Finite Group Databasebeta

← #34180910

Subgroups — #34180910

Order 20224.

1166 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 147 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
1#23763480C212yesmaximal, complemented
2#23761727C212yesmaximal, complemented
3#23761727C212yesmaximal, complemented
4C22×#123×C79C211yesmaximal, characteristic
5#19098527C2214yes—
6#19098332—24no—
7#19098287—24no—
8#19098287—24no—
9C23×D4×C79C411yescharacteristic
10C24×C4×C79C411yescharacteristic
11C2×#123×C79—24no—
12C2×#123×C79—24no—
13C2×#123×C79—24no—
14C22×C4⋊C4×C79C411yescharacteristic
15C22×C4⋊C4×C79C2211yescharacteristic
16C22×C22⋊C4×C79C411yescharacteristic
17C22×C22⋊C4×C79C2211yescharacteristic
18C22×C22⋊C4×C79C2211yescharacteristic
19C2×#11929608—24no—
20C22⋊(C79⋊C8)—24no—
21C22⋊(C79⋊C8)—24no—
22C25×C79C2×C411yescharacteristic
23C25×C79C2×C411yescharacteristic
24C22×D4×C79—22no—
25C22×D4×C79—22no—
26C22×D4×C79—22no—
27C22×D4×C79—44no—
28C23×C4×C79—22no—
29C23×C4×C79—22no—
30C23×C4×C79—22no—
31C23×C4×C79—44no—
32C23×C4×C79C2×C411yescharacteristic
33C23×C4×C79C2×C411yescharacteristic
34C23×C4×C79C2×C411yescharacteristic
35C23×C4×C79C2×C411yescharacteristic
36C23×C4×C79C2311yescharacteristic
37#123×C79—232no—
38C2×C4⋊C4×C79—22no—
39C2×C4⋊C4×C79—22no—
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79D414yes—
44C2×C22⋊C4×C79—22no—
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79—22no—
47C2×C22⋊C4×C79—22no—
48C2×C22⋊C4×C79—22no—
49C2×C22⋊C4×C79—22no—
50C2×C22⋊C4×C79—22no—
51C2×C22⋊C4×C79D414yes—
52C2×C22⋊C4×C79D414yes—
53#11929608—48no—
54C2×C79⋊C8—44no—
55C24×C79—24no—
56C24×C79—21no—
57C24×C79—21no—
58C24×C79—21no—
59C24×C79—21no—
60C24×C79—42no—
61C24×C79—42no—
62C24×C79—42no—
63C24×C79—42no—
64C24×C79—42no—
65C24×C79C22⋊C412yes—
66C24×C79C22⋊C412yes—
67C24×C79C22×C411yescharacteristic
68C2×D4×C79—28no—
69C2×D4×C79—48no—
70C2×D4×C79—44no—
71C2×D4×C79—48no—
72C2×D4×C79—44no—
73C22×C4×C79—28no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—42no—
83C22×C4×C79—48no—
84C22×C4×C79—48no—
85C22×C4×C79—44no—
86C22×C4×C79—44no—
87C22×C4×C79—42no—
88C22×C4×C79—42no—
89C22×C4×C79—42no—
90C22×C4×C79—42no—
91C22×C4×C79C22⋊C412yes—
92C22×C4×C79C22⋊C412yes—
93C22×C4×C79C22⋊C412yes—
94C22×C4×C79C22⋊C412yes—
95C22×C4×C79C2×D412yes—
96C22×C4×C79C2×D412yes—
97C22×C4×C79C2×D412yes—
98C4⋊C4×C79—28no—
99C4⋊C4×C79—28no—
100C22⋊C4×C79—28no—
101C22⋊C4×C79—28no—
102C22⋊C4×C79—28no—
103C23×C79—22no—
104C23×C79—22no—
105C23×C79—22no—
106C23×C79—24no—
107C23×C79—24no—
108C23×C79—22no—
109C23×C79—22no—
110C23×C79—22no—
111C23×C79—24no—
112C23×C79—42no—
113C23×C79—42no—
114C23×C79—416no—
115C23×C79—42no—
116C23×C79—42no—
117C23×C79—42no—
118C23×C79—42no—
119C23×C79—42no—
120C23×C79—42no—
121C23×C79—42no—
122C23×C79—42no—
123C23×C79—42no—
124C23×C79—42no—
125C23×C79—42no—
126C23×C79—42no—
127C23×C79—88no—
128C23×C79#10014yes—
129C23×C79C2×C22⋊C411yescharacteristic
130C23×C79C2×C22⋊C411yescharacteristic
131C23×C79C2×C22⋊C411yescharacteristic
132D4×C79—416no—
133C2×C4×C79—24no—
134C2×C4×C79—22no—
135C2×C4×C79—22no—
136C2×C4×C79—22no—
137C2×C4×C79—22no—
138C2×C4×C79—42no—
139C2×C4×C79—42no—
140C2×C4×C79—42no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79—416no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79—42no—
147C2×C4×C79—42no—
148C2×C4×C79—42no—
149C2×C4×C79—42no—
150C2×C4×C79—42no—
151C2×C4×C79—42no—
152C2×C4×C79—42no—
153C2×C4×C79#12014yes—
154C79⋊C8—84no—
155C22×C79—28no—
156C22×C79—21no—
157C22×C79—21no—
158C22×C79—21no—
159C22×C79—21no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—24no—
163C22×C79—44no—
164C22×C79—48no—
165C22×C79—44no—
166C22×C79—48no—
167C22×C79—42no—
168C22×C79—42no—
169C22×C79—42no—
170C22×C79—42no—
171C22×C79—42no—
172C22×C79—84no—
173C22×C79—84no—
174C22×C79—84no—
175C22×C79#37911yescharacteristic
176C22×C79#41012yes—
177C22×C79C2×#10012yes—
178C22×C79#41312yes—
179C4×C79—24no—
180C4×C79—42no—
181C4×C79—42no—
182C4×C79—42no—
183C4×C79—42no—
184#8553—791nomaximal, Sylow
185C2×C79—22no—
186C2×C79—22no—
187C2×C79—22no—
188C2×C79—44no—
189C2×C79—82no—
190C2×C79#178211yescharacteristic
191C2×C79#178311yescharacteristic
192C2×C79#178511yescharacteristic
193C22×#123D7911yescharacteristic
194#1821—792no—
195#1462—792no—
196#1462—792no—
197C79#855311yescharacteristic, complemented, Sylow
198C23×D4Dic7911yescharacteristic
199C24×C4Dic7911yescharacteristic
200C2×#123—24no—
201C2×#123—24no—
202C2×#123—24no—
203C22×C4⋊C4Dic7911yescharacteristic
204C22×C4⋊C4C2×D7911yescharacteristic
205C22×C22⋊C4Dic7911yescharacteristic
206C22×C22⋊C4C2×D7911yescharacteristic
207C22×C22⋊C4C2×D7911yescharacteristic
208#407—794no—
209#343—1584no—
210#323—1584no—
211#323—1584no—
212C25C2×Dic7911yescharacteristic
213C25C2×Dic7911yescharacteristic
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—44no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4C2×Dic7911yescharacteristic
223C23×C4C2×Dic7911yescharacteristic
224C23×C4C2×Dic7911yescharacteristic
225C23×C4C2×Dic7911yescharacteristic
226C23×C4C22×D7911yescharacteristic
227C2×M16—1584no—
228#123—232no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C22⋊D7914yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4C22⋊D7914yes—
243C2×C22⋊C4C22⋊D7914yes—
244C22⋊C8—1584no—
245C22⋊C8—1584no—
246C24—24no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C22⋊Dic7912yes—
257C24C22⋊Dic7912yes—
258C24C22×Dic7911yescharacteristic
259C2×D4—28no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C22×C4—28no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C22⋊Dic7912yes—
283C22×C4C22⋊Dic7912yes—
284C22×C4C22⋊Dic7912yes—
285C22×C4C22⋊Dic7912yes—
286C22×C4C2×C22⋊D7912yes—
287C22×C4C2×C22⋊D7912yes—
288C22×C4C2×C22⋊D7912yes—
289M16—3168no—
290C2×C8—3164no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—416no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—88no—
321C23#1457907814yes—
322C23C2×C22⋊Dic7911yescharacteristic
323C23C2×C22⋊Dic7911yescharacteristic
324C23C2×C22⋊Dic7911yescharacteristic
325D4—416no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—416no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4#1457913014yes—
347C8—6324no—
348C22—28no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—24no—
356C22—44no—
357C22—48no—
358C22—44no—
359C22—48no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22#1909841911yescharacteristic
369C22#1909853712yes—
370C22C2×#1457907812yes—
371C22#1909854712yes—
372C4—24no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—44no—
381C2—82no—
382C2#2376331811yescharacteristic
383C2#2376332111yescharacteristic
384C2#2376332811yescharacteristic
subgroupG/H
centerC22#19098419
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC22×#123×C79C2
socleC22×C79#379
prime radicalC23×D4×C79C4
cosocleC23×C4×C79C23
hypercenterC22×#123D79
hypocenterC79#8553
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. #23763480C2semidirect
2. #23761727C2semidirect
3. #23761727C2semidirect
197. C79#8553semidirect

Maximal subgroups

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

subgroupindex
#237634802
#237617272
C22×#123×C792
#855379