Finite Group Databasebeta

← #34197917

Subgroups — #34197917

Order 20224.

424 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 87 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#23768573C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761343C214yesmaximal, complemented
4C79×#1465C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic
6C2×C22⋊(C79⋊C8)—21no—
7#19098475—21no—
8#19098475—21no—
9#19098466C412yes—
10#19098428C2212yes—
11#19098423—21no—
12#19098423—21no—
13C4×C4⋊Dic79C411yescharacteristic
14#19098289C2212yes—
15#19098287—21no—
16C2×C4×D4×C79C2211yescharacteristic
17#407×C79—21no—
18#324×C79C2212yes—
19#323×C79—21no—
20C22×Dic158—21no—
21C22×C79⋊C8—41no—
22#14579161—41no—
23#14579161—41no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C412yes—
27C2×C22⋊Dic79—22no—
28C2×C22⋊Dic79—22no—
29C2×C4×Dic79C2×C412yes—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—41no—
32C79⋊#95—21no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C22×D4×C79C2×C411yescharacteristic
36C23×C4×C79D411yescharacteristic
37C23×C4×C79D411yescharacteristic
38C2×M16×C79—41no—
39C4×D4×C79—41no—
40C4×D4×C79—41no—
41C2×C4⋊C4×C79C2311yescharacteristic
42C2×C22⋊C4×C79D411yescharacteristic
43C2×C22⋊C4×C79D411yescharacteristic
44C2×C42×C79C2×C411yescharacteristic
45C22⋊C8×C79—22no—
46C22⋊C8×C79—41no—
47C2×Dic158—42no—
48C2×Dic158—82no—
49C22×Dic79—22no—
50C22×Dic79—22no—
51C2×C79⋊C8—42no—
52C2×C79⋊C8—82no—
53C4⋊Dic79—42no—
54C4⋊Dic79—42no—
55C4⋊Dic79—41no—
56C4⋊Dic79—41no—
57C22⋊Dic79—82no—
58C22⋊Dic79—82no—
59C4×Dic79—24no—
60C24×C79C22⋊C411yescharacteristic
61C24×C79C22⋊C411yescharacteristic
62C2×D4×C79—22no—
63C2×D4×C79—22no—
64C2×D4×C79—41no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—42no—
70C22×C4×C79—42no—
71C22×C4×C79C22⋊C411yescharacteristic
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22×C411yescharacteristic
74C22×C4×C79C2×D411yescharacteristic
75C22×C4×C79C2×D411yescharacteristic
76M16×C79—82no—
77C2×C8×C79—42no—
78C4⋊C4×C79—41no—
79C22⋊C4×C79—41no—
80C22⋊C4×C79—41no—
81C42×C79—21no—
82C42×C79—21no—
83C23×C79—21no—
84C23×C79—21no—
85C23×C79—21no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79—41no—
89C23×C79—41no—
90C23×C79—41no—
91C23×C79—41no—
92C23×C79#10112yes—
93C23×C79C2×C22⋊C411yescharacteristic
94D4×C79—44no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—22no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79—81no—
107C2×C4×C79—81no—
108C2×C4×C79#9911yescharacteristic
109C2×C4×C79#9911yescharacteristic
110C2×C4×C79#13112yes—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117Dic158—84no—
118C8×C79—82no—
119C79⋊C8—82no—
120C22×C79—21no—
121C22×C79—21no—
122C22×C79—22no—
123C22×C79—42no—
124C22×C79—42no—
125C22×C79—41no—
126C22×C79—41no—
127C22×C79—41no—
128C22×C79—41no—
129C22×C79—41no—
130C22×C79—81no—
131C22×C79—81no—
132C22×C79#40811yescharacteristic
133C22×C79C2×#9911yescharacteristic
134C22×C79C2×#10111yescharacteristic
135C4×C79—21no—
136C4×C79—22no—
137C4×C79—41no—
138C4×C79—41no—
139C4×C79—41no—
140C4×C79#35412yes—
141Dic79—82no—
142Dic79—82no—
143#11569—791nomaximal, Sylow
144C2×C79—21no—
145C2×C79—21no—
146C2×C79—42no—
147C2×C79—81no—
148C2×C79#147211yescharacteristic
149C2×C79C2×#35411yescharacteristic
150C2×C79#213411yescharacteristic
151#2563—791no—
152#1465D7911yescharacteristic
153#1459—791no—
154#1324—794no—
155C79#1156911yescharacteristic, complemented, Sylow
156C2×C4⋊Q8—791no—
157C2×C4×D4C2×D7911yescharacteristic
158#407—21no—
159C2×C22⋊C8—1581no—
160#393—1581no—
161#393—1581no—
162#390—792no—
163#382—792no—
164#380—1581no—
165#380—1581no—
166C4×C4⋊C4—791no—
167#324—792no—
168#324C2×D7912yes—
169#323—21no—
170#323—1581no—
171C22×Q8—1581no—
172C22×D4C4×D7911yescharacteristic
173C23×C4D31611yescharacteristic
174C23×C4C22⋊D7911yescharacteristic
175C2×M16—41no—
176C22×C8—3161no—
177C4⋊Q8—3161no—
178C4⋊Q8—3161no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—792no—
182C2×C4⋊C4—1581no—
183C2×C4⋊C4—1581no—
184C2×C4⋊C4C22×D7911yescharacteristic
185C2×C22⋊C4—1582no—
186C2×C22⋊C4—1582no—
187C2×C22⋊C4D31611yescharacteristic
188C2×C22⋊C4C22⋊D7911yescharacteristic
189C2×C42—792no—
190C2×C42C4×D7911yescharacteristic
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—3161no—
194C22⋊C8—1582no—
195#95—1581no—
196#95—1582no—
197#95—1582no—
198C24(C2×D79)⋊C411yescharacteristic
199C24(C2×D79)⋊C411yescharacteristic
200C2×Q8—3162no—
201C2×Q8—6322no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—1582no—
212C22×C4—1582no—
213C22×C4(C2×D79)⋊C411yescharacteristic
214C22×C4(C2×D79)⋊C411yescharacteristic
215C22×C4C2×C4×D7911yescharacteristic
216C22×C4C2×C22⋊D7911yescharacteristic
217C22×C4C2×D31611yescharacteristic
218M16—82no—
219C2×C8—42no—
220C2×C8—3162no—
221C2×C8—6322no—
222C4⋊C4—41no—
223C4⋊C4—3162no—
224C4⋊C4—3162no—
225C4⋊C4—3161no—
226C4⋊C4—3161no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—6322no—
230C22⋊C4—6322no—
231C42—21no—
232C42—21no—
233C42—1584no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#1457907912yes—
244C23C2×(C2×D79)⋊C411yescharacteristic
245Q8—6324no—
246D4—44no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—3162no—
261C2×C4—3162no—
262C2×C4—3162no—
263C2×C4—3162no—
264C2×C4—3162no—
265C2×C4—3162no—
266C2×C4#1457907511yescharacteristic
267C2×C4#1457907511yescharacteristic
268C2×C4#1457916912yes—
269C8—82no—
270C8—6322no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22#1909853111yescharacteristic
284C22C2×#1457907511yescharacteristic
285C22C2×#1457907911yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—6322no—
292C4—6322no—
293C4#1909835712yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2376176511yescharacteristic
299C2C2×#1909835711yescharacteristic
300C2#2376509111yescharacteristic
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#11569
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. #23761343C2semidirect
155. C79#11569semidirect

Maximal subgroups

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

subgroupindex
#237685732
#237617182
#237613432
C79×#14652
#1156979