Finite Group Databasebeta

← #34199289

Subgroups — #34199289

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#23767708C211yesmaximal, characteristic
2#23763388C211yesmaximal, characteristic
3#23761401C214yesmaximal, complemented
4C79×#1460C211yesmaximal, characteristic
5C2×#14579142—21no—
6C22×C4⋊Dic79C411yescharacteristic, complemented
7C2×#14579076—21no—
8#19098493C412yescomplemented
9#19098488C2212yes—
10#19098461—21no—
11#19098424—21no—
12#19098423—21no—
13#19098423—21no—
14#19098423C411yescharacteristic, complemented
15#19098330C2212yes—
16C2×C4×D4×C79C2211yescharacteristic
17#431×C79—21no—
18#323×C79—21no—
19#323×C79C2212yes—
20C23×Dic79—21no—
21#14579142—42no—
22#14579142—42no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—41no—
26C2×C4⋊Dic79—41no—
27C2×C4⋊Dic79—41no—
28C2×C4⋊Dic79C2×C412yes—
29C2×Dic79⋊C4—22no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79—22no—
33#14579076—42no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—41no—
37C79⋊#95C2×C412yes—
38C22×D4×C79D411yescharacteristic
39C23×C4×C79C2×C411yescharacteristic
40C23×C4×C79D411yescharacteristic
41C4×D4×C79—41no—
42C4×D4×C79—41no—
43C2×C4⋊C4×C79C2×C411yescharacteristic
44C2×C22⋊C4×C79D411yescharacteristic
45C2×C22⋊C4×C79C2311yescharacteristic
46C2×C42×C79D411yescharacteristic
47C4⋊C8×C79—41no—
48C22⋊C8×C79—22no—
49C4×C8×C79—41no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52C22×Dic79—42no—
53C22×Dic79—42no—
54C22×Dic79—42no—
55C22×Dic79—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—41no—
59C4⋊Dic79—41no—
60C4⋊Dic79—82no—
61Dic79⋊C4—44no—
62C22⋊Dic79—42no—
63C22⋊Dic79—42no—
64C22⋊Dic79—42no—
65C22⋊Dic79—42no—
66C22⋊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—
85C4⋊C4×C79—41no—
86C22⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C42×C79—21no—
89C42×C79—21no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79#10212yes—
99C23×C79C2×C22⋊C411yescharacteristic
100C23×C79C2×D811yescharacteristic
101C23×C79C2×SD1611yescharacteristic
102D4×C79—42no—
103D4×C79—42no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—22no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—42no—
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#9911yescharacteristic
118C2×C4×C79#9911yescharacteristic
119C2×C4×C79#10212yes—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—84no—
127C2×Dic79—82no—
128C2×Dic79—82no—
129C2×Dic79—82no—
130C8×C79—82no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—22no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—42no—
137C22×C79—41no—
138C22×C79—41no—
139C22×C79—42no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—41no—
143C22×C79—41no—
144C22×C79—81no—
145C22×C79C2×#9911yescharacteristic
146C22×C79C2×#10211yescharacteristic
147C22×C79#42111yescharacteristic
148C4×C79—21no—
149C4×C79—22no—
150C4×C79—41no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79#35212yes—
154Dic79—82no—
155Dic79—82no—
156Dic79—162no—
157#11874—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—41no—
161C2×C79—42no—
162C2×C79—41no—
163C2×C79#151911yescharacteristic
164C2×C79C2×#35211yescharacteristic
165C2×C79#212911yescharacteristic
166#2486—791no—
167#1799—791no—
168#1460D7911yescharacteristic
169#1344—794no—
170C79#1187411yescharacteristic, complemented, Sylow
171C2×#123—1581no—
172C2×C4×D4C2×D7911yescharacteristic
173C22×C4⋊C4—791no—
174#431—21no—
175C2×#99—1581no—
176#398—792no—
177#397—792no—
178#389—1581no—
179#381—1581no—
180#380—791no—
181#380—1581no—
182#380—1581no—
183#342—792no—
184#323—21no—
185#323C2×D7912yes—
186C22×D4D31611yescharacteristic
187C23×C4—1581no—
188C23×C4C4×D7911yescharacteristic
189C23×C4D31611yescharacteristic
190#123—3162no—
191#123—3162no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—3161no—
195C2×C4⋊C4—3161no—
196C2×C4⋊C4—3161no—
197C2×C4⋊C4—792no—
198C2×C4⋊C4—1582no—
199C2×C4⋊C4—1581no—
200C2×C4⋊C4—1581no—
201C2×C4⋊C4C4×D7911yescharacteristic
202C2×C22⋊C4—1582no—
203C2×C22⋊C4—1581no—
204C2×C22⋊C4—1582no—
205C2×C22⋊C4C22⋊D7911yescharacteristic
206C2×C22⋊C4C22×D7911yescharacteristic
207C2×C42C22⋊D7911yescharacteristic
208C4⋊C8—41no—
209#99—3162no—
210C22⋊C8—22no—
211C4×C8—41no—
212#95—3161no—
213#95—792no—
214#95—1582no—
215#95—1582no—
216C24(C2×D79)⋊C411yescharacteristic
217C24C2×D31611yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4D63212yes—
221C2×D4#1192961312yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—3162no—
229C22×C4—3162no—
230C22×C4—3162no—
231C22×C4—3162no—
232C22×C4—1582no—
233C22×C4—1582no—
234C22×C4(C2×D79)⋊C411yescharacteristic
235C22×C4(C2×D79)⋊C411yescharacteristic
236C22×C4(C2×D79)⋊C411yescharacteristic
237C22×C4C2×C4×D7911yescharacteristic
238C22×C4C2×C22⋊D7911yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—3162no—
242C4⋊C4—3162no—
243C4⋊C4—3161no—
244C4⋊C4—3161no—
245C4⋊C4—3164no—
246C4⋊C4—6322no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—3162no—
250C22⋊C4—3162no—
251C22⋊C4—3162no—
252C22⋊C4—3162no—
253C22⋊C4—3162no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#1457908312yes—
265C23C2×(C2×D79)⋊C411yescharacteristic
266C23C2×D63211yescharacteristic
267C23C2×#1192961311yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—3162no—
284C2×C4—3162no—
285C2×C4—3162no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—3162no—
289C2×C4—6324no—
290C2×C4—6322no—
291C2×C4—6322no—
292C2×C4—6322no—
293C2×C4#1457907511yescharacteristic
294C2×C4#1457907511yescharacteristic
295C2×C4#1457908312yes—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1457907511yescharacteristic
312C22C2×#1457908311yescharacteristic
313C22#1909857911yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—6322no—
320C4—6322no—
321C4—12642no—
322C4#1909835412yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2376196911yescharacteristic
329C2C2×#1909835411yescharacteristic
330C2#2376507411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1460C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1460D79
hypocenterC79#11874
LCSG ▷ C22×C4×C79 ▷ C23×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. #23761401C2semidirect
6. C22×C4⋊Dic79C4semidirect
8. #19098493C4semidirect
14. #19098423C4semidirect
170. C79#11874semidirect

Maximal subgroups

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

subgroupindex
#237677082
#237633882
#237614012
C79×#14602
#1187479