Finite Group Databasebeta

← #34199180

Subgroups — #34199180

Order 20224.

396 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 83 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#23767687C211yesmaximal, characteristic
2C2×#19098289C211yesmaximal, characteristic
3#23761426C212yesmaximal, complemented
4#23761405C212yesmaximal, complemented
5C79×#1821C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic
7C2×C22⋊(C79⋊C8)—21no—
8#19098502C411yescharacteristic
9#19098502C2211yescharacteristic
10#19098493C411yescharacteristic
11#19098493C2211yescharacteristic
12#19098444—21no—
13#19098444—21no—
14#19098436—21no—
15#19098426—21no—
16#19098426C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C22×C4⋊C4×C79C2211yescharacteristic
20#407×C79—21no—
21#343×C79—21no—
22#343×C79C2212yes—
23C22×C79⋊C8—41no—
24#14579120—41no—
25#14579120—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—21no—
32C2×C4×Dic79—21no—
33C22⋊(C79⋊C8)—22no—
34C22⋊(C79⋊C8)—41no—
35C79⋊#95—21no—
36C79⋊#95—21no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79C2×C411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—22no—
48C2×C4⋊C4×C79—41no—
49C2×C4⋊C4×C79—41no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79C2311yescharacteristic
54C22⋊C8×C79—41no—
55C22⋊C8×C79—41no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C2×C79⋊C8—42no—
61C2×C79⋊C8—82no—
62C4⋊Dic79—41no—
63C4⋊Dic79—41no—
64Dic79⋊C4—42no—
65Dic79⋊C4—42no—
66C22⋊Dic79—42no—
67C22⋊Dic79—41no—
68C22⋊Dic79—41no—
69C4×Dic79—42no—
70C24×C79C22⋊C411yescharacteristic
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—22no—
78C22×C4×C79—41no—
79C22×C4×C79—42no—
80C22×C4×C79—42no—
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87M16×C79—42no—
88M16×C79—42no—
89C2×C8×C79—42no—
90C4⋊C4×C79—41no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—41no—
94C23×C79—22no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—41no—
98C23×C79#9911yescharacteristic
99C23×C79#9911yescharacteristic
100C23×C79C2×C22⋊C411yescharacteristic
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—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—81no—
120C2×C4×C79#13512yes—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—42no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C8×C79—82no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—42no—
142C22×C79—42no—
143C22×C79#35212yes—
144C22×C79C2×#9911yescharacteristic
145C22×C79#41611yescharacteristic
146C22×C79#42111yescharacteristic
147C4×C79—22no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—41no—
151C4×C79—41no—
152C4×C79—41no—
153Dic79—81no—
154Dic79—81no—
155Dic79—81no—
156Dic79—81no—
157#11850—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—22no—
161C2×C79—41no—
162C2×C79#151611yescharacteristic
163C2×C79C2×#35211yescharacteristic
164C2×C79#213311yescharacteristic
165#2484—791no—
166#1821D7911yescharacteristic
167C2×#324—791no—
168#1353—792no—
169#1346—792no—
170C79#1185011yescharacteristic, complemented, Sylow
171C2×#117—791no—
172C22×C4⋊C4C2×D7911yescharacteristic
173#407—21no—
174C2×C22⋊C8—1581no—
175#400—791no—
176#400—791no—
177#398—791no—
178#398—791no—
179#387—1581no—
180#387—1581no—
181#385—1581no—
182#381—791no—
183#381—1581no—
184#343—21no—
185#343C2×D7912yes—
186#324—792no—
187#324—1581no—
188C23×C4C4×D7911yescharacteristic
189C23×C4D31611yescharacteristic
190C23×C4C22⋊D7911yescharacteristic
191C2×M16—22no—
192C22×C8—3161no—
193#117—3161no—
194#117—3161no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—791no—
198C2×C4⋊C4—791no—
199C2×C4⋊C4—1581no—
200C2×C4⋊C4—1581no—
201C2×C4⋊C4C4×D7911yescharacteristic
202C2×C4⋊C4D31611yescharacteristic
203C2×C4⋊C4C22⋊D7911yescharacteristic
204C2×C4⋊C4C22×D7911yescharacteristic
205C2×C22⋊C4—1581no—
206C2×C22⋊C4—1581no—
207C2×C42—1581no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—3161no—
211C22⋊C8—1582no—
212#95—791no—
213#95—791no—
214#95—1581no—
215#95—1581no—
216#95—1581no—
217#95—1581no—
218#95—1581no—
219#95—1581no—
220#95—1581no—
221C24(C2×D79)⋊C411yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—1581no—
233C22×C4—1581no—
234C22×C4—1581no—
235C22×C4—1581no—
236C22×C4(C2×D79)⋊C411yescharacteristic
237C22×C4(C2×D79)⋊C411yescharacteristic
238C22×C4(C2×D79)⋊C411yescharacteristic
239C22×C4C2×C4×D7911yescharacteristic
240C22×C4C2×C22⋊D7911yescharacteristic
241C22×C4C2×D31611yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—3162no—
246C2×C8—6322no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—3161no—
252C4⋊C4—3161no—
253C4⋊C4—3162no—
254C4⋊C4—3162no—
255C22⋊C4—3162no—
256C22⋊C4—3161no—
257C22⋊C4—3161no—
258C42—3162no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—41no—
263C23#1457907511yescharacteristic
264C23#1457907511yescharacteristic
265C23C2×(C2×D79)⋊C411yescharacteristic
266C2×C4—21no—
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—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3162no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4#1457918112yes—
297C8—82no—
298C8—6322no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#1909835412yes—
309C22C2×#1457907511yescharacteristic
310C22#1909855711yescharacteristic
311C22#1909858111yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—6321no—
319C4—6321no—
320C4—6321no—
321C4—6321no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#2376195411yescharacteristic
327C2C2×#1909835411yescharacteristic
328C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1821C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1821D79
hypocenterC79#11850
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #23761426C2semidirect
4. #23761405C2semidirect
170. C79#11850semidirect

Maximal subgroups

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

subgroupindex
#237676872
C2×#190982892
#237614262
#237614052
C79×#18212
#1185079