Finite Group Databasebeta

← #34185939

Subgroups — #34185939

Order 20224.

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

normal subgroups only

This group has 490 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#23772429C211yesmaximal, characteristic
2#23762027C212yesmaximal, complemented
3#23761741C211yesmaximal, characteristic
4#23761256C212yesmaximal, complemented
5C79×#2566C211yesmaximal, characteristic
6#19098635—24no—
7#19098620C2212yes—
8#19098620C2212yes—
9#19098527—21no—
10C2×C22⋊(C79⋊C8)—21no—
11#19098305—24no—
12#19098287—21no—
13#19098287—21no—
14C2×#125×C79C2211yescharacteristic
15C2×C4×D4×C79C2211yescharacteristic
16#396×C79—21no—
17#396×C79—21no—
18#390×C79C411yescharacteristic
19#390×C79C411yescharacteristic
20#385×C79—21no—
21#385×C79—21no—
22#384×C79C411yescharacteristic
23#382×C79C411yescharacteristic
24C4×C4⋊C4×C79C2211yescharacteristic
25C2×#11929608—22no—
26C22×C79⋊C8—22no—
27C4⋊(C79⋊C8)—22no—
28C4⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33#14579072—22no—
34#14579072D414yes—
35C22×D4×C79C2×C411yescharacteristic
36C23×C4×C79D411yescharacteristic
37C23×C4×C79D411yescharacteristic
38#125×C79—22no—
39#125×C79—22no—
40C4×D4×C79—22no—
41C4×D4×C79—22no—
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79C2×C411yescharacteristic
48C2×C4⋊C4×C79C2×C411yescharacteristic
49C2×C4⋊C4×C79C2×C411yescharacteristic
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C22⋊C4×C79D411yescharacteristic
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79C2×C411yescharacteristic
58C2×C42×C79C2311yescharacteristic
59#95×C79—21no—
60#95×C79—21no—
61#95×C79—21no—
62#11929608—82no—
63C2×C79⋊C8—24no—
64C2×C79⋊C8—42no—
65C2×C79⋊C8—82no—
66C24×C79C22⋊C411yescharacteristic
67C24×C79C22⋊C411yescharacteristic
68C2×D4×C79—22no—
69C2×D4×C79—22no—
70C2×D4×C79—41no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—81no—
80C22×C4×C79—81no—
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C4⋊C4×C79—22no—
87C4⋊C4×C79—22no—
88C4⋊C4×C79—22no—
89C4⋊C4×C79—42no—
90C4⋊C4×C79—41no—
91C4⋊C4×C79—42no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—81no—
94C4⋊C4×C79—81no—
95C22⋊C4×C79—22no—
96C22⋊C4×C79—22no—
97C22⋊C4×C79—42no—
98C22⋊C4×C79—42no—
99C22⋊C4×C79—81no—
100C22⋊C4×C79—81no—
101C42×C79—41no—
102C42×C79—41no—
103C42×C79C2×D412yes—
104C42×C79#4112yes—
105C23×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—41no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79#10112yes—
115C23×C79C2×C22⋊C411yescharacteristic
116D4×C79—44no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—22no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
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—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—81no—
140C2×C4×C79—81no—
141C2×C4×C79C4×D412yes—
142C2×C4×C79#12011yescharacteristic
143C2×C4×C79#12112yes—
144C2×C4×C79#12311yescharacteristic
145C79⋊C8—44no—
146C79⋊C8—82no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79—22no—
150C22×C79—42no—
151C22×C79—42no—
152C22×C79—41no—
153C22×C79—41no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—81no—
158C22×C79—81no—
159C22×C79#38611yescharacteristic
160C22×C79C2×#10111yescharacteristic
161C22×C79#41311yescharacteristic
162C4×C79—22no—
163C4×C79—22no—
164C4×C79—21no—
165C4×C79—21no—
166C4×C79—42no—
167C4×C79—41no—
168C4×C79—41no—
169C4×C79—81no—
170C4×C79—81no—
171C4×C79—81no—
172#9449—791nomaximal, Sylow
173C2×C79—21no—
174C2×C79—21no—
175C2×C79—42no—
176C2×C79—81no—
177C2×C79#188411yescharacteristic
178C2×C79#190111yescharacteristic
179C2×C79#191111yescharacteristic
180#2973—791no—
181#2566D7911yescharacteristic
182#1534—792no—
183#1466—791no—
184#1285—792no—
185C79#944911yescharacteristic, complemented, Sylow
186C2×#125C2×D7911yescharacteristic
187C2×C4×D4C2×D7911yescharacteristic
188#435—1584no—
189#432—792no—
190#432—792no—
191#407—1581no—
192C2×C22⋊C8—1581no—
193#396—21no—
194#396—21no—
195#390Dic7911yescharacteristic
196#390Dic7911yescharacteristic
197#385—21no—
198#385—21no—
199#384Dic7911yescharacteristic
200#382Dic7911yescharacteristic
201C4×C4⋊C4C2×D7911yescharacteristic
202#330—1584no—
203#323—1581no—
204#323—1581no—
205C22×D4C2×Dic7911yescharacteristic
206C23×C4C22⋊D7911yescharacteristic
207C23×C4C22⋊D7911yescharacteristic
208C2×M16—1582no—
209C22×C8—1582no—
210#125—22no—
211#125—22no—
212C4×D4—22no—
213C4×D4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C2×Dic7911yescharacteristic
220C2×C4⋊C4C2×Dic7911yescharacteristic
221C2×C4⋊C4C2×Dic7911yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4C22⋊D7911yescharacteristic
227C2×C22⋊C4C22⋊D7911yescharacteristic
228C2×C42C2×Dic7911yescharacteristic
229C2×C42C2×Dic7911yescharacteristic
230C2×C42C22×D7911yescharacteristic
231C4⋊C8—1582no—
232C4⋊C8—1582no—
233C22⋊C8—1582no—
234C22⋊C8—1582no—
235C22⋊C8—1582no—
236C22⋊C8—1582no—
237#97—794no—
238#97—1582no—
239#95—21no—
240#95—21no—
241#95—21no—
242C24C22⋊Dic7911yescharacteristic
243C24C22⋊Dic7911yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4C22⋊Dic7911yescharacteristic
258C22×C4C22⋊Dic7911yescharacteristic
259C22×C4C22×Dic7911yescharacteristic
260C22×C4C2×C22⋊D7911yescharacteristic
261C22×C4C2×C22⋊D7911yescharacteristic
262M16—6322no—
263C2×C8—3162no—
264C2×C8—6322no—
265C2×C8—1584no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C42—41no—
282C42—41no—
283C42D4×D7912yes—
284C42#1192962312yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1457908012yes—
295C23C2×C22⋊Dic7911yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4D4×Dic7912yes—
322C2×C4#1457912911yescharacteristic
323C2×C4#1457913312yes—
324C2×C4#1457914211yescharacteristic
325C8—3164no—
326C8—6322no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1909844311yescharacteristic
340C22C2×#1457908011yescharacteristic
341C22#1909854711yescharacteristic
342C4—22no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#2376378411yescharacteristic
357C2#2376386911yescharacteristic
358C2#2376393311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2566C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC2×C42×C79C23
hypercenter#2566D79
hypocenterC79#9449
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2566

Complements

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

NH ≅ G/Ntype
2. #23762027C2semidirect
4. #23761256C2semidirect
185. C79#9449semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237620272
#237617412
#237612562
C79×#25662
#944979