Finite Group Databasebeta

← #34197759

Subgroups — #34197759

Order 20224.

440 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 83 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#23766112C211yesmaximal, characteristic
2#23761738C211yesmaximal, characteristic
3#23761345C212yesmaximal, complemented
4#23761333C212yesmaximal, complemented
5C79×#1465C211yesmaximal, characteristic
6C2×#14579148C411yescharacteristic
7C2×C4×Dic158C411yescharacteristic
8#19098527—21no—
9#19098468C2211yescharacteristic
10#19098466C411yescharacteristic
11#19098451—21no—
12#19098451—21no—
13#19098436—21no—
14#19098436—21no—
15#19098432C2211yescharacteristic
16#19098409C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C2×C4×D4×C79C2211yescharacteristic
20#407×C79—21no—
21#324×C79C2212yes—
22#323×C79—21no—
23C22×Dic158C2×C411yescharacteristic
24C2×#11929608—41no—
25#14579148—22no—
26#14579148—22no—
27C4×Dic158—41no—
28C4×Dic158—41no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—21no—
36C2×C22⋊Dic79—21no—
37C2×C4×Dic79—21no—
38C2×C4×Dic79C2×C411yescharacteristic
39C2×C4×Dic79C2×C411yescharacteristic
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—41no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C22×D4×C79C2×C411yescharacteristic
47C23×C4×C79D411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C2×M16×C79—41no—
50C4×D4×C79—41no—
51C4×D4×C79—41no—
52C2×C4⋊C4×C79C2311yescharacteristic
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C42×C79C2×C411yescharacteristic
56C22⋊C8×C79—22no—
57C22⋊C8×C79—41no—
58C2×Dic158—21no—
59C2×Dic158—41no—
60C2×Dic158—41no—
61C2×Dic158M1612yescomplemented
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66#11929608—82no—
67C2×C79⋊C8—42no—
68C4⋊Dic79—41no—
69Dic79⋊C4—41no—
70Dic79⋊C4—41no—
71Dic79⋊C4—81no—
72Dic79⋊C4—81no—
73C22⋊Dic79—24no—
74C22⋊Dic79—24no—
75C22⋊Dic79—81no—
76C22⋊Dic79—81no—
77C4×Dic79—21no—
78C4×Dic79—41no—
79C4×Dic79—42no—
80C4×Dic79M1612yescomplemented
81C24×C79C22⋊C411yescharacteristic
82C24×C79C22⋊C411yescharacteristic
83C2×D4×C79—21no—
84C2×D4×C79—21no—
85C2×D4×C79—41no—
86C2×D4×C79—41no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—81no—
92C22×C4×C79—81no—
93C22×C4×C79C22⋊C411yescharacteristic
94C22×C4×C79C22⋊C411yescharacteristic
95C22×C4×C79C22×C411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98M16×C79—82no—
99C2×C8×C79—42no—
100C4⋊C4×C79—41no—
101C22⋊C4×C79—41no—
102C22⋊C4×C79—41no—
103C42×C79—21no—
104C42×C79—21no—
105C23×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—21no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79C2×C22⋊C411yescharacteristic
116D4×C79—42no—
117D4×C79—42no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—41no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—81no—
130C2×C4×C79—81no—
131C2×C4×C79#9911yescharacteristic
132C2×C4×C79#9911yescharacteristic
133C2×C4×C79C2×M1611yescharacteristic
134C2×C4×C79C2×M1611yescharacteristic
135C2×Dic79—22no—
136C2×Dic79—22no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147Dic158—42no—
148Dic158—42no—
149C8×C79—82no—
150C79⋊C8—82no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—42no—
154C22×C79—42no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—41no—
158C22×C79—41no—
159C22×C79—41no—
160C22×C79—41no—
161C22×C79—81no—
162C22×C79—81no—
163C22×C79#40711yescharacteristic
164C22×C79C2×#9911yescharacteristic
165C22×C79#41311yescharacteristic
166C4×C79—21no—
167C4×C79—21no—
168C4×C79—21no—
169C4×C79—21no—
170C4×C79—41no—
171C4×C79—41no—
172C4×C79—41no—
173Dic79—42no—
174Dic79—42no—
175Dic79—81no—
176Dic79—81no—
177#11535—791nomaximal, Sylow
178C2×C79—21no—
179C2×C79—21no—
180C2×C79—81no—
181C2×C79—81no—
182C2×C79#146511yescharacteristic
183C2×C79#212711yescharacteristic
184C2×C79#213511yescharacteristic
185#2326—791no—
186#1465—791no—
187#1465D7911yescharacteristic
188#1325—792no—
189#1319—792no—
190C79#1153511yescharacteristic, complemented, Sylow
191C2×#124—791no—
192C2×C4×Q8—791no—
193C2×C4×D4C2×D7911yescharacteristic
194#407—21no—
195#407—1581no—
196#391—791no—
197#390—791no—
198#388—1581no—
199#388—1581no—
200#385—1581no—
201#385—1581no—
202#384—791no—
203#376—791no—
204#324—792no—
205#324C2×D7912yes—
206#323—21no—
207#323—1581no—
208C22×Q8—791no—
209C22×D4C4×D7911yescharacteristic
210C23×C4D31611yescharacteristic
211C23×C4C22⋊D7911yescharacteristic
212C2×M16—41no—
213C2×M16—3161no—
214#124—1582no—
215#124—1582no—
216C4×Q8—3161no—
217C4×Q8—3161no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—791no—
221C2×C4⋊C4—1581no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4—1581no—
224C2×C4⋊C4C22×D7911yescharacteristic
225C2×C22⋊C4—1581no—
226C2×C22⋊C4—1581no—
227C2×C22⋊C4—1581no—
228C2×C22⋊C4—1581no—
229C2×C22⋊C4D31611yescharacteristic
230C2×C22⋊C4C22⋊D7911yescharacteristic
231C2×C42—791no—
232C2×C42—791no—
233C2×C42—1581no—
234C2×C42C4×D7911yescharacteristic
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—3161no—
238C22⋊C8—1582no—
239#95—1581no—
240#95—1581no—
241#95—1581no—
242#95—1581no—
243C24(C2×D79)⋊C411yescharacteristic
244C24(C2×D79)⋊C411yescharacteristic
245C2×Q8—3161no—
246C2×Q8—3161no—
247C2×Q8—792no—
248C2×Q8—1581no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—1581no—
260C22×C4—1581no—
261C22×C4—1581no—
262C22×C4—1581no—
263C22×C4(C2×D79)⋊C411yescharacteristic
264C22×C4(C2×D79)⋊C411yescharacteristic
265C22×C4C2×C4×D7911yescharacteristic
266C22×C4C2×C22⋊D7911yescharacteristic
267C22×C4C2×D31611yescharacteristic
268M16—82no—
269M16—6322no—
270C2×C8—42no—
271C2×C8—3162no—
272C4⋊C4—41no—
273C4⋊C4—3161no—
274C4⋊C4—3161no—
275C4⋊C4—3161no—
276C4⋊C4—6321no—
277C4⋊C4—6321no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—6321no—
281C22⋊C4—6321no—
282C22⋊C4—1584no—
283C22⋊C4—1584no—
284C42—21no—
285C42—21no—
286C42—3161no—
287C42—3162no—
288C42—792no—
289C42—1581no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×(C2×D79)⋊C411yescharacteristic
301Q8—3162no—
302Q8—3162no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3161no—
319C2×C4—3161no—
320C2×C4—3161no—
321C2×C4—3161no—
322C2×C4—3161no—
323C2×C4—3161no—
324C2×C4—3161no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—1582no—
329C2×C4—1582no—
330C2×C4#1457907511yescharacteristic
331C2×C4#1457907511yescharacteristic
332C2×C4M16×D7911yescharacteristic
333C2×C4M16×D7911yescharacteristic
334C8—82no—
335C8—6322no—
336C22—21no—
337C22—21no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22#1909852611yescharacteristic
349C22C2×#1457907511yescharacteristic
350C22#1909854511yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3162no—
359C4—3162no—
360C4—6321no—
361C4—6321no—
362C2—21no—
363C2—21no—
364C2—81no—
365C2—81no—
366C2#2376173611yescharacteristic
367C2#2376506711yescharacteristic
368C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1465C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1465D79
hypocenterC79#11535
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #23761345C2semidirect
4. #23761333C2semidirect
61. C2×Dic158M16semidirect
80. C4×Dic79M16semidirect
190. C79#11535semidirect

Maximal subgroups

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

subgroupindex
#237661122
#237617382
#237613452
#237613332
C79×#14652
#1153579