Finite Group Databasebeta

← #34198656

Subgroups — #34198656

Order 20224.

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

normal subgroups only

This group has 400 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#23769260C211yesmaximal, characteristic
2#23761724C211yesmaximal, characteristic, complemented
3#23761429C212yesmaximal, complemented
4#23761402C212yesmaximal
5C4×#101×C79C211yesmaximal, characteristic, complemented
6C2×#14579149C411yescharacteristic
7#19098616—21no—
8#19098498—21no—
9#19098483C411yescharacteristic
10#19098483C2211yescharacteristic
11#19098480C411yescharacteristic
12#19098480C2211yescharacteristic
13#19098474—21no—
14#19098454—21no—
15#19098454—21no—
16#19098431C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C2×C4×Q8×C79C2211yescharacteristic
20C2×#101×C79—21no—
21C4×M16×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C22×D316C2×C411yescharacteristic
25#14579149—41no—
26#14579149—41no—
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×(C2×D79)⋊C4—21no—
30C2×(C2×D79)⋊C4—21no—
31C2×(C2×D79)⋊C4—21no—
32C2×(C2×D79)⋊C4—21no—
33C2×(C2×D79)⋊C4—21no—
34C2×(C2×D79)⋊C4—21no—
35C4⋊(C79⋊C8)—41no—
36C22⋊(C79⋊C8)—22no—
37C4×C79⋊C8—41no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C22×Q8×C79D411yescharacteristic
45C2×M16×C79—22no—
46C4×Q8×C79—41no—
47C4×Q8×C79—41no—
48C2×C4⋊C4×C79C2×C411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C42×C79C2×C411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54#101×C79—24no—
55#97×C79—41no—
56C4×C8×C79—41no—
57C23×D79—21no—
58C2×Dic158—41no—
59C2×Dic158—81no—
60C2×D316—41no—
61C2×D316—81no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C2×C79⋊C8—42no—
66Dic79⋊C4—81no—
67Dic79⋊C4—81no—
68(C2×D79)⋊C4—42no—
69(C2×D79)⋊C4—41no—
70(C2×D79)⋊C4—41no—
71(C2×D79)⋊C4—42no—
72(C2×D79)⋊C4—42no—
73(C2×D79)⋊C4—81no—
74(C2×D79)⋊C4—81no—
75C2×Q8×C79—22no—
76C2×Q8×C79—22no—
77C2×Q8×C79—41no—
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85M16×C79—24no—
86M16×C79—42no—
87C2×C8×C79—42no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—41no—
90C4⋊C4×C79—41no—
91C42×C79—21no—
92C42×C79—21no—
93C42×C79—21no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C23×C79C2×C22⋊C411yescharacteristic
98C22×D79—41no—
99C22×D79—41no—
100C22×D79—41no—
101C22×D79—41no—
102C22×D79—41no—
103C22×D79—41no—
104C22×D79—41no—
105Q8×C79—44no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79#9911yescharacteristic
121C2×C4×C79#9911yescharacteristic
122C2×C4×C79#10412yes—
123C2×C4×C79#13512yes—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133D316—82no—
134Dic158—82no—
135C8×C79—44no—
136C79⋊C8—82no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79C2×#9911yescharacteristic
140C22×C79#41611yescharacteristic
141C22×C79C2×#10411yescharacteristic
142C2×D79—81no—
143C2×D79—81no—
144C2×D79—81no—
145C2×D79—81no—
146C2×D79—81no—
147C2×D79—81no—
148C2×D79—81no—
149C4×C79—21no—
150C4×C79—21no—
151C4×C79—22no—
152C4×C79—22no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79—41no—
157Dic79—81no—
158Dic79—81no—
159Dic79—81no—
160#11738—791nomaximal, Sylow
161C2×C79—21no—
162C2×C79—21no—
163C2×C79#150111yescharacteristic
164C2×C79#212211yescharacteristic
165C2×C79#213311yescharacteristic
166D79—161no—
167#2632—791no—
168C4×#101D7911yescharacteristic, complemented
169#1461—791no—
170#1354—792no—
171#1345—792no—
172C79#1173811yescharacteristic, complemented, Sylow
173C2×#124—791no—
174C2×C4×Q8C2×D7911yescharacteristic
175#431—1581no—
176C2×#101—21no—
177C4×M16—21no—
178#399—1581no—
179#395—791no—
180#395—791no—
181#394—791no—
182#394—791no—
183#393—1581no—
184#388—1581no—
185#388—1581no—
186#383—791no—
187#343C2×D7912yes—
188#324—792no—
189#324—1581no—
190C22×Q8—791no—
191C22×Q8C22⋊D7911yescharacteristic
192C22×D4—791no—
193C2×M16—22no—
194#124—3161no—
195#124—3161no—
196C4×Q8—41no—
197C4×Q8—41no—
198C2×C4⋊C4—1581no—
199C2×C4⋊C4—1581no—
200C2×C4⋊C4C4×D7911yescharacteristic
201C2×C4⋊C4D31611yescharacteristic
202C2×C4⋊C4C22×D7911yescharacteristic
203C2×C22⋊C4—1581no—
204C2×C22⋊C4—1581no—
205C2×C22⋊C4—1581no—
206C2×C22⋊C4—1581no—
207C2×C22⋊C4—1581no—
208C2×C22⋊C4—1581no—
209C2×C42C4×D7911yescharacteristic
210C2×C42D31611yescharacteristic
211C2×C42C22⋊D7911yescharacteristic
212C4⋊C8—3161no—
213#101—24no—
214C22⋊C8—1582no—
215#97—41no—
216C4×C8—41no—
217C4×C8—3161no—
218#95—791no—
219#95—791no—
220#95—1581no—
221#95—1581no—
222#95—1581no—
223#95—1581no—
224C24—1581no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—3161no—
229C2×Q8—6321no—
230C2×D4—3161no—
231C2×D4—6321no—
232C22×C4—1581no—
233C22×C4—1581no—
234C22×C4—1581no—
235C22×C4(C2×D79)⋊C411yescharacteristic
236C22×C4(C2×D79)⋊C411yescharacteristic
237C22×C4(C2×D79)⋊C411yescharacteristic
238C22×C4(C2×D79)⋊C411yescharacteristic
239C22×C4C2×C4×D7911yescharacteristic
240C22×C4C2×C22⋊D7911yescharacteristic
241C22×C4C2×D31611yescharacteristic
242M16—24no—
243M16—42no—
244C2×C8—42no—
245C2×C8—3162no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—6321no—
250C4⋊C4—6321no—
251C22⋊C4—3162no—
252C22⋊C4—3161no—
253C22⋊C4—3161no—
254C22⋊C4—3162no—
255C22⋊C4—3162no—
256C22⋊C4—6321no—
257C22⋊C4—6321no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—3161no—
265C23—3161no—
266C23—3161no—
267C23—3161no—
268C23—3161no—
269C23—3161no—
270C23—3161no—
271C23C2×(C2×D79)⋊C411yescharacteristic
272Q8—44no—
273Q8—6322no—
274D4—6322no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—3161no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4—3161no—
297C2×C4—3161no—
298C2×C4#1457907511yescharacteristic
299C2×C4#1457907511yescharacteristic
300C2×C4#1457908912yes—
301C2×C4#1457918112yes—
302C8—44no—
303C8—6322no—
304C22—21no—
305C22—21no—
306C22—6321no—
307C22—6321no—
308C22—6321no—
309C22—6321no—
310C22—6321no—
311C22—6321no—
312C22—6321no—
313C22C2×#1457907511yescharacteristic
314C22#1909855711yescharacteristic
315C22C2×#1457908911yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—6321no—
325C4—6321no—
326C4—6321no—
327C2—21no—
328C2—21no—
329C2—12641no—
330C2#2376188911yescharacteristic
331C2#2376505111yescharacteristic
332C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC4×#101×C79C2
socleC22×C79C2×#99
prime radicalC22×D316C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenterC4×#101D79
hypocenterC79#11738
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #23761724C2semidirect
3. #23761429C2semidirect
5. C4×#101×C79C2semidirect
168. C4×#101D79semidirect
172. C79#11738semidirect

Maximal subgroups

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

subgroupindex
#237692602
#237617242
#237614292
#237614022
C4×#101×C792
#1173879