Finite Group Databasebeta

← #34197943

Subgroups — #34197943

Order 20224.

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

normal subgroups only

This group has 554 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#23766765C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761349C214yesmaximal, complemented
4C79×#1466C211yesmaximal, characteristic
5C23×Dic158C411yescharacteristic
6C2×#14579140—21no—
7C2×C22⋊(C79⋊C8)—21no—
8#19098483—21no—
9#19098475—21no—
10#19098469—21no—
11#19098443C412yes—
12#19098435C2212yes—
13#19098423C411yescharacteristic
14#19098289—21no—
15#19098287C2212yes—
16C2×C4×D4×C79C2211yescharacteristic
17#432×C79—21no—
18#323×C79—21no—
19#323×C79C2212yes—
20C22×Dic158—21no—
21C22×Dic158—21no—
22C22×Dic158—44no—
23C22×Dic158—41no—
24C22×Dic158—41no—
25C23×Dic79C2×C412yes—
26C22×C79⋊C8—41no—
27#14579140—44no—
28C2×C4⋊Dic79—21no—
29C2×Dic79⋊C4—22no—
30C2×C22⋊Dic79—22no—
31C2×C22⋊Dic79C2×C412yes—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—41no—
34C79⋊#95—21no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C22×D4×C79C2×C411yescharacteristic
38C23×C4×C79C2×C411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C4×D4×C79—41no—
41C4×D4×C79—41no—
42C2×C4⋊C4×C79D411yescharacteristic
43C2×C22⋊C4×C79D411yescharacteristic
44C2×C22⋊C4×C79C2311yescharacteristic
45C2×C42×C79D411yescharacteristic
46C4⋊C8×C79—41no—
47C22⋊C8×C79—22no—
48#97×C79—41no—
49C2×Dic158—24no—
50C2×Dic158—42no—
51C2×Dic158—42no—
52C2×Dic158—44no—
53C2×Dic158—48no—
54C2×Dic158—44no—
55C2×Dic158—44no—
56C2×Dic158—41no—
57C2×Dic158—41no—
58C22×Dic79—22no—
59C22×Dic79—22no—
60C22×Dic79—42no—
61C22×Dic79—44no—
62C2×C79⋊C8—42no—
63C2×C79⋊C8—82no—
64C4⋊Dic79—42no—
65Dic79⋊C4—44no—
66C22⋊Dic79—24no—
67C22⋊Dic79—44no—
68C24×C79C22⋊C411yescharacteristic
69C24×C79C22×C411yescharacteristic
70C2×D4×C79—22no—
71C2×D4×C79—22no—
72C2×D4×C79—41no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—42no—
78C22×C4×C79—42no—
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C2×C8×C79—42no—
85C4⋊C4×C79—41no—
86C22⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C42×C79—21no—
89C42×C79—21no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79#10112yes—
99C23×C79C2×C22⋊C411yescharacteristic
100C23×C79#13112yes—
101D4×C79—44no—
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—42no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—81no—
116C2×C4×C79#9911yescharacteristic
117C2×C4×C79#9911yescharacteristic
118C2×Dic79—24no—
119C2×Dic79—42no—
120C2×Dic79—44no—
121C2×Dic79—42no—
122C2×Dic79—44no—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126Dic158—44no—
127Dic158—48no—
128Dic158—44no—
129C8×C79—82no—
130C79⋊C8—82no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—22no—
134C22×C79—22no—
135C22×C79—42no—
136C22×C79—42no—
137C22×C79—42no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—81no—
143C22×C79#40811yescharacteristic
144C22×C79C2×#9911yescharacteristic
145C22×C79C2×#10111yescharacteristic
146C4×C79—21no—
147C4×C79—41no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—41no—
151C4×C79AΣL(1,𝔽16)12yes—
152Dic79—44no—
153Dic79—82no—
154#11575—791nomaximal, Sylow
155C2×C79—21no—
156C2×C79—21no—
157C2×C79—42no—
158C2×C79—42no—
159C2×C79#147211yescharacteristic
160C2×C79C2×AΣL(1,𝔽16)11yescharacteristic
161C2×C79#212311yescharacteristic
162#2409—791no—
163#1466D7911yescharacteristic
164#1459—791no—
165#1326—794no—
166C79#1157511yescharacteristic, complemented, Sylow
167C23×Q8—791no—
168C2×C22⋊Q8—1581no—
169C2×C4×D4C2×D7911yescharacteristic
170#432—21no—
171C2×C22⋊C8—1581no—
172#395—1581no—
173#393—1581no—
174#391—1581no—
175#386—792no—
176#385—792no—
177#380—791no—
178#324—1581no—
179#323—21no—
180#323—792no—
181#323C2×D7912yes—
182C22×Q8—3161no—
183C22×Q8—3161no—
184C22×Q8—3164no—
185C22×Q8—1581no—
186C22×Q8—1581no—
187C22×D4C4×D7911yescharacteristic
188C23×C4—792no—
189C23×C4C4×D7911yescharacteristic
190C23×C4D31611yescharacteristic
191C22×C8—3161no—
192C22⋊Q8—3164no—
193C4×D4—41no—
194C4×D4—41no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4—1582no—
197C2×C4⋊C4D31611yescharacteristic
198C2×C22⋊C4—792no—
199C2×C22⋊C4—1582no—
200C2×C22⋊C4C22⋊D7911yescharacteristic
201C2×C22⋊C4C22×D7911yescharacteristic
202C2×C42C22⋊D7911yescharacteristic
203C4⋊C8—41no—
204C22⋊C8—22no—
205C22⋊C8—3161no—
206C22⋊C8—1582no—
207#97—41no—
208#95—1581no—
209#95—1582no—
210#95—1582no—
211C24(C2×D79)⋊C411yescharacteristic
212C24C2×C4×D7911yescharacteristic
213C2×Q8—3162no—
214C2×Q8—3162no—
215C2×Q8—3164no—
216C2×Q8—3168no—
217C2×Q8—3164no—
218C2×Q8—3164no—
219C2×Q8—3161no—
220C2×Q8—3161no—
221C2×Q8—1584no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—3162no—
232C22×C4—3164no—
233C22×C4—1582no—
234C22×C4—1582no—
235C22×C4(C2×D79)⋊C411yescharacteristic
236C22×C4(C2×D79)⋊C411yescharacteristic
237C22×C4(C2×D79)⋊C411yescharacteristic
238C22×C4C2×C22⋊D7911yescharacteristic
239C22×C4C2×D31611yescharacteristic
240C2×C8—42no—
241C2×C8—3162no—
242C2×C8—6322no—
243C4⋊C4—41no—
244C4⋊C4—3162no—
245C4⋊C4—3164no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—3164no—
249C22⋊C4—1584no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#1457907912yes—
261C23C2×(C2×D79)⋊C411yescharacteristic
262C23#1457916912yes—
263Q8—3164no—
264Q8—3168no—
265Q8—3164no—
266D4—44no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—3162no—
282C2×C4—3164no—
283C2×C4—3162no—
284C2×C4—3164no—
285C2×C4—3162no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—1584no—
289C2×C4#1457907511yescharacteristic
290C2×C4#1457907511yescharacteristic
291C8—82no—
292C8—6322no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#1909853111yescharacteristic
306C22C2×#1457907511yescharacteristic
307C22C2×#1457907911yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3164no—
314C4—6322no—
315C4#1909835112yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2#2376176511yescharacteristic
321C2C2×#1909835111yescharacteristic
322C2#2376505311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1466C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#1466D79
hypocenterC79#11575
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #23761349C2semidirect
166. C79#11575semidirect

Maximal subgroups

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

subgroupindex
#237667652
#237617182
#237613492
C79×#14662
#1157579