Finite Group Databasebeta

← #34199017

Subgroups — #34199017

Order 20224.

386 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 87 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#23763480C211yesmaximal, characteristic
2#23761730C211yesmaximal, characteristic
3#23761430C212yesmaximal, complemented
4#23761427C212yesmaximal, complemented
5C79×#2627C211yesmaximal, characteristic
6#19098527—21no—
7#19098527—21no—
8#19098332—21no—
9#19098332C2212yes—
10#19098289—21no—
11#19098289C2212yes—
12C2×C22⋊Q8×C79C411yescharacteristic
13C22×C4⋊C4×C79C2211yescharacteristic
14#400×C79C411yescharacteristic
15#400×C79C2211yescharacteristic
16#398×C79—21no—
17#398×C79—21no—
18#395×C79C411yescharacteristic
19#395×C79C2211yescharacteristic
20#385×C79—21no—
21#381×C79—21no—
22#380×C79C411yescharacteristic
23C2×#11929608—22no—
24C2×#11929608—41no—
25C22⋊(C79⋊C8)—22no—
26C22⋊(C79⋊C8)—41no—
27C22⋊(C79⋊C8)—41no—
28C22⋊(C79⋊C8)—41no—
29C22×Q8×C79C2×C411yescharacteristic
30C23×C4×C79C2×C411yescharacteristic
31C23×C4×C79D411yescharacteristic
32C23×C4×C79D411yescharacteristic
33C22⋊Q8×C79—41no—
34C22⋊Q8×C79—41no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79—41no—
40C2×C4⋊C4×C79—41no—
41C2×C4⋊C4×C79C2×C411yescharacteristic
42C2×C4⋊C4×C79C2×C411yescharacteristic
43C2×C4⋊C4×C79D411yescharacteristic
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C4⋊C4×C79C2311yescharacteristic
46C2×C22⋊C4×C79—21no—
47C2×C22⋊C4×C79—21no—
48#95×C79—21no—
49#95×C79—21no—
50#95×C79—21no—
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
54#95×C79C2×C411yescharacteristic
55#95×C79C2×C411yescharacteristic
56#11929608—42no—
57#11929608—42no—
58#11929608—82no—
59C2×C79⋊C8—42no—
60C2×C79⋊C8—42no—
61C24×C79C22⋊C411yescharacteristic
62C2×Q8×C79—41no—
63C2×Q8×C79—81no—
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—22no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—41no—
76C22×C4×C79—81no—
77C22×C4×C79—81no—
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22×C411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C4⋊C4×C79—41no—
85C4⋊C4×C79—41no—
86C4⋊C4×C79—41no—
87C4⋊C4×C79—41no—
88C4⋊C4×C79—42no—
89C4⋊C4×C79—41no—
90C4⋊C4×C79—81no—
91C4⋊C4×C79—81no—
92C4⋊C4×C79—81no—
93C22⋊C4×C79—42no—
94C22⋊C4×C79—41no—
95C22⋊C4×C79—41no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79#9911yescharacteristic
101C23×C79#9911yescharacteristic
102C23×C79C2×C22⋊C411yescharacteristic
103Q8×C79—82no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—41no—
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—42no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—81no—
132C2×C4×C79—81no—
133C2×C4×C79—81no—
134C2×C4×C79#13712yes—
135C79⋊C8—82no—
136C79⋊C8—82no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—42no—
146C22×C79—42no—
147C22×C79C2×#9911yescharacteristic
148C22×C79#41911yescharacteristic
149C22×C79#42111yescharacteristic
150C4×C79—22no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79—81no—
157C4×C79—81no—
158C4×C79—81no—
159C4×C79—81no—
160#11817—791nomaximal, Sylow
161C2×C79—21no—
162C2×C79—21no—
163C2×C79—41no—
164C2×C79—41no—
165C2×C79#151111yescharacteristic
166C2×C79#212111yescharacteristic
167C2×C79#213311yescharacteristic
168#2627D7911yescharacteristic
169#1821—791no—
170#1463—791no—
171#1354—792no—
172#1353—792no—
173C79#1181711yescharacteristic, complemented, Sylow
174C2×C22⋊Q8Dic7911yescharacteristic
175C22×C4⋊C4C2×D7911yescharacteristic
176#407—1581no—
177#407—1581no—
178#400Dic7911yescharacteristic
179#400C2×D7911yescharacteristic
180#398—21no—
181#398—21no—
182#395Dic7911yescharacteristic
183#395C2×D7911yescharacteristic
184#385—21no—
185#381—21no—
186#380Dic7911yescharacteristic
187#343—792no—
188#343—1581no—
189#324—792no—
190#324—1581no—
191C22×Q8C2×Dic7911yescharacteristic
192C23×C4C2×Dic7911yescharacteristic
193C23×C4C22⋊D7911yescharacteristic
194C23×C4C22⋊D7911yescharacteristic
195C2×M16—3161no—
196C2×M16—1582no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×Dic7911yescharacteristic
206C2×C4⋊C4C2×Dic7911yescharacteristic
207C2×C4⋊C4C22⋊D7911yescharacteristic
208C2×C4⋊C4C22⋊D7911yescharacteristic
209C2×C4⋊C4C22×D7911yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C22⋊C8—3161no—
213C22⋊C8—3161no—
214C22⋊C8—3161no—
215C22⋊C8—1582no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C2×Dic7911yescharacteristic
223#95C2×Dic7911yescharacteristic
224C24C22⋊Dic7911yescharacteristic
225C2×Q8—41no—
226C2×Q8—81no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4C22⋊Dic7911yescharacteristic
242C22×C4C22⋊Dic7911yescharacteristic
243C22×C4C22⋊Dic7911yescharacteristic
244C22×C4C22×Dic7911yescharacteristic
245C22×C4C2×C22⋊D7911yescharacteristic
246C22×C4C2×C22⋊D7911yescharacteristic
247M16—3162no—
248M16—3162no—
249M16—6322no—
250C2×C8—3162no—
251C2×C8—3162no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1457907611yescharacteristic
269C23#1457907611yescharacteristic
270C23C2×C22⋊Dic7911yescharacteristic
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
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—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4#1457919212yes—
303C8—6322no—
304C8—6322no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1457907611yescharacteristic
316C22#1909857211yescharacteristic
317C22#1909858011yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#2376193311yescharacteristic
333C2#2376504611yescharacteristic
334C2#2376508911yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2627C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2627D79
hypocenterC79#11817
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
3. #23761430C2semidirect
4. #23761427C2semidirect
173. C79#11817semidirect

Maximal subgroups

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

subgroupindex
#237634802
#237617302
#237614302
#237614272
C79×#26272
#1181779