupG/H
center, socleC2×C23#2403
commutator, Frattini, cosocleC24C24×C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3485C2semidirect
2. C23×#3026C2semidirect
3. C23×#2022C2semidirect
4. C23×#1913C2semidirect
5. C23×#1782C2semidirect
6. C2×C23×H(5,𝔽2)C22semidirect
16. C2×C23×#123C22semidirect
17. C23×#518C22semidirect
20. C2×C23×#99C22semidirect
21. C2×C23×#99C22semidirect
22. C23×#381C22semidirect
23. C23×#342C22semidirect
30. C22×Q8×C23D4semidirect
33. C22×D4×C23D4semidirect
36. C23×C4×C23D4semidirect
47. C23×#123D4semidirect
57. C23×#120D4semidirect
127. #23896C23direct (N × H)
170. #3485C2×C23semidirect
171. #3026C2×C23semidirect
172. #2022C2×C23semidirect
173. #1913C2×C23semidirect
174. #1782C2×C23semidirect
195. C2×H(5,𝔽2)C22×C23semidirect
205. C2×#123C22×C23semidirect
206. #518C22×C23semidirect
209. C2×#99C22×C23semidirect
210. C2×#99C22×C23semidirect
211. #381C22×C23semidirect
212. #342C22×C23semidirect
225. C22×Q8D4×C23semidirect
228. C22×D4D4×C23semidirect
231. C23×C4D4×C23semidirect
242. #123D4×C23semidirect
252. #120D4×C23semidirect
266. C23#23896direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34852
C23×#30262
C23×#20222
C23×#19132
C23×#17822
#2389623