Finite Group Databasebeta

← #34197869

Subgroups — #34197869

Order 20224.

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

normal subgroups only

This group has 446 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#23767708C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761329C214yesmaximal, complemented
4C79×#1466C211yesmaximal, characteristic
5C2×#14579142C412yes—
6C22×C4⋊Dic79C411yescharacteristic
7C2×C22⋊(C79⋊C8)—21no—
8#19098493—21no—
9#19098488—21no—
10#19098461—21no—
11#19098424C2212yes—
12#19098423—21no—
13#19098423C411yescharacteristic
14#19098289—21no—
15#19098287C2212yes—
16C2×C4×D4×C79C2211yescharacteristic
17#432×C79—21no—
18#323×C79—21no—
19#323×C79C2212yes—
20C23×Dic79C2×C412yes—
21C22×C79⋊C8—41no—
22#14579142—44no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79—41no—
27C2×C4⋊Dic79—41no—
28C2×Dic79⋊C4—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79C2×C412yes—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—41no—
33C79⋊#95—21no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C22×D4×C79C2×C411yescharacteristic
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C4×D4×C79—41no—
40C4×D4×C79—41no—
41C2×C4⋊C4×C79D411yescharacteristic
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—24no—
51C22×Dic79—42no—
52C22×Dic79—42no—
53C2×C79⋊C8—42no—
54C2×C79⋊C8—82no—
55C4⋊Dic79—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—41no—
58C4⋊Dic79—41no—
59Dic79⋊C4—44no—
60C22⋊Dic79—42no—
61C22⋊Dic79—44no—
62C24×C79C22⋊C411yescharacteristic
63C24×C79C22×C411yescharacteristic
64C2×D4×C79—22no—
65C2×D4×C79—22no—
66C2×D4×C79—41no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—42no—
72C22×C4×C79—42no—
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C2×C8×C79—42no—
79C4⋊C4×C79—41no—
80C22⋊C4×C79—41no—
81C22⋊C4×C79—41no—
82C42×C79—21no—
83C42×C79—21no—
84C23×C79—21no—
85C23×C79—21no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79—41no—
89C23×C79—41no—
90C23×C79—41no—
91C23×C79—41no—
92C23×C79#10012yes—
93C23×C79C2×C22⋊C411yescharacteristic
94C23×C79#13112yes—
95D4×C79—44no—
96C2×C4×C79—22no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—81no—
110C2×C4×C79#9911yescharacteristic
111C2×C4×C79#9911yescharacteristic
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—84no—
119C8×C79—82no—
120C79⋊C8—82no—
121C22×C79—21no—
122C22×C79—21no—
123C22×C79—22no—
124C22×C79—22no—
125C22×C79—42no—
126C22×C79—42no—
127C22×C79—42no—
128C22×C79—41no—
129C22×C79—41no—
130C22×C79—41no—
131C22×C79—41no—
132C22×C79—81no—
133C22×C79#40811yescharacteristic
134C22×C79C2×#9911yescharacteristic
135C22×C79C2×#10011yescharacteristic
136C4×C79—21no—
137C4×C79—41no—
138C4×C79—41no—
139C4×C79—41no—
140C4×C79—41no—
141C4×C79#35212yes—
142Dic79—82no—
143Dic79—82no—
144#11558—791nomaximal, Sylow
145C2×C79—21no—
146C2×C79—21no—
147C2×C79—42no—
148C2×C79—42no—
149C2×C79#147111yescharacteristic
150C2×C79C2×#35211yescharacteristic
151C2×C79#212311yescharacteristic
152#2486—791no—
153#1466D7911yescharacteristic
154#1459—791no—
155#1317—794no—
156C79#1155811yescharacteristic, complemented, Sylow
157C2×#123—792no—
158C2×C4×D4C2×D7911yescharacteristic
159C22×C4⋊C4—791no—
160#432—21no—
161C2×C22⋊C8—1581no—
162#398—1581no—
163#397—1581no—
164#389—1581no—
165#381—792no—
166#380—791no—
167#380—1581no—
168#324—1581no—
169#323—21no—
170#323—792no—
171#323C2×D7912yes—
172C22×D4C4×D7911yescharacteristic
173C23×C4—792no—
174C23×C4C4×D7911yescharacteristic
175C23×C4D31611yescharacteristic
176C22×C8—3161no—
177#123—3164no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—3161no—
181C2×C4⋊C4—3161no—
182C2×C4⋊C4—1581no—
183C2×C4⋊C4—1582no—
184C2×C4⋊C4—1581no—
185C2×C4⋊C4—1581no—
186C2×C4⋊C4D31611yescharacteristic
187C2×C22⋊C4—792no—
188C2×C22⋊C4—1582no—
189C2×C22⋊C4C22⋊D7911yescharacteristic
190C2×C22⋊C4C22×D7911yescharacteristic
191C2×C42C22⋊D7911yescharacteristic
192C4⋊C8—41no—
193C22⋊C8—22no—
194C22⋊C8—3161no—
195C22⋊C8—1582no—
196#97—41no—
197#95—1581no—
198#95—1582no—
199#95—1582no—
200C24(C2×D79)⋊C411yescharacteristic
201C24C2×C4×D7911yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—3162no—
212C22×C4—3162no—
213C22×C4—1582no—
214C22×C4—1582no—
215C22×C4—1584no—
216C22×C4(C2×D79)⋊C411yescharacteristic
217C22×C4(C2×D79)⋊C411yescharacteristic
218C22×C4(C2×D79)⋊C411yescharacteristic
219C22×C4C2×C22⋊D7911yescharacteristic
220C22×C4C2×D31611yescharacteristic
221C2×C8—42no—
222C2×C8—3162no—
223C2×C8—6322no—
224C4⋊C4—41no—
225C4⋊C4—3164no—
226C4⋊C4—3162no—
227C4⋊C4—3162no—
228C4⋊C4—3161no—
229C4⋊C4—3161no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—3164no—
233C22⋊C4—3162no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#1457907712yes—
245C23C2×(C2×D79)⋊C411yescharacteristic
246C23#1457916912yes—
247D4—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—3162no—
263C2×C4—3162no—
264C2×C4—3162no—
265C2×C4—3162no—
266C2×C4—3162no—
267C2×C4—3162no—
268C2×C4—6324no—
269C2×C4#1457907511yescharacteristic
270C2×C4#1457907511yescharacteristic
271C8—82no—
272C8—6322no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#1909853111yescharacteristic
286C22C2×#1457907511yescharacteristic
287C22C2×#1457907711yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—6322no—
294C4—6322no—
295C4#1909835412yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—42no—
300C2#2376176111yescharacteristic
301C2C2×#1909835411yescharacteristic
302C2#2376505311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1466C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#1466D79
hypocenterC79#11558
LCSG ▷ C24×C79 ▷ C22×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. #23761329C2semidirect
156. C79#11558semidirect

Maximal subgroups

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

subgroupindex
#237677082
#237617182
#237613292
C79×#14662
#1155879