Finite Group Databasebeta

← #34191450

Subgroups — #34191450

Order 20224.

360 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 75 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#23769049C211yesmaximal, characteristic
2#23764345C211yesmaximal, characteristic
3#23761912C211yesmaximal, characteristic
4#23761292C211yesmaximal, characteristic
5#23761264C211yesmaximal, characteristic
6#23761255C211yesmaximal, characteristic
7C79×#1515C211yesmaximal, characteristic
8#19098620C2211yescharacteristic
9C4×#11929608C2211yescharacteristic
10#19098505—21no—
11#19098484—21no—
12#19098467—21no—
13#19098467—21no—
14#19098450—21no—
15#19098446—21no—
16C4×C4⋊Dic79C2211yescharacteristic
17#19098409C2211yescharacteristic
18#19098322—21no—
19#19098322—21no—
20#19098300—21no—
21#19098300—21no—
22#533×C79C2211yescharacteristic
23#432×C79C2211yescharacteristic
24#383×C79C2211yescharacteristic
25#328×C79—21no—
26#328×C79—21no—
27C22×Dic158—21no—
28C2×#11929608—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C4×Dic79—21no—
34C2×C4×Dic79D411yescharacteristic
35C2×C4×Dic79D411yescharacteristic
36C4⋊(C79⋊C8)—21no—
37C22⋊(C79⋊C8)—21no—
38#14579072—21no—
39#14579072—21no—
40C4×C79⋊C8—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95D411yescharacteristic
48C79⋊#95D411yescharacteristic
49C4⋊Q8×C79Q811yescharacteristic, complemented
50C4⋊Q8×C79Q811yescharacteristic, complemented
51#125×C79—21no—
52C22⋊Q8×C79—21no—
53C22⋊Q8×C79—21no—
54C4×Q8×C79—21no—
55C4×Q8×C79—21no—
56#117×C79—21no—
57C2×C42×C79C2311yescharacteristic
58C4⋊C8×C79—21no—
59C22⋊C8×C79—21no—
60#97×C79—21no—
61#95×C79—21no—
62#95×C79—21no—
63#95×C79D411yescharacteristic
64#95×C79D411yescharacteristic
65C2×Dic158—42no—
66C2×Dic158—42no—
67C2×Dic158—81no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70C22×Dic79—21no—
71C22×Dic79—21no—
72#11929608—22no—
73#11929608—41no—
74C2×C79⋊C8—41no—
75C2×C79⋊C8—41no—
76C4⋊Dic79—42no—
77C4⋊Dic79—42no—
78Dic79⋊C4—42no—
79Dic79⋊C4—81no—
80C4×Dic79—41no—
81C4×Dic79—41no—
82C4×Dic79—42no—
83C2×Q8×C79—21no—
84C2×Q8×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C2×C8×C79—41no—
91C4⋊C4×C79—21no—
92C4⋊C4×C79—21no—
93C4⋊C4×C79—21no—
94C4⋊C4×C79—21no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C22⋊C4×C79—41no—
98C42×C79—41no—
99C42×C79C2×Q811yescharacteristic
100C42×C79#4111yescharacteristic
101C42×C79#4111yescharacteristic
102C42×C79#4111yescharacteristic
103C23×C79#12011yescharacteristic
104Q8×C79—81no—
105Q8×C79—81no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79C22⋊Q811yescharacteristic
119C2×C4×C79C22⋊Q811yescharacteristic
120C2×C4×C79C22⋊Q811yescharacteristic
121C2×C4×C79#12411yescharacteristic
122C2×C4×C79#12411yescharacteristic
123C2×C4×C79#12411yescharacteristic
124C2×Dic79—22no—
125C2×Dic79—22no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136Dic158—84no—
137C8×C79—81no—
138C79⋊C8—42no—
139C79⋊C8—81no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79#39311yescharacteristic
143C22×C79#45511yescharacteristic
144C22×C79#45611yescharacteristic
145C4×C79—21no—
146C4×C79—21no—
147C4×C79—21no—
148C4×C79—21no—
149C4×C79—21no—
150C4×C79—21no—
151C4×C79—81no—
152C4×C79—81no—
153C4×C79—81no—
154C4×C79—81no—
155Dic79—42no—
156Dic79—42no—
157Dic79—81no—
158Dic79—81no—
159#10379—791nomaximal, Sylow
160C2×C79—21no—
161C2×C79—21no—
162C2×C79#202811yescharacteristic
163C2×C79#203111yescharacteristic
164C2×C79#203111yescharacteristic
165#2605—791no—
166#1986—791no—
167#1515D7911yescharacteristic
168#1506—791no—
169#1300—791no—
170#1288—791no—
171#1285—791no—
172C79#1037911yescharacteristic, complemented, Sylow
173#533C2×D7911yescharacteristic
174#432—791no—
175#432C2×D7911yescharacteristic
176C4×M16—791no—
177#400—1581no—
178#395—1581no—
179#391—1581no—
180#391—1581no—
181#387—1581no—
182#387—1581no—
183#383C2×D7911yescharacteristic
184C4×C4⋊C4—791no—
185#376—791no—
186#339—1581no—
187#339—1581no—
188#328—21no—
189#328—21no—
190#328—1581no—
191#328—1581no—
192C22×Q8—1581no—
193C2×M16—1581no—
194C4⋊Q8Dic15811yescharacteristic, complemented
195C4⋊Q8Dic15811yescharacteristic, complemented
196#125—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201#117—21no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C42—791no—
207C2×C42—791no—
208C2×C42—1581no—
209C2×C42C22×D7911yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—1581no—
212C22⋊C8—21no—
213C22⋊C8—1581no—
214#97—21no—
215#97—1581no—
216#97—1581no—
217C4×C8—1581no—
218#95—21no—
219#95—21no—
220#95—791no—
221#95—791no—
222#95—1581no—
223#95—1581no—
224#95—1581no—
225#95—1581no—
226#95—1581no—
227#95—1581no—
228#95C22⋊D7911yescharacteristic
229#95C22⋊D7911yescharacteristic
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—3162no—
233C2×Q8—3162no—
234C2×Q8—6321no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—1581no—
238C22×C4—1581no—
239C22×C4—1581no—
240C22×C4—1581no—
241C22×C4D4×D7911yescharacteristic
242C22×C4D4×D7911yescharacteristic
243C22×C4C2×C22⋊D7911yescharacteristic
244M16—3161no—
245M16—1582no—
246C2×C8—41no—
247C2×C8—3161no—
248C2×C8—3161no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—3162no—
256C4⋊C4—3162no—
257C4⋊C4—3162no—
258C4⋊C4—6321no—
259C22⋊C4—41no—
260C42—41no—
261C42—3161no—
262C42—3161no—
263C42—3162no—
264C42C2×Dic15811yescharacteristic
265C42#1192962311yescharacteristic
266C42#1192962311yescharacteristic
267C42#1192962411yescharacteristic
268C23#1457912911yescharacteristic
269Q8—81no—
270Q8—81no—
271Q8—6324no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—1582no—
295C2×C4—1582no—
296C2×C4#1457913811yescharacteristic
297C2×C4#1457913811yescharacteristic
298C2×C4#1457914011yescharacteristic
299C2×C4#1457914711yescharacteristic
300C2×C4#1457914711yescharacteristic
301C2×C4#1457914811yescharacteristic
302C8—81no—
303C8—3162no—
304C8—6321no—
305C22—21no—
306C22—21no—
307C22#1909847511yescharacteristic
308C22#1909873911yescharacteristic
309C22#1909874411yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—3162no—
321C4—3162no—
322C4—6321no—
323C4—6321no—
324C2—21no—
325C2—21no—
326C2#2376455911yescharacteristic
327C2#2376457011yescharacteristic
328C2#2376457411yescharacteristic
subgroupG/H
centerC22#19098475
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1515C2
socleC22×C79#393
prime radicalC23×C79#120
hypercenter#1515D79
hypocenterC79#10379
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
49. C4⋊Q8×C79Q8semidirect
50. C4⋊Q8×C79Q8semidirect
172. C79#10379semidirect
194. C4⋊Q8Dic158semidirect
195. C4⋊Q8Dic158semidirect

Maximal subgroups

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

subgroupindex
#237690492
#237643452
#237619122
#237612922
#237612642
#237612552
C79×#15152
#1037979