Finite Group Databasebeta

← #34199142

Subgroups — #34199142

Order 20224.

390 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 87 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#23763477C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic
3#23761421C212yesmaximal
4#23761400C212yesmaximal
5C79×#2599C211yesmaximal, characteristic
6#19098616—21no—
7#19098426—21no—
8#19098330—21no—
9#19098330C2212yes—
10#19098289—21no—
11#19098287C2212yes—
12C2×C4×Q8×C79C411yescharacteristic, complemented
13C2×#117×C79C2211yescharacteristic
14#398×C79C411yescharacteristic, complemented
15#397×C79C2211yescharacteristic
16#395×C79C411yescharacteristic, complemented
17#393×C79C2211yescharacteristic
18#388×C79—21no—
19#387×C79—21no—
20#387×C79—21no—
21#385×C79—21no—
22#383×C79C411yescharacteristic, complemented
23C2×C22⋊Dic79—22no—
24C4⋊(C79⋊C8)—41no—
25C22⋊(C79⋊C8)—22no—
26C4×C79⋊C8—41no—
27C79⋊#95—41no—
28C79⋊#95—41no—
29C22×Q8×C79C2×C411yescharacteristic
30C23×C4×C79D411yescharacteristic
31C4×Q8×C79—41no—
32C4×Q8×C79—41no—
33#117×C79—41no—
34#117×C79—41no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79C2×C411yescharacteristic
39C2×C4⋊C4×C79C2×C411yescharacteristic
40C2×C4⋊C4×C79D411yescharacteristic
41C2×C22⋊C4×C79—21no—
42C2×C22⋊C4×C79—21no—
43C2×C22⋊C4×C79D411yescharacteristic
44C2×C22⋊C4×C79C2311yescharacteristic
45C2×C42×C79—21no—
46C2×C42×C79C2×C411yescharacteristic
47C2×C42×C79D411yescharacteristic
48#95×C79—21no—
49#95×C79—21no—
50#95×C79—21no—
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
54#95×C79C2×C411yescharacteristic
55#95×C79C2×C411yescharacteristic
56C22×Dic79—42no—
57C2×C79⋊C8—42no—
58C22⋊Dic79—42no—
59C22⋊Dic79—42no—
60C24×C79C2×D411yescharacteristic
61C2×Q8×C79—42no—
62C2×Q8×C79—41no—
63C22×C4×C79—21no—
64C22×C4×C79—22no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—41no—
71C22×C4×C79C22⋊C411yescharacteristic
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22×C411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77C4⋊C4×C79—41no—
78C4⋊C4×C79—41no—
79C4⋊C4×C79—42no—
80C4⋊C4×C79—42no—
81C4⋊C4×C79—41no—
82C4⋊C4×C79—41no—
83C4⋊C4×C79—41no—
84C22⋊C4×C79—41no—
85C22⋊C4×C79—41no—
86C22⋊C4×C79—42no—
87C22⋊C4×C79—42no—
88C42×C79—21no—
89C42×C79—21no—
90C42×C79—21no—
91C42×C79—21no—
92C42×C79—42no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79C2×C22⋊C411yescharacteristic
98C23×C79#13512yes—
99Q8×C79—42no—
100Q8×C79—42no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—22no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79#9911yescharacteristic
129C2×C4×C79#9911yescharacteristic
130C2×Dic79—82no—
131C2×Dic79—82no—
132C2×Dic79—82no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—22no—
137C22×C79—42no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—81no—
141C22×C79C2×#9911yescharacteristic
142C22×C79#41611yescharacteristic
143C22×C79#42111yescharacteristic
144C4×C79—21no—
145C4×C79—21no—
146C4×C79—41no—
147C4×C79—42no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—41no—
151C4×C79—41no—
152C4×C79—81no—
153C4×C79—81no—
154C4×C79—81no—
155Dic79—162no—
156#11842—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—21no—
159C2×C79—42no—
160C2×C79#151611yescharacteristic
161C2×C79#212011yescharacteristic
162C2×C79#212911yescharacteristic
163#2599D7911yescharacteristic
164#1820—791no—
165#1458—791no—
166#1351—792no—
167#1344—792no—
168C79#1184211yescharacteristic, complemented, Sylow
169C2×C4×Q8Dic7911yescharacteristic, complemented
170C2×#117C2×D7911yescharacteristic
171#431—1581no—
172#398Dic7911yescharacteristic, complemented
173#397C2×D7911yescharacteristic
174#395Dic7911yescharacteristic, complemented
175#393C2×D7911yescharacteristic
176#388—21no—
177#387—21no—
178#387—21no—
179#385—21no—
180#383Dic7911yescharacteristic, complemented
181#381—1581no—
182#342—792no—
183#342—1581no—
184#324—1581no—
185#323—792no—
186C22×Q8C2×Dic7911yescharacteristic
187C23×C4C22⋊D7911yescharacteristic
188C4×Q8—41no—
189C4×Q8—41no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic7911yescharacteristic
196C2×C4⋊C4C2×Dic7911yescharacteristic
197C2×C4⋊C4C22⋊D7911yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—1582no—
201C2×C22⋊C4C22⋊D7911yescharacteristic
202C2×C22⋊C4C22×D7911yescharacteristic
203C2×C42—21no—
204C2×C42C2×Dic7911yescharacteristic
205C2×C42C22⋊D7911yescharacteristic
206C4⋊C8—3161no—
207C22⋊C8—1582no—
208C4×C8—3161no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—3161no—
216#95—3161no—
217#95C2×Dic7911yescharacteristic
218#95C2×Dic7911yescharacteristic
219C24C2×C22⋊D7911yescharacteristic
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—3162no—
231C22×C4C22⋊Dic7911yescharacteristic
232C22×C4C22⋊Dic7911yescharacteristic
233C22×C4C22⋊Dic7911yescharacteristic
234C22×C4C22⋊Dic7911yescharacteristic
235C22×C4C22×Dic7911yescharacteristic
236C22×C4C2×C22⋊D7911yescharacteristic
237C2×C8—3162no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—3162no—
250C22⋊C4—3162no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—42no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23C2×C22⋊Dic7911yescharacteristic
261C23#1457918112yes—
262Q8—42no—
263Q8—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—6322no—
292C2×C4—6322no—
293C2×C4—6322no—
294C2×C4#1457907611yescharacteristic
295C2×C4#1457907611yescharacteristic
296C8—6322no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1457907611yescharacteristic
305C22#1909855911yescharacteristic
306C22#1909858011yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—12642no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#2376195611yescharacteristic
323C2#2376504111yescharacteristic
324C2#2376507511yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2599C2
socleC22×C79C2×#99
prime radicalC24×C79C2×D4
cosocleC2×C22⋊C4×C79C23
hypercenter#2599D79
hypocenterC79#11842
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C79C4semidirect
14. #398×C79C4semidirect
16. #395×C79C4semidirect
22. #383×C79C4semidirect
168. C79#11842semidirect
169. C2×C4×Q8Dic79semidirect
172. #398Dic79semidirect
174. #395Dic79semidirect
180. #383Dic79semidirect

Maximal subgroups

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

subgroupindex
#237634772
#237617142
#237614212
#237614002
C79×#25992
#1184279