Finite Group Databasebeta

← #34198861

Subgroups — #34198861

Order 20224.

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

normal subgroups only

This group has 482 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#23768399C211yesmaximal, characteristic
2C4×#14579080C211yesmaximal, characteristic, complemented
3#23761404C214yesmaximal, complemented
4C79×#1467C211yesmaximal, characteristic, complemented
5C2×#14579158C411yescharacteristic
6C2×#14579150—21no—
7C2×#14579080—21no—
8C4×#11929608—21no—
9#19098480C412yes—
10#19098480C2212yes—
11#19098465—21no—
12#19098465—21no—
13#19098464—21no—
14C4×C4⋊Dic79C411yescharacteristic
15#19098332C2212yes—
16C2×C4×Q8×C79C2211yescharacteristic
17#432×C79—21no—
18#324×C79—21no—
19#324×C79C2212yes—
20C22×D316—21no—
21C22×D316—21no—
22C22×D316C2×C412yes—
23C2×#11929608—22no—
24#14579158—41no—
25#14579158—41no—
26#14579150—44no—
27C2×C4⋊Dic79—21no—
28C2×(C2×D79)⋊C4—22no—
29C2×(C2×D79)⋊C4—22no—
30C2×(C2×D79)⋊C4—22no—
31C2×(C2×D79)⋊C4—22no—
32C2×C4×Dic79—21no—
33#14579080—24no—
34#14579072—41no—
35C4×C79⋊C8—41no—
36C79⋊#95C2×C412yes—
37C22×Q8×C79D411yescharacteristic
38C4×Q8×C79—41no—
39C4×Q8×C79—41no—
40C2×C4⋊C4×C79C2×C411yescharacteristic
41C2×C4⋊C4×C79D411yescharacteristic
42C2×C4⋊C4×C79C2311yescharacteristic
43C2×C42×C79C2×C411yescharacteristic
44C2×C42×C79D411yescharacteristic
45C2×C42×C79D411yescharacteristic
46C4⋊C8×C79—41no—
47C22⋊C8×C79—22no—
48#97×C79—41no—
49C23×D79—22no—
50C2×D316—42no—
51C2×D316—42no—
52C2×D316—42no—
53C2×D316—42no—
54C2×D316—42no—
55C2×D316—41no—
56C2×D316—41no—
57C2×D316—42no—
58C2×D316—82no—
59C22×Dic79—22no—
60#11929608—24no—
61#11929608—42no—
62C2×C79⋊C8—42no—
63C4⋊Dic79—42no—
64(C2×D79)⋊C4—44no—
65(C2×D79)⋊C4—44no—
66(C2×D79)⋊C4—82no—
67(C2×D79)⋊C4—82no—
68C4×Dic79—42no—
69C2×Q8×C79—22no—
70C2×Q8×C79—22no—
71C2×Q8×C79—41no—
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22×C411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C2×C8×C79—42no—
80C4⋊C4×C79—41no—
81C4⋊C4×C79—41no—
82C4⋊C4×C79—41no—
83C42×C79—21no—
84C42×C79—21no—
85C42×C79—21no—
86C42×C79—21no—
87C42×C79—21no—
88C42×C79—21no—
89C23×C79C2×C22⋊C411yescharacteristic
90C22×D79—42no—
91C22×D79—42no—
92C22×D79—42no—
93C22×D79—42no—
94C22×D79—42no—
95C22×D79—42no—
96C22×D79—42no—
97Q8×C79—44no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
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—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79#9911yescharacteristic
113C2×C4×C79#9911yescharacteristic
114C2×C4×C79#10412yes—
115C2×C4×C79AGL(1,ℤ/8)12yes—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119D316—82no—
120D316—84no—
121D316—84no—
122D316—82no—
123C8×C79—82no—
124C79⋊C8—44no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79C2×#9911yescharacteristic
128C22×C79#41811yescharacteristic
129C22×C79C2×#10411yescharacteristic
130C2×D79—82no—
131C2×D79—82no—
132C2×D79—82no—
133C2×D79—82no—
134C2×D79—82no—
135C2×D79—82no—
136C2×D79—82no—
137C4×C79—21no—
138C4×C79—22no—
139C4×C79—22no—
140C4×C79—41no—
141C4×C79—41no—
142C4×C79—41no—
143C4×C79—41no—
144C4×C79#35512yes—
145Dic79—82no—
146#11781—791nomaximal, Sylow
147C2×C79—21no—
148C2×C79—21no—
149C2×C79#150711yescharacteristic
150C2×C79#212211yescharacteristic
151C2×C79C2×#35511yescharacteristic
152D79—162no—
153#2548—791no—
154C4×#101—791no—
155#1467D7911yescharacteristic, complemented
156#1345—794no—
157C79#1178111yescharacteristic, complemented, Sylow
158C2×#127—791no—
159C2×#124—1581no—
160C2×C4×Q8C2×D7911yescharacteristic
161#432—21no—
162C2×#101—1581no—
163C4×M16—1581no—
164#394—792no—
165#394—792no—
166#390—1581no—
167#390—1581no—
168#390—1581no—
169C4×C4⋊C4—791no—
170#343—792no—
171#324—21no—
172#324C2×D7912yes—
173C22×Q8D31611yescharacteristic
174C22×D4—792no—
175C22×D4—1581no—
176C22×D4—1581no—
177C2×M16—1582no—
178#127—3161no—
179#127—3161no—
180#124—3164no—
181C4×Q8—41no—
182C4×Q8—41no—
183C2×C4⋊C4—1581no—
184C2×C4⋊C4C4×D7911yescharacteristic
185C2×C4⋊C4C22⋊D7911yescharacteristic
186C2×C4⋊C4C22×D7911yescharacteristic
187C2×C22⋊C4—1582no—
188C2×C22⋊C4—1582no—
189C2×C22⋊C4—1582no—
190C2×C22⋊C4—1582no—
191C2×C42—1581no—
192C2×C42C4×D7911yescharacteristic
193C2×C42D31611yescharacteristic
194C2×C42C22⋊D7911yescharacteristic
195C4⋊C8—41no—
196#101—1584no—
197C22⋊C8—22no—
198#97—41no—
199#97—3161no—
200C4×C8—3161no—
201#95—792no—
202C24—1582no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×D4—3162no—
207C2×D4—3162no—
208C2×D4—3162no—
209C2×D4—3162no—
210C2×D4—3162no—
211C2×D4—3161no—
212C2×D4—3161no—
213C2×D4—3162no—
214C2×D4—6322no—
215C22×C4—1582no—
216C22×C4(C2×D79)⋊C411yescharacteristic
217C22×C4(C2×D79)⋊C411yescharacteristic
218C22×C4(C2×D79)⋊C411yescharacteristic
219C22×C4(C2×D79)⋊C411yescharacteristic
220C22×C4C2×C4×D7911yescharacteristic
221C22×C4C2×C22⋊D7911yescharacteristic
222C22×C4C2×D31611yescharacteristic
223M16—3162no—
224M16—1584no—
225C2×C8—42no—
226C2×C8—3162no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—3162no—
231C22⋊C4—3164no—
232C22⋊C4—3164no—
233C22⋊C4—6322no—
234C22⋊C4—6322no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—3162no—
242C23—3162no—
243C23—3162no—
244C23—3162no—
245C23—3162no—
246C23—3162no—
247C23—3162no—
248C23—3162no—
249C23C2×(C2×D79)⋊C411yescharacteristic
250Q8—44no—
251D4—6322no—
252D4—6324no—
253D4—6324no—
254D4—6322no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
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—3162no—
270C2×C4—3162no—
271C2×C4—3162no—
272C2×C4#1457907511yescharacteristic
273C2×C4#1457907511yescharacteristic
274C2×C4#1457908712yes—
275C2×C4#1457918912yes—
276C8—82no—
277C8—3164no—
278C22—21no—
279C22—21no—
280C22—6322no—
281C22—6322no—
282C22—6322no—
283C22—6322no—
284C22—6322no—
285C22—6322no—
286C22—6322no—
287C22C2×#1457907511yescharacteristic
288C22#1909856911yescharacteristic
289C22C2×#1457908711yescharacteristic
290C4—21no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—6322no—
298C4#1909835912yes—
299C2—21no—
300C2—21no—
301C2—12642no—
302C2#2376191711yescharacteristic
303C2#2376504911yescharacteristic
304C2C2×#1909835911yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1467C2
socleC22×C79C2×#99
prime radicalC2×#14579158C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1467D79
hypocenterC79#11781
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. C4×#14579080C2semidirect
3. #23761404C2semidirect
4. C79×#1467C2semidirect
155. #1467D79semidirect
157. C79#11781semidirect

Maximal subgroups

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

subgroupindex
#237683992
C4×#145790802
#237614042
C79×#14672
#1178179