Finite Group Databasebeta

← #34198996

Subgroups — #34198996

Order 20224.

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

normal subgroups only

This group has 382 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#23769221C211yesmaximal, characteristic
2#23763477C211yesmaximal, characteristic
3#23761422C214yesmaximal
4C79×#1464C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic, complemented
6#19098502—21no—
7#19098499—21no—
8#19098483C412yescomplemented
9#19098475C2212yes—
10#19098461—21no—
11#19098429C411yescharacteristic, complemented
12#19098426—21no—
13#19098423—21no—
14#19098330—21no—
15#19098330C2212yes—
16C2×#117×C79C2211yescharacteristic
17#432×C79—21no—
18#324×C79—21no—
19#323×C79C2212yes—
20C22×Dic158C2×C412yes—
21#14579161—41no—
22#14579161—41no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—21no—
26C2×Dic79⋊C4—22no—
27C2×C22⋊Dic79—22no—
28C2×C22⋊Dic79—22no—
29C79⋊#95—22no—
30C79⋊#95—21no—
31C79⋊#95—22no—
32C79⋊#95—41no—
33C79⋊#95—41no—
34C79⋊#95C2×C412yes—
35C23×C4×C79D411yescharacteristic
36#117×C79—41no—
37#117×C79—41no—
38C2×C4⋊C4×C79C2×C411yescharacteristic
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C22⋊C4×C79D411yescharacteristic
41C2×C22⋊C4×C79C2311yescharacteristic
42C2×C42×C79C2×C411yescharacteristic
43C2×C42×C79D411yescharacteristic
44C4⋊C8×C79—41no—
45C22⋊C8×C79—22no—
46#97×C79—41no—
47C2×Dic158—42no—
48C2×Dic158—82no—
49C22×Dic79—22no—
50C22×Dic79—22no—
51C22×Dic79—42no—
52C4⋊Dic79—42no—
53C4⋊Dic79—42no—
54C4⋊Dic79—41no—
55C4⋊Dic79—41no—
56Dic79⋊C4—82no—
57C22⋊Dic79—42no—
58C22⋊Dic79—42no—
59C22⋊Dic79—82no—
60C24×C79C2×D411yescharacteristic
61C22×C4×C79—22no—
62C22×C4×C79—21no—
63C22×C4×C79—21no—
64C22×C4×C79—41no—
65C22×C4×C79C22⋊C411yescharacteristic
66C22×C4×C79C22⋊C411yescharacteristic
67C22×C4×C79C22⋊C411yescharacteristic
68C22×C4×C79C22⋊C411yescharacteristic
69C22×C4×C79C22×C411yescharacteristic
70C22×C4×C79C2×D411yescharacteristic
71C2×C8×C79—42no—
72C4⋊C4×C79—41no—
73C4⋊C4×C79—41no—
74C22⋊C4×C79—41no—
75C22⋊C4×C79—41no—
76C42×C79—21no—
77C42×C79—21no—
78C42×C79—21no—
79C42×C79—21no—
80C23×C79—21no—
81C23×C79—21no—
82C23×C79—41no—
83C23×C79—41no—
84C23×C79C2×C22⋊C411yescharacteristic
85C23×C79#13712yes—
86C2×C4×C79—21no—
87C2×C4×C79—21no—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—21no—
91C2×C4×C79—21no—
92C2×C4×C79—21no—
93C2×C4×C79—21no—
94C2×C4×C79—21no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—41no—
98C2×C4×C79—41no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79#9911yescharacteristic
102C2×C4×C79#9911yescharacteristic
103C2×Dic79—42no—
104C2×Dic79—42no—
105C2×Dic79—42no—
106C2×Dic79—42no—
107C2×Dic79—42no—
108C2×Dic79—42no—
109C2×Dic79—82no—
110C2×Dic79—82no—
111C2×Dic79—82no—
112Dic158—84no—
113C8×C79—82no—
114C22×C79—21no—
115C22×C79—21no—
116C22×C79—22no—
117C22×C79—42no—
118C22×C79—41no—
119C22×C79—41no—
120C22×C79—81no—
121C22×C79C2×#9911yescharacteristic
122C22×C79#41911yescharacteristic
123C22×C79#42111yescharacteristic
124C4×C79—21no—
125C4×C79—21no—
126C4×C79—41no—
127C4×C79—41no—
128C4×C79—41no—
129C4×C79—41no—
130C4×C79—41no—
131Dic79—82no—
132Dic79—82no—
133Dic79—162no—
134#11812—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79—42no—
138C2×C79#151111yescharacteristic
139C2×C79#212211yescharacteristic
140C2×C79#212911yescharacteristic
141#2628—791no—
142#1820—791no—
143#1464D7911yescharacteristic
144#1351—794no—
145C79#1181211yescharacteristic, complemented, Sylow
146C2×C4⋊Q8—791no—
147C2×#117C2×D7911yescharacteristic
148#432—21no—
149#400—1581no—
150#399—1581no—
151#395—792no—
152#393—792no—
153#389—1581no—
154#382—791no—
155#381—1581no—
156#380—1581no—
157#342—792no—
158#342—1581no—
159#324—21no—
160#323C2×D7912yes—
161C22×Q8—792no—
162C23×C4D31611yescharacteristic
163C4⋊Q8—3161no—
164C4⋊Q8—3161no—
165#117—41no—
166#117—41no—
167C2×C4⋊C4—1581no—
168C2×C4⋊C4—1581no—
169C2×C4⋊C4—1581no—
170C2×C4⋊C4—1582no—
171C2×C4⋊C4C4×D7911yescharacteristic
172C2×C4⋊C4C22⋊D7911yescharacteristic
173C2×C22⋊C4—1582no—
174C2×C22⋊C4—1582no—
175C2×C22⋊C4D31611yescharacteristic
176C2×C22⋊C4C22×D7911yescharacteristic
177C2×C42C4×D7911yescharacteristic
178C2×C42C22⋊D7911yescharacteristic
179C4⋊C8—41no—
180C22⋊C8—22no—
181#97—41no—
182#95—3161no—
183#95—3161no—
184#95—792no—
185#95—1582no—
186#95—1581no—
187#95—1582no—
188C24C2×D31611yescharacteristic
189C2×Q8—3162no—
190C2×Q8—6322no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—3162no—
196C22×C4—1582no—
197C22×C4—1582no—
198C22×C4(C2×D79)⋊C411yescharacteristic
199C22×C4(C2×D79)⋊C411yescharacteristic
200C22×C4(C2×D79)⋊C411yescharacteristic
201C22×C4(C2×D79)⋊C411yescharacteristic
202C22×C4C2×C4×D7911yescharacteristic
203C22×C4C2×C22⋊D7911yescharacteristic
204C2×C8—42no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—3162no—
208C4⋊C4—3162no—
209C4⋊C4—3161no—
210C4⋊C4—3161no—
211C4⋊C4—6322no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—3162no—
215C22⋊C4—3162no—
216C22⋊C4—6322no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23C2×(C2×D79)⋊C411yescharacteristic
226C23#1457919412yes—
227Q8—6324no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—3162no—
244C2×C4—3162no—
245C2×C4—3162no—
246C2×C4—3162no—
247C2×C4—3162no—
248C2×C4—3162no—
249C2×C4—6322no—
250C2×C4—6322no—
251C2×C4—6322no—
252C2×C4#1457907511yescharacteristic
253C2×C4#1457907511yescharacteristic
254C8—82no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22C2×#1457907511yescharacteristic
263C22#1909857311yescharacteristic
264C22#1909857911yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—6322no—
273C4—6322no—
274C4—12642no—
275C2—21no—
276C2—21no—
277C2—42no—
278C2#2376193411yescharacteristic
279C2#2376505111yescharacteristic
280C2#2376507411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1464C2
socleC22×C79C2×#99
prime radicalC24×C79C2×D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1464D79
hypocenterC79#11812
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
5. C2×#14579161C4semidirect
8. #19098483C4semidirect
11. #19098429C4semidirect
145. C79#11812semidirect

Maximal subgroups

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

subgroupindex
#237692212
#237634772
#237614222
C79×#14642
#1181279