Finite Group Databasebeta

← #34183784

Subgroups — #34183784

Order 20224.

538 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 119 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#23766112C211yesmaximal, characteristic
2#23761796C212yesmaximal, complemented
3#23761721C211yesmaximal, characteristic
4#23761228C212yesmaximal, complemented
5C79×#2975C211yesmaximal, characteristic
6C2×#14579148C411yescharacteristic
7C2×C4×Dic158C2211yescharacteristic
8#19098616C2212yes—
9#19098468C411yescharacteristic
10#19098466C411yescharacteristic
11#19098451—22no—
12#19098436—22no—
13#19098432C411yescharacteristic
14#19098409C2211yescharacteristic
15#19098295—24no—
16#19098287—22no—
17C2×C4×D4×C79C2211yescharacteristic
18#436×C79—24no—
19#431×C79C2212yes—
20#407×C79—22no—
21C22×Dic158C2×C411yescharacteristic
22#14579148—22no—
23#14579148—22no—
24C4×Dic158—22no—
25C4×Dic158—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79—22no—
31C2×C4×Dic79—21no—
32C2×C4×Dic79C2×C411yescharacteristic
33C2×C4×Dic79C2×C411yescharacteristic
34C4⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—22no—
36C4×C79⋊C8—22no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C22×D4×C79C2×C411yescharacteristic
41C23×C4×C79D412yes—
42C2×M16×C79—24no—
43C4×D4×C79—22no—
44C4×D4×C79—22no—
45C2×C4⋊C4×C79C2×C411yescharacteristic
46C2×C22⋊C4×C79D412yes—
47C2×C42×C79C2311yescharacteristic
48C4⋊C8×C79—22no—
49C22⋊C8×C79—24no—
50C22⋊C8×C79—22no—
51C4×C8×C79D414yes—
52C2×Dic158—22no—
53C2×Dic158—21no—
54C2×Dic158—22no—
55C2×Dic158M1612yescomplemented
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—22no—
62Dic79⋊C4—41no—
63Dic79⋊C4—41no—
64Dic79⋊C4—82no—
65C22⋊Dic79—28no—
66C22⋊Dic79—82no—
67C4×Dic79—21no—
68C4×Dic79—22no—
69C4×Dic79—42no—
70C4×Dic79M1612yescomplemented
71C24×C79C22⋊C412yes—
72C2×D4×C79—22no—
73C2×D4×C79—21no—
74C2×D4×C79—21no—
75C2×D4×C79—22no—
76C22×C4×C79—22no—
77C22×C4×C79—22no—
78C22×C4×C79—82no—
79C22×C4×C79C22⋊C412yes—
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83M16×C79—84no—
84C2×C8×C79—24no—
85C4⋊C4×C79—22no—
86C22⋊C4×C79—24no—
87C42×C79C2×D412yes—
88C42×C79#4112yes—
89C23×C79—22no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79—42no—
93C23×C79—42no—
94C23×C79C2×C22⋊C411yescharacteristic
95D4×C79—42no—
96D4×C79—42no—
97C2×C4×C79—22no—
98C2×C4×C79—41no—
99C2×C4×C79—41no—
100C2×C4×C79—44no—
101C2×C4×C79—42no—
102C2×C4×C79—82no—
103C2×C4×C79C4×D412yes—
104C2×C4×C79#12011yescharacteristic
105C2×C4×C79#12112yes—
106C2×C4×C79#12311yescharacteristic
107C2×C4×C79C2×M1611yescharacteristic
108C2×C4×C79C2×M1611yescharacteristic
109C2×C4×C79#13112yes—
110C2×Dic79—22no—
111C2×Dic79—22no—
112C2×Dic79—41no—
113C2×Dic79—41no—
114C2×Dic79—41no—
115C2×Dic79—41no—
116C2×Dic79—41no—
117C2×Dic79—42no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121Dic158—24no—
122Dic158—42no—
123C8×C79—44no—
124C79⋊C8—82no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79—44no—
128C22×C79—42no—
129C22×C79—42no—
130C22×C79—42no—
131C22×C79—82no—
132C22×C79#38611yescharacteristic
133C22×C79#40711yescharacteristic
134C22×C79#40811yescharacteristic
135C4×C79—22no—
136C4×C79—21no—
137C4×C79—21no—
138C4×C79—42no—
139C4×C79#43512yes—
140C4×C79#43612yes—
141Dic79—42no—
142Dic79—42no—
143Dic79—81no—
144Dic79—81no—
145#9103—791nomaximal, Sylow
146C2×C79—21no—
147C2×C79—21no—
148C2×C79—82no—
149C2×C79#187011yescharacteristic
150C2×C79#190911yescharacteristic
151C2×C79#191111yescharacteristic
152#2975D7911yescharacteristic
153#2326—791no—
154#1479—792no—
155#1460—791no—
156#1274—792no—
157C79#910311yescharacteristic, complemented, Sylow
158C2×#124—791no—
159C2×C4×Q8—791no—
160C2×C4×D4C2×D7911yescharacteristic
161#436—24no—
162#431—792no—
163#431C2×D7912yes—
164#407—22no—
165#391—791no—
166#390—791no—
167#388—1582no—
168#385—1582no—
169#384—791no—
170#376—791no—
171Q8⋊C8—1584no—
172#323—1582no—
173C22×Q8—791no—
174C22×D4C4×D7911yescharacteristic
175C23×C4C22⋊D7912yes—
176C2×M16—24no—
177#124—1582no—
178#124—1582no—
179C4×Q8—1582no—
180C4×Q8—1582no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—791no—
184C2×C4⋊C4—1581no—
185C2×C4⋊C4—1582no—
186C2×C4⋊C4C4×D7911yescharacteristic
187C2×C22⋊C4—1582no—
188C2×C22⋊C4—1582no—
189C2×C22⋊C4D31612yes—
190C2×C42—791no—
191C2×C42—791no—
192C2×C42—1581no—
193C2×C42C22×D7911yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—1582no—
196C22⋊C8—24no—
197C22⋊C8—22no—
198C22⋊C8—1582no—
199C4×C8—1582no—
200C4×C8C22⋊D7914yes—
201#95—1582no—
202#95—1581no—
203#95—1581no—
204C24(C2×D79)⋊C412yes—
205C2×Q8—792no—
206C2×Q8—1582no—
207C2×Q8—1582no—
208C2×Q8—1581no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—82no—
216C22×C4—1581no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4—1581no—
220C22×C4(C2×D79)⋊C412yes—
221C22×C4C2×C4×D7911yescharacteristic
222C22×C4C2×C22⋊D7911yescharacteristic
223C22×C4C2×D31611yescharacteristic
224M16—84no—
225C2×C8—24no—
226C2×C8—3162no—
227C4⋊C4—22no—
228C4⋊C4—3161no—
229C4⋊C4—3161no—
230C4⋊C4—6322no—
231C4⋊C4—1582no—
232C22⋊C4—24no—
233C22⋊C4—6322no—
234C22⋊C4—1588no—
235C42—3162no—
236C42—792no—
237C42—1582no—
238C42—1581no—
239C42C2×C22⋊D7912yes—
240C42#1192962412yes—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23C2×(C2×D79)⋊C411yescharacteristic
247Q8—3162no—
248Q8—1584no—
249D4—42no—
250D4—42no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—3161no—
258C2×C4—3161no—
259C2×C4—3161no—
260C2×C4—3161no—
261C2×C4—3161no—
262C2×C4—3162no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—3161no—
266C2×C4—1582no—
267C2×C4—1582no—
268C2×C4C4×C22⋊D7912yes—
269C2×C4#1457913011yescharacteristic
270C2×C4#1457913512yes—
271C2×C4#1457914411yescharacteristic
272C2×C4M16×D7911yescharacteristic
273C2×C4M16×D7911yescharacteristic
274C2×C4#1457916912yes—
275C8—44no—
276C8—6322no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1909844211yescharacteristic
285C22#1909852611yescharacteristic
286C22#1909853111yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—3162no—
292C4—3162no—
293C4—6321no—
294C4—6321no—
295C4#1909863212yes—
296C4#1909863912yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2376369711yescharacteristic
301C2#2376392211yescharacteristic
302C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2975C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC2×C42×C79C23
hypercenter#2975D79
hypocenterC79#9103
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #23761796C2semidirect
4. #23761228C2semidirect
55. C2×Dic158M16semidirect
70. C4×Dic79M16semidirect
157. C79#9103semidirect

Maximal subgroups

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

subgroupindex
#237661122
#237617962
#237617212
#237612282
C79×#29752
#910379