Finite Group Databasebeta

← #34185735

Subgroups — #34185735

Order 20224.

720 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 107 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#23765998C211yesmaximal, characteristic, complemented
2#23761916C212yesmaximal, complemented
3#23761727C211yesmaximal, characteristic, complemented
4#23761286C212yesmaximal, complemented
5C79×#2539C211yesmaximal, characteristic
6#19098527C2212yes—
7#19098456—24no—
8#19098423—21no—
9#19098419—21no—
10C4×C22⋊Dic79C2212yes—
11#19098297—24no—
12#19098287—22no—
13C2×#123×C79—21no—
14C2×#121×C79C2211yescharacteristic
15C2×#120×C79—21no—
16C22×C22⋊C4×C79C2211yescharacteristic
17#399×C79C412yescomplemented
18#392×C79C412yescomplemented
19#390×C79—22no—
20C4×C22⋊C4×C79C2211yescharacteristic
21C2×#11929608—22no—
22C2×C4⋊Dic79—22no—
23C2×C22⋊Dic79—22no—
24C2×C22⋊Dic79—22no—
25C2×C22⋊Dic79—22no—
26C2×C4×Dic79—22no—
27C22⋊(C79⋊C8)—22no—
28C22⋊(C79⋊C8)—22no—
29C79⋊#95—22no—
30C79⋊#95D414yescomplemented
31C25×C79D411yescharacteristic
32C22×D4×C79—21no—
33C22×D4×C79—22no—
34C22×D4×C79C2×C411yescharacteristic
35C23×C4×C79D411yescharacteristic
36C23×C4×C79C2311yescharacteristic
37#123×C79—44no—
38#121×C79—22no—
39#121×C79—22no—
40#120×C79—44no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79C2×C411yescharacteristic
43C2×C22⋊C4×C79—21no—
44C2×C22⋊C4×C79—22no—
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79—22no—
47C2×C22⋊C4×C79—22no—
48C2×C22⋊C4×C79—21no—
49C2×C22⋊C4×C79C2×C412yes—
50C2×C22⋊C4×C79D412yes—
51C2×C42×C79—21no—
52#95×C79C2×C412yes—
53C22×Dic79—24no—
54#11929608—82no—
55C2×C79⋊C8—42no—
56C4⋊Dic79—82no—
57C22⋊Dic79—44no—
58C22⋊Dic79—82no—
59C22⋊Dic79—82no—
60C4×Dic79—82no—
61C24×C79—21no—
62C24×C79—21no—
63C24×C79—21no—
64C24×C79—42no—
65C24×C79—42no—
66C24×C79—81no—
67C24×C79C22⋊C412yes—
68C24×C79C2×D411yescharacteristic
69C2×D4×C79—22no—
70C2×D4×C79—44no—
71C2×D4×C79—42no—
72C2×D4×C79—42no—
73C2×D4×C79—44no—
74C2×D4×C79—41no—
75C2×D4×C79—41no—
76C2×D4×C79—82no—
77C2×D4×C79—81no—
78C22×C4×C79—22no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—42no—
83C22×C4×C79—81no—
84C22×C4×C79C22⋊C412yes—
85C22×C4×C79C22×C411yescharacteristic
86C22×C4×C79C2×D412yes—
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79#4112yes—
89C4⋊C4×C79—22no—
90C4⋊C4×C79—44no—
91C22⋊C4×C79—24no—
92C22⋊C4×C79—24no—
93C22⋊C4×C79—24no—
94C22⋊C4×C79—44no—
95C22⋊C4×C79—44no—
96C22⋊C4×C79—42no—
97C42×C79—81no—
98C23×C79—22no—
99C23×C79—22no—
100C23×C79—22no—
101C23×C79—22no—
102C23×C79—22no—
103C23×C79—41no—
104C23×C79—42no—
105C23×C79—42no—
106C23×C79—42no—
107C23×C79—42no—
108C23×C79—42no—
109C23×C79—41no—
110C23×C79—41no—
111C23×C79—42no—
112C23×C79—42no—
113C23×C79—41no—
114C23×C79—42no—
115C23×C79—81no—
116C23×C79—81no—
117C23×C79—88no—
118C23×C79C2×C22⋊C411yescharacteristic
119C23×C79#12011yescharacteristic
120C23×C79#12311yescharacteristic
121D4×C79—82no—
122D4×C79—88no—
123D4×C79—82no—
124D4×C79—82no—
125C2×C4×C79—24no—
126C2×C4×C79—22no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—42no—
132C2×C4×C79—42no—
133C2×C4×C79—42no—
134C2×C4×C79—42no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—81no—
139C2×C4×C79C4×D412yes—
140C2×C4×C79#12112yes—
141C2×Dic79—44no—
142C2×Dic79—44no—
143C2×Dic79—44no—
144C79⋊C8—82no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—44no—
153C22×C79—44no—
154C22×C79—42no—
155C22×C79—42no—
156C22×C79—42no—
157C22×C79—82no—
158C22×C79—84no—
159C22×C79—82no—
160C22×C79—82no—
161C22×C79—84no—
162C22×C79—81no—
163C22×C79—81no—
164C22×C79—81no—
165C22×C79#38611yescharacteristic
166C22×C79#42112yes—
167C22×C79#45312yes—
168C4×C79—22no—
169C4×C79—22no—
170C4×C79—44no—
171C4×C79—42no—
172C4×C79—81no—
173Dic79—84no—
174#9413—791nomaximal, Sylow
175C2×C79—22no—
176C2×C79—22no—
177C2×C79—41no—
178C2×C79—82no—
179C2×C79—161no—
180C2×C79#188711yescharacteristic
181C2×C79#189711yescharacteristic
182C2×C79#191311yescharacteristic
183#2539D7911yescharacteristic
184#2312—791no—
185#1507—792no—
186#1462—791no—
187#1297—792no—
188C79#941311yescharacteristic, complemented, Sylow
189C2×#123—21no—
190C2×#121C2×D7911yescharacteristic
191C2×#120—21no—
192C22×C22⋊C4C2×D7911yescharacteristic
193#407—792no—
194#399Dic7912yescomplemented
195#392Dic7912yescomplemented
196#390—22no—
197#388—1584no—
198#380—1581no—
199#379—1581no—
200C4×C22⋊C4—792no—
201C4×C22⋊C4C2×D7911yescharacteristic
202#327—1584no—
203#323—1582no—
204C25C22⋊D7911yescharacteristic
205C22×D4—21no—
206C22×D4—22no—
207C22×D4C2×Dic7911yescharacteristic
208C23×C4C22⋊D7911yescharacteristic
209C23×C4C22×D7911yescharacteristic
210C2×M16—1582no—
211#123—44no—
212#121—22no—
213#121—22no—
214#120—44no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—1582no—
217C2×C4⋊C4C2×Dic7911yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—1582no—
225C2×C22⋊C4—1582no—
226C2×C22⋊C4—1582no—
227C2×C22⋊C4C2×Dic7912yes—
228C2×C22⋊C4C22⋊D7912yes—
229C2×C42—21no—
230C2×C42—1582no—
231C22⋊C8—1582no—
232C22⋊C8—1582no—
233#95—794no—
234#95—1582no—
235#95C2×Dic7912yes—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—42no—
240C24—42no—
241C24—81no—
242C24C22⋊Dic7912yes—
243C24C2×C22⋊D7911yescharacteristic
244C2×D4—22no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—82no—
252C2×D4—81no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—1584no—
260C22×C4C22⋊Dic7912yes—
261C22×C4C22×Dic7911yescharacteristic
262C22×C4D4×D7912yes—
263C22×C4C2×C22⋊D7911yescharacteristic
264C22×C4#1192962312yes—
265M16—6322no—
266C2×C8—3162no—
267C4⋊C4—22no—
268C4⋊C4—44no—
269C4⋊C4—6322no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—3164no—
277C22⋊C4—6322no—
278C22⋊C4—6322no—
279C42—81no—
280C42—6322no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—41no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23—88no—
301C23C2×C22⋊Dic7911yescharacteristic
302C23#1457912911yescharacteristic
303C23#1457914211yescharacteristic
304D4—82no—
305D4—88no—
306D4—82no—
307D4—82no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—3164no—
323C2×C4—3164no—
324C2×C4—3164no—
325C2×C4D4×Dic7912yes—
326C2×C4#1457913312yes—
327C8—6322no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—84no—
342C22—82no—
343C22—82no—
344C22—84no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1909844311yescharacteristic
349C22#1909858012yes—
350C22#1909873012yes—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—42no—
355C4—81no—
356C4—6324no—
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—161no—
362C2#2376379911yescharacteristic
363C2#2376384911yescharacteristic
364C2#2376394311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
Fitting, prime radicalC79×#2539C2
socleC22×C79#386
cosocleC23×C4×C79C23
hypercenter#2539D79
hypocenterC79#9413
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. #23765998C2semidirect
2. #23761916C2semidirect
3. #23761727C2semidirect
4. #23761286C2semidirect
17. #399×C79C4semidirect
18. #392×C79C4semidirect
30. C79⋊#95D4semidirect
188. C79#9413semidirect
194. #399Dic79semidirect
195. #392Dic79semidirect

Maximal subgroups

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

subgroupindex
#237659982
#237619162
#237617272
#237612862
C79×#25392
#941379