Finite Group Databasebeta

← #34186003

Subgroups — #34186003

Order 20224.

700 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 111 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#23767040C211yesmaximal, characteristic
2#23763324C211yesmaximal, characteristic
3#23761932C212yesmaximal
4#23761252C212yesmaximal
5C79×#2310C211yesmaximal, characteristic
6C2×#14579159C412yescomplemented
7C2×#14579140C2211yescharacteristic
8C2×#14579080—22no—
9#19098527C2212yes—
10#19098482—22no—
11#19098470C412yescomplemented
12#19098436—21no—
13#19098436—21no—
14C4×C22⋊Dic79C2211yescharacteristic
15#19098299—24no—
16C23×Q8×C79C2211yescharacteristic
17C2×#117×C79—21no—
18#391×C79—24no—
19#381×C79—21no—
20#381×C79C2212yes—
21C22×Dic158C2×C411yescharacteristic
22C2×#11929608—22no—
23#14579159—44no—
24#14579140—22no—
25#14579140—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—22no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—21no—
32C2×C4×Dic79C2×C412yes—
33#14579080—44no—
34C22⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—22no—
36C79⋊#95—22no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C22×Q8×C79—22no—
40C22×Q8×C79—22no—
41C22×Q8×C79—44no—
42C22×Q8×C79C2×C412yes—
43C22×Q8×C79D412yes—
44C23×C4×C79D411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C23×C4×C79C2311yescharacteristic
47#117×C79—44no—
48C2×C4⋊C4×C79—22no—
49C2×C22⋊C4×C79—22no—
50C2×C42×C79—22no—
51#95×C79—24no—
52#95×C79—22no—
53#95×C79D414yes—
54C2×Dic158—22no—
55C2×Dic158—42no—
56C22×Dic79—22no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59#11929608—44no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—22no—
62Dic79⋊C4—44no—
63Dic79⋊C4—44no—
64Dic79⋊C4—42no—
65C22⋊Dic79—24no—
66C22⋊Dic79—41no—
67C22⋊Dic79—41no—
68C4×Dic79—42no—
69C24×C79C2×D411yescharacteristic
70C2×Q8×C79—24no—
71C2×Q8×C79—24no—
72C2×Q8×C79—24no—
73C2×Q8×C79—24no—
74C2×Q8×C79—42no—
75C2×Q8×C79—42no—
76C2×Q8×C79—44no—
77C2×Q8×C79—44no—
78C2×Q8×C79—48no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—24no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—24no—
85C22×C4×C79—42no—
86C22×C4×C79—42no—
87C22×C4×C79C22⋊C412yes—
88C22×C4×C79C22⋊C412yes—
89C22×C4×C79C22×C411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D412yes—
92C22×C4×C79#4112yes—
93C4⋊C4×C79—44no—
94C22⋊C4×C79—44no—
95C42×C79—44no—
96C23×C79—22no—
97C23×C79—22no—
98C23×C79—41no—
99C23×C79C2×C22⋊C411yescharacteristic
100C23×C79#12011yescharacteristic
101C23×C79#12311yescharacteristic
102Q8×C79—48no—
103Q8×C79—48no—
104C2×C4×C79—22no—
105C2×C4×C79—22no—
106C2×C4×C79—24no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—44no—
119C2×C4×C79—44no—
120C2×C4×C79—44no—
121C2×C4×C79#9914yes—
122C2×C4×C79C4×D412yes—
123C2×C4×C79#12112yes—
124C2×Dic79—24no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—42no—
131Dic158—82no—
132C79⋊C8—82no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—44no—
141C22×C79#38611yescharacteristic
142C22×C79C2×#9912yes—
143C22×C79#45612yes—
144C4×C79—24no—
145C4×C79—22no—
146C4×C79—22no—
147C4×C79—42no—
148C4×C79—84no—
149Dic79—44no—
150Dic79—81no—
151Dic79—81no—
152#9462—791nomaximal, Sylow
153C2×C79—22no—
154C2×C79—22no—
155C2×C79—41no—
156C2×C79#188611yescharacteristic
157C2×C79#189811yescharacteristic
158C2×C79#191211yescharacteristic
159#2434—791no—
160#2310D7911yescharacteristic
161#1784—791no—
162#1511—792no—
163#1284—792no—
164C79#946211yescharacteristic, complemented, Sylow
165C23×Q8C2×D7911yescharacteristic
166C2×C4⋊Q8—792no—
167C2×C22⋊Q8—791no—
168C2×#117—21no—
169C2×#101—1582no—
170#407—792no—
171#395—1582no—
172#391—24no—
173#391—792no—
174#385—1581no—
175#385—1581no—
176#381—21no—
177#381C2×D7912yes—
178C4×C22⋊C4—791no—
179#328—1584no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8—791no—
184C22×Q8C4×D7912yes—
185C22×Q8D31612yes—
186C23×C4C22⋊D7911yescharacteristic
187C23×C4C22⋊D7911yescharacteristic
188C23×C4C22×D7911yescharacteristic
189C2×M16—1582no—
190C4⋊Q8—3164no—
191C22⋊Q8—1582no—
192C22⋊Q8—1582no—
193#117—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—791no—
196C2×C4⋊C4—1582no—
197C2×C4⋊C4—1582no—
198C2×C4⋊C4—1581no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1581no—
201C2×C22⋊C4—1581no—
202C2×C42—22no—
203C2×C42—792no—
204#101—3164no—
205C22⋊C8—1582no—
206C22⋊C8—1582no—
207#95—24no—
208#95—22no—
209#95—1582no—
210#95—1582no—
211#95—1581no—
212#95C22⋊D7914yes—
213C24C2×C22⋊D7911yescharacteristic
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—3162no—
224C2×Q8—1582no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—1581no—
234C22×C4—1581no—
235C22×C4—1582no—
236C22×C4(C2×D79)⋊C412yes—
237C22×C4(C2×D79)⋊C412yes—
238C22×C4C2×C4×D7911yescharacteristic
239C22×C4C2×C22⋊D7912yes—
240C22×C4C2×D31611yescharacteristic
241C22×C4#1192962412yes—
242M16—3164no—
243C2×C8—3162no—
244C4⋊C4—44no—
245C4⋊C4—3164no—
246C4⋊C4—3164no—
247C4⋊C4—3162no—
248C4⋊C4—1582no—
249C22⋊C4—44no—
250C22⋊C4—3161no—
251C22⋊C4—3161no—
252C22⋊C4—1584no—
253C42—44no—
254C42—3162no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23C2×(C2×D79)⋊C411yescharacteristic
259C23#1457913011yescharacteristic
260C23#1457914411yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—6322no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—3162no—
282C2×C4—3162no—
283C2×C4—3162no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3162no—
287C2×C4—1584no—
288C2×C4#1457907514yes—
289C2×C4C4×C22⋊D7912yes—
290C2×C4#1457913512yes—
291C8—6322no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#1909844211yescharacteristic
301C22C2×#1457907512yes—
302C22#1909874612yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—42no—
307C4—84no—
308C4—3164no—
309C4—6321no—
310C4—6321no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#2376379311yescharacteristic
315C2#2376385311yescharacteristic
316C2#2376393711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2310C2
socleC22×C79#386
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#2310D79
hypocenterC79#9462
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
6. C2×#14579159C4semidirect
11. #19098470C4semidirect
164. C79#9462semidirect

Maximal subgroups

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

subgroupindex
#237670402
#237633242
#237619322
#237612522
C79×#23102
#946279