Finite Group Databasebeta

← #34177405

Subgroups — #34177405

Order 20224.

338 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 79 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#23771391C211yesmaximal, characteristic
2#23764877C211yesmaximal, characteristic
3#23764872C211yesmaximal, characteristic
4#23761224C211yesmaximal, characteristic
5#23761223C211yesmaximal, characteristic
6#23761222C211yesmaximal, characteristic
7C79×#2099C211yesmaximal, characteristic
8C2×#14579094—21no—
9C2×#14579092—21no—
10#19098503—21no—
11#19098502—21no—
12#19098501—21no—
13#19098461C2211yescharacteristic
14#19098461C2211yescharacteristic
15#19098450—21no—
16#19098449—21no—
17#19098444—21no—
18#19098325—21no—
19#19098325—21no—
20#19098318C2211yescharacteristic
21#19098318C2211yescharacteristic
22#389×C79C2211yescharacteristic
23#384×C79C2211yescharacteristic
24#340×C79—21no—
25#340×C79—21no—
26#336×C79C2211yescharacteristic
27C22×C79⋊C8—21no—
28C22×C79⋊C8—21no—
29C2×C4⋊Dic79D411yescharacteristic
30C2×C4⋊Dic79D411yescharacteristic
31C2×C4⋊Dic79D411yescharacteristic
32C2×C4⋊Dic79D411yescharacteristic
33C2×Dic79⋊C4—21no—
34C2×Dic79⋊C4—21no—
35C2×C4×Dic79—21no—
36#14579094—42no—
37#14579092—42no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C22×C8×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79—21no—
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C42×C79C2311yescharacteristic
55#95×C79—21no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C2×C79⋊C8—22no—
61C2×C79⋊C8—22no—
62C2×C79⋊C8—41no—
63C2×C79⋊C8—41no—
64C2×C79⋊C8—41no—
65C2×C79⋊C8—41no—
66C4⋊Dic79—21no—
67C4⋊Dic79—21no—
68C4⋊Dic79—21no—
69C4⋊Dic79—21no—
70C4⋊Dic79—41no—
71C4⋊Dic79—41no—
72Dic79⋊C4—41no—
73Dic79⋊C4—41no—
74Dic79⋊C4—41no—
75Dic79⋊C4—41no—
76C4×Dic79—81no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—41no—
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C2×C8×C79—41no—
84C2×C8×C79—41no—
85C2×C8×C79—41no—
86C4⋊C4×C79—21no—
87C4⋊C4×C79—21no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—42no—
90C4⋊C4×C79—41no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—41no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—41no—
96C42×C79—41no—
97C23×C79#12011yescharacteristic
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—41no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—81no—
111C2×C4×C79—81no—
112C2×C4×C79—81no—
113C2×C4×C79#13511yescharacteristic
114C2×C4×C79#13511yescharacteristic
115C2×C4×C79#13511yescharacteristic
116C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C79#13711yescharacteristic
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C8×C79—81no—
132C79⋊C8—42no—
133C79⋊C8—42no—
134C22×C79#44911yescharacteristic
135C22×C79#44911yescharacteristic
136C22×C79#45211yescharacteristic
137C22×C79#45411yescharacteristic
138C22×C79#45611yescharacteristic
139C22×C79#45611yescharacteristic
140C22×C79#45811yescharacteristic
141C4×C79—21no—
142C4×C79—21no—
143C4×C79—21no—
144C4×C79—21no—
145C4×C79—21no—
146C4×C79—21no—
147C4×C79—81no—
148C4×C79—81no—
149C4×C79—161no—
150Dic79—81no—
151Dic79—81no—
152Dic79—81no—
153Dic79—81no—
154#7803—791nomaximal, Sylow
155C2×C79#160911yescharacteristic
156C2×C79#160911yescharacteristic
157C2×C79#161511yescharacteristic
158C2×C79#162811yescharacteristic
159C2×C79#165011yescharacteristic
160C2×C79#165211yescharacteristic
161C2×C79#166311yescharacteristic
162#2810—791no—
163#2099D7911yescharacteristic
164#2084—791no—
165#2083—791no—
166#1271—791no—
167#1271—791no—
168#1271—791no—
169C79#780311yescharacteristic, complemented, Sylow
170C2×#107—1581no—
171C2×#106—1581no—
172#400—1581no—
173#400—1581no—
174#400—1581no—
175#389—791no—
176#389—791no—
177#389C2×D7911yescharacteristic
178#387—1581no—
179#387—1581no—
180#387—1581no—
181#384C2×D7911yescharacteristic
182#340—21no—
183#340—21no—
184#340—1581no—
185#340—1581no—
186#336—791no—
187#336—791no—
188#336C2×D7911yescharacteristic
189C22×C8—21no—
190C22×C8—1581no—
191C22×C8—1581no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—791no—
197C2×C4⋊C4—791no—
198C2×C4⋊C4—791no—
199C2×C4⋊C4—791no—
200C2×C4⋊C4—1581no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4C22⋊D7911yescharacteristic
203C2×C4⋊C4C22⋊D7911yescharacteristic
204C2×C42—1581no—
205C2×C42C22×D7911yescharacteristic
206#107—3162no—
207#106—3162no—
208#95—21no—
209#95—1581no—
210#95—1581no—
211#95—1581no—
212#95—1581no—
213#95—1581no—
214#95—1581no—
215#95—1581no—
216#95—1581no—
217#95—1581no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—1581no—
222C22×C4—1581no—
223C22×C4—1581no—
224C22×C4—1581no—
225C22×C4D4×D7911yescharacteristic
226C22×C4D4×D7911yescharacteristic
227C22×C4C2×C22⋊D7911yescharacteristic
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—3161no—
232C2×C8—3161no—
233C2×C8—3161no—
234C2×C8—3161no—
235C2×C8—1582no—
236C2×C8—1582no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—3161no—
248C4⋊C4—3161no—
249C4⋊C4—3161no—
250C4⋊C4—3161no—
251C4⋊C4—3161no—
252C4⋊C4—3161no—
253C4⋊C4—1581no—
254C4⋊C4—1581no—
255C4⋊C4—1581no—
256C4⋊C4—1581no—
257C42—41no—
258C42—6321no—
259C23#1457912911yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—3161no—
276C2×C4—3161no—
277C2×C4—3161no—
278C2×C4—3161no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3161no—
283C2×C4—3161no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4#1457918011yescharacteristic
288C2×C4#1457918111yescharacteristic
289C2×C4#1457918211yescharacteristic
290C2×C4#1457918511yescharacteristic
291C2×C4#1457918811yescharacteristic
292C2×C4#1457919211yescharacteristic
293C8—81no—
294C8—3162no—
295C8—3162no—
296C22#1909870511yescharacteristic
297C22#1909870711yescharacteristic
298C22#1909872311yescharacteristic
299C22#1909873511yescharacteristic
300C22#1909874411yescharacteristic
301C22#1909874411yescharacteristic
302C22#1909875211yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—81no—
310C4—81no—
311C4—161no—
312C4—6321no—
313C4—6321no—
314C4—6321no—
315C4—6321no—
316C2#2376239411yescharacteristic
317C2#2376239611yescharacteristic
318C2#2376242911yescharacteristic
319C2#2376250711yescharacteristic
320C2#2376265511yescharacteristic
321C2#2376266911yescharacteristic
322C2#2376272611yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2099C2
socle, prime radicalC23×C79#120
hypercenter#2099D79
hypocenterC79#7803
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
169. C79#7803semidirect

Maximal subgroups

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

subgroupindex
#237713912
#237648772
#237648722
#237612242
#237612232
#237612222
C79×#20992
#780379