Finite Group Databasebeta

← #34199006

Subgroups — #34199006

Order 20224.

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

normal subgroups only

This group has 586 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
1C2×#19098483C211yesmaximal, characteristic
2#23761730C211yesmaximal, characteristic
3#23761429C214yesmaximal, complemented
4C79×#1821C211yesmaximal, characteristic
5C23×Dic158C411yescharacteristic
6C22×Dic79⋊C4—21no—
7#19098527—21no—
8#19098483—21no—
9#19098483—21no—
10#19098483C412yes—
11#19098483C2212yes—
12C2×C79⋊#95—21no—
13C2×C79⋊#95C411yescharacteristic
14#19098289—21no—
15#19098289C2212yes—
16C22×C4⋊C4×C79C2211yescharacteristic
17#407×C79—21no—
18#343×C79—21no—
19#343×C79C2212yes—
20C22×Dic158—21no—
21C22×Dic158—44no—
22C22×Dic158—41no—
23C22×Dic158—41no—
24C22×Dic158C2×C412yes—
25C23×Dic79—21no—
26C23×Dic79—21no—
27C2×#11929608—41no—
28C2×Dic79⋊C4—22no—
29C2×Dic79⋊C4—22no—
30C2×Dic79⋊C4—44no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—41no—
33C79⋊#95—22no—
34C79⋊#95—21no—
35C79⋊#95—22no—
36C79⋊#95C2×C412yes—
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C2×M16×C79—22no—
41C2×C4⋊C4×C79—41no—
42C2×C4⋊C4×C79—41no—
43C2×C4⋊C4×C79C2×C411yescharacteristic
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C4⋊C4×C79C2311yescharacteristic
47C22⋊C8×C79—41no—
48C22⋊C8×C79—41no—
49C2×Dic158—42no—
50C2×Dic158—42no—
51C2×Dic158—44no—
52C2×Dic158—44no—
53C2×Dic158—48no—
54C2×Dic158—44no—
55C2×Dic158—44no—
56C22×Dic79—24no—
57C22×Dic79—24no—
58C22×Dic79—22no—
59C22×Dic79—22no—
60C22×Dic79—42no—
61C22×Dic79—42no—
62C22×Dic79—42no—
63C22×Dic79—41no—
64C22×Dic79—41no—
65#11929608—82no—
66C2×C79⋊C8—42no—
67Dic79⋊C4—44no—
68Dic79⋊C4—44no—
69C24×C79C22⋊C411yescharacteristic
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—22no—
77C22×C4×C79—41no—
78C22×C4×C79—42no—
79C22×C4×C79—42no—
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86M16×C79—42no—
87M16×C79—42no—
88C2×C8×C79—42no—
89C4⋊C4×C79—41no—
90C4⋊C4×C79—41no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C23×C79—22no—
94C23×C79—22no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79#9911yescharacteristic
98C23×C79#9911yescharacteristic
99C23×C79C2×C22⋊C411yescharacteristic
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
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—42no—
113C2×C4×C79—42no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—81no—
119C2×C4×C79#13712yes—
120C2×Dic79—24no—
121C2×Dic79—24no—
122C2×Dic79—42no—
123C2×Dic79—48no—
124C2×Dic79—42no—
125C2×Dic79—44no—
126C2×Dic79—44no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129Dic158—44no—
130Dic158—48no—
131Dic158—44no—
132C8×C79—82no—
133C79⋊C8—82no—
134C22×C79—24no—
135C22×C79—24no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79AΣL(1,𝔽16)12yes—
143C22×C79#35612yes—
144C22×C79C2×#9911yescharacteristic
145C22×C79#41911yescharacteristic
146C22×C79#42111yescharacteristic
147C4×C79—22no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—41no—
151C4×C79—41no—
152C4×C79—41no—
153Dic79—44no—
154Dic79—44no—
155#11815—791nomaximal, Sylow
156C2×C79—21no—
157C2×C79—22no—
158C2×C79—22no—
159C2×C79—21no—
160C2×C79#151111yescharacteristic
161C2×C79C2×AΣL(1,𝔽16)11yescharacteristic
162C2×C79C2×#35611yescharacteristic
163C2×#395—791no—
164#1821D7911yescharacteristic
165#1463—791no—
166#1354—794no—
167C79#1181511yescharacteristic, complemented, Sylow
168C23×Q8—791no—
169C22×C4⋊C4—1581no—
170C22×C4⋊C4C2×D7911yescharacteristic
171#407—21no—
172#407—1581no—
173#395—792no—
174#395—792no—
175#395—1581no—
176#395—1581no—
177C2×#95—791no—
178C2×#95—1581no—
179#343—21no—
180#343C2×D7912yes—
181#324—792no—
182#324—1581no—
183C22×Q8—3161no—
184C22×Q8—3161no—
185C22×Q8—3164no—
186C22×Q8—792no—
187C22×Q8—1581no—
188C23×C4—1581no—
189C23×C4—1581no—
190C23×C4C4×D7911yescharacteristic
191C23×C4D31611yescharacteristic
192C23×C4C22⋊D7911yescharacteristic
193C2×M16—22no—
194C2×M16—3161no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—3164no—
198C2×C4⋊C4—1582no—
199C2×C4⋊C4—1582no—
200C2×C4⋊C4C4×D7911yescharacteristic
201C2×C4⋊C4D31611yescharacteristic
202C2×C4⋊C4C22⋊D7911yescharacteristic
203C2×C4⋊C4C22×D7911yescharacteristic
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—3161no—
207C22⋊C8—1582no—
208#95—792no—
209#95—1581no—
210#95—1582no—
211#95—1582no—
212C24(C2×D79)⋊C411yescharacteristic
213C2×Q8—3162no—
214C2×Q8—3162no—
215C2×Q8—3168no—
216C2×Q8—3164no—
217C2×Q8—3164no—
218C2×Q8—3164no—
219C2×Q8—3164no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—3162no—
231C22×C4—3162no—
232C22×C4—3162no—
233C22×C4—3161no—
234C22×C4—3161no—
235C22×C4—1584no—
236C22×C4—1584no—
237C22×C4—1582no—
238C22×C4—1582no—
239C22×C4(C2×D79)⋊C411yescharacteristic
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4C2×C4×D7911yescharacteristic
243C22×C4C2×C22⋊D7911yescharacteristic
244C22×C4C2×D31611yescharacteristic
245M16—42no—
246M16—42no—
247M16—6322no—
248C2×C8—42no—
249C2×C8—3162no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—3164no—
255C4⋊C4—3164no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1457907511yescharacteristic
261C23#1457907511yescharacteristic
262C23C2×(C2×D79)⋊C411yescharacteristic
263Q8—3164no—
264Q8—3168no—
265Q8—3164no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—3162no—
286C2×C4—3162no—
287C2×C4—3164no—
288C2×C4—3168no—
289C2×C4—3162no—
290C2×C4—3162no—
291C2×C4—3164no—
292C2×C4—1584no—
293C2×C4—1584no—
294C2×C4#1457919112yes—
295C8—82no—
296C8—6322no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1909835112yes—
306C22#1909836112yes—
307C22C2×#1457907511yescharacteristic
308C22#1909857111yescharacteristic
309C22#1909858111yescharacteristic
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—3164no—
317C4—3164no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2376193211yescharacteristic
323C2C2×#1909835111yescharacteristic
324C2C2×#1909836111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1821C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1821D79
hypocenterC79#11815
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
167. C79#11815semidirect

Maximal subgroups

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

subgroupindex
C2×#190984832
#237617302
#237614292
C79×#18212
#1181579