Finite Group Databasebeta

← #34203450

Subgroups — #34203450

Order 20224.

500 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 115 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#23765969C211yesmaximal, characteristic
2C2×#19098361C212yesmaximal
3#101×Dic79C212yesmaximal
4#23761527C211yesmaximal, characteristic
5C79×#1394C211yesmaximal, characteristic
6C2×#14579159C2211yescharacteristic
7C2×Q8×Dic79C2211yescharacteristic
8C2×#14579126C411yescharacteristic
9C2×#14579080C2212yes—
10M16×Dic79—22no—
11#19098469C411yescharacteristic
12#19098460—22no—
13#19098432C411yescharacteristic
14#19098414—22no—
15C42×Dic79C411yescharacteristic
16#19098361C418yescomplemented
17#19098331—22no—
18C2×#101×C79C2212yes—
19#391×C79C2211yescharacteristic
20C22×Dic158C2×C411yescharacteristic
21C2×#11929608—22no—
22#14579159—21no—
23#14579159—21no—
24#14579159C2×C414yes—
25Q8×Dic79—41no—
26Q8×Dic79—41no—
27#14579126—41no—
28#14579126—41no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C4⋊Dic79D412yes—
31C2×Dic79⋊C4—22no—
32C2×Dic79⋊C4—22no—
33C2×Dic79⋊C4—21no—
34C2×C4×Dic79—21no—
35C2×C4×Dic79—21no—
36C2×C4×Dic79—21no—
37C2×C4×Dic79C2×C411yescharacteristic
38C2×C4×Dic79C2×C411yescharacteristic
39C2×C4×Dic79D412yes—
40#14579080—22no—
41#14579080C2×C414yes—
42#14579071—42no—
43C8×Dic79—42no—
44C79⋊#95—22no—
45C22×Q8×C79C2311yescharacteristic
46C2×M16×C79—22no—
47C2×C4⋊C4×C79C2×C411yescharacteristic
48C2×C42×C79C2×C411yescharacteristic
49#101×C79—22no—
50#101×C79C2×C414yes—
51#95×C79—22no—
52C2×Dic158—21no—
53C2×Dic158—41no—
54C2×Dic158—41no—
55C2×Dic158C4212yes—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79C22⋊C411yescharacteristic
59C22×Dic79C22⋊C412yes—
60C22×Dic79C22⋊C411yescharacteristic
61#11929608—24no—
62#11929608—42no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—41no—
65C4⋊Dic79—42no—
66Dic79⋊C4—24no—
67Dic79⋊C4—24no—
68Dic79⋊C4—44no—
69Dic79⋊C4—42no—
70C4×Dic79—24no—
71C4×Dic79—22no—
72C4×Dic79—22no—
73C4×Dic79—21no—
74C4×Dic79—21no—
75C4×Dic79—21no—
76C4×Dic79—42no—
77C4×Dic79—41no—
78C4×Dic79—41no—
79C4×Dic79C4212yes—
80C2×Q8×C79—41no—
81C2×Q8×C79—41no—
82C2×Q8×C79C22×C412yes—
83C2×Q8×C79#4111yescharacteristic
84C2×Q8×C79#4111yescharacteristic
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79C22×C411yescharacteristic
88C22×C4×C79C2×D412yes—
89M16×C79—24no—
90M16×C79—42no—
91C2×C8×C79—42no—
92C4⋊C4×C79—41no—
93C42×C79—41no—
94C23×C79C2×C22⋊C411yescharacteristic
95Q8×C79—42no—
96Q8×C79—41no—
97Q8×C79—41no—
98C2×C4×C79—22no—
99C2×C4×C79—21no—
100C2×C4×C79—22no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—42no—
106C2×C4×C79—41no—
107C2×C4×C79C2×C4211yescharacteristic
108C2×C4×C79#11711yescharacteristic
109C2×C4×C79C4×D412yes—
110C2×C4×C79C4×D412yes—
111C2×Dic79—22no—
112C2×Dic79—22no—
113C2×Dic79—22no—
114C2×Dic79—22no—
115C2×Dic79—22no—
116C2×Dic79—21no—
117C2×Dic79—22no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—42no—
122C2×Dic79#9911yescharacteristic
123C2×Dic79#9911yescharacteristic
124Dic158—42no—
125Dic158—42no—
126C8×C79—44no—
127C79⋊C8—44no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79C4×C22⋊C411yescharacteristic
131C22×C79C2×#9911yescharacteristic
132C22×C79#41011yescharacteristic
133C4×C79—21no—
134C4×C79—21no—
135C4×C79—22no—
136C4×C79—22no—
137C4×C79—42no—
138C4×C79—81no—
139Dic79—24no—
140Dic79—21no—
141Dic79—41no—
142Dic79—42no—
143Dic79—42no—
144Dic79#35612yescomplemented
145C4×#356—791nomaximal, Sylow
146C2×C79—21no—
147C2×C79—21no—
148C2×C79C4×#9911yescharacteristic
149C2×C79C2×#35611yescharacteristic
150C2×C79#213211yescharacteristic
151C4×C4⋊Q8—791no—
152C2×#356—792no—
153C4×#101—792no—
154#1394—791no—
155#1394D7911yescharacteristic
156C79C4×#35611yescharacteristic, complemented, Sylow
157C2×C4⋊Q8—791no—
158C2×C4×Q8—791no—
159C2×C4×Q8—791no—
160C2×#101—792no—
161C2×#101C2×D7912yes—
162C4×M16—1582no—
163#391—791no—
164#391C2×D7911yescharacteristic
165#389—1582no—
166#384—791no—
167C4×C4⋊C4—1582no—
168C43—791no—
169#356—798no—
170#343—1582no—
171C22×Q8—791no—
172C22×Q8C22×D7911yescharacteristic
173C2×M16—22no—
174C2×M16—1582no—
175C4⋊Q8—794no—
176C4⋊Q8—1581no—
177C4⋊Q8—1581no—
178C4×Q8—3161no—
179C4×Q8—3161no—
180C4×Q8—3161no—
181C4×Q8—3161no—
182C2×C4⋊C4—791no—
183C2×C4⋊C4—792no—
184C2×C4⋊C4—1581no—
185C2×C4⋊C4—1582no—
186C2×C4⋊C4—1582no—
187C2×C4⋊C4C4×D7911yescharacteristic
188C2×C42—791no—
189C2×C42—791no—
190C2×C42—792no—
191C2×C42—1581no—
192C2×C42—1581no—
193C2×C42—1581no—
194C2×C42C4×D7911yescharacteristic
195#101—22no—
196#101—794no—
197#101—1582no—
198#101C4×D7914yes—
199#97—3162no—
200C4×C8—3162no—
201#95—22no—
202#95—1582no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—3161no—
206C2×Q8—3161no—
207C2×Q8—792no—
208C2×Q8—1581no—
209C2×Q8C2×C4×D7912yes—
210C2×Q8#1192962311yescharacteristic
211C2×Q8#1192962511yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—792no—
215C22×C4—791no—
216C22×C4—791no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4C2×C4×D7911yescharacteristic
220C22×C4D4×D7912yes—
221M16—24no—
222M16—42no—
223M16—3162no—
224M16—1584no—
225C2×C8—42no—
226C2×C8—3162no—
227C4⋊C4—41no—
228C4⋊C4—3161no—
229C4⋊C4—3162no—
230C4⋊C4—3164no—
231C4⋊C4—3162no—
232C4⋊C4—1584no—
233C4⋊C4—1584no—
234C42—41no—
235C42—3161no—
236C42—3161no—
237C42—3162no—
238C42—792no—
239C42—1582no—
240C42—1582no—
241C42—1581no—
242C42—1581no—
243C42—1584no—
244C42—1581no—
245C23C22⋊C4×D7911yescharacteristic
246Q8—42no—
247Q8—41no—
248Q8—41no—
249Q8—3162no—
250Q8—3162no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—3161no—
261C2×C4—3161no—
262C2×C4—3161no—
263C2×C4—3162no—
264C2×C4—791no—
265C2×C4—791no—
266C2×C4—1582no—
267C2×C4—1581no—
268C2×C4—1582no—
269C2×C4—1582no—
270C2×C4—1582no—
271C2×C4—1582no—
272C2×C4—1582no—
273C2×C4C42×D7911yescharacteristic
274C2×C4#1457911911yescharacteristic
275C2×C4(C22⋊D79)⋊C412yes—
276C2×C4#1457912312yes—
277C8—44no—
278C8—3164no—
279C22—21no—
280C22—21no—
281C22#1909841011yescharacteristic
282C22#99×D7911yescharacteristic
283C22#1909853511yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—81no—
290C4—3161no—
291C4—3162no—
292C4—3162no—
293C4—792no—
294C4—1584no—
295C4—1581no—
296C2—21no—
297C2—21no—
298C2#2376318711yescharacteristic
299C2#356×D7911yescharacteristic
300C2#2376508211yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC2×C4×C79C2×C42
FrattiniC22×Q8C22×D79
FittingC79×#1394C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC22×Q8×C79C23
hypercenter#1394D79
hypocenterC79C4×#356
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
16. #19098361C4semidirect
144. Dic79#356semidirect
156. C79C4×#356semidirect

Maximal subgroups

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

subgroupindex
#237659692
C2×#190983612
#101×Dic792
#237615272
C79×#13942
C4×#35679