Finite Group Databasebeta

← #34197931

Subgroups — #34197931

Order 20224.

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

normal subgroups only

This group has 416 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#23768596C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761345C212yesmaximal, complemented
4#23761343C212yesmaximal, complemented
5C79×#1465C211yesmaximal, characteristic
6C2×#14579153C411yescharacteristic
7C2×C22⋊(C79⋊C8)—21no—
8#19098486—21no—
9#19098486—21no—
10#19098466C411yescharacteristic
11#19098466C411yescharacteristic
12#19098436—21no—
13#19098436—21no—
14#19098432C2211yescharacteristic
15#19098428C2211yescharacteristic
16C4×C4⋊Dic79C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C2×C4×D4×C79C2211yescharacteristic
20#407×C79—21no—
21#324×C79C2212yes—
22#323×C79—21no—
23C22×C79⋊C8—41no—
24#14579153—41no—
25#14579153—41no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—21no—
36C2×C22⋊Dic79—21no—
37C2×C4×Dic79C2×C411yescharacteristic
38C2×C4×Dic79C2×C411yescharacteristic
39C22⋊(C79⋊C8)—22no—
40C22⋊(C79⋊C8)—41no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C22×D4×C79C2×C411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—41no—
48C4×D4×C79—41no—
49C4×D4×C79—41no—
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C22⋊C4×C79D411yescharacteristic
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C42×C79C2×C411yescharacteristic
54C22⋊C8×C79—22no—
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—
64C4⋊Dic79—42no—
65Dic79⋊C4—42no—
66Dic79⋊C4—41no—
67Dic79⋊C4—41no—
68Dic79⋊C4—81no—
69Dic79⋊C4—81no—
70C22⋊Dic79—42no—
71C22⋊Dic79—42no—
72C22⋊Dic79—81no—
73C22⋊Dic79—81no—
74C4×Dic79—22no—
75C4×Dic79—22no—
76C24×C79C22⋊C411yescharacteristic
77C24×C79C22⋊C411yescharacteristic
78C2×D4×C79—22no—
79C2×D4×C79—22no—
80C2×D4×C79—41no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—81no—
86C22×C4×C79—81no—
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22×C411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92M16×C79—82no—
93C2×C8×C79—42no—
94C4⋊C4×C79—41no—
95C22⋊C4×C79—41no—
96C22⋊C4×C79—41no—
97C42×C79—21no—
98C42×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—21no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79—41no—
108C23×C79#10112yes—
109C23×C79C2×C22⋊C411yescharacteristic
110D4×C79—44no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—41no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—81no—
123C2×C4×C79—81no—
124C2×C4×C79#9911yescharacteristic
125C2×C4×C79#9911yescharacteristic
126C2×C4×C79#13112yes—
127C2×Dic79—22no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C8×C79—82no—
140C79⋊C8—82no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—42no—
145C22×C79—42no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—41no—
150C22×C79—41no—
151C22×C79—81no—
152C22×C79—81no—
153C22×C79#40811yescharacteristic
154C22×C79C2×#9911yescharacteristic
155C22×C79C2×#10111yescharacteristic
156C4×C79—21no—
157C4×C79—21no—
158C4×C79—22no—
159C4×C79—41no—
160C4×C79—41no—
161C4×C79—41no—
162Dic79—42no—
163Dic79—81no—
164Dic79—81no—
165Dic79—81no—
166#11572—791nomaximal, Sylow
167C2×C79—21no—
168C2×C79—21no—
169C2×C79—42no—
170C2×C79—81no—
171C2×C79#147211yescharacteristic
172C2×C79#212711yescharacteristic
173C2×C79#213211yescharacteristic
174#2566—791no—
175#1465D7911yescharacteristic
176#1459—791no—
177#1325—792no—
178#1324—792no—
179C79#1157211yescharacteristic, complemented, Sylow
180C2×#125—791no—
181C2×C4×D4C2×D7911yescharacteristic
182#407—21no—
183C2×C22⋊C8—1581no—
184#396—1581no—
185#396—1581no—
186#390—791no—
187#390—791no—
188#385—1581no—
189#385—1581no—
190#384—791no—
191#382—791no—
192C4×C4⋊C4—791no—
193#324—792no—
194#324C2×D7912yes—
195#323—21no—
196#323—1581no—
197C22×D4C4×D7911yescharacteristic
198C23×C4D31611yescharacteristic
199C23×C4C22⋊D7911yescharacteristic
200C2×M16—41no—
201C22×C8—3161no—
202#125—3161no—
203#125—3161no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—791no—
207C2×C4⋊C4—791no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4—1581no—
212C2×C4⋊C4—1581no—
213C2×C4⋊C4C22×D7911yescharacteristic
214C2×C22⋊C4—1581no—
215C2×C22⋊C4—1581no—
216C2×C22⋊C4—1581no—
217C2×C22⋊C4—1581no—
218C2×C22⋊C4D31611yescharacteristic
219C2×C22⋊C4C22⋊D7911yescharacteristic
220C2×C42—791no—
221C2×C42—791no—
222C2×C42C4×D7911yescharacteristic
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—3161no—
226C22⋊C8—1582no—
227#95—1581no—
228#95—1581no—
229#95—1581no—
230C24(C2×D79)⋊C411yescharacteristic
231C24(C2×D79)⋊C411yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—1581no—
242C22×C4—1581no—
243C22×C4—1581no—
244C22×C4—1581no—
245C22×C4(C2×D79)⋊C411yescharacteristic
246C22×C4(C2×D79)⋊C411yescharacteristic
247C22×C4C2×C4×D7911yescharacteristic
248C22×C4C2×C22⋊D7911yescharacteristic
249C22×C4C2×D31611yescharacteristic
250M16—82no—
251C2×C8—42no—
252C2×C8—3162no—
253C2×C8—6322no—
254C4⋊C4—41no—
255C4⋊C4—3161no—
256C4⋊C4—3161no—
257C4⋊C4—3162no—
258C4⋊C4—3162no—
259C4⋊C4—3161no—
260C4⋊C4—3161no—
261C4⋊C4—6321no—
262C4⋊C4—6321no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—3162no—
266C22⋊C4—3162no—
267C22⋊C4—6321no—
268C22⋊C4—6321no—
269C42—21no—
270C42—21no—
271C42—1582no—
272C42—1582no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#1457907912yes—
283C23C2×(C2×D79)⋊C411yescharacteristic
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—3161no—
304C2×C4—3161no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—1582no—
310C2×C4#1457907511yescharacteristic
311C2×C4#1457907511yescharacteristic
312C2×C4#1457916912yes—
313C8—82no—
314C8—6322no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22#1909853111yescharacteristic
328C22C2×#1457907511yescharacteristic
329C22C2×#1457907911yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—3162no—
337C4—6321no—
338C4—6321no—
339C4—6321no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2376176511yescharacteristic
345C2#2376506711yescharacteristic
346C2#2376508311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1465C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1465D79
hypocenterC79#11572
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #23761345C2semidirect
4. #23761343C2semidirect
179. C79#11572semidirect

Maximal subgroups

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

subgroupindex
#237685962
#237617182
#237613452
#237613432
C79×#14652
#1157279