Finite Group Databasebeta

← #34201436

Subgroups — #34201436

Order 20224.

364 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 59 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#23764561C211yesmaximal, characteristic
2#23762407C211yesmaximal, characteristic
3#23762373C211yesmaximal, characteristic
4#23761430C211yesmaximal, characteristic
5#23761406C211yesmaximal, characteristic
6#23761334C211yesmaximal, characteristic
7C79×#2586C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098893—21no—
10#19098719—21no—
11#19098572—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18#398×C79C2211yescharacteristic
19#395×C79—21no—
20#395×C79C2211yescharacteristic
21#391×C79—21no—
22#391×C79C2211yescharacteristic
23#389×C79—21no—
24#387×C79—21no—
25#387×C79—21no—
26C4×C4⋊C4×C79—21no—
27C2×#11929614—41no—
28C2×#11929608—21no—
29#14579160—41no—
30#14579152—41no—
31#14579140—21no—
32#14579140—21no—
33#14579120—21no—
34#14579092—41no—
35#14579085—41no—
36#14579085—41no—
37#14579085—41no—
38#14579084—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22×Q8×C79D411yescharacteristic
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×C79D411yescharacteristic
49C2×C4⋊C4×C79C2311yescharacteristic
50C2×C42×C79—21no—
51C2×C42×C79—21no—
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×C79D411yescharacteristic
61#95×C79D411yescharacteristic
62C2×Dic158—21no—
63C2×Dic158—21no—
64#11929614—82no—
65#11929608—41no—
66#11929608—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—41no—
70C4⋊Dic79—21no—
71C4⋊Dic79—21no—
72C4⋊Dic79—21no—
73C4⋊Dic79—21no—
74Dic79⋊C4—41no—
75Dic79⋊C4—41no—
76C22⋊Dic79—41no—
77C4×Dic79—41no—
78C2×Q8×C79—21no—
79C2×Q8×C79—21no—
80C2×Q8×C79—42no—
81C2×Q8×C79—41no—
82C2×Q8×C79—41no—
83C2×Q8×C79—41no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C4⋊C4×C79—24no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—42no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—81no—
100C4⋊C4×C79—81no—
101C42×C79—42no—
102C42×C79—41no—
103C42×C79—81no—
104C23×C79#12011yescharacteristic
105Q8×C79—42no—
106Q8×C79—42no—
107Q8×C79—81no—
108Q8×C79—81no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—22no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
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—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79#13511yescharacteristic
134C2×C4×C79#13711yescharacteristic
135C2×Dic79—41no—
136C2×Dic79—41no—
137Dic158—81no—
138Dic158—81no—
139C79⋊C8—81no—
140C79⋊C8—81no—
141C79⋊C8—81no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79#45211yescharacteristic
145C22×C79#45511yescharacteristic
146C22×C79B(4,𝔽2)11yescharacteristic
147C4×C79—21no—
148C4×C79—21no—
149C4×C79—42no—
150C4×C79—42no—
151C4×C79—42no—
152C4×C79—42no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79—81no—
156C4×C79—81no—
157Dic79—81no—
158Dic79—81no—
159#12273—791nomaximal, Sylow
160C2×C79—21no—
161C2×C79—21no—
162C2×C79#160611yescharacteristic
163C2×C79#219711yescharacteristic
164C2×C79#220511yescharacteristic
165#2586D7911yescharacteristic
166#2028—791no—
167#1611—791no—
168#1605—791no—
169#1354—791no—
170#1346—791no—
171#1319—791no—
172C79#1227311yescharacteristic, complemented, Sylow
173#544—791no—
174#483—1581no—
175#451—1581no—
176#419—1581no—
177#398C2×D7911yescharacteristic
178#395—21no—
179#395C2×D7911yescharacteristic
180#391—21no—
181#391C2×D7911yescharacteristic
182#389—21no—
183#387—21no—
184#387—21no—
185C4×C4⋊C4—21no—
186#343—791no—
187#339—1581no—
188#328—1581no—
189#328—1581no—
190#324—791no—
191#324—791no—
192C22×Q8C22⋊D7911yescharacteristic
193C22×Q8C22⋊D7911yescharacteristic
194C2×Q16—3161no—
195C2×M16—1581no—
196C4⋊Q8—3161no—
197#125—3161no—
198C22⋊Q8—1581no—
199C22⋊Q8—1581no—
200#117—1581no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C22⋊D7911yescharacteristic
207C2×C4⋊C4C22×D7911yescharacteristic
208C2×C42—21no—
209C2×C42—21no—
210C2×C42C22⋊D7911yescharacteristic
211#106—3161no—
212#103—3161no—
213#103—3161no—
214#103—3161no—
215#103—3161no—
216C22⋊C8—1581no—
217C22⋊C8—1581no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95C22⋊D7911yescharacteristic
226#95C22⋊D7911yescharacteristic
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—1581no—
234C2×Q8—1581no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4C2×C22⋊D7911yescharacteristic
242C22×C4C2×C22⋊D7911yescharacteristic
243C22×C4C2×C22⋊D7911yescharacteristic
244Q16—6322no—
245M16—3161no—
246M16—3161no—
247C2×C8—3161no—
248C2×C8—3161no—
249C2×C8—3161no—
250C4⋊C4—24no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—3161no—
259C4⋊C4—3161no—
260C4⋊C4—1581no—
261C4⋊C4—1581no—
262C4⋊C4—1581no—
263C4⋊C4—1581no—
264C22⋊C4—3161no—
265C42—41no—
266C42—42no—
267C42—81no—
268C42—3161no—
269C23#1457913011yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—81no—
273Q8—81no—
274Q8—6321no—
275Q8—6321no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4#1457918111yescharacteristic
303C2×C4#1457919111yescharacteristic
304C8—6321no—
305C8—6321no—
306C8—6321no—
307C22—21no—
308C22—21no—
309C22#1909872511yescharacteristic
310C22#1909874111yescharacteristic
311C22#1909875011yescharacteristic
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—6321no—
323C4—6321no—
324C2—21no—
325C2—21no—
326C2#2376238011yescharacteristic
327C2#2376539011yescharacteristic
328C2#2376543011yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2586C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2586D79
hypocenterC79#12273
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2586

Complements

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

NH ≅ G/Ntype
172. C79#12273semidirect

Maximal subgroups

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

subgroupindex
#237645612
#237624072
#237623732
#237614302
#237614062
#237613342
C79×#25862
#1227379