Finite Group Databasebeta

← #34197841

Subgroups — #34197841

Order 20224.

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

normal subgroups only

This group has 366 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#23767849C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic
3#23761345C212yesmaximal, complemented
4#23761333C212yesmaximal, complemented
5C79×#1464C211yesmaximal, characteristic
6C2×#14579140C411yescharacteristic
7#19098616—21no—
8#19098491—21no—
9#19098491—21no—
10#19098468C411yescharacteristic
11#19098468C2211yescharacteristic
12#19098447—21no—
13#19098447—21no—
14#19098432C2211yescharacteristic
15#19098428C411yescharacteristic
16C4×C22⋊Dic79C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C2×#117×C79C2211yescharacteristic
20#432×C79—21no—
21#324×C79C2212yes—
22#323×C79—21no—
23C22×Dic158C2×C411yescharacteristic
24#14579140—41no—
25#14579140—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—21no—
32C2×C4×Dic79C2×C411yescharacteristic
33C2×C4×Dic79C2×C411yescharacteristic
34C4⋊(C79⋊C8)—41no—
35C22⋊(C79⋊C8)—22no—
36C4×C79⋊C8—41no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C23×C4×C79C2×C411yescharacteristic
45#117×C79—41no—
46#117×C79—41no—
47C2×C4⋊C4×C79C2×C411yescharacteristic
48C2×C4⋊C4×C79C2311yescharacteristic
49C2×C22⋊C4×C79D411yescharacteristic
50C2×C22⋊C4×C79D411yescharacteristic
51C2×C42×C79D411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C4⋊C8×C79—41no—
54C22⋊C8×C79—22no—
55#97×C79—41no—
56C2×Dic158—22no—
57C2×Dic158—41no—
58C2×Dic158—41no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—41no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67Dic79⋊C4—41no—
68Dic79⋊C4—41no—
69C22⋊Dic79—42no—
70C22⋊Dic79—42no—
71C4×Dic79—22no—
72C4×Dic79—41no—
73C24×C79C22⋊C411yescharacteristic
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—22no—
77C22×C4×C79—41no—
78C22×C4×C79C22⋊C411yescharacteristic
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—
86C4⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C22⋊C4×C79—41no—
89C42×C79—21no—
90C42×C79—21no—
91C42×C79—21no—
92C42×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79C2×C22⋊C411yescharacteristic
99C2×C4×C79—21no—
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—42no—
108C2×C4×C79—42no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79#9911yescharacteristic
114C2×C4×C79#9911yescharacteristic
115C2×C4×C79#13112yes—
116C2×Dic79—22no—
117C2×Dic79—41no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128Dic158—82no—
129C8×C79—82no—
130C79⋊C8—82no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—41no—
136C22×C79—81no—
137C22×C79—81no—
138C22×C79#40811yescharacteristic
139C22×C79C2×#9911yescharacteristic
140C22×C79#41311yescharacteristic
141C4×C79—21no—
142C4×C79—21no—
143C4×C79—22no—
144C4×C79—41no—
145C4×C79—41no—
146C4×C79—41no—
147C4×C79—41no—
148Dic79—42no—
149Dic79—81no—
150Dic79—81no—
151Dic79—81no—
152#11552—791nomaximal, Sylow
153C2×C79—21no—
154C2×C79—21no—
155C2×C79—81no—
156C2×C79#146811yescharacteristic
157C2×C79#212611yescharacteristic
158C2×C79#213511yescharacteristic
159#2499—791no—
160#1464D7911yescharacteristic
161#1458—791no—
162#1325—792no—
163#1319—792no—
164C79#1155211yescharacteristic, complemented, Sylow
165C2×C22⋊Q8—791no—
166C2×#117C2×D7911yescharacteristic
167#432—21no—
168#431—1581no—
169#397—1581no—
170#397—1581no—
171#391—791no—
172#391—791no—
173#387—1581no—
174#387—1581no—
175#384—791no—
176#382—791no—
177C4×C22⋊C4—791no—
178#324—792no—
179#324C2×D7912yes—
180#323—21no—
181#323—1581no—
182C22×Q8—791no—
183C23×C4C4×D7911yescharacteristic
184C22⋊Q8—3161no—
185C22⋊Q8—3161no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—791no—
189C2×C4⋊C4—1581no—
190C2×C4⋊C4—1581no—
191C2×C4⋊C4—1581no—
192C2×C4⋊C4C4×D7911yescharacteristic
193C2×C4⋊C4C22×D7911yescharacteristic
194C2×C22⋊C4—1581no—
195C2×C22⋊C4—1581no—
196C2×C22⋊C4D31611yescharacteristic
197C2×C22⋊C4C22⋊D7911yescharacteristic
198C2×C42—791no—
199C2×C42—791no—
200C2×C42D31611yescharacteristic
201C2×C42C22⋊D7911yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—3161no—
204C22⋊C8—22no—
205C22⋊C8—1582no—
206#97—41no—
207C4×C8—3161no—
208#95—1581no—
209#95—1581no—
210#95—1581no—
211#95—1581no—
212#95—1581no—
213#95—1581no—
214#95—1581no—
215C24(C2×D79)⋊C411yescharacteristic
216C2×Q8—3161no—
217C2×Q8—3161no—
218C2×Q8—1582no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—1581no—
224C22×C4—1581no—
225C22×C4—1581no—
226C22×C4—1581no—
227C22×C4(C2×D79)⋊C411yescharacteristic
228C22×C4(C2×D79)⋊C411yescharacteristic
229C22×C4(C2×D79)⋊C411yescharacteristic
230C22×C4C2×C4×D7911yescharacteristic
231C22×C4C2×C22⋊D7911yescharacteristic
232C22×C4C2×D31611yescharacteristic
233C2×C8—42no—
234C2×C8—3162no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—3161no—
238C4⋊C4—3162no—
239C4⋊C4—3162no—
240C4⋊C4—3161no—
241C4⋊C4—3161no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—3162no—
245C22⋊C4—3162no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—3161no—
251C42—1582no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×(C2×D79)⋊C411yescharacteristic
258Q8—6322no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—3161no—
274C2×C4—3161no—
275C2×C4—3161no—
276C2×C4—3161no—
277C2×C4—3161no—
278C2×C4—3161no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3161no—
283C2×C4—3161no—
284C2×C4—1582no—
285C2×C4#1457907511yescharacteristic
286C2×C4#1457907511yescharacteristic
287C2×C4#1457916912yes—
288C8—82no—
289C8—6322no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1909853111yescharacteristic
298C22C2×#1457907511yescharacteristic
299C22#1909854511yescharacteristic
300C4—21no—
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—3162no—
308C4—6321no—
309C4—6321no—
310C4—6321no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#2376174911yescharacteristic
315C2#2376506311yescharacteristic
316C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1464C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1464D79
hypocenterC79#11552
LCSG ▷ C22×C4×C79 ▷ C22×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
3. #23761345C2semidirect
4. #23761333C2semidirect
164. C79#11552semidirect

Maximal subgroups

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

subgroupindex
#237678492
#237617142
#237613452
#237613332
C79×#14642
#1155279