Finite Group Databasebeta

← #34183273

Subgroups — #34183273

Order 20224.

836 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19098475C211yesmaximal, characteristic
2#23763334C212yesmaximal, complemented
3C2×#19098287C211yesmaximal, characteristic
4#23761278C212yesmaximal, complemented
5C79×#2312C211yesmaximal, characteristic
6C23×Dic158C2211yescharacteristic
7C22×C22⋊Dic79—21no—
8C2×#14579084—24no—
9C2×C22⋊(C79⋊C8)C2212yes—
10#19098475—22no—
11#19098475C414yescomplemented
12C2×C79⋊#95—21no—
13C2×C79⋊#95C2211yescharacteristic
14#19098287—22no—
15C22×C22⋊C4×C79C2211yescharacteristic
16#388×C79—24no—
17#380×C79C2212yes—
18#379×C79—21no—
19C4×C22⋊C4×C79—21no—
20C22×Dic158—21no—
21C22×Dic158—22no—
22C22×Dic158—22no—
23C22×Dic158—44no—
24C22×Dic158C2×C412yes—
25C23×Dic79—21no—
26C23×Dic79—21no—
27C22×C79⋊C8—22no—
28C2×C22⋊Dic79—24no—
29C2×C22⋊Dic79—44no—
30#14579084—48no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C79⋊#95—24no—
34C79⋊#95—21no—
35C79⋊#95C2×C412yes—
36C25×C79D411yescharacteristic
37C23×C4×C79D411yescharacteristic
38C23×C4×C79C2311yescharacteristic
39C2×C4⋊C4×C79D414yes—
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—22no—
42C2×C22⋊C4×C79—22no—
43C2×C22⋊C4×C79—22no—
44C2×C22⋊C4×C79—22no—
45C2×C22⋊C4×C79C2×C412yes—
46C2×C22⋊C4×C79D412yes—
47C2×C42×C79—22no—
48#95×C79—22no—
49#95×C79—22no—
50C2×Dic158—28no—
51C2×Dic158—24no—
52C2×Dic158—44no—
53C2×Dic158—44no—
54C2×Dic158—48no—
55C2×Dic158—44no—
56C2×Dic158—42no—
57C22×Dic79—24no—
58C22×Dic79—22no—
59C22×Dic79—22no—
60C22×Dic79—42no—
61C22×Dic79—42no—
62C22×Dic79—42no—
63C22×Dic79—41no—
64C22×Dic79—41no—
65C22×Dic79—42no—
66C2×C79⋊C8—44no—
67C2×C79⋊C8—42no—
68C22⋊Dic79—48no—
69C24×C79—21no—
70C24×C79—22no—
71C24×C79—21no—
72C24×C79—22no—
73C24×C79—42no—
74C24×C79—42no—
75C24×C79C22⋊C412yes—
76C24×C79C2×D411yescharacteristic
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—24no—
80C22×C4×C79—42no—
81C22×C4×C79—42no—
82C22×C4×C79C22⋊C412yes—
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D412yes—
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79#4112yes—
87C4⋊C4×C79—44no—
88C22⋊C4×C79—24no—
89C22⋊C4×C79—24no—
90C22⋊C4×C79—42no—
91C22⋊C4×C79—42no—
92C22⋊C4×C79—82no—
93C22⋊C4×C79—82no—
94C42×C79—82no—
95C23×C79—22no—
96C23×C79—22no—
97C23×C79—22no—
98C23×C79—22no—
99C23×C79—22no—
100C23×C79—42no—
101C23×C79—42no—
102C23×C79—48no—
103C23×C79—42no—
104C23×C79—42no—
105C23×C79—42no—
106C23×C79—42no—
107C23×C79—81no—
108C23×C79—81no—
109C23×C79—84no—
110C23×C79C2×C22⋊C411yescharacteristic
111C23×C79#12011yescharacteristic
112C23×C79#12311yescharacteristic
113C23×C79#13512yes—
114C23×C79#13712yes—
115C2×C4×C79—24no—
116C2×C4×C79—22no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—44no—
123C2×C4×C79—44no—
124C2×C4×C79—44no—
125C2×C4×C79—81no—
126C2×C4×C79C4×D412yes—
127C2×C4×C79#12112yes—
128C2×Dic79—24no—
129C2×Dic79—42no—
130C2×Dic79—44no—
131C2×Dic79—42no—
132C2×Dic79—44no—
133C2×Dic79—42no—
134C2×Dic79—42no—
135C2×Dic79—42no—
136C2×Dic79—84no—
137Dic158—28no—
138Dic158—44no—
139Dic158—48no—
140C79⋊C8—82no—
141C22×C79—24no—
142C22×C79—24no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—44no—
148C22×C79—42no—
149C22×C79—42no—
150C22×C79—42no—
151C22×C79—82no—
152C22×C79—84no—
153C22×C79—84no—
154C22×C79#38611yescharacteristic
155C22×C79#41611yescharacteristic
156C22×C79#41911yescharacteristic
157C22×C79#45212yes—
158C22×C79#45512yes—
159C22×C79B(4,𝔽2)12yes—
160C22×C79#48512yes—
161C4×C79—22no—
162C4×C79—22no—
163C4×C79—42no—
164C4×C79—84no—
165Dic79—44no—
166Dic79—82no—
167#9016—791nomaximal, Sylow
168C2×C79—21no—
169C2×C79—21no—
170C2×C79—22no—
171C2×C79—82no—
172C2×C79#160614yes—
173C2×C79#187311yescharacteristic
174C2×C79#188511yescharacteristic
175C2×C79#188911yescharacteristic
176C2×#393—791no—
177#2312D7911yescharacteristic
178#1787—792no—
179C2×#323—791no—
180#1294—792no—
181C79#901611yescharacteristic, complemented, Sylow
182C23×Q8—791no—
183C22×C22⋊C4—1581no—
184C22×C22⋊C4C2×D7911yescharacteristic
185C2×#103—1584no—
186C2×C22⋊C8—792no—
187#393—794no—
188#393—1582no—
189#388—24no—
190#380C2×D7912yes—
191#379—21no—
192C4×C22⋊C4—21no—
193C2×#95—791no—
194C2×#95—1581no—
195#323—1582no—
196C25C22⋊D7911yescharacteristic
197C22×Q8—3164no—
198C22×Q8—792no—
199C22×Q8—1582no—
200C22×Q8—1582no—
201C22×Q8—1581no—
202C23×C4—1581no—
203C23×C4—1581no—
204C23×C4C22⋊D7911yescharacteristic
205C23×C4C22×D7911yescharacteristic
206C22×C8—1582no—
207C2×C4⋊C4C22⋊D7914yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—3164no—
214C2×C22⋊C4—1584no—
215C2×C22⋊C4C4×D7912yes—
216C2×C22⋊C4D31612yes—
217C2×C42—22no—
218#103—3168no—
219C22⋊C8—1582no—
220C22⋊C8—1582no—
221#95—22no—
222#95—22no—
223#95—792no—
224#95—1584no—
225#95—1581no—
226C24—21no—
227C24—22no—
228C24—21no—
229C24—22no—
230C24—42no—
231C24—42no—
232C24(C2×D79)⋊C412yes—
233C24C2×C22⋊D7911yescharacteristic
234C2×Q8—3164no—
235C2×Q8—3168no—
236C2×Q8—3164no—
237C2×Q8—3164no—
238C2×Q8—3162no—
239C2×Q8—1588no—
240C2×Q8—1584no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—3162no—
247C22×C4—3161no—
248C22×C4—3161no—
249C22×C4—3162no—
250C22×C4—3162no—
251C22×C4—3162no—
252C22×C4—1582no—
253C22×C4—1584no—
254C22×C4—1582no—
255C22×C4(C2×D79)⋊C412yes—
256C22×C4C2×C4×D7911yescharacteristic
257C22×C4C2×C22⋊D7912yes—
258C22×C4C2×D31611yescharacteristic
259C22×C4#1192962412yes—
260C2×C8—3164no—
261C2×C8—3162no—
262C4⋊C4—44no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C22⋊C4—3168no—
270C42—82no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—84no—
286C23C2×(C2×D79)⋊C411yescharacteristic
287C23#1457913011yescharacteristic
288C23#1457914411yescharacteristic
289C23#1457918112yes—
290C23#1457919112yes—
291Q8—3168no—
292Q8—3164no—
293Q8—1588no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—81no—
305C2×C4—3162no—
306C2×C4—3164no—
307C2×C4—3162no—
308C2×C4—3164no—
309C2×C4—3162no—
310C2×C4—3162no—
311C2×C4—3162no—
312C2×C4—6324no—
313C2×C4—1584no—
314C2×C4C4×C22⋊D7912yes—
315C2×C4#1457913512yes—
316C8—6322no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—24no—
322C22—24no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#1909844211yescharacteristic
331C22#1909855711yescharacteristic
332C22#1909857111yescharacteristic
333C22#1909872512yes—
334C22#1909874112yes—
335C22#1909875012yes—
336C22#1909890412yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—84no—
341C4—3164no—
342C4—6322no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#2376238014yes—
348C2#2376371211yescharacteristic
349C2#2376378811yescharacteristic
350C2#2376380811yescharacteristic
subgroupG/H
centerC23#14579130
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2312C2
socleC23×C79#120
prime radicalC25×C79D4
cosocleC23×C4×C79C23
hypercenter#2312D79
hypocenterC79#9016
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #23763334C2semidirect
4. #23761278C2semidirect
11. #19098475C4semidirect
181. C79#9016semidirect

Maximal subgroups

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

subgroupindex
C2×#190984752
#237633342
C2×#190982872
#237612782
C79×#23122
#901679