Finite Group Databasebeta

← #34200858

Subgroups — #34200858

Order 20224.

362 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764624C211yesmaximal, characteristic
2#23762436C211yesmaximal, characteristic
3#23762368C211yesmaximal, characteristic
4#23761430C211yesmaximal, characteristic, complemented
5#23761427C211yesmaximal, characteristic, complemented
6#23761346C211yesmaximal, characteristic, complemented
7C79×#2843C211yesmaximal, characteristic, complemented
8#19099144C2211yescharacteristic
9#19098881—21no—
10#19098702—21no—
11C2×#14579089—21no—
12#19098562—21no—
13#19098332C2211yescharacteristic
14#19098296—21no—
15#19098296—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18#400×C79—21no—
19#400×C79C2211yescharacteristic
20#398×C79—21no—
21#398×C79—21no—
22#398×C79—21no—
23#395×C79—21no—
24#395×C79C2211yescharacteristic
25#384×C79C2211yescharacteristic
26#383×C79—21no—
27C2×#11929624—21no—
28C2×#11929611—41no—
29C2×#11929608—21no—
30#14579141—41no—
31#14579136—41no—
32#14579135—21no—
33#14579135—21no—
34#14579092—41no—
35#14579089—42no—
36#14579085—41no—
37#14579084—41no—
38#14579081—41no—
39#14579081—41no—
40C22⋊(C79⋊C8)—21no—
41C22⋊(C79⋊C8)—21no—
42C22×Q8×C79D411yescharacteristic
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C4⋊C4×C79C2311yescharacteristic
52C2×C42×C79D411yescharacteristic
53#95×C79—21no—
54#95×C79—21no—
55#95×C79—21no—
56#95×C79—21no—
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61#95×C79D411yescharacteristic
62#95×C79D411yescharacteristic
63#11929624—41no—
64#11929624—41no—
65#11929624—41no—
66#11929624—41no—
67C2×Dic158—21no—
68C2×D316—21no—
69C2×D316—21no—
70C2×D316—21no—
71C2×C22⋊D79—41no—
72C2×C4×D79—41no—
73#11929611—82no—
74#11929608—41no—
75#11929608—41no—
76C2×C79⋊C8—41no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8—41no—
79C4⋊Dic79—21no—
80C4⋊Dic79—21no—
81(C2×D79)⋊C4—41no—
82(C2×D79)⋊C4—41no—
83C2×Q8×C79—41no—
84C2×Q8×C79—81no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C4⋊C4×C79—41no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—81no—
98C4⋊C4×C79—81no—
99C4⋊C4×C79—81no—
100C4⋊C4×C79—81no—
101C4⋊C4×C79—81no—
102C4⋊C4×C79—81no—
103C42×C79—21no—
104C42×C79SD1612yescomplemented
105C23×C79#12011yescharacteristic
106C22×D79—41no—
107Q8×C79—42no—
108Q8×C79—81no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—22no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—42no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79C2×SD1611yescharacteristic
133C2×C4×C79#13711yescharacteristic
134C22⋊D79—81no—
135C22⋊D79—81no—
136C2×Dic79—41no—
137D316—41no—
138D316—41no—
139D316—81no—
140D316—81no—
141C4×D79—81no—
142C4×D79—81no—
143Dic158—41no—
144Dic158—41no—
145C79⋊C8—81no—
146C79⋊C8—81no—
147C79⋊C8—81no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79#44811yescharacteristic
151C22×C79#45411yescharacteristic
152C22×C79B(4,𝔽2)11yescharacteristic
153C2×D79—81no—
154C2×D79—81no—
155C2×D79—81no—
156C4×C79—21no—
157C4×C79—21no—
158C4×C79—42no—
159C4×C79—41no—
160C4×C79—41no—
161C4×C79—81no—
162C4×C79—81no—
163C4×C79—81no—
164C4×C79—81no—
165C4×C79—81no—
166Dic79—81no—
167#12183—791nomaximal, Sylow
168C2×C79—21no—
169C2×C79—21no—
170C2×C79#159911yescharacteristic
171C2×C79#220011yescharacteristic
172C2×C79#220311yescharacteristic
173D79—161no—
174#2843D7911yescharacteristic, complemented
175#2039—791no—
176#1616—791no—
177#1604—791no—
178#1354—791no—
179#1353—791no—
180#1325—791no—
181C79#1218311yescharacteristic, complemented, Sylow
182#537—791no—
183#481—1581no—
184#448—1581no—
185C2×#104—1581no—
186#417—1581no—
187#400—21no—
188#400C2×D7911yescharacteristic
189#398—21no—
190#398—21no—
191#398—21no—
192#395—21no—
193#395C2×D7911yescharacteristic
194#384C2×D7911yescharacteristic
195#383—21no—
196#343—791no—
197#327—1581no—
198#327—1581no—
199#324—791no—
200#324—791no—
201C2×#41—1581no—
202C22×Q8C22⋊D7911yescharacteristic
203C2×SD16—3161no—
204C2×M16—1581no—
205C22⋊Q8—3161no—
206#121—3161no—
207#121—1581no—
208#121—1581no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22⋊D7911yescharacteristic
216C2×C4⋊C4C22⋊D7911yescharacteristic
217C2×C4⋊C4C22×D7911yescharacteristic
218C2×C42C22⋊D7911yescharacteristic
219#106—3161no—
220#104—3162no—
221#103—3161no—
222#103—3161no—
223#102—3161no—
224#102—3161no—
225C22⋊C8—1581no—
226C22⋊C8—1581no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C22⋊D7911yescharacteristic
236#95C22⋊D7911yescharacteristic
237#41—3161no—
238#41—3161no—
239#41—3161no—
240#41—3161no—
241C2×Q8—41no—
242C2×Q8—81no—
243C2×Q8—1581no—
244C2×D4—3161no—
245C2×D4—1581no—
246C2×D4—1581no—
247C2×D4—1581no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—3161no—
255C22×C4C2×C22⋊D7911yescharacteristic
256C22×C4C2×C22⋊D7911yescharacteristic
257C22×C4C2×C22⋊D7911yescharacteristic
258SD16—6322no—
259M16—3161no—
260M16—3161no—
261C2×C8—3161no—
262C2×C8—3161no—
263C2×C8—3161no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—1581no—
274C4⋊C4—1581no—
275C22⋊C4—3161no—
276C22⋊C4—3161no—
277C42—21no—
278C42#1192961112yescomplemented
279C23—3161no—
280C23#1457913011yescharacteristic
281Q8—42no—
282Q8—81no—
283Q8—3161no—
284Q8—3161no—
285D4—3161no—
286D4—3161no—
287D4—6321no—
288D4—6321no—
289D4—6321no—
290D4—6321no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—3161no—
315C2×C4—6321no—
316C2×C4—6321no—
317C2×C4C2×#1192961111yescharacteristic
318C2×C4#1457919211yescharacteristic
319C8—6321no—
320C8—6321no—
321C8—6321no—
322C22—21no—
323C22—21no—
324C22—6321no—
325C22—6321no—
326C22—6321no—
327C22#1909870211yescharacteristic
328C22#1909873611yescharacteristic
329C22#1909875011yescharacteristic
330C4—21no—
331C4—21no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—6321no—
341C2—21no—
342C2—21no—
343C2—12641no—
344C2#2376234011yescharacteristic
345C2#2376540711yescharacteristic
346C2#2376542111yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2843C2
socleC22×C79B(4,𝔽2)
prime radical#19099144C22
hypercenter#2843D79
hypocenterC79#12183
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2843

Complements

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

NH ≅ G/Ntype
4. #23761430C2semidirect
5. #23761427C2semidirect
6. #23761346C2semidirect
7. C79×#2843C2semidirect
104. C42×C79SD16semidirect
174. #2843D79semidirect
181. C79#12183semidirect
278. C42#11929611semidirect

Maximal subgroups

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

subgroupindex
#237646242
#237624362
#237623682
#237614302
#237614272
#237613462
C79×#28432
#1218379