Finite Group Databasebeta

← #34198883

Subgroups — #34198883

Order 20224.

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

normal subgroups only

This group has 462 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#23767717C211yesmaximal, characteristic
2C4×#14579076C211yesmaximal, characteristic
3#23761425C214yesmaximal, complemented
4C79×#1466C211yesmaximal, characteristic
5C22×C4⋊Dic79C411yescharacteristic, complemented
6C2×#14579076—21no—
7#19098502C412yescomplemented
8#19098499C2212yes—
9#19098461—21no—
10#19098443—21no—
11#19098435—21no—
12#19098423—21no—
13#19098423C411yescharacteristic, complemented
14C4×C22⋊Dic79—21no—
15#19098330C2212yes—
16C2×C4×D4×C79C2211yescharacteristic
17#432×C79—21no—
18#323×C79—21no—
19#323×C79C2212yes—
20C23×Dic79—21no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—41no—
24C2×C4⋊Dic79—41no—
25C2×C4⋊Dic79C2×C412yes—
26C2×Dic79⋊C4—22no—
27C2×C22⋊Dic79—22no—
28C2×C22⋊Dic79—21no—
29C2×C22⋊Dic79—22no—
30C2×C4×Dic79—41no—
31#14579076—24no—
32C79⋊#95—22no—
33C79⋊#95—22no—
34C79⋊#95—41no—
35C79⋊#95C2×C412yes—
36C22×D4×C79D411yescharacteristic
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C4×D4×C79—41no—
40C4×D4×C79—41no—
41C2×C4⋊C4×C79C2×C411yescharacteristic
42C2×C22⋊C4×C79D411yescharacteristic
43C2×C22⋊C4×C79C2311yescharacteristic
44C2×C42×C79D411yescharacteristic
45C4⋊C8×C79—41no—
46C22⋊C8×C79—22no—
47#97×C79—41no—
48C22×Dic79—22no—
49C22×Dic79—22no—
50C22×Dic79—42no—
51C22×Dic79—42no—
52C22×Dic79—82no—
53C4⋊Dic79—42no—
54C4⋊Dic79—42no—
55C4⋊Dic79—41no—
56C4⋊Dic79—41no—
57Dic79⋊C4—82no—
58C22⋊Dic79—24no—
59C22⋊Dic79—42no—
60C22⋊Dic79—42no—
61C22⋊Dic79—82no—
62C4×Dic79—82no—
63C24×C79C22⋊C411yescharacteristic
64C24×C79C2×D411yescharacteristic
65C2×D4×C79—22no—
66C2×D4×C79—22no—
67C2×D4×C79—41no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—42no—
73C22×C4×C79—42no—
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C2×C8×C79—42no—
80C4⋊C4×C79—41no—
81C22⋊C4×C79—41no—
82C22⋊C4×C79—41no—
83C42×C79—21no—
84C42×C79—21no—
85C23×C79—21no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79—21no—
89C23×C79—41no—
90C23×C79—41no—
91C23×C79—41no—
92C23×C79—41no—
93C23×C79#10412yes—
94C23×C79C2×C22⋊C411yescharacteristic
95C23×C79#13712yes—
96D4×C79—44no—
97C2×C4×C79—22no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—81no—
111C2×C4×C79#9911yescharacteristic
112C2×C4×C79#9911yescharacteristic
113C2×Dic79—44no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—82no—
121C2×Dic79—82no—
122C2×Dic79—84no—
123C8×C79—82no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—22no—
127C22×C79—22no—
128C22×C79—42no—
129C22×C79—42no—
130C22×C79—42no—
131C22×C79—41no—
132C22×C79—41no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—81no—
136C22×C79C2×#9911yescharacteristic
137C22×C79#41911yescharacteristic
138C22×C79C2×#10411yescharacteristic
139C4×C79—21no—
140C4×C79—41no—
141C4×C79—41no—
142C4×C79—41no—
143C4×C79—41no—
144C4×C79#35212yes—
145Dic79—82no—
146Dic79—84no—
147Dic79—82no—
148#11786—791nomaximal, Sylow
149C2×C79—21no—
150C2×C79—21no—
151C2×C79—42no—
152C2×C79—42no—
153C2×C79#150811yescharacteristic
154C2×C79C2×#35211yescharacteristic
155C2×C79#212811yescharacteristic
156#2487—791no—
157C4×#99—791no—
158#1466D7911yescharacteristic
159#1352—794no—
160C79#1178611yescharacteristic, complemented, Sylow
161C2×C4×D4C2×D7911yescharacteristic
162C22×C4⋊C4—791no—
163#432—21no—
164C2×#99—1581no—
165#400—792no—
166#399—792no—
167#389—1581no—
168#386—1581no—
169#385—1581no—
170#380—791no—
171#380—1581no—
172C4×C22⋊C4—1581no—
173#342—792no—
174#323—21no—
175#323C2×D7912yes—
176C22×D4D31611yescharacteristic
177C23×C4—1581no—
178C23×C4C4×D7911yescharacteristic
179C23×C4D31611yescharacteristic
180C4×D4—41no—
181C4×D4—41no—
182C2×C4⋊C4—3161no—
183C2×C4⋊C4—3161no—
184C2×C4⋊C4—792no—
185C2×C4⋊C4—1581no—
186C2×C4⋊C4—1581no—
187C2×C4⋊C4—1582no—
188C2×C4⋊C4C4×D7911yescharacteristic
189C2×C22⋊C4—1581no—
190C2×C22⋊C4—1582no—
191C2×C22⋊C4—1582no—
192C2×C22⋊C4C22⋊D7911yescharacteristic
193C2×C22⋊C4C22×D7911yescharacteristic
194C2×C42—3161no—
195C2×C42C22⋊D7911yescharacteristic
196C4⋊C8—41no—
197#99—1584no—
198C22⋊C8—22no—
199#97—41no—
200#95—3161no—
201#95—792no—
202#95—1582no—
203#95—1582no—
204C24(C2×D79)⋊C411yescharacteristic
205C24C2×D31611yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—3162no—
216C22×C4—3162no—
217C22×C4—6322no—
218C22×C4—1582no—
219C22×C4—1582no—
220C22×C4(C2×D79)⋊C411yescharacteristic
221C22×C4(C2×D79)⋊C411yescharacteristic
222C22×C4(C2×D79)⋊C411yescharacteristic
223C22×C4C2×C4×D7911yescharacteristic
224C22×C4C2×C22⋊D7911yescharacteristic
225C2×C8—42no—
226C4⋊C4—41no—
227C4⋊C4—3162no—
228C4⋊C4—3162no—
229C4⋊C4—3161no—
230C4⋊C4—3161no—
231C4⋊C4—6322no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—3162no—
235C22⋊C4—3162no—
236C22⋊C4—6322no—
237C22⋊C4—1584no—
238C42—21no—
239C42—21no—
240C42—6322no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#1457908712yes—
250C23C2×(C2×D79)⋊C411yescharacteristic
251C23#1457919412yes—
252D4—44no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—3164no—
268C2×C4—3162no—
269C2×C4—3162no—
270C2×C4—3162no—
271C2×C4—3162no—
272C2×C4—3162no—
273C2×C4—3162no—
274C2×C4—6322no—
275C2×C4—6324no—
276C2×C4—6322no—
277C2×C4#1457907511yescharacteristic
278C2×C4#1457907511yescharacteristic
279C8—82no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×#1457907511yescharacteristic
293C22#1909857311yescharacteristic
294C22C2×#1457908711yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—6322no—
301C4—6324no—
302C4—6322no—
303C4#1909835412yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2376192111yescharacteristic
309C2C2×#1909835411yescharacteristic
310C2#2376507111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1466C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1466D79
hypocenterC79#11786
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #23761425C2semidirect
5. C22×C4⋊Dic79C4semidirect
7. #19098502C4semidirect
13. #19098423C4semidirect
160. C79#11786semidirect

Maximal subgroups

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

subgroupindex
#237677172
C4×#145790762
#237614252
C79×#14662
#1178679