Finite Group Databasebeta

← #34181660

Subgroups — #34181660

Order 20224.

588 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23763468C211yesmaximal, characteristic
2#23761627C211yesmaximal, characteristic
3#23761278C214yesmaximal
4C79×#2285C211yesmaximal, characteristic
5C2×C22⋊(C79⋊C8)C2212yes—
6C2×C79⋊#95—21no—
7C2×C79⋊#95—21no—
8C2×C79⋊#95C2212yes—
9#19098318—21no—
10#19098318—21no—
11C2×#117×C79—21no—
12C22×C42×C79C2211yescharacteristic
13#384×C79C411yescharacteristic, complemented
14#384×C79C411yescharacteristic, complemented
15#382×C79—21no—
16#380×C79—21no—
17#380×C79C2212yes—
18C4×C4⋊C4×C79—21no—
19C4×C4⋊C4×C79C412yescomplemented
20C23×Dic79—22no—
21C22×C79⋊C8—22no—
22C22⋊(C79⋊C8)—24no—
23C79⋊#95—24no—
24C79⋊#95—24no—
25C79⋊#95—42no—
26C23×C4×C79D411yescharacteristic
27C23×C4×C79D411yescharacteristic
28C23×C4×C79C2311yescharacteristic
29#117×C79—42no—
30#117×C79—42no—
31C2×C4⋊C4×C79—21no—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79—21no—
35C2×C4⋊C4×C79C2×C412yes—
36C2×C4⋊C4×C79C2×C412yes—
37C2×C22⋊C4×C79—22no—
38C2×C42×C79—24no—
39C2×C42×C79—22no—
40C2×C42×C79C2×C411yescharacteristic
41C2×C42×C79C2×C411yescharacteristic
42C2×C42×C79D411yescharacteristic
43C2×C42×C79D411yescharacteristic
44#95×C79—22no—
45#95×C79—22no—
46C22×Dic79—24no—
47C22×Dic79—42no—
48C22×Dic79—42no—
49C22×Dic79—42no—
50C22×Dic79—42no—
51C22×Dic79—42no—
52C22×Dic79—42no—
53C2×C79⋊C8—42no—
54C2×C79⋊C8—42no—
55C2×C79⋊C8—42no—
56C24×C79C2×D411yescharacteristic
57C22×C4×C79—22no—
58C22×C4×C79—22no—
59C22×C4×C79—22no—
60C22×C4×C79—22no—
61C22×C4×C79—22no—
62C22×C4×C79—22no—
63C22×C4×C79—41no—
64C22×C4×C79—41no—
65C22×C4×C79—41no—
66C22×C4×C79C22⋊C411yescharacteristic
67C22×C4×C79C22⋊C411yescharacteristic
68C22×C4×C79C22⋊C411yescharacteristic
69C22×C4×C79C22⋊C411yescharacteristic
70C22×C4×C79D812yes—
71C22×C4×C79SD1612yes—
72C22×C4×C79C22×C411yescharacteristic
73C22×C4×C79C2×D411yescharacteristic
74C22×C4×C79#4114yes—
75C4⋊C4×C79—42no—
76C4⋊C4×C79—42no—
77C4⋊C4×C79—42no—
78C4⋊C4×C79—42no—
79C4⋊C4×C79—42no—
80C4⋊C4×C79—42no—
81C22⋊C4×C79—42no—
82C22⋊C4×C79—42no—
83C42×C79—22no—
84C42×C79—22no—
85C42×C79—22no—
86C42×C79—22no—
87C42×C79—44no—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79C2×C22⋊C411yescharacteristic
93C23×C79#12312yes—
94C23×C79C2×D811yescharacteristic
95C23×C79C2×SD1611yescharacteristic
96C23×C79#13712yes—
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—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—24no—
108C2×C4×C79—24no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—44no—
113C2×C4×C79—44no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79#10212yes—
121C2×C4×C79#10212yes—
122C2×C4×C79#10412yes—
123C2×C4×C79#11712yes—
124C2×C4×C79#12312yes—
125C2×Dic79—44no—
126C2×Dic79—44no—
127C2×Dic79—44no—
128C2×Dic79—82no—
129C2×Dic79—82no—
130C2×Dic79—82no—
131C2×Dic79—82no—
132C79⋊C8—82no—
133C22×C79—22no—
134C22×C79—22no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—22no—
138C22×C79—22no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79#38111yescharacteristic
142C22×C79#41011yescharacteristic
143C22×C79#41011yescharacteristic
144C22×C79C2×#10211yescharacteristic
145C22×C79#41911yescharacteristic
146C22×C79C2×#10411yescharacteristic
147C22×C79#42111yescharacteristic
148C22×C79#48012yes—
149C22×C79#48312yes—
150C4×C79—22no—
151C4×C79—22no—
152C4×C79—22no—
153C4×C79—22no—
154C4×C79—22no—
155C4×C79—41no—
156C4×C79—82no—
157C4×C79—82no—
158Dic79—84no—
159#8716—791nomaximal, Sylow
160C2×C79—22no—
161C2×C79—22no—
162C2×C79#150211yescharacteristic
163C2×C79#150611yescharacteristic
164C2×C79#151511yescharacteristic
165C2×C79#151711yescharacteristic
166C2×C79#182011yescharacteristic
167C2×C79#182311yescharacteristic
168C2×C79#182611yescharacteristic
169#2285D7911yescharacteristic
170#1817—791no—
171#1434—791no—
172#1294—794no—
173C79#871611yescharacteristic, complemented, Sylow
174C2×#117—21no—
175C22×C42C2×D7911yescharacteristic
176C2×C22⋊C8—792no—
177#384Dic7911yescharacteristic, complemented
178#384Dic7911yescharacteristic, complemented
179#382—21no—
180#380—21no—
181#380C2×D7912yes—
182C4×C4⋊C4—21no—
183C4×C4⋊C4Dic7912yescomplemented
184C2×#95—792no—
185C2×#95—1581no—
186C2×#95—1581no—
187#336—1581no—
188#336—1581no—
189C23×C4—1582no—
190C23×C4C22⋊D7911yescharacteristic
191C23×C4C22⋊D7911yescharacteristic
192C23×C4C22×D7911yescharacteristic
193C22×C8—1582no—
194#117—42no—
195#117—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C2×Dic7912yes—
201C2×C4⋊C4C2×Dic7912yes—
202C2×C22⋊C4—22no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42C2×Dic7911yescharacteristic
206C2×C42C2×Dic7911yescharacteristic
207C2×C42C22⋊D7911yescharacteristic
208C2×C42C22⋊D7911yescharacteristic
209C22⋊C8—1584no—
210#95—22no—
211#95—22no—
212#95—3162no—
213#95—1584no—
214#95—1584no—
215C24C2×C22⋊D7911yescharacteristic
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—3162no—
226C22×C4—3162no—
227C22×C4—3162no—
228C22×C4—3162no—
229C22×C4—3162no—
230C22×C4—3162no—
231C22×C4—1584no—
232C22×C4C22⋊Dic7911yescharacteristic
233C22×C4C22⋊Dic7911yescharacteristic
234C22×C4C22⋊Dic7911yescharacteristic
235C22×C4C22⋊Dic7911yescharacteristic
236C22×C4#1192960912yes—
237C22×C4#1192961212yes—
238C22×C4C22×Dic7911yescharacteristic
239C22×C4C2×C22⋊D7911yescharacteristic
240C22×C4#1192962414yes—
241C2×C8—3162no—
242C2×C8—3162no—
243C2×C8—3162no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—44no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23C2×C22⋊Dic7911yescharacteristic
262C23#1457914412yes—
263C23C2×#1192960911yescharacteristic
264C23C2×#1192961211yescharacteristic
265C23#1457919212yes—
266C2×C4—24no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—3164no—
290C2×C4—3164no—
291C2×C4—3164no—
292C2×C4—6322no—
293C2×C4—6322no—
294C2×C4—6322no—
295C2×C4—6322no—
296C2×C4#1457908212yes—
297C2×C4#1457908212yes—
298C2×C4#1457908812yes—
299C2×C4#1457912012yes—
300C2×C4#1457914412yes—
301C8—6322no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22#1909842611yescharacteristic
311C22#1909853711yescharacteristic
312C22#1909853711yescharacteristic
313C22C2×#1457908211yescharacteristic
314C22#1909857211yescharacteristic
315C22C2×#1457908811yescharacteristic
316C22#1909858011yescharacteristic
317C22#1909887612yes—
318C22#1909889312yes—
319C4—22no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—82no—
326C4—82no—
327C4—6324no—
328C2—22no—
329C2—22no—
330C2#2376189511yescharacteristic
331C2#2376191211yescharacteristic
332C2#2376195111yescharacteristic
333C2#2376196011yescharacteristic
334C2#2376347711yescharacteristic
335C2#2376348611yescharacteristic
336C2#2376349911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic79
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2285C2
socleC23×C79C2×C22⋊C4
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#2285D79
hypocenterC79#8716
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
13. #384×C79C4semidirect
14. #384×C79C4semidirect
19. C4×C4⋊C4×C79C4semidirect
173. C79#8716semidirect
177. #384Dic79semidirect
178. #384Dic79semidirect
183. C4×C4⋊C4Dic79semidirect

Maximal subgroups

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

subgroupindex
#237634682
#237616272
#237612782
C79×#22852
#871679