Finite Group Databasebeta

← #34198960

Subgroups — #34198960

Order 20224.

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

normal subgroups only

This group has 472 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#23763483C211yesmaximal, characteristic, complemented
2#23761734C211yesmaximal, characteristic, complemented
3#23761406C212yesmaximal, complemented
4#23761403C212yesmaximal, complemented
5C79×#2392C211yesmaximal, characteristic
6#19098620—21no—
7#19098527—21no—
8#19098332—21no—
9#19098332C2212yes—
10#19098289C2212yes—
11#19098287—21no—
12C2×C4×D4×C79—21no—
13C2×C4×D4×C79C411yescharacteristic
14C2×#117×C79C2211yescharacteristic
15#398×C79C2211yescharacteristic
16#397×C79C411yescharacteristic
17#396×C79—21no—
18#394×C79—21no—
19#394×C79C2211yescharacteristic
20#392×C79C411yescharacteristic
21C4×C4⋊C4×C79C411yescharacteristic
22C4×C22⋊C4×C79—21no—
23C2×#11929608—22no—
24C4⋊(C79⋊C8)—41no—
25C22⋊(C79⋊C8)—22no—
26C22⋊(C79⋊C8)—41no—
27C22⋊(C79⋊C8)—41no—
28#14579072—41no—
29C22×D4×C79—21no—
30C22×D4×C79C2×C411yescharacteristic
31C23×C4×C79—21no—
32C23×C4×C79D411yescharacteristic
33C4×D4×C79—41no—
34C4×D4×C79—41no—
35C4×D4×C79—44no—
36#117×C79—41no—
37#117×C79—41no—
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×C79—21no—
46C2×C22⋊C4×C79—21no—
47C2×C22⋊C4×C79—21no—
48C2×C22⋊C4×C79C2×C411yescharacteristic
49C2×C22⋊C4×C79D411yescharacteristic
50C2×C42×C79—21no—
51C2×C42×C79C2×C411yescharacteristic
52C2×C42×C79D411yescharacteristic
53#95×C79—21no—
54#95×C79—21no—
55#95×C79C2×C411yescharacteristic
56#95×C79C2×C411yescharacteristic
57#11929608—42no—
58#11929608—42no—
59C2×C79⋊C8—42no—
60C2×C79⋊C8—42no—
61C24×C79—21no—
62C24×C79C22⋊C411yescharacteristic
63C2×D4×C79—42no—
64C2×D4×C79—41no—
65C2×D4×C79—42no—
66C2×D4×C79—42no—
67C2×D4×C79—42no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—22no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—41no—
75C22×C4×C79—42no—
76C22×C4×C79—42no—
77C22×C4×C79—41no—
78C22×C4×C79—41no—
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C4⋊C4×C79—41no—
86C4⋊C4×C79—41no—
87C4⋊C4×C79—42no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—42no—
90C22⋊C4×C79—41no—
91C22⋊C4×C79—41no—
92C22⋊C4×C79—42no—
93C22⋊C4×C79—42no—
94C22⋊C4×C79—42no—
95C22⋊C4×C79—42no—
96C22⋊C4×C79—42no—
97C42×C79—21no—
98C42×C79—21no—
99C42×C79—21no—
100C42×C79—21no—
101C42×C79—42no—
102C23×C79—22no—
103C23×C79—21no—
104C23×C79—21no—
105C23×C79—21no—
106C23×C79—41no—
107C23×C79—41no—
108C23×C79—41no—
109C23×C79—41no—
110C23×C79—41no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79C2×C22⋊C411yescharacteristic
115D4×C79—42no—
116D4×C79—48no—
117D4×C79—42no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—22no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—21no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
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—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—82no—
142C2×C4×C79—82no—
143C2×C4×C79#9911yescharacteristic
144C2×C4×C79#9911yescharacteristic
145C2×C4×C79AGL(1,ℤ/8)12yes—
146C79⋊C8—82no—
147C79⋊C8—82no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—42no—
151C22×C79—42no—
152C22×C79—42no—
153C22×C79—42no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—81no—
158C22×C79—81no—
159C22×C79—81no—
160C22×C79—81no—
161C22×C79—81no—
162C22×C79C2×#9911yescharacteristic
163C22×C79#41811yescharacteristic
164C22×C79#42111yescharacteristic
165C4×C79—21no—
166C4×C79—21no—
167C4×C79—22no—
168C4×C79—42no—
169C4×C79—41no—
170C4×C79—41no—
171C4×C79—41no—
172C4×C79—41no—
173C4×C79—81no—
174C4×C79—81no—
175#11804—791nomaximal, Sylow
176C2×C79—21no—
177C2×C79—21no—
178C2×C79—81no—
179C2×C79—82no—
180C2×C79#151011yescharacteristic
181C2×C79#212011yescharacteristic
182C2×C79#213211yescharacteristic
183#2392D7911yescharacteristic
184#1822—791no—
185#1464—791no—
186#1346—792no—
187#1345—792no—
188C79#1180411yescharacteristic, complemented, Sylow
189C2×C4×D4—21no—
190C2×C4×D4Dic7911yescharacteristic
191C2×#117C2×D7911yescharacteristic
192#432—1581no—
193#407—1581no—
194#398C2×D7911yescharacteristic
195#397Dic7911yescharacteristic
196#396—21no—
197#394—21no—
198#394C2×D7911yescharacteristic
199#392Dic7911yescharacteristic
200C4×C4⋊C4Dic7911yescharacteristic
201C4×C22⋊C4—21no—
202#343—792no—
203#343—1581no—
204#324—792no—
205#323—1581no—
206C22×D4—21no—
207C22×D4C2×Dic7911yescharacteristic
208C23×C4—21no—
209C23×C4C22⋊D7911yescharacteristic
210C2×M16—1582no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—44no—
214#117—41no—
215#117—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C2×Dic7911yescharacteristic
219C2×C4⋊C4C22⋊D7911yescharacteristic
220C2×C4⋊C4C22×D7911yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4C2×Dic7911yescharacteristic
227C2×C22⋊C4C22⋊D7911yescharacteristic
228C2×C42—21no—
229C2×C42C2×Dic7911yescharacteristic
230C2×C42C22⋊D7911yescharacteristic
231C4⋊C8—3161no—
232C22⋊C8—3161no—
233C22⋊C8—3161no—
234C22⋊C8—1582no—
235#97—3161no—
236#95—21no—
237#95—21no—
238#95C2×Dic7911yescharacteristic
239#95C2×Dic7911yescharacteristic
240C24—21no—
241C24C22⋊Dic7911yescharacteristic
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22⋊Dic7911yescharacteristic
259C22×C4C22⋊Dic7911yescharacteristic
260C22×C4C22⋊Dic7911yescharacteristic
261C22×C4C22×Dic7911yescharacteristic
262C22×C4C2×C22⋊D7911yescharacteristic
263C22×C4C2×C22⋊D7911yescharacteristic
264M16—3162no—
265M16—3162no—
266C2×C8—3162no—
267C2×C8—3162no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—42no—
285C23—22no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic7911yescharacteristic
298D4—42no—
299D4—48no—
300D4—42no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#1457907611yescharacteristic
327C2×C4#1457907611yescharacteristic
328C2×C4#1457918712yes—
329C8—6322no—
330C8—6322no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#1457907611yescharacteristic
346C22#1909856811yescharacteristic
347C22#1909858011yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—81no—
357C4—81no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—82no—
362C2#2376192911yescharacteristic
363C2#2376504111yescharacteristic
364C2#2376508511yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2392C2
socleC22×C79C2×#99
prime radical#392×C79C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2392D79
hypocenterC79#11804
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #23763483C2semidirect
2. #23761734C2semidirect
3. #23761406C2semidirect
4. #23761403C2semidirect
188. C79#11804semidirect

Maximal subgroups

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

subgroupindex
#237634832
#237617342
#237614062
#237614032
C79×#23922
#1180479