Finite Group Databasebeta

← #34197949

Subgroups — #34197949

Order 20224.

424 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 83 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#23768743C211yesmaximal, characteristic
2#23761738C211yesmaximal, characteristic
3#23761349C212yesmaximal, complemented
4#23761347C212yesmaximal, complemented
5C79×#1460C211yesmaximal, characteristic
6C22×Dic79⋊C4C411yescharacteristic
7#19098527—21no—
8#19098496—21no—
9#19098486—21no—
10#19098447—21no—
11#19098443C411yescharacteristic
12#19098443C411yescharacteristic
13#19098436—21no—
14#19098435C2211yescharacteristic
15#19098435C2211yescharacteristic
16#19098423C411yescharacteristic
17#19098289—21no—
18#19098287C2212yes—
19C2×C4×D4×C79C2211yescharacteristic
20#431×C79—21no—
21#323×C79—21no—
22#323×C79C2212yes—
23C23×Dic79C2×C411yescharacteristic
24C23×Dic79C2×C411yescharacteristic
25C2×#11929608—41no—
26C2×C4⋊Dic79—21no—
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—41no—
32C2×Dic79⋊C4—41no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79C2×C411yescharacteristic
36C2×C22⋊Dic79C2×C411yescharacteristic
37C22⋊(C79⋊C8)—22no—
38C22⋊(C79⋊C8)—41no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C22×D4×C79C2×C411yescharacteristic
45C23×C4×C79C2×C411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C4×D4×C79—41no—
48C4×D4×C79—41no—
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C22⋊C4×C79D411yescharacteristic
51C2×C22⋊C4×C79C2311yescharacteristic
52C2×C42×C79D411yescharacteristic
53C4⋊C8×C79—41no—
54C22⋊C8×C79—22no—
55C4×C8×C79—41no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—41no—
61C22×Dic79—41no—
62C22×Dic79—42no—
63C22×Dic79—81no—
64#11929608—82no—
65C2×C79⋊C8—42no—
66C4⋊Dic79—81no—
67Dic79⋊C4—42no—
68Dic79⋊C4—42no—
69Dic79⋊C4—41no—
70Dic79⋊C4—41no—
71Dic79⋊C4—81no—
72C22⋊Dic79—22no—
73C22⋊Dic79—22no—
74C22⋊Dic79—42no—
75C22⋊Dic79—81no—
76C24×C79C22⋊C411yescharacteristic
77C24×C79C22×C411yescharacteristic
78C2×D4×C79—22no—
79C2×D4×C79—22no—
80C2×D4×C79—41no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—81no—
86C22×C4×C79—81no—
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C2×C8×C79—42no—
93C4⋊C4×C79—41no—
94C22⋊C4×C79—41no—
95C22⋊C4×C79—41no—
96C42×C79—21no—
97C42×C79—21no—
98C23×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79C2×C22⋊C411yescharacteristic
108C23×C79#13112yes—
109D4×C79—44no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—41no—
116C2×C4×C79—42no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—81no—
121C2×C4×C79—81no—
122C2×C4×C79—81no—
123C2×C4×C79#9911yescharacteristic
124C2×C4×C79#9911yescharacteristic
125C2×C4×C79#10112yes—
126C2×Dic79—22no—
127C2×Dic79—41no—
128C2×Dic79—42no—
129C2×Dic79—41no—
130C2×Dic79—42no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—82no—
141C8×C79—82no—
142C79⋊C8—82no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—22no—
146C22×C79—42no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79—41no—
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—22no—
161C4×C79—41no—
162C4×C79—41no—
163C4×C79—41no—
164Dic79—42no—
165Dic79—81no—
166Dic79—81no—
167Dic79—81no—
168#11577—791nomaximal, Sylow
169C2×C79—21no—
170C2×C79—21no—
171C2×C79—42no—
172C2×C79—81no—
173C2×C79#147211yescharacteristic
174C2×C79#212011yescharacteristic
175C2×C79#212111yescharacteristic
176#2579—791no—
177#1465—791no—
178#1460D7911yescharacteristic
179#1326—792no—
180#1326—792no—
181C79#1157711yescharacteristic, complemented, Sylow
182C2×C4×D4C2×D7911yescharacteristic
183C22×C4⋊C4—791no—
184#431—21no—
185#407—1581no—
186#398—1581no—
187#396—1581no—
188#387—1581no—
189#386—791no—
190#386—791no—
191#385—791no—
192#385—791no—
193#385—1581no—
194#380—791no—
195#324—1581no—
196#323—21no—
197#323—792no—
198#323C2×D7912yes—
199C22×D4C4×D7911yescharacteristic
200C23×C4—791no—
201C23×C4—791no—
202C23×C4C4×D7911yescharacteristic
203C23×C4D31611yescharacteristic
204C2×M16—3161no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—3161no—
208C2×C4⋊C4—3161no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4—1581no—
212C2×C4⋊C4—1581no—
213C2×C4⋊C4—1581no—
214C2×C4⋊C4D31611yescharacteristic
215C2×C22⋊C4—791no—
216C2×C22⋊C4—791no—
217C2×C22⋊C4—1581no—
218C2×C22⋊C4—1581no—
219C2×C22⋊C4C22⋊D7911yescharacteristic
220C2×C22⋊C4C22×D7911yescharacteristic
221C2×C42C22⋊D7911yescharacteristic
222C4⋊C8—41no—
223C22⋊C8—22no—
224C22⋊C8—3161no—
225C22⋊C8—1582no—
226C4×C8—41no—
227#95—1581no—
228#95—1581no—
229#95—1581no—
230#95—1581no—
231#95—1581no—
232C24(C2×D79)⋊C411yescharacteristic
233C24C2×C4×D7911yescharacteristic
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—3161no—
244C22×C4—3162no—
245C22×C4—3161no—
246C22×C4—6321no—
247C22×C4—1581no—
248C22×C4—1581no—
249C22×C4—1581no—
250C22×C4—1581no—
251C22×C4(C2×D79)⋊C411yescharacteristic
252C22×C4(C2×D79)⋊C411yescharacteristic
253C22×C4(C2×D79)⋊C411yescharacteristic
254C22×C4C2×C22⋊D7911yescharacteristic
255C22×C4C2×D31611yescharacteristic
256M16—6322no—
257C2×C8—42no—
258C2×C8—3162no—
259C4⋊C4—41no—
260C4⋊C4—3162no—
261C4⋊C4—3162no—
262C4⋊C4—3161no—
263C4⋊C4—3161no—
264C4⋊C4—6321no—
265C4⋊C4—6321no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—3162no—
269C22⋊C4—6321no—
270C22⋊C4—1582no—
271C22⋊C4—1582no—
272C42—21no—
273C42—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×(C2×D79)⋊C411yescharacteristic
284C23#1457916712yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—3161no—
300C2×C4—3162no—
301C2×C4—3161no—
302C2×C4—3162no—
303C2×C4—3161no—
304C2×C4—3161no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—6322no—
313C2×C4—1582no—
314C2×C4#1457907511yescharacteristic
315C2×C4#1457907511yescharacteristic
316C2×C4#1457907912yes—
317C8—82no—
318C8—6322no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1909852911yescharacteristic
332C22C2×#1457907511yescharacteristic
333C22C2×#1457907911yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—3162no—
341C4—6321no—
342C4—6321no—
343C4—6321no—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2#2376176311yescharacteristic
349C2#2376503911yescharacteristic
350C2#2376504711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1460C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#1460D79
hypocenterC79#11577
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #23761349C2semidirect
4. #23761347C2semidirect
181. C79#11577semidirect

Maximal subgroups

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

subgroupindex
#237687432
#237617382
#237613492
#237613472
C79×#14602
#1157779