Finite Group Databasebeta

← #34197956

Subgroups — #34197956

Order 20224.

426 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 91 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#23761741C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761348C212yesmaximal, complemented
4#23761348C212yesmaximal, complemented
5C79×#2579C211yesmaximal, characteristic
6#19098620—21no—
7C2×C22⋊(C79⋊C8)—21no—
8#19098289—21no—
9#19098287—21no—
10#19098287C2212yes—
11#19098287C2212yes—
12C2×C4×D4×C79C2211yescharacteristic
13C22×C4⋊C4×C79C411yescharacteristic
14#398×C79—21no—
15#396×C79—21no—
16#387×C79—21no—
17#386×C79C411yescharacteristic
18#386×C79C411yescharacteristic
19#385×C79—21no—
20#385×C79C2211yescharacteristic
21#385×C79C2211yescharacteristic
22#380×C79C411yescharacteristic
23C22×C79⋊C8—41no—
24C4⋊(C79⋊C8)—41no—
25C22⋊(C79⋊C8)—22no—
26C22⋊(C79⋊C8)—22no—
27C22⋊(C79⋊C8)—41no—
28#14579072—41no—
29C22×D4×C79C2×C411yescharacteristic
30C23×C4×C79C2×C411yescharacteristic
31C23×C4×C79C2×C411yescharacteristic
32C23×C4×C79C2×C411yescharacteristic
33C23×C4×C79D411yescharacteristic
34C4×D4×C79—41no—
35C4×D4×C79—41no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—21no—
41C2×C4⋊C4×C79—41no—
42C2×C4⋊C4×C79—41no—
43C2×C4⋊C4×C79D411yescharacteristic
44C2×C22⋊C4×C79—21no—
45C2×C22⋊C4×C79—21no—
46C2×C22⋊C4×C79C2×C411yescharacteristic
47C2×C22⋊C4×C79C2×C411yescharacteristic
48C2×C22⋊C4×C79D411yescharacteristic
49C2×C22⋊C4×C79C2311yescharacteristic
50C2×C42×C79D411yescharacteristic
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
54#95×C79—21no—
55#95×C79—21no—
56C2×C79⋊C8—42no—
57C2×C79⋊C8—42no—
58C2×C79⋊C8—82no—
59C24×C79C22⋊C411yescharacteristic
60C24×C79C22×C411yescharacteristic
61C2×D4×C79—22no—
62C2×D4×C79—22no—
63C2×D4×C79—41no—
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—42no—
73C22×C4×C79—41no—
74C22×C4×C79—41no—
75C22×C4×C79—81no—
76C22×C4×C79—81no—
77C22×C4×C79—81no—
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C4⋊C4×C79—41no—
84C4⋊C4×C79—42no—
85C4⋊C4×C79—42no—
86C4⋊C4×C79—41no—
87C4⋊C4×C79—41no—
88C4⋊C4×C79—81no—
89C4⋊C4×C79—81no—
90C22⋊C4×C79—22no—
91C22⋊C4×C79—22no—
92C22⋊C4×C79—41no—
93C22⋊C4×C79—41no—
94C22⋊C4×C79—42no—
95C22⋊C4×C79—81no—
96C42×C79—21no—
97C42×C79—21no—
98C23×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—41no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79#10112yes—
107C23×C79C2×C22⋊C411yescharacteristic
108C23×C79#13112yes—
109D4×C79—44no—
110C2×C4×C79—22no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—41no—
118C2×C4×C79—42no—
119C2×C4×C79—41no—
120C2×C4×C79—42no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—42no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—81no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79—82no—
139C2×C4×C79#9911yescharacteristic
140C2×C4×C79#9911yescharacteristic
141C79⋊C8—82no—
142C79⋊C8—82no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—22no—
146C22×C79—22no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79—42no—
150C22×C79—41no—
151C22×C79—41no—
152C22×C79—41no—
153C22×C79—41no—
154C22×C79—81no—
155C22×C79#40811yescharacteristic
156C22×C79C2×#9911yescharacteristic
157C22×C79C2×#10111yescharacteristic
158C4×C79—21no—
159C4×C79—21no—
160C4×C79—42no—
161C4×C79—41no—
162C4×C79—41no—
163C4×C79—41no—
164C4×C79—41no—
165C4×C79—81no—
166C4×C79—81no—
167C4×C79—81no—
168#11578—791nomaximal, Sylow
169C2×C79—21no—
170C2×C79—21no—
171C2×C79—42no—
172C2×C79—42no—
173C2×C79#147211yescharacteristic
174C2×C79#212011yescharacteristic
175C2×C79#212111yescharacteristic
176#2579D7911yescharacteristic
177#1466—791no—
178#1459—791no—
179#1326—792no—
180#1326—792no—
181C79#1157811yescharacteristic, complemented, Sylow
182C2×C4×D4C2×D7911yescharacteristic
183C22×C4⋊C4Dic7911yescharacteristic
184#432—1581no—
185C2×C22⋊C8—1581no—
186#398—21no—
187#396—21no—
188#387—21no—
189#386Dic7911yescharacteristic
190#386Dic7911yescharacteristic
191#385—21no—
192#385C2×D7911yescharacteristic
193#385C2×D7911yescharacteristic
194#380Dic7911yescharacteristic
195#324—1581no—
196#323—792no—
197#323—792no—
198#323—1581no—
199C22×D4C2×Dic7911yescharacteristic
200C23×C4C2×Dic7911yescharacteristic
201C23×C4C2×Dic7911yescharacteristic
202C23×C4C2×Dic7911yescharacteristic
203C23×C4C22⋊D7911yescharacteristic
204C22×C8—3161no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4C22⋊D7911yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4C2×Dic7911yescharacteristic
218C2×C22⋊C4C2×Dic7911yescharacteristic
219C2×C22⋊C4C22⋊D7911yescharacteristic
220C2×C22⋊C4C22×D7911yescharacteristic
221C2×C42C22⋊D7911yescharacteristic
222C4⋊C8—3161no—
223C22⋊C8—3161no—
224C22⋊C8—1582no—
225C22⋊C8—1582no—
226#97—3161no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232C24C22⋊Dic7911yescharacteristic
233C24C22×Dic7911yescharacteristic
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C22⋊Dic7911yescharacteristic
252C22×C4C22⋊Dic7911yescharacteristic
253C22×C4C22⋊Dic7911yescharacteristic
254C22×C4C2×C22⋊D7911yescharacteristic
255C22×C4C2×C22⋊D7911yescharacteristic
256C2×C8—3162no—
257C2×C8—3162no—
258C2×C8—6322no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—81no—
272C42—21no—
273C42—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#1457908012yes—
283C23C2×C22⋊Dic7911yescharacteristic
284C23#1457916812yes—
285D4—44no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4#1457907611yescharacteristic
316C2×C4#1457907611yescharacteristic
317C8—6322no—
318C8—6322no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1909853011yescharacteristic
332C22C2×#1457907611yescharacteristic
333C22C2×#1457908011yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—42no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—42no—
348C2#2376176411yescharacteristic
349C2#2376504111yescharacteristic
350C2#2376504611yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC24×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2579C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#2579D79
hypocenterC79#11578
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2579

Complements

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

NH ≅ G/Ntype
3. #23761348C2semidirect
4. #23761348C2semidirect
181. C79#11578semidirect

Maximal subgroups

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

subgroupindex
#237617412
#237617182
#237613482
C79×#25792
#1157879