Finite Group Databasebeta

← #34180900

Subgroups — #34180900

Order 20224.

1134 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 155 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#23763324C212yesmaximal, complemented
2#23761730C212yesmaximal, complemented
3#23761727C212yesmaximal
4C22×C22⋊Q8×C79C211yesmaximal, characteristic
5C2×#14579080—24no—
6#19098527C2214yes—
7#19098289—24no—
8#19098287—24no—
9C23×Q8×C79C2211yescharacteristic
10C24×C4×C79C411yescharacteristic
11C2×C22⋊Q8×C79—24no—
12C2×C22⋊Q8×C79—24no—
13C2×C22⋊Q8×C79—24no—
14C22×C4⋊C4×C79C411yescharacteristic
15C22×C4⋊C4×C79C411yescharacteristic
16C22×C4⋊C4×C79C2211yescharacteristic
17C22×C22⋊C4×C79C411yescharacteristic
18C22×C22⋊C4×C79C2211yescharacteristic
19C2×#11929608—24no—
20#14579080—48no—
21C22⋊(C79⋊C8)—24no—
22C22⋊(C79⋊C8)—24no—
23C25×C79C2×C411yescharacteristic
24C22×Q8×C79—22no—
25C22×Q8×C79—22no—
26C22×Q8×C79—44no—
27C22×Q8×C79D414yes—
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×C79C2×C411yescharacteristic
37C23×C4×C79C2311yescharacteristic
38C22⋊Q8×C79—232no—
39C2×C4⋊C4×C79—22no—
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79—22no—
45C2×C4⋊C4×C79—22no—
46C2×C4⋊C4×C79—22no—
47C2×C4⋊C4×C79D414yes—
48C2×C22⋊C4×C79—22no—
49C2×C22⋊C4×C79—22no—
50C2×C22⋊C4×C79—22no—
51C2×C22⋊C4×C79—22no—
52C2×C22⋊C4×C79—22no—
53C2×C22⋊C4×C79D414yes—
54#11929608—48no—
55C2×C79⋊C8—44no—
56C24×C79—24no—
57C24×C79—21no—
58C24×C79—21no—
59C24×C79—42no—
60C24×C79—42no—
61C24×C79C22⋊C412yes—
62C24×C79C22×C411yescharacteristic
63C2×Q8×C79—28no—
64C2×Q8×C79—28no—
65C2×Q8×C79—44no—
66C2×Q8×C79—48no—
67C2×Q8×C79—48no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—24no—
71C22×C4×C79—28no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
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—42no—
81C22×C4×C79—42no—
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×C79C22⋊C412yes—
90C22×C4×C79C22⋊C412yes—
91C22×C4×C79C22⋊C412yes—
92C22×C4×C79C22⋊C412yes—
93C22×C4×C79C22⋊C412yes—
94C22×C4×C79C2×D412yes—
95C22×C4×C79C2×D412yes—
96C22×C4×C79C2×D412yes—
97C4⋊C4×C79—28no—
98C4⋊C4×C79—28no—
99C4⋊C4×C79—28no—
100C22⋊C4×C79—28no—
101C22⋊C4×C79—28no—
102C23×C79—22no—
103C23×C79—22no—
104C23×C79—22no—
105C23×C79—24no—
106C23×C79—24no—
107C23×C79—22no—
108C23×C79—22no—
109C23×C79—42no—
110C23×C79—42no—
111C23×C79—42no—
112C23×C79—416no—
113C23×C79—42no—
114C23×C79—42no—
115C23×C79—42no—
116C23×C79#10114yescomplemented
117C23×C79C2×C22⋊C411yescharacteristic
118C23×C79C2×C22⋊C411yescharacteristic
119C23×C79C2×C22⋊C411yescharacteristic
120Q8×C79—416no—
121C2×C4×C79—24no—
122C2×C4×C79—24no—
123C2×C4×C79—22no—
124C2×C4×C79—24no—
125C2×C4×C79—22no—
126C2×C4×C79—22no—
127C2×C4×C79—22no—
128C2×C4×C79—22no—
129C2×C4×C79—22no—
130C2×C4×C79—24no—
131C2×C4×C79—24no—
132C2×C4×C79—42no—
133C2×C4×C79—42no—
134C2×C4×C79—416no—
135C2×C4×C79—42no—
136C2×C4×C79—42no—
137C2×C4×C79—42no—
138C2×C4×C79—42no—
139C2×C4×C79—42no—
140C2×C4×C79—42no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79—42no—
144C2×C4×C79—42no—
145C2×C4×C79#9914yes—
146C2×C4×C79#12014yes—
147C79⋊C8—84no—
148C22×C79—28no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—24no—
156C22×C79—44no—
157C22×C79—48no—
158C22×C79—44no—
159C22×C79—48no—
160C22×C79—42no—
161C22×C79—42no—
162C22×C79#37911yescharacteristic
163C22×C79C2×#9912yes—
164C22×C79C2×#10112yes—
165C22×C79#41312yes—
166C4×C79—24no—
167C4×C79—24no—
168C4×C79—42no—
169C4×C79—42no—
170C4×C79—42no—
171C4×C79—42no—
172#8550—791nomaximal, Sylow
173C2×C79—22no—
174C2×C79—22no—
175C2×C79—22no—
176C2×C79—44no—
177C2×C79#178111yescharacteristic
178C2×C79#178411yescharacteristic
179C2×C79#178511yescharacteristic
180C22×C22⋊Q8D7911yescharacteristic
181#1784—792no—
182#1463—792no—
183#1462—792no—
184C79#855011yescharacteristic, complemented, Sylow
185C23×Q8C2×D7911yescharacteristic
186C24×C4Dic7911yescharacteristic
187C2×C22⋊Q8—24no—
188C2×C22⋊Q8—24no—
189C2×C22⋊Q8—24no—
190C22×C4⋊C4Dic7911yescharacteristic
191C22×C4⋊C4Dic7911yescharacteristic
192C22×C4⋊C4C2×D7911yescharacteristic
193C22×C22⋊C4Dic7911yescharacteristic
194C22×C22⋊C4C2×D7911yescharacteristic
195C2×#101—1584no—
196#407—794no—
197#324—1584no—
198#323—1584no—
199C25C2×Dic7911yescharacteristic
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—44no—
203C22×Q8C22⋊D7914yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—44no—
208C23×C4C2×Dic7911yescharacteristic
209C23×C4C2×Dic7911yescharacteristic
210C23×C4C2×Dic7911yescharacteristic
211C23×C4C2×Dic7911yescharacteristic
212C23×C4C2×Dic7911yescharacteristic
213C23×C4C22×D7911yescharacteristic
214C2×M16—1584no—
215C22⋊Q8—232no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C22⋊D7914yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C22⋊D7914yes—
231#101—3168no—
232C22⋊C8—1584no—
233C22⋊C8—1584no—
234C24—24no—
235C24—21no—
236C24—21no—
237C24—42no—
238C24—42no—
239C24C22⋊Dic7912yes—
240C24C22×Dic7911yescharacteristic
241C2×Q8—28no—
242C2×Q8—28no—
243C2×Q8—44no—
244C2×Q8—48no—
245C2×Q8—48no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C22⋊Dic7912yes—
268C22×C4C22⋊Dic7912yes—
269C22×C4C22⋊Dic7912yes—
270C22×C4C22⋊Dic7912yes—
271C22×C4C22⋊Dic7912yes—
272C22×C4C2×C22⋊D7912yes—
273C22×C4C2×C22⋊D7912yes—
274C22×C4C2×C22⋊D7912yes—
275M16—3168no—
276C2×C8—3164no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—24no—
286C23—24no—
287C23—22no—
288C23—22no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—416no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23#1457908014yescomplemented
297C23C2×C22⋊Dic7911yescharacteristic
298C23C2×C22⋊Dic7911yescharacteristic
299C23C2×C22⋊Dic7911yescharacteristic
300Q8—416no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—416no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4#1457907614yes—
326C2×C4#1457913014yes—
327C8—6324no—
328C22—28no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—44no—
337C22—48no—
338C22—44no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22#1909841911yescharacteristic
343C22C2×#1457907612yes—
344C22C2×#1457908012yes—
345C22#1909854712yes—
346C4—24no—
347C4—24no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C2—22no—
353C2—22no—
354C2—22no—
355C2—44no—
356C2#2376331511yescharacteristic
357C2#2376332411yescharacteristic
358C2#2376332811yescharacteristic
subgroupG/H
centerC22#19098419
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC22×C22⋊Q8×C79C2
socleC22×C79#379
prime radicalC25×C79C2×C4
cosocleC23×C4×C79C23
hypercenterC22×C22⋊Q8D79
hypocenterC79#8550
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23763324C2semidirect
2. #23761730C2semidirect
116. C23×C79#101semidirect
184. C79#8550semidirect
296. C23#14579080semidirect

Maximal subgroups

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

subgroupindex
#237633242
#237617302
#237617272
C22×C22⋊Q8×C792
#855079