Finite Group Databasebeta

← #34197443

Subgroups — #34197443

Order 20224.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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#23765882C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic
3#23761349C212yesmaximal
4#23761347C212yesmaximal
5C79×#1458C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic
7C22×C4×Dic79C411yescharacteristic
8#19098616—21no—
9#19098459—21no—
10#19098459—21no—
11#19098435C411yescharacteristic
12#19098435C411yescharacteristic
13#19098435C2211yescharacteristic
14#19098435C2211yescharacteristic
15C4×Dic79⋊C4—21no—
16C4×Dic79⋊C4—21no—
17#19098289—21no—
18#19098287C2212yes—
19C2×#117×C79C2211yescharacteristic
20#431×C79—21no—
21#324×C79—21no—
22#323×C79C2212yes—
23C23×Dic79C2×C411yescharacteristic
24C23×Dic79C2×C411yescharacteristic
25#14579120—22no—
26#14579120C814yescomplemented
27C2×C4⋊Dic79—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×C22⋊Dic79C2×C411yescharacteristic
33C2×C22⋊Dic79C2×C411yescharacteristic
34C2×C4×Dic79—21no—
35C2×C4×Dic79—21no—
36C2×C4×Dic79—21no—
37C2×C4×Dic79—41no—
38C2×C4×Dic79—41no—
39C4⋊(C79⋊C8)—41no—
40C22⋊(C79⋊C8)—22no—
41C4×C79⋊C8—41no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C23×C4×C79C2×C411yescharacteristic
47#117×C79—41no—
48#117×C79—41no—
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79C2311yescharacteristic
53C2×C42×C79D411yescharacteristic
54C2×C42×C79D411yescharacteristic
55C4⋊C8×C79—41no—
56C22⋊C8×C79—22no—
57C4×C8×C79—41no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—41no—
63C22×Dic79—41no—
64C22×Dic79—42no—
65C22×Dic79—42no—
66C2×C79⋊C8—42no—
67C4⋊Dic79—24no—
68Dic79⋊C4—42no—
69Dic79⋊C4—42no—
70Dic79⋊C4—42no—
71Dic79⋊C4—42no—
72C22⋊Dic79—21no—
73C22⋊Dic79—21no—
74C22⋊Dic79C2×C812yes—
75C22⋊Dic79C2×C812yes—
76C4×Dic79—24no—
77C4×Dic79—42no—
78C4×Dic79—41no—
79C4×Dic79—41no—
80C24×C79C22×C411yescharacteristic
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—22no—
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C22⋊C411yescharacteristic
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C2×C812yes—
89C22×C4×C79M1612yes—
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C2×C8×C79—42no—
93C4⋊C4×C79—41no—
94C4⋊C4×C79—41no—
95C22⋊C4×C79—41no—
96C22⋊C4×C79—41no—
97C42×C79—21no—
98C42×C79—21no—
99C42×C79—21no—
100C42×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79C2×C22⋊C411yescharacteristic
106C23×C79C22×C811yescharacteristic
107C23×C79C2×M1611yescharacteristic
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79C22⋊C812yes—
123C2×C4×C79C22⋊C812yes—
124C2×C4×C79#9911yescharacteristic
125C2×C4×C79#9911yescharacteristic
126C2×C4×C79#10112yes—
127C2×Dic79—22no—
128C2×Dic79—22no—
129C2×Dic79—22no—
130C2×Dic79—22no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—42no—
138C2×Dic79—42no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—42no—
142C2×Dic79—42no—
143C8×C79—82no—
144C79⋊C8—82no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79—41no—
150C22×C79—41no—
151C22×C79—42no—
152C22×C79—41no—
153C22×C79—41no—
154C22×C79C2×C22⋊C811yescharacteristic
155C22×C79C2×#9911yescharacteristic
156C22×C79C2×#10111yescharacteristic
157C4×C79—22no—
158C4×C79—41no—
159C4×C79—41no—
160C4×C79—41no—
161C4×C79—41no—
162C4×C79#32412yes—
163C4×C79#32412yes—
164Dic79—42no—
165Dic79—42no—
166Dic79—42no—
167Dic79—42no—
168#11452—791nomaximal, Sylow
169C2×C79—21no—
170C2×C79—21no—
171C2×C79—41no—
172C2×C79—41no—
173C2×C79C2×#32411yescharacteristic
174C2×C79#212011yescharacteristic
175C2×C79#212011yescharacteristic
176#2296—791no—
177#1458—791no—
178#1458D7911yescharacteristic
179#1326—792no—
180#1326—792no—
181C79#1145211yescharacteristic, complemented, Sylow
182C2×#117—791no—
183C2×#117C2×D7911yescharacteristic
184C22×C42—791no—
185#431—21no—
186#431—1581no—
187#389—1581no—
188#389—1581no—
189#385—791no—
190#385—791no—
191#385—791no—
192#385—791no—
193C4×C4⋊C4—1581no—
194C4×C4⋊C4—1581no—
195#324—21no—
196#324—1581no—
197#323—792no—
198#323C2×D7912yes—
199C23×C4—791no—
200C23×C4—791no—
201C23×C4C4×D7911yescharacteristic
202#117—41no—
203#117—41no—
204#117—794no—
205#117—1582no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4D31611yescharacteristic
212C2×C4⋊C4C22⋊D7911yescharacteristic
213C2×C22⋊C4—791no—
214C2×C22⋊C4—791no—
215C2×C22⋊C4C4×D7911yescharacteristic
216C2×C22⋊C4C22×D7911yescharacteristic
217C2×C42—3161no—
218C2×C42—3161no—
219C2×C42—1581no—
220C2×C42—1581no—
221C2×C42—1581no—
222C2×C42D31611yescharacteristic
223C2×C42C22⋊D7911yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—3161no—
226C22⋊C8—22no—
227C22⋊C8—1582no—
228C4×C8—41no—
229C4×C8—3161no—
230#95—1581no—
231#95—1581no—
232#95—1581no—
233#95—1581no—
234C24C2×C4×D7911yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—3161no—
239C22×C4—3161no—
240C22×C4—3162no—
241C22×C4—3162no—
242C22×C4—1581no—
243C22×C4—1581no—
244C22×C4—1581no—
245C22×C4—1581no—
246C22×C4(C2×D79)⋊C411yescharacteristic
247C22×C4(C2×D79)⋊C411yescharacteristic
248C22×C4(C2×D79)⋊C411yescharacteristic
249C22×C4(C2×D79)⋊C411yescharacteristic
250C22×C4C8×D7912yes—
251C22×C4#1192960712yes—
252C22×C4C2×C22⋊D7911yescharacteristic
253C22×C4C2×D31611yescharacteristic
254C2×C8—42no—
255C2×C8—3162no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—3162no—
259C4⋊C4—3162no—
260C4⋊C4—3162no—
261C4⋊C4—3162no—
262C4⋊C4—1584no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—792no—
266C22⋊C4—792no—
267C22⋊C4—1581no—
268C22⋊C4—1581no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—3162no—
274C42—3161no—
275C42—3161no—
276C42—1584no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23C2×(C2×D79)⋊C411yescharacteristic
282C23C2×C8×D7911yescharacteristic
283C23C2×#1192960711yescharacteristic
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—3161no—
304C2×C4—3162no—
305C2×C4—3162no—
306C2×C4—3161no—
307C2×C4—3161no—
308C2×C4—3162no—
309C2×C4—3162no—
310C2×C4—1582no—
311C2×C4—1582no—
312C2×C4—1582no—
313C2×C4—1582no—
314C2×C4(C2×D79)⋊C812yes—
315C2×C4(C2×D79)⋊C812yes—
316C2×C4#1457907511yescharacteristic
317C2×C4#1457907511yescharacteristic
318C2×C4#1457907912yes—
319C8—82no—
320C8—6322no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22C2×(C2×D79)⋊C811yescharacteristic
331C22C2×#1457907511yescharacteristic
332C22C2×#1457907911yescharacteristic
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—3162no—
339C4—3162no—
340C4—3162no—
341C4—3162no—
342C4#1909828812yes—
343C4#1909828812yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1909828811yescharacteristic
349C2#2376503911yescharacteristic
350C2#2376504211yescharacteristic
subgroupG/H
centerC2×C4#14579075
commutatorC23×C79C22×C8
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1458C2
socleC22×C79C2×#99
prime radicalC24×C79C22×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#1458D79
hypocenterC79#11452
LCSG ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
26. #14579120C8semidirect
181. C79#11452semidirect

Maximal subgroups

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

subgroupindex
#237658822
#237617142
#237613492
#237613472
C79×#14582
#1145279