Finite Group Databasebeta

← #34184930

Subgroups — #34184930

Order 20224.

422 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 107 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#23774348C211yesmaximal, characteristic
2#23772337C211yesmaximal, characteristic
3#23764374C211yesmaximal, characteristic
4#23761965C211yesmaximal, characteristic
5#23761260C211yesmaximal, characteristic
6#23761230C211yesmaximal, characteristic
7C79×#1549C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098942C412yescomplemented
10#19098913C412yescomplemented
11#19098770—22no—
12#19098719—22no—
13#19098692—22no—
14#19098692—22no—
15#19098616C2211yescharacteristic
16#19098591—21no—
17#19098521—21no—
18C2×#14579072C2211yescharacteristic
19#19098429C2211yescharacteristic
20#19098325—21no—
21#19098325—21no—
22#19098301—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24#431×C79C2211yescharacteristic
25#423×C79C2211yescharacteristic
26#332×C79—22no—
27C2×#11929614—24no—
28C22×C79⋊C8D411yescharacteristic
29C22×C79⋊C8D411yescharacteristic
30#14579161C2×C411yescharacteristic
31#14579161C2×C411yescharacteristic
32#14579153—21no—
33#14579140—21no—
34#14579140D412yes—
35C4×Dic158—21no—
36C4×Dic158D412yes—
37#14579120—21no—
38C2×C4⋊Dic79—21no—
39C4⋊(C79⋊C8)—21no—
40Dic79⋊C8—24no—
41#14579085—24no—
42C22⋊(C79⋊C8)—21no—
43#14579072—21no—
44#14579072—21no—
45#14579071—22no—
46C4×C79⋊C8C2×C412yes—
47C8×Dic79—22no—
48C79⋊#95—21no—
49C79⋊#95—21no—
50C22×Q8×C79—21no—
51C2×M16×C79—21no—
52C4⋊Q8×C79—21no—
53C4⋊Q8×C79—21no—
54C4⋊Q8×C79—21no—
55C4⋊Q8×C79C2×C412yes—
56C2×C4⋊C4×C79—21no—
57C2×C4⋊C4×C79—21no—
58C2×C42×C79C2311yescharacteristic
59C4⋊C8×C79—21no—
60C4⋊C8×C79—21no—
61C4⋊C8×C79D412yes—
62C22⋊C8×C79—21no—
63C4×C8×C79—21no—
64C2×Dic158—21no—
65C2×Dic158C22⋊C412yes—
66C22×Dic79—41no—
67#11929614—84no—
68C2×C79⋊C8—22no—
69C2×C79⋊C8—21no—
70C2×C79⋊C8—21no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8Q1612yescomplemented
75C4⋊Dic79—21no—
76C4⋊Dic79—21no—
77C4⋊Dic79—21no—
78C4⋊Dic79—41no—
79C4⋊Dic79—41no—
80C4⋊Dic79C22⋊C412yes—
81Dic79⋊C4—22no—
82Dic79⋊C4—22no—
83C22⋊Dic79—22no—
84C4×Dic79—22no—
85C2×Q8×C79—22no—
86C2×Q8×C79—41no—
87C2×Q8×C79—82no—
88C22×C4×C79—41no—
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79#4111yescharacteristic
92M16×C79—41no—
93M16×C79—41no—
94C2×C8×C79—22no—
95C2×C8×C79—41no—
96C4⋊C4×C79—22no—
97C4⋊C4×C79—22no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C42×C79C22×C411yescharacteristic
101C42×C79C2×D411yescharacteristic
102C42×C79C2×D411yescharacteristic
103C42×C79#4111yescharacteristic
104C23×C79#12111yescharacteristic
105Q8×C79—44no—
106Q8×C79—82no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—42no—
110C2×C4×C79—82no—
111C2×C4×C79C2×C22⋊C411yescharacteristic
112C2×C4×C79C4×D411yescharacteristic
113C2×C4×C79C4×D411yescharacteristic
114C2×C4×C79#12011yescharacteristic
115C2×C4×C79#12111yescharacteristic
116C2×C4×C79#12311yescharacteristic
117C2×C4×C79C2×Q1611yescharacteristic
118C2×C4×C79#13511yescharacteristic
119C2×C4×C79#13711yescharacteristic
120C2×C4×C79#13711yescharacteristic
121C2×Dic79—22no—
122C2×Dic79—22no—
123C2×Dic79—81no—
124C2×Dic79—81no—
125C2×Dic79—81no—
126Dic158—42no—
127Dic158—81no—
128C8×C79—42no—
129C8×C79—81no—
130C79⋊C8—22no—
131C79⋊C8—42no—
132C79⋊C8—41no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79#38611yescharacteristic
136C22×C79#46211yescharacteristic
137C22×C79#46411yescharacteristic
138C4×C79—21no—
139C4×C79—21no—
140C4×C79—21no—
141C4×C79—21no—
142C4×C79—82no—
143C4×C79C4×Q1611yescharacteristic
144C4×C79#44111yescharacteristic
145C4×C79#45111yescharacteristic
146C4×C79#45211yescharacteristic
147Dic79—42no—
148Dic79—42no—
149Dic79—161no—
150#9290—791nomaximal, Sylow
151C2×C79—21no—
152C2×C79—21no—
153C2×C79#187811yescharacteristic
154C2×C79#190211yescharacteristic
155C2×C79#191011yescharacteristic
156#3209—791no—
157#2961—791no—
158#1993—791no—
159#1549D7911yescharacteristic
160#1518—791no—
161#1286—791no—
162#1275—791no—
163C79#929011yescharacteristic, complemented, Sylow
164#533—791no—
165C2×C4⋊Q8C2×D7911yescharacteristic
166#494—792no—
167#487—792no—
168#462—1582no—
169#451—1582no—
170#446—1582no—
171#446—1582no—
172#431—791no—
173#431C2×D7911yescharacteristic
174#424—1581no—
175#423C2×D7911yescharacteristic
176#405—1581no—
177C2×#97—791no—
178#382—791no—
179#340—1581no—
180#340—1581no—
181#332—22no—
182#328—1582no—
183C22×Q8—21no—
184C2×Q16—1584no—
185C2×M16—21no—
186C22×C8—791no—
187C22×C8—791no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—791no—
192C4⋊Q8—791no—
193C4⋊Q8C4×D7912yes—
194#125—1581no—
195C22⋊Q8—792no—
196C22⋊Q8—1581no—
197C4×Q8—792no—
198C4×Q8—1581no—
199#117—1581no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—1581no—
203C2×C42C22×D7911yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—1584no—
207C4⋊C8—1581no—
208C4⋊C8C22⋊D7912yes—
209#103—1584no—
210C22⋊C8—21no—
211C22⋊C8—1581no—
212#97—1582no—
213#97—1581no—
214#97—1581no—
215C4×C8—21no—
216C4×C8—792no—
217C4×C8—1582no—
218#95—1581no—
219#95—1581no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—792no—
224C2×Q8—1581no—
225C22×C4—41no—
226C22×C4—3161no—
227C22×C4D4×D7911yescharacteristic
228C22×C4D4×D7911yescharacteristic
229C22×C4#1192962411yescharacteristic
230Q16—6324no—
231M16—41no—
232M16—41no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—792no—
236C2×C8—1582no—
237C2×C8—1581no—
238C2×C8—1581no—
239C2×C8—1581no—
240C2×C8—1581no—
241C2×C8—1581no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—3161no—
247C4⋊C4—3161no—
248C4⋊C4—792no—
249C4⋊C4—1581no—
250C4⋊C4—1581no—
251C4⋊C4—1581no—
252C4⋊C4—1582no—
253C4⋊C4—1582no—
254C22⋊C4—1582no—
255C42—1582no—
256C42C2×C4×D7911yescharacteristic
257C42D4×D7911yescharacteristic
258C42C2×C22⋊D7911yescharacteristic
259C42#1192962311yescharacteristic
260C23#1457913411yescharacteristic
261Q8—44no—
262Q8—82no—
263Q8—3162no—
264Q8—6321no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—6321no—
270C2×C4—6321no—
271C2×C4—6321no—
272C2×C4—1582no—
273C2×C4—1582no—
274C2×C4C22⋊C4×D7911yescharacteristic
275C2×C4(C22⋊D79)⋊C411yescharacteristic
276C2×C4C4×C22⋊D7911yescharacteristic
277C2×C4#1457912911yescharacteristic
278C2×C4#1457913211yescharacteristic
279C2×C4#1457914311yescharacteristic
280C2×C4Q16×D7911yescharacteristic
281C2×C4#1457918211yescharacteristic
282C2×C4#1457919011yescharacteristic
283C2×C4#1457919311yescharacteristic
284C8—42no—
285C8—81no—
286C8—3162no—
287C8—3161no—
288C8—1582no—
289C22—21no—
290C22—21no—
291C22#1909843911yescharacteristic
292C22#1909877211yescharacteristic
293C22#1909878811yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—3162no—
300C4—3162no—
301C4—12641no—
302C4#1909865511yescharacteristic
303C4#1909866711yescharacteristic
304C4#1909871711yescharacteristic
305C4#1909872211yescharacteristic
306C2—21no—
307C2—21no—
308C2#2376373811yescharacteristic
309C2#2376387611yescharacteristic
310C2#2376392411yescharacteristic
subgroupG/H
centerC22#19098439
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1549C2
socleC22×C79#386
prime radicalC23×C79#121
cosocleC2×C42×C79C23
hypercenter#1549D79
hypocenterC79#9290
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
9. #19098942C4semidirect
10. #19098913C4semidirect
74. C2×C79⋊C8Q16semidirect
163. C79#9290semidirect

Maximal subgroups

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

subgroupindex
#237743482
#237723372
#237643742
#237619652
#237612602
#237612302
C79×#15492
#929079