Finite Group Databasebeta

← #19212281

Subgroups — #19212281

Order 5888.

846 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 117 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
1D23×#390C211yesmaximal, characteristic
2#15673464C212yesmaximal, complemented
3#15671218C211yesmaximal, characteristic, complemented
4#15671217C212yesmaximal, complemented
5C23×#1270C211yesmaximal, characteristic, complemented
6C22×D4×D23C411yescharacteristic
7C2×C4⋊C4×D23C411yescharacteristic
8C2×C22⋊C4×D23—22no—
9C2×C42×D23C2211yescharacteristic
10C2×(C2×D23)⋊C8—22no—
11#13092320C411yescharacteristic
12#13092319C2211yescharacteristic
13#13092318C411yescharacteristic
14#13092317—22no—
15#13092172C2212yescomplemented
16#13092171—22no—
17C23×#390C2211yescharacteristic
18C23×#336C2212yescomplemented
19C24×D23—21no—
20C22×D4×C23C2×C411yescharacteristic
21C22×C22⋊D23—21no—
22C2×D4×D23—22no—
23C2×D4×D23—42no—
24C2×D4×D23—42no—
25C2×D4×D23C814yescomplemented
26C22×D92C2×C411yescharacteristic
27C22×C4×D23—21no—
28C22×C4×D23C2×C411yescharacteristic
29C22×C4×D23D412yes—
30C22×C8×C23—22no—
31C2×C4⋊C4×C23C2×C411yescharacteristic
32C2×C22⋊C4×C23—22no—
33C2×C42×C23C2311yescharacteristic
34C2×C22⋊Dic23—22no—
35C2×(C2×D23)⋊C4—22no—
36C2×(C2×D23)⋊C4—22no—
37C2×C4⋊Dic23C2×C411yescharacteristic
38C2×Dic23⋊C4—21no—
39C2×C4×Dic23C2×C411yescharacteristic
40C2×C4×Dic23D412yes—
41C22×C23⋊C8—22no—
42C4⋊C4×D23—42no—
43C22⋊C4×D23—44no—
44C22⋊C4×D23—44no—
45C42×D23—42no—
46(C2×D23)⋊C8—44no—
47C24×C23—21no—
48C23×D23—21no—
49C23×D23—24no—
50C23×D23—44no—
51C23×D23—44no—
52C23×D23—42no—
53C23×D23—42no—
54C23×D23C22⋊C411yescharacteristic
55C2×D4×C23—21no—
56C2×D4×C23—41no—
57C2×D4×C23—41no—
58C2×D4×C23C2×C812yes—
59C22×C4×C23—21no—
60C22×C4×C23C22×C411yescharacteristic
61C22×C4×C23C2×D412yes—
62C2×C22⋊D23—24no—
63C2×C22⋊D23—42no—
64C2×C22⋊D23—42no—
65C22×Dic23—21no—
66C22×Dic23C22⋊C412yes—
67C22×Dic23C22⋊C411yescharacteristic
68D4×D23—48no—
69D4×D23—48no—
70C2×D92—21no—
71C2×D92—41no—
72C2×D92—41no—
73C2×D92C2×C812yes—
74C2×C4×D23—22no—
75C2×C4×D23—22no—
76C2×C4×D23—24no—
77C2×C4×D23—22no—
78C2×C4×D23—22no—
79C2×C4×D23—42no—
80C2×C4×D23—44no—
81C2×C4×D23—42no—
82C2×C4×D23C2×C812yes—
83C2×C4×D23M1612yes—
84C2×C8×C23—42no—
85C2×C8×C23—42no—
86C2×C8×C23—42no—
87C4⋊C4×C23—41no—
88C22⋊C4×C23—42no—
89C22⋊C4×C23—42no—
90C42×C23—41no—
91C22⋊Dic23—44no—
92(C2×D23)⋊C4—44no—
93(C2×D23)⋊C4—42no—
94(C2×D23)⋊C4—42no—
95C4⋊Dic23—41no—
96Dic23⋊C4—42no—
97C4×Dic23—41no—
98C4×Dic23—42no—
99C2×C23⋊C8—42no—
100C2×C23⋊C8—42no—
101C2×C23⋊C8—42no—
102#7592—231nomaximal, Sylow
103C23×C23—22no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—42no—
107C23×C23—42no—
108C23×C23C2×C22⋊C411yescharacteristic
109C22×D23—22no—
110C22×D23—22no—
111C22×D23—22no—
112C22×D23—416no—
113C22×D23—41no—
114C22×D23—41no—
115C22×D23—42no—
116C22×D23—42no—
117C22×D23—48no—
118C22×D23—84no—
119C22×D23—88no—
120C22×D23—84no—
121C22×D23C22⋊C812yes—
122C22×D23#9912yes—
123C22×D23#10012yes—
124D4×C23—42no—
125D4×C23—42no—
126C2×C4×C23—22no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—22no—
130C2×C4×C23—41no—
131C2×C4×C23—42no—
132C2×C4×C23C22×C811yescharacteristic
133C2×C4×C23C2×M1611yescharacteristic
134C2×C4×C23AGL(1,ℤ/8)12yes—
135C2×C4×C23AGL(1,ℤ/8)12yes—
136C22⋊D23—48no—
137C2×Dic23—21no—
138C2×Dic23—21no—
139C2×Dic23—22no—
140C2×Dic23—22no—
141C2×Dic23—22no—
142C2×Dic23—41no—
143C2×Dic23—42no—
144C2×Dic23C22⋊C812yes—
145D92—42no—
146D92—42no—
147C4×D23—24no—
148C4×D23—24no—
149C4×D23—44no—
150C4×D23—44no—
151C4×D23—84no—
152C8×C23—82no—
153C23⋊C8—82no—
154C2×#390—231no—
155#1817—232no—
156#1270—231no—
157#1270—232no—
158#1270D2311yescharacteristic, complemented
159C22×C23—42no—
160C22×C23—44no—
161C22×C23—81no—
162C22×C23—82no—
163C22×C23—81no—
164C22×C23C2×C22⋊C811yescharacteristic
165C22×C23C2×#9911yescharacteristic
166C22×C23C2×#10011yescharacteristic
167C22×C23#41812yes—
168C22×C23#42112yes—
169C2×D23—22no—
170C2×D23—24no—
171C2×D23—24no—
172C2×D23—22no—
173C2×D23—42no—
174C2×D23—44no—
175C2×D23—816no—
176C2×D23—81no—
177C2×D23—82no—
178C2×D23—81no—
179C2×D23#32314yes—
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—22no—
183C4×C23—22no—
184C4×C23—81no—
185Dic23—22no—
186Dic23—42no—
187Dic23—81no—
188C23×D4—231no—
189C22×C4⋊C4—231no—
190C22×C22⋊C4—462no—
191C22×C42—231no—
192C2×C22⋊C8—462no—
193#390—231no—
194#390—231no—
195#390—231no—
196#390—462no—
197#390C2×D2311yescharacteristic
198#336—232no—
199#336—462no—
200#336C2×D2312yescomplemented
201C2×C23—82no—
202C2×C23C2×#32311yescharacteristic
203C2×C23#147812yes—
204C2×C23#151012yes—
205C2×C23#153311yescharacteristic
206C2×C23#155511yescharacteristic
207D23—24no—
208D23—82no—
209C25—461no—
210C22×D4—234no—
211C22×D4—231no—
212C22×D4—462no—
213C22×D4—461no—
214C22×D4—922no—
215C22×D4—922no—
216C22×D4C4×D2311yescharacteristic
217C23×C4—231no—
218C23×C4—232no—
219C23×C4—461no—
220C22×C8—22no—
221C22×C8—462no—
222C2×C4⋊C4—231no—
223C2×C4⋊C4—461no—
224C2×C4⋊C4—922no—
225C2×C4⋊C4C4×D2311yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—462no—
228C2×C22⋊C4—462no—
229C2×C22⋊C4—462no—
230C2×C22⋊C4—924no—
231C2×C22⋊C4—924no—
232C2×C42—231no—
233C2×C42—232no—
234C2×C42—922no—
235C2×C42C22×D2311yescharacteristic
236C22⋊C8—924no—
237C23#759211yescharacteristic, complemented, Sylow
238C24—21no—
239C24—231no—
240C24—464no—
241C24—461no—
242C24—922no—
243C24—922no—
244C24—924no—
245C24—924no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—232no—
250C2×D4—464no—
251C2×D4—461no—
252C2×D4—921no—
253C2×D4—921no—
254C2×D4—928no—
255C2×D4—928no—
256C2×D4—922no—
257C2×D4—922no—
258C2×D4C8×D2312yes—
259C22×C4—21no—
260C22×C4—231no—
261C22×C4—232no—
262C22×C4—232no—
263C22×C4—232no—
264C22×C4—462no—
265C22×C4—462no—
266C22×C4—464no—
267C22×C4—462no—
268C22×C4—462no—
269C22×C4—461no—
270C22×C4—922no—
271C22×C4—922no—
272C22×C4—924no—
273C22×C4C2×C4×D2311yescharacteristic
274C22×C4D4×D2312yes—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—922no—
279C2×C8—922no—
280C2×C8—922no—
281C4⋊C4—41no—
282C4⋊C4—921no—
283C4⋊C4—922no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—924no—
287C22⋊C4—924no—
288C22⋊C4—922no—
289C22⋊C4—922no—
290C42—41no—
291C42—922no—
292C42—921no—
293C23—22no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—42no—
298C23—232no—
299C23—232no—
300C23—232no—
301C23—462no—
302C23—462no—
303C23—462no—
304C23—928no—
305C23—9216no—
306C23—921no—
307C23—921no—
308C23—922no—
309C23—922no—
310C23—1844no—
311C23—1848no—
312C23—1844no—
313C23C22⋊C4×D2311yescharacteristic
314D4—42no—
315D4—42no—
316D4—922no—
317D4—922no—
318D4—928no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—232no—
326C2×C4—464no—
327C2×C4—464no—
328C2×C4—461no—
329C2×C4—461no—
330C2×C4—462no—
331C2×C4—462no—
332C2×C4—462no—
333C2×C4—924no—
334C2×C4—924no—
335C2×C4—921no—
336C2×C4—922no—
337C2×C4—1844no—
338C2×C4C2×C8×D2311yescharacteristic
339C2×C4M16×D2311yescharacteristic
340C2×C4#1062508212yes—
341C2×C4#1062508512yes—
342C8—82no—
343C8—1842no—
344C22—42no—
345C22—44no—
346C22—81no—
347C22—82no—
348C22—81no—
349C22—234no—
350C22—462no—
351C22—464no—
352C22—462no—
353C22—464no—
354C22—922no—
355C22—924no—
356C22—18416no—
357C22—1841no—
358C22—1842no—
359C22—1841no—
360C22C22⋊C8×D2311yescharacteristic
361C22#99×D2311yescharacteristic
362C22#100×D2311yescharacteristic
363C22#1309242012yes—
364C22#1309243212yes—
365C4—22no—
366C4—22no—
367C4—21no—
368C4—21no—
369C4—81no—
370C4—462no—
371C4—922no—
372C4—1841no—
373C2—82no—
374C2—464no—
375C2—1842no—
376C2D23×#32311yescharacteristic
377C2#1567178612yes—
378C2#1567192412yes—
379C2#1567201911yescharacteristic
380C2#1567210911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D23
commutatorC2×C4×C23C22×C8
FrattiniC2×C42C22×D23
FittingC23×#1270C2
socleC23×C23C2×C22⋊C4
hypercenter#1270D23
hypocenterC23#7592
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #15673464C2semidirect
3. #15671218C2semidirect
4. #15671217C2semidirect
5. C23×#1270C2semidirect
15. #13092172C22semidirect
18. C23×#336C22semidirect
25. C2×D4×D23C8semidirect
158. #1270D23semidirect
200. #336C2×D23semidirect
237. C23#7592semidirect