Finite Group Databasebeta

← #34199307

Subgroups — #34199307

Order 20224.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#23768392C211yesmaximal, characteristic
2#23763388C211yesmaximal, characteristic
3#23761401C214yesmaximal, complemented
4C79×#1459C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic, complemented
6C2×#14579148—21no—
7C2×#14579076—21no—
8#19098488C412yescomplemented
9#19098488C2212yes—
10#19098468—21no—
11#19098423—21no—
12#19098423—21no—
13#19098423—21no—
14C4×C4⋊Dic79C411yescharacteristic, complemented
15#19098330C2212yes—
16C2×C4×D4×C79C2211yescharacteristic
17C2×C22⋊C8×C79—21no—
18#324×C79—21no—
19#323×C79C2212yes—
20C22×Dic158—21no—
21#14579161—41no—
22#14579161—41no—
23#14579148—42no—
24#14579148—42no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—41no—
28C2×C4⋊Dic79C2×C412yes—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79—22no—
31C2×C22⋊Dic79—22no—
32C2×C4×Dic79—21no—
33#14579076—42no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—41no—
37C79⋊#95C2×C412yes—
38C22×D4×C79D411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C23×C4×C79D411yescharacteristic
41C22×C8×C79—41no—
42C4×D4×C79—41no—
43C4×D4×C79—41no—
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C22⋊C4×C79C2×C411yescharacteristic
46C2×C22⋊C4×C79C2311yescharacteristic
47C2×C42×C79C2×C411yescharacteristic
48C22⋊C8×C79—22no—
49C22⋊C8×C79—41no—
50C2×Dic158—42no—
51C2×Dic158—42no—
52C2×Dic158—42no—
53C22×Dic79—22no—
54C22×Dic79—22no—
55C22×Dic79—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—41no—
59C4⋊Dic79—41no—
60C4⋊Dic79—82no—
61C22⋊Dic79—42no—
62C22⋊Dic79—44no—
63C22⋊Dic79—42no—
64C22⋊Dic79—42no—
65C22⋊Dic79—42no—
66C4×Dic79—42no—
67C24×C79C22⋊C411yescharacteristic
68C24×C79C2×D411yescharacteristic
69C2×D4×C79—41no—
70C2×D4×C79—41no—
71C2×D4×C79D812yes—
72C2×D4×C79SD1612yes—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—42no—
78C22×C4×C79—42no—
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C2×C8×C79—42no—
85C2×C8×C79—82no—
86C4⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C22⋊C4×C79—41no—
89C42×C79—21no—
90C42×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79#10212yes—
100C23×C79C2×C22⋊C411yescharacteristic
101C23×C79C2×D811yescharacteristic
102C23×C79C2×SD1611yescharacteristic
103D4×C79—42no—
104D4×C79—42no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—22no—
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—81no—
117C2×C4×C79—81no—
118C2×C4×C79#9911yescharacteristic
119C2×C4×C79#9911yescharacteristic
120C2×C4×C79#10212yes—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—82no—
128C2×Dic79—82no—
129C2×Dic79—82no—
130Dic158—84no—
131C8×C79—82no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—22no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—42no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—41no—
143C22×C79—41no—
144C22×C79—81no—
145C22×C79—81no—
146C22×C79C2×#9911yescharacteristic
147C22×C79C2×#10211yescharacteristic
148C22×C79#42111yescharacteristic
149C4×C79—21no—
150C4×C79—22no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79—41no—
154C4×C79#35412yes—
155Dic79—82no—
156Dic79—82no—
157Dic79—162no—
158#11878—791nomaximal, Sylow
159C2×C79—21no—
160C2×C79—21no—
161C2×C79—41no—
162C2×C79—42no—
163C2×C79—41no—
164C2×C79#151911yescharacteristic
165C2×C79#212311yescharacteristic
166C2×C79C2×#35411yescharacteristic
167#2547—791no—
168#1799—791no—
169#1459D7911yescharacteristic
170#1344—794no—
171C79#1187811yescharacteristic, complemented, Sylow
172C2×C4⋊Q8—791no—
173C2×#124—1581no—
174C2×C4×D4C2×D7911yescharacteristic
175C2×#99—1581no—
176C2×C22⋊C8—21no—
177#397—792no—
178#397—792no—
179#391—1581no—
180#380—1581no—
181#380—1581no—
182#380—1581no—
183C4×C4⋊C4—791no—
184#342—792no—
185#324—21no—
186#323C2×D7912yes—
187C22×Q8—1581no—
188C22×D4D31611yescharacteristic
189C23×C4D31611yescharacteristic
190C23×C4C22⋊D7911yescharacteristic
191C22×C8—41no—
192C4⋊Q8—3161no—
193C4⋊Q8—3161no—
194#124—3162no—
195#124—3162no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—3161no—
199C2×C4⋊C4—792no—
200C2×C4⋊C4—1581no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4C22⋊D7911yescharacteristic
203C2×C22⋊C4—1582no—
204C2×C22⋊C4—1582no—
205C2×C22⋊C4—1582no—
206C2×C22⋊C4C4×D7911yescharacteristic
207C2×C22⋊C4C22×D7911yescharacteristic
208C2×C42—1581no—
209C2×C42C4×D7911yescharacteristic
210#99—3162no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213#95—3161no—
214#95—792no—
215#95—1582no—
216#95—1582no—
217C24(C2×D79)⋊C411yescharacteristic
218C24C2×D31611yescharacteristic
219C2×Q8—3162no—
220C2×Q8—3162no—
221C2×Q8—3162no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4D63212yes—
225C2×D4#1192961312yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—3162no—
233C22×C4—1582no—
234C22×C4—1582no—
235C22×C4(C2×D79)⋊C411yescharacteristic
236C22×C4(C2×D79)⋊C411yescharacteristic
237C22×C4(C2×D79)⋊C411yescharacteristic
238C22×C4C2×C4×D7911yescharacteristic
239C22×C4C2×C22⋊D7911yescharacteristic
240C2×C8—42no—
241C2×C8—82no—
242C4⋊C4—41no—
243C4⋊C4—3162no—
244C4⋊C4—3162no—
245C4⋊C4—3161no—
246C4⋊C4—3161no—
247C4⋊C4—6322no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—3164no—
251C22⋊C4—3162no—
252C22⋊C4—3162no—
253C22⋊C4—3162no—
254C22⋊C4—3162no—
255C42—21no—
256C42—21no—
257C42—3162no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#1457908312yes—
267C23C2×(C2×D79)⋊C411yescharacteristic
268C23C2×D63211yescharacteristic
269C23C2×#1192961311yescharacteristic
270Q8—6324no—
271D4—42no—
272D4—42no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—3162no—
289C2×C4—3162no—
290C2×C4—3162no—
291C2×C4—3162no—
292C2×C4—6322no—
293C2×C4—6322no—
294C2×C4—6322no—
295C2×C4#1457907511yescharacteristic
296C2×C4#1457907511yescharacteristic
297C2×C4#1457908312yes—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22C2×#1457907511yescharacteristic
314C22C2×#1457908311yescharacteristic
315C22#1909857911yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—6322no—
322C4—6322no—
323C4—12642no—
324C4#1909835712yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—42no—
329C2—41no—
330C2#2376196911yescharacteristic
331C2#2376505511yescharacteristic
332C2C2×#1909835711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1459C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1459D79
hypocenterC79#11878
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #23761401C2semidirect
5. C2×#14579161C4semidirect
8. #19098488C4semidirect
14. C4×C4⋊Dic79C4semidirect
171. C79#11878semidirect

Maximal subgroups

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

subgroupindex
#237683922
#237633882
#237614012
C79×#14592
#1187879