Finite Group Databasebeta

← #34186149

Subgroups — #34186149

Order 20224.

544 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 111 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#23771819C211yesmaximal, characteristic
2#23761847C212yesmaximal, complemented
3#23761734C211yesmaximal, characteristic
4#23761264C212yesmaximal, complemented
5C79×#2599C211yesmaximal, characteristic
6C22×#11929608—21no—
7#19098620C2212yes—
8#19098519—24no—
9C4×#11929608—21no—
10C4×#11929608C2212yes—
11Q8⋊(C79⋊C8)—24no—
12#19098289—21no—
13#19098287—21no—
14C2×C4×Q8×C79C2211yescharacteristic
15C2×#117×C79C2211yescharacteristic
16#398×C79C411yescharacteristic
17#397×C79C411yescharacteristic
18#395×C79C411yescharacteristic
19#393×C79C411yescharacteristic
20#388×C79—21no—
21#387×C79—21no—
22#387×C79—21no—
23#385×C79—21no—
24#383×C79C2211yescharacteristic
25C2×#11929608—22no—
26C2×#11929608—24no—
27C2×#11929608—24no—
28C2×#11929608—22no—
29C22×C79⋊C8—22no—
30C4⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32#14579072—22no—
33#14579072—22no—
34#14579072—22no—
35C4×C79⋊C8—22no—
36C4×C79⋊C8D414yes—
37C22×Q8×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C4×Q8×C79—22no—
40C4×Q8×C79—22no—
41#117×C79—22no—
42#117×C79—22no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79C2×C411yescharacteristic
47C2×C4⋊C4×C79C2×C411yescharacteristic
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C22⋊C4×C79—21no—
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C42×C79—21no—
54C2×C42×C79D411yescharacteristic
55C2×C42×C79C2311yescharacteristic
56#95×C79—21no—
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61#95×C79—21no—
62#95×C79C2×C411yescharacteristic
63#95×C79C2×C411yescharacteristic
64#11929608—28no—
65#11929608—24no—
66#11929608—24no—
67C2×C79⋊C8—28no—
68C2×C79⋊C8—24no—
69C2×C79⋊C8—42no—
70C24×C79C22⋊C411yescharacteristic
71C2×Q8×C79—22no—
72C2×Q8×C79—42no—
73C22×C4×C79—22no—
74C22×C4×C79—22no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C4⋊C4×C79—22no—
88C4⋊C4×C79—22no—
89C4⋊C4×C79—22no—
90C4⋊C4×C79—42no—
91C4⋊C4×C79—42no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—41no—
94C22⋊C4×C79—22no—
95C22⋊C4×C79—22no—
96C22⋊C4×C79—42no—
97C22⋊C4×C79—42no—
98C42×C79—21no—
99C42×C79—21no—
100C42×C79—42no—
101C42×C79C2×D412yes—
102C42×C79#4112yes—
103C23×C79—21no—
104C23×C79—21no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79#10412yes—
108C23×C79C2×C22⋊C411yescharacteristic
109Q8×C79—24no—
110Q8×C79—42no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—24no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—22no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
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—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79#10412yes—
137C2×C4×C79C4×D412yes—
138C2×C4×C79#12011yescharacteristic
139C2×C4×C79#12112yes—
140C2×C4×C79#12311yescharacteristic
141C79⋊C8—28no—
142C79⋊C8—82no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—22no—
146C22×C79—42no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—81no—
150C22×C79#38611yescharacteristic
151C22×C79C2×#10411yescharacteristic
152C22×C79C2×#10411yescharacteristic
153C4×C79—22no—
154C4×C79—22no—
155C4×C79—21no—
156C4×C79—21no—
157C4×C79—22no—
158C4×C79—41no—
159C4×C79—42no—
160C4×C79—41no—
161C4×C79—81no—
162C4×C79—81no—
163C4×C79—81no—
164#9491—791nomaximal, Sylow
165C2×C79—21no—
166C2×C79—21no—
167C2×C79—42no—
168C2×C79#189111yescharacteristic
169C2×C79#189611yescharacteristic
170C2×C79#190911yescharacteristic
171#2873—791no—
172#2599D7911yescharacteristic
173#1491—792no—
174#1464—791no—
175#1288—792no—
176C79#949111yescharacteristic, complemented, Sylow
177C22×M16—1581no—
178C2×C4×Q8C2×D7911yescharacteristic
179C2×#117C2×D7911yescharacteristic
180#432—792no—
181#405—1584no—
182C4×M16—792no—
183C4×M16—1581no—
184#398Dic7911yescharacteristic
185#397Dic7911yescharacteristic
186#395Dic7911yescharacteristic
187#393Dic7911yescharacteristic
188#388—21no—
189#387—21no—
190#387—21no—
191#385—21no—
192#383C2×D7911yescharacteristic
193Q8⋊C8—1584no—
194#324—1581no—
195#323—1581no—
196C22×Q8C2×Dic7911yescharacteristic
197C23×C4C22⋊D7911yescharacteristic
198C2×M16—1582no—
199C2×M16—1584no—
200C2×M16—1584no—
201C2×M16—1582no—
202C22×C8—1582no—
203C4×Q8—22no—
204C4×Q8—22no—
205#117—22no—
206#117—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C2×Dic7911yescharacteristic
211C2×C4⋊C4C2×Dic7911yescharacteristic
212C2×C4⋊C4C22⋊D7911yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4C2×Dic7911yescharacteristic
216C2×C22⋊C4C22⋊D7911yescharacteristic
217C2×C42—21no—
218C2×C42C22⋊D7911yescharacteristic
219C2×C42C22×D7911yescharacteristic
220C4⋊C8—1582no—
221C22⋊C8—1582no—
222#97—1582no—
223#97—1582no—
224#97—1582no—
225C4×C8—794no—
226C4×C8—1582no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95C2×Dic7911yescharacteristic
234#95C2×Dic7911yescharacteristic
235C24C22⋊Dic7911yescharacteristic
236C2×Q8—22no—
237C2×Q8—42no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4C22⋊Dic7911yescharacteristic
247C22×C4C22⋊Dic7911yescharacteristic
248C22×C4C22⋊Dic7911yescharacteristic
249C22×C4C22×Dic7911yescharacteristic
250C22×C4C2×C22⋊D7911yescharacteristic
251C22×C4C2×C22⋊D7911yescharacteristic
252M16—1588no—
253M16—1584no—
254M16—1584no—
255C2×C8—3162no—
256C2×C8—1588no—
257C2×C8—1584no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C42—21no—
270C42—21no—
271C42—42no—
272C42D4×D7912yes—
273C42#1192962312yes—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23#1457908812yes—
279C23C2×C22⋊Dic7911yescharacteristic
280Q8—24no—
281Q8—42no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
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—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4#1457908812yes—
308C2×C4D4×Dic7912yes—
309C2×C4#1457912911yescharacteristic
310C2×C4#1457913312yes—
311C2×C4#1457914211yescharacteristic
312C8—6322no—
313C8—1588no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1909844311yescharacteristic
322C22C2×#1457908811yescharacteristic
323C22C2×#1457908811yescharacteristic
324C4—22no—
325C4—21no—
326C4—22no—
327C4—21no—
328C4—22no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2376381911yescharacteristic
339C2#2376384411yescharacteristic
340C2#2376392311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2599C2
socleC22×C79#386
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#2599D79
hypocenterC79#9491
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2599

Complements

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

NH ≅ G/Ntype
2. #23761847C2semidirect
4. #23761264C2semidirect
176. C79#9491semidirect

Maximal subgroups

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

subgroupindex
#237718192
#237618472
#237617342
#237612642
C79×#25992
#949179