Finite Group Databasebeta

← #34183874

Subgroups — #34183874

Order 20224.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771710C211yesmaximal, characteristic
2#23763162C211yesmaximal, characteristic
3C2×#19098304C212yesmaximal, complemented
4#23761282C212yesmaximal, complemented
5C22×#97×C79C211yesmaximal, characteristic
6C2×#14579153C2211yescharacteristic
7C2×C22⋊(C79⋊C8)—21no—
8C2×#14579072C2212yes—
9#19098461C411yescharacteristic
10#19098461C411yescharacteristic
11#19098447—21no—
12#19098447—21no—
13#19098436—21no—
14#19098436—21no—
15#19098429C2211yescharacteristic
16#19098423C411yescharacteristic
17#19098423C411yescharacteristic
18#19098318—21no—
19#19098304—24no—
20C23×C8×C79—21no—
21C22×C42×C79C2211yescharacteristic
22C2×#97×C79—21no—
23C2×#97×C79—24no—
24C2×#97×C79C2212yes—
25C22×C79⋊C8—22no—
26#14579153—22no—
27#14579153—22no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C4⋊Dic79C2×C411yescharacteristic
31C2×C4⋊Dic79C2×C411yescharacteristic
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—21no—
34C2×Dic79⋊C4—21no—
35C2×Dic79⋊C4—21no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79—21no—
38C22⋊(C79⋊C8)—24no—
39#14579072—22no—
40#14579072—22no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C23×C4×C79C2×C411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C23×C4×C79D411yescharacteristic
50C22×C8×C79—24no—
51C22×C8×C79—24no—
52C22×C8×C79—24no—
53C22×C8×C79—22no—
54C2×C42×C79—22no—
55C2×C42×C79—22no—
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79D411yescharacteristic
58C2×C42×C79D411yescharacteristic
59C2×C42×C79C2311yescharacteristic
60#97×C79—24no—
61#97×C79—22no—
62#97×C79D414yes—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C2×C79⋊C8—24no—
68C2×C79⋊C8—42no—
69C2×C79⋊C8—42no—
70C4⋊Dic79—22no—
71C4⋊Dic79—22no—
72C4⋊Dic79—41no—
73C4⋊Dic79—41no—
74Dic79⋊C4—42no—
75Dic79⋊C4—42no—
76Dic79⋊C4—41no—
77Dic79⋊C4—42no—
78Dic79⋊C4—41no—
79C22⋊Dic79—42no—
80C22⋊Dic79—42no—
81C24×C79C22⋊C411yescharacteristic
82C22×C4×C79—22no—
83C22×C4×C79—22no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—22no—
87C22×C4×C79—22no—
88C22×C4×C79—22no—
89C22×C4×C79—22no—
90C22×C4×C79—41no—
91C22×C4×C79—41no—
92C22×C4×C79C22⋊C411yescharacteristic
93C22×C4×C79C22⋊C411yescharacteristic
94C22×C4×C79C22⋊C411yescharacteristic
95C22×C4×C79C22×C411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98C2×C8×C79—216no—
99C2×C8×C79—24no—
100C2×C8×C79—24no—
101C2×C8×C79—24no—
102C42×C79—22no—
103C42×C79—21no—
104C42×C79—22no—
105C42×C79—21no—
106C42×C79C2×D412yes—
107C42×C79#4112yes—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—21no—
111C23×C79—21no—
112C23×C79#10112yes—
113C23×C79#10412yes—
114C23×C79#10412yes—
115C23×C79C2×C22⋊C411yescharacteristic
116C2×C4×C79—24no—
117C2×C4×C79—24no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—44no—
133C2×C4×C79—44no—
134C2×C4×C79#10412yes—
135C2×C4×C79C4×D412yes—
136C2×C4×C79#12011yescharacteristic
137C2×C4×C79#12112yes—
138C2×C4×C79#12311yescharacteristic
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C8×C79—28no—
152C79⋊C8—44no—
153C22×C79—22no—
154C22×C79—22no—
155C22×C79—22no—
156C22×C79—22no—
157C22×C79—22no—
158C22×C79—22no—
159C22×C79#33014yescomplemented
160C22×C79#38611yescharacteristic
161C22×C79C2×#10111yescharacteristic
162C22×C79#41311yescharacteristic
163C22×C79C2×#10411yescharacteristic
164C22×C79C2×#10411yescharacteristic
165C22×C79C2×#10411yescharacteristic
166C22×C79#42111yescharacteristic
167C4×C79—22no—
168C4×C79—22no—
169C4×C79—21no—
170C4×C79—21no—
171C4×C79—22no—
172C4×C79—41no—
173C4×C79—41no—
174Dic79—81no—
175Dic79—81no—
176Dic79—81no—
177Dic79—81no—
178#9121—791nomaximal, Sylow
179C2×C79—24no—
180C2×C79C2×#33011yescharacteristic
181C2×C79#152411yescharacteristic
182C2×C79#153111yescharacteristic
183C2×C79#153211yescharacteristic
184C2×C79#187111yescharacteristic
185C2×C79#189611yescharacteristic
186C2×C79#190111yescharacteristic
187C22×#97D7911yescharacteristic
188#2851—791no—
189#1749—791no—
190C2×#330—792no—
191#1295—792no—
192C79#912111yescharacteristic, complemented, Sylow
193C23×C8—21no—
194C2×#125—791no—
195C22×C42C2×D7911yescharacteristic
196C2×C22⋊C8—1581no—
197C2×#97—21no—
198C2×#97—24no—
199C2×#97—792no—
200C2×#97C2×D7912yes—
201#389—791no—
202#389—791no—
203#387—1581no—
204#387—1581no—
205#385—1581no—
206#385—1581no—
207#382—791no—
208#380—791no—
209#380—791no—
210#336—1581no—
211#330—1584no—
212C23×C4C4×D7911yescharacteristic
213C23×C4D31611yescharacteristic
214C23×C4C22⋊D7911yescharacteristic
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—22no—
219C22×C8—1582no—
220#125—1582no—
221#125—1582no—
222C2×C4⋊C4—791no—
223C2×C4⋊C4—791no—
224C2×C4⋊C4—791no—
225C2×C4⋊C4—791no—
226C2×C4⋊C4—1581no—
227C2×C4⋊C4—1581no—
228C2×C4⋊C4—1581no—
229C2×C4⋊C4—1581no—
230C2×C22⋊C4—1581no—
231C2×C22⋊C4—1581no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C4×D7911yescharacteristic
235C2×C42D31611yescharacteristic
236C2×C42C22⋊D7911yescharacteristic
237C2×C42C22×D7911yescharacteristic
238C22⋊C8—1584no—
239#97—24no—
240#97—22no—
241#97—1582no—
242#97—1582no—
243#97C22⋊D7914yes—
244#95—1581no—
245#95—1581no—
246#95—1581no—
247#95—1581no—
248#95—1581no—
249#95—1581no—
250C24(C2×D79)⋊C411yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—1581no—
262C22×C4—1581no—
263C22×C4—1581no—
264C22×C4—1581no—
265C22×C4(C2×D79)⋊C411yescharacteristic
266C22×C4(C2×D79)⋊C411yescharacteristic
267C22×C4(C2×D79)⋊C411yescharacteristic
268C22×C4C2×C4×D7911yescharacteristic
269C22×C4C2×C22⋊D7911yescharacteristic
270C22×C4C2×D31611yescharacteristic
271C2×C8—216no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—3162no—
276C2×C8—3162no—
277C2×C8—1584no—
278C4⋊C4—3161no—
279C4⋊C4—3161no—
280C4⋊C4—3162no—
281C4⋊C4—3162no—
282C4⋊C4—3161no—
283C4⋊C4—3162no—
284C4⋊C4—3161no—
285C4⋊C4—1582no—
286C4⋊C4—1582no—
287C22⋊C4—3162no—
288C22⋊C4—3162no—
289C42—22no—
290C42—21no—
291C42—22no—
292C42—21no—
293C42C2×C22⋊D7912yes—
294C42#1192962412yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1457907912yes—
300C23#1457908712yes—
301C23#1457908912yes—
302C23C2×(C2×D79)⋊C411yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—3161no—
322C2×C4—3161no—
323C2×C4—3161no—
324C2×C4—3161no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—3161no—
329C2×C4—3161no—
330C2×C4—3161no—
331C2×C4—3161no—
332C2×C4—3161no—
333C2×C4#1457908912yes—
334C2×C4C4×C22⋊D7912yes—
335C2×C4#1457913011yescharacteristic
336C2×C4#1457913512yes—
337C2×C4#1457914411yescharacteristic
338C8—28no—
339C8—3164no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#1909830414yescomplemented
347C22#1909844211yescharacteristic
348C22C2×#1457907911yescharacteristic
349C22#1909854511yescharacteristic
350C22C2×#1457908711yescharacteristic
351C22C2×#1457908911yescharacteristic
352C22C2×#1457908911yescharacteristic
353C22#1909857911yescharacteristic
354C4—21no—
355C4—22no—
356C4—21no—
357C4—22no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—6321no—
362C4—6321no—
363C4—6321no—
364C4—6321no—
365C2—24no—
366C2C2×#1909830411yescharacteristic
367C2#2376198611yescharacteristic
368C2#2376201411yescharacteristic
369C2#2376201811yescharacteristic
370C2#2376370211yescharacteristic
371C2#2376384311yescharacteristic
372C2#2376386811yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC22×#97×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenterC22×#97D79
hypocenterC79#9121
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
3. C2×#19098304C2semidirect
4. #23761282C2semidirect
159. C22×C79#330semidirect
192. C79#9121semidirect
346. C22#19098304semidirect

Maximal subgroups

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

subgroupindex
#237717102
#237631622
C2×#190983042
#237612822
C22×#97×C792
#912179