Finite Group Databasebeta

← #34198775

Subgroups — #34198775

Order 20224.

428 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 91 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#23763139C211yesmaximal, characteristic
2#23761734C211yesmaximal, characteristic
3#23761430C212yesmaximal, complemented
4#23761406C212yesmaximal, complemented
5C79×#2599C211yesmaximal, characteristic
6C22×#11929608—21no—
7#19098620—21no—
8#19098332—21no—
9#19098332C2212yes—
10#19098289C2212yes—
11#19098287—21no—
12C2×C4×Q8×C79C411yescharacteristic
13C2×#117×C79C2211yescharacteristic
14#398×C79C2211yescharacteristic
15#397×C79C411yescharacteristic
16#395×C79C2211yescharacteristic
17#393×C79C411yescharacteristic
18#388×C79—21no—
19#387×C79—21no—
20#387×C79—21no—
21#385×C79—21no—
22#383×C79C411yescharacteristic
23C2×#11929608—24no—
24C2×#11929608—22no—
25C2×#11929608—41no—
26C2×#11929608—42no—
27C22×C79⋊C8—41no—
28C4⋊(C79⋊C8)—41no—
29C22⋊(C79⋊C8)—22no—
30#14579072—41no—
31C22×Q8×C79C2×C411yescharacteristic
32C23×C4×C79D411yescharacteristic
33C4×Q8×C79—41no—
34C4×Q8×C79—41no—
35#117×C79—41no—
36#117×C79—41no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79C2×C411yescharacteristic
41C2×C4⋊C4×C79D411yescharacteristic
42C2×C4⋊C4×C79C2311yescharacteristic
43C2×C22⋊C4×C79—21no—
44C2×C22⋊C4×C79—21no—
45C2×C22⋊C4×C79C2×C411yescharacteristic
46C2×C22⋊C4×C79D411yescharacteristic
47C2×C42×C79—21no—
48C2×C42×C79C2×C411yescharacteristic
49C2×C42×C79D411yescharacteristic
50#95×C79—21no—
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
54#95×C79—21no—
55#95×C79—21no—
56#95×C79C2×C411yescharacteristic
57#95×C79C2×C411yescharacteristic
58#11929608—24no—
59#11929608—44no—
60#11929608—42no—
61C2×C79⋊C8—44no—
62C2×C79⋊C8—42no—
63C2×C79⋊C8—42no—
64C24×C79C22⋊C411yescharacteristic
65C2×Q8×C79—42no—
66C2×Q8×C79—41no—
67C22×C4×C79—22no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22×C411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C4⋊C4×C79—41no—
83C4⋊C4×C79—41no—
84C4⋊C4×C79—42no—
85C4⋊C4×C79—41no—
86C4⋊C4×C79—41no—
87C4⋊C4×C79—41no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—41no—
90C22⋊C4×C79—41no—
91C22⋊C4×C79—41no—
92C22⋊C4×C79—42no—
93C22⋊C4×C79—42no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79—42no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79#10412yes—
104C23×C79C2×C22⋊C411yescharacteristic
105Q8×C79—42no—
106Q8×C79—42no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—22no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
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—41no—
134C2×C4×C79—41no—
135C2×C4×C79#9911yescharacteristic
136C2×C4×C79#9911yescharacteristic
137C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C79#13711yescharacteristic
139C79⋊C8—44no—
140C79⋊C8—82no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—42no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—81no—
148C22×C79C2×#9911yescharacteristic
149C22×C79#41711yescharacteristic
150C22×C79C2×#10411yescharacteristic
151C4×C79—21no—
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—42no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79—41no—
159C4×C79—41no—
160C4×C79—81no—
161C4×C79—81no—
162C4×C79—81no—
163#11763—791nomaximal, Sylow
164C2×C79—21no—
165C2×C79—21no—
166C2×C79—42no—
167C2×C79#150411yescharacteristic
168C2×C79#212011yescharacteristic
169C2×C79#213511yescharacteristic
170#2599D7911yescharacteristic
171#1742—791no—
172#1464—791no—
173#1354—792no—
174#1346—792no—
175C79#1176311yescharacteristic, complemented, Sylow
176C22×M16—1581no—
177C2×C4×Q8Dic7911yescharacteristic
178C2×#117C2×D7911yescharacteristic
179#432—1581no—
180#398C2×D7911yescharacteristic
181#397Dic7911yescharacteristic
182#395C2×D7911yescharacteristic
183#393Dic7911yescharacteristic
184#388—21no—
185#387—21no—
186#387—21no—
187#385—21no—
188#383Dic7911yescharacteristic
189#343—792no—
190#343—1581no—
191#324—792no—
192#323—1581no—
193C22×Q8C2×Dic7911yescharacteristic
194C23×C4C22⋊D7911yescharacteristic
195C2×M16—3161no—
196C2×M16—3162no—
197C2×M16—1584no—
198C2×M16—1582no—
199C22×C8—3161no—
200C4×Q8—41no—
201C4×Q8—41no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2×Dic7911yescharacteristic
208C2×C4⋊C4C22⋊D7911yescharacteristic
209C2×C4⋊C4C22×D7911yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4C2×Dic7911yescharacteristic
213C2×C22⋊C4C22⋊D7911yescharacteristic
214C2×C42—21no—
215C2×C42C2×Dic7911yescharacteristic
216C2×C42C22⋊D7911yescharacteristic
217C4⋊C8—3161no—
218C22⋊C8—1582no—
219#97—3161no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95C2×Dic7911yescharacteristic
227#95C2×Dic7911yescharacteristic
228C24C22⋊Dic7911yescharacteristic
229C2×Q8—42no—
230C2×Q8—41no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4C22⋊Dic7911yescharacteristic
241C22×C4C22⋊Dic7911yescharacteristic
242C22×C4C22⋊Dic7911yescharacteristic
243C22×C4C22×Dic7911yescharacteristic
244C22×C4C2×C22⋊D7911yescharacteristic
245C22×C4C2×C22⋊D7911yescharacteristic
246M16—3164no—
247M16—3162no—
248M16—1584no—
249C2×C8—3164no—
250C2×C8—3162no—
251C2×C8—3162no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#1457908812yes—
274C23C2×C22⋊Dic7911yescharacteristic
275Q8—42no—
276Q8—42no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
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—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4#1457907611yescharacteristic
306C2×C4#1457907611yescharacteristic
307C2×C4#1457918611yescharacteristic
308C2×C4#1457919111yescharacteristic
309C8—3164no—
310C8—6322no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1457907611yescharacteristic
319C22#1909856411yescharacteristic
320C22C2×#1457908811yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2376190411yescharacteristic
337C2#2376504111yescharacteristic
338C2#2376509511yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2599C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2599D79
hypocenterC79#11763
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
3. #23761430C2semidirect
4. #23761406C2semidirect
175. C79#11763semidirect

Maximal subgroups

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

subgroupindex
#237631392
#237617342
#237614302
#237614062
C79×#25992
#1176379