Finite Group Databasebeta

← #34201273

Subgroups — #34201273

Order 20224.

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

normal subgroups only

This group has 428 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#23764750C211yesmaximal, characteristic, complemented
2#23762407C211yesmaximal, characteristic, complemented
3#23762339C211yesmaximal, characteristic
4#23761406C211yesmaximal, characteristic, complemented
5#23761403C211yesmaximal, characteristic
6#23761344C211yesmaximal, characteristic, complemented
7C79×#2609C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098893—21no—
10#19098714—21no—
11#19098603—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18C2×#117×C79—21no—
19#398×C79C2211yescharacteristic
20#396×C79—21no—
21#394×C79C2211yescharacteristic
22#388×C79—21no—
23#387×C79—21no—
24#386×C79—21no—
25#385×C79—21no—
26#382×C79C2211yescharacteristic
27C2×#11929612—41no—
28C2×#11929608—21no—
29#14579160—41no—
30#14579152—41no—
31#14579140—21no—
32#14579140—21no—
33#14579120—21no—
34#14579094—41no—
35#14579092—41no—
36#14579092—41no—
37#14579084—41no—
38#14579082—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22×D4×C79D411yescharacteristic
42C23×C4×C79—21no—
43#117×C79—42no—
44#117×C79—42no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79—21no—
56C2×C42×C79—21no—
57C2×C42×C79D411yescharacteristic
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61#95×C79—21no—
62#95×C79D411yescharacteristic
63#95×C79D411yescharacteristic
64C2×Dic158—21no—
65C2×Dic158—21no—
66#11929612—82no—
67#11929608—22no—
68#11929608—41no—
69C2×C79⋊C8—41no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C4⋊Dic79—21no—
73C4⋊Dic79—21no—
74C4⋊Dic79—21no—
75C4⋊Dic79—21no—
76Dic79⋊C4—41no—
77Dic79⋊C4—41no—
78C22⋊Dic79—41no—
79C4×Dic79—41no—
80C24×C79—21no—
81C2×D4×C79—41no—
82C2×D4×C79—42no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—42no—
89C22×C4×C79—42no—
90C22×C4×C79—42no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C4⋊C4×C79—41no—
95C4⋊C4×C79—42no—
96C4⋊C4×C79—42no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79—41no—
101C22⋊C4×C79—42no—
102C22⋊C4×C79—42no—
103C22⋊C4×C79—42no—
104C22⋊C4×C79—41no—
105C22⋊C4×C79—41no—
106C22⋊C4×C79—81no—
107C42×C79—21no—
108C42×C79—42no—
109C42×C79SD1612yescomplemented
110C23×C79—22no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79#12011yescharacteristic
118D4×C79—42no—
119D4×C79—81no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—22no—
123C2×C4×C79—22no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—48no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79C2×SD1611yescharacteristic
143C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
144C2×Dic79—41no—
145C2×Dic79—41no—
146Dic158—81no—
147Dic158—81no—
148C79⋊C8—42no—
149C79⋊C8—81no—
150C79⋊C8—81no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—42no—
154C22×C79—42no—
155C22×C79—42no—
156C22×C79—81no—
157C22×C79—81no—
158C22×C79—81no—
159C22×C79—81no—
160C22×C79#45011yescharacteristic
161C22×C79#45611yescharacteristic
162C22×C79B(4,𝔽2)11yescharacteristic
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—42no—
166C4×C79—42no—
167C4×C79—41no—
168C4×C79—41no—
169C4×C79—81no—
170C4×C79—81no—
171C4×C79—81no—
172Dic79—81no—
173Dic79—81no—
174#12248—791nomaximal, Sylow
175C2×C79—21no—
176C2×C79—21no—
177C2×C79—82no—
178C2×C79#160411yescharacteristic
179C2×C79#219711yescharacteristic
180C2×C79#220411yescharacteristic
181#2609D7911yescharacteristic
182#2061—791no—
183#1611—791no—
184#1599—791no—
185#1346—791no—
186#1345—791no—
187#1324—791no—
188C79#1224811yescharacteristic, complemented, Sylow
189#544—791no—
190C2×#117—21no—
191#483—1581no—
192#450—1581no—
193#428—1581no—
194#398C2×D7911yescharacteristic
195#396—21no—
196#394C2×D7911yescharacteristic
197#388—21no—
198#387—21no—
199#386—21no—
200#385—21no—
201#382C2×D7911yescharacteristic
202#343—791no—
203#339—1581no—
204#328—1581no—
205#328—1581no—
206#324—791no—
207#324—791no—
208C22×D4C22⋊D7911yescharacteristic
209C23×C4—21no—
210C2×SD16—3161no—
211C2×M16—1581no—
212C4⋊Q8—3161no—
213#125—3161no—
214C22⋊Q8—1581no—
215C22⋊Q8—1581no—
216#117—42no—
217#117—42no—
218#117—1581no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C22⋊D7911yescharacteristic
223C2×C4⋊C4C22⋊D7911yescharacteristic
224C2×C4⋊C4C22×D7911yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42C22⋊D7911yescharacteristic
232#107—3161no—
233#106—3161no—
234#106—3161no—
235#103—3161no—
236#102—3161no—
237C22⋊C8—1581no—
238C22⋊C8—1581no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95C22⋊D7911yescharacteristic
244#95C22⋊D7911yescharacteristic
245C24—21no—
246C2×Q8—1581no—
247C2×Q8—1581no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×C22⋊D7911yescharacteristic
259C22×C4C2×C22⋊D7911yescharacteristic
260C22×C4C2×C22⋊D7911yescharacteristic
261SD16—6322no—
262M16—3161no—
263M16—1582no—
264C2×C8—3161no—
265C2×C8—3161no—
266C2×C8—3161no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—3161no—
275C4⋊C4—3161no—
276C4⋊C4—1581no—
277C4⋊C4—1581no—
278C4⋊C4—1581no—
279C4⋊C4—1581no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—81no—
286C22⋊C4—3161no—
287C42—21no—
288C42—42no—
289C42—3161no—
290C42#1192961212yescomplemented
291C23—22no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#1457913011yescharacteristic
299Q8—6321no—
300Q8—6321no—
301D4—42no—
302D4—81no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—48no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4C2×#1192961211yescharacteristic
328C2×C4#1457918711yescharacteristic
329C8—3162no—
330C8—6321no—
331C8—6321no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1909871411yescharacteristic
342C22#1909874611yescharacteristic
343C22#1909875011yescharacteristic
344C4—21no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—6321no—
354C4—6321no—
355C2—21no—
356C2—21no—
357C2—82no—
358C2#2376236911yescharacteristic
359C2#2376539011yescharacteristic
360C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2609C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
hypercenter#2609D79
hypocenterC79#12248
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2609

Complements

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

NH ≅ G/Ntype
1. #23764750C2semidirect
2. #23762407C2semidirect
4. #23761406C2semidirect
6. #23761344C2semidirect
109. C42×C79SD16semidirect
188. C79#12248semidirect
290. C42#11929612semidirect

Maximal subgroups

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

subgroupindex
#237647502
#237624072
#237623392
#237614062
#237614032
#237613442
C79×#26092
#1224879