Finite Group Databasebeta

← #34258803

Subgroups — #34258803

Order 20224.

630 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773968C211yesmaximal, characteristic
2#23772372C212yesmaximal, complemented
3#23762296C211yesmaximal, characteristic, complemented
4#23762293C212yesmaximal, complemented
5#23762285C211yesmaximal, characteristic, complemented
6C8×Dic316C211yesmaximal, characteristic, complemented
7C8×#11929612C212yesmaximal, complemented
8#23762162C211yesmaximal, characteristic, complemented
9#23761857C212yesmaximal, complemented
10C82×D79C211yesmaximal, characteristic, complemented
11C8×D8×C79C211yesmaximal, characteristic, complemented
12#19099095C2212yes—
13#19098964C411yescharacteristic
14#19098946C411yescharacteristic
15#19098916C412yes—
16#19098805C411yescharacteristic
17#19098796C411yescharacteristic
18#19098791C412yes—
19#19098690C2212yes—
20C8×Dic158C2212yes—
21C4×Dic316C2211yescharacteristic
22C4×#11929612C2212yes—
23#19098641C2212yes—
24#19098637—22no—
25C8×C22⋊D79—22no—
26#19098621—22no—
27#19098619C2212yes—
28#19098518—22no—
29C4×C8×D79C2211yescharacteristic
30C4×C8×D79C2211yescharacteristic
31C4×C8×D79C2211yescharacteristic
32#19098316C2211yescharacteristic
33#19098313C2212yes—
34#19098295C2212yes—
35#19098293C2212yes—
36D4⋊(C79⋊C8)C2212yes—
37C8×C79⋊C8C2211yescharacteristic
38C8×C79⋊C8C2211yescharacteristic
39C8×C79⋊C8C2211yescharacteristic
40C4×D8×C79C2211yescharacteristic
41C8×D4×C79C2212yes—
42#335×C79C2211yescharacteristic
43D4⋊C8×C79C2212yes—
44C82×C79C2211yescharacteristic
45C2×Dic316C2×C411yescharacteristic
46C2×#11929612C2×C412yes—
47C2×#11929608—22no—
48C2×C8×D79—21no—
49C2×C8×D79C2×C411yescharacteristic
50C2×C8×D79C2×C411yescharacteristic
51C2×C8×D79D411yescharacteristic
52C2×C8×D79D411yescharacteristic
53#14579154—22no—
54#14579152C2×C412yes—
55#14579148C2×C412yes—
56#14579143—22no—
57#14579141C2×C412yes—
58#14579132C2×C412yes—
59C4×Dic158C2312yes—
60C4×C22⋊D79—22no—
61#14579120—22no—
62C42×D79C2311yescharacteristic
63#14579095C2×C411yescharacteristic
64#14579092C2×C412yes—
65C4⋊(C79⋊C8)C2312yes—
66Dic79⋊C8—22no—
67Dic79⋊C8—22no—
68#14579086C2×C412yes—
69#14579084C2×C412yes—
70#14579082C2×C412yes—
71C22⋊(C79⋊C8)—22no—
72(C2×D79)⋊C8—22no—
73#14579071—22no—
74#14579071—22no—
75C4×C79⋊C8C2311yescharacteristic
76C4×C79⋊C8C2311yescharacteristic
77C4×C79⋊C8C2311yescharacteristic
78C8×Dic79—22no—
79C8×Dic79—21no—
80C8×Dic79C2×C411yescharacteristic
81C8×Dic79C2×C411yescharacteristic
82C8×Dic79D411yescharacteristic
83C8×Dic79D411yescharacteristic
84C2×D8×C79C2×C411yescharacteristic
85C22×C8×C79—22no—
86C4×D4×C79C2312yes—
87#107×C79C2×C411yescharacteristic
88C4⋊C8×C79C2312yes—
89#102×C79C2×C412yes—
90C22⋊C8×C79—22no—
91C4×C8×C79C2311yescharacteristic
92C4×C8×C79C2311yescharacteristic
93C4×C8×C79C2311yescharacteristic
94C2×Dic158C22×C412yes—
95C2×C22⋊D79—22no—
96C2×C4×D79C22×C411yescharacteristic
97C2×C4×D79C2×D411yescharacteristic
98C2×C4×D79#4111yescharacteristic
99Dic316—22no—
100#11929612—24no—
101#11929608—44no—
102C2×C79⋊C8—22no—
103C2×C79⋊C8—21no—
104C2×C79⋊C8C22×C411yescharacteristic
105C2×C79⋊C8C22×C411yescharacteristic
106C2×C79⋊C8C2×D411yescharacteristic
107C2×C79⋊C8C2×D411yescharacteristic
108C8×D79—24no—
109C8×D79—22no—
110C8×D79—22no—
111C8×D79—22no—
112C8×D79—22no—
113C4⋊Dic79—22no—
114C4⋊Dic79—22no—
115C4⋊Dic79C22×C412yes—
116Dic79⋊C4—22no—
117Dic79⋊C4—22no—
118C22⋊Dic79—22no—
119C22⋊Dic79—22no—
120C22⋊Dic79—22no—
121(C2×D79)⋊C4—22no—
122C4×Dic79—22no—
123C4×Dic79C22×C411yescharacteristic
124C4×Dic79C2×D411yescharacteristic
125C4×Dic79#4111yescharacteristic
126C2×D4×C79C22×C412yes—
127C22×C4×C79—22no—
128D8×C79—22no—
129C2×C8×C79—22no—
130C2×C8×C79—21no—
131C2×C8×C79—44no—
132C2×C8×C79C22×C411yescharacteristic
133C2×C8×C79C22×C411yescharacteristic
134C2×C8×C79C2×D411yescharacteristic
135C2×C8×C79C2×D411yescharacteristic
136C4⋊C4×C79C22×C412yes—
137C22⋊C4×C79—22no—
138C42×C79C2411yescharacteristic
139C23×C79—22no—
140C22×D79C4×D411yescharacteristic
141D4×C79—22no—
142D4×C79—22no—
143C2×C4×C79—22no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79C23×C411yescharacteristic
147C2×C4×C79C22×D411yescharacteristic
148C2×C4×C79C2×#4111yescharacteristic
149C22⋊D79—44no—
150C2×Dic79—22no—
151C2×Dic79—22no—
152C2×Dic79C4×D411yescharacteristic
153C2×Dic79C4×D411yescharacteristic
154C2×Dic79C4×D411yescharacteristic
155C4×D79—22no—
156C4×D79—21no—
157C4×D79—21no—
158C4×D79—21no—
159C4×D79—21no—
160Dic158—22no—
161Dic158—22no—
162C8×C79—22no—
163C8×C79—21no—
164C8×C79—21no—
165C8×C79—42no—
166C8×C79#13112yes—
167C8×C79#13512yes—
168C79⋊C8—22no—
169C79⋊C8—21no—
170C79⋊C8—21no—
171C79⋊C8—42no—
172C79⋊C8#13112yes—
173C79⋊C8#13512yes—
174C22×C79—42no—
175C22×C79—42no—
176C22×C79—42no—
177C22×C79C2×C4×D411yescharacteristic
178C2×D79—21no—
179C2×D79—21no—
180C2×D79#44312yes—
181C4×C79—21no—
182C4×C79—42no—
183C4×C79C2×#13111yescharacteristic
184C4×C79C2×#13111yescharacteristic
185C4×C79C2×#13511yescharacteristic
186C4×C79C2×#13511yescharacteristic
187Dic79—21no—
188Dic79—21no—
189Dic79—21no—
190Dic79—42no—
191Dic79—42no—
192#17959—791nomaximal, Sylow
193C2×C79—82no—
194C2×C79#293111yescharacteristic
195C2×C79C4×#13511yescharacteristic
196C2×C79C2×#44311yescharacteristic
197D79—22no—
198#3168—791no—
199#2966—792no—
200#1591—791no—
201#1590—792no—
202#1589—791no—
203C8×Q16—791no—
204C8×SD16—792no—
205C8×D8D7911yescharacteristic, complemented
206#1568—791no—
207#1494—792no—
208C2×C82—791no—
209C79#1795911yescharacteristic, complemented, Sylow
210#529—792no—
211#499—791no—
212#495—791no—
213#488—792no—
214#467—791no—
215#466—791no—
216#465—792no—
217#446—792no—
218C8×Q8—792no—
219C4×Q16—791no—
220C4×SD16—792no—
221C4×D8C2×D7911yescharacteristic
222#436—792no—
223#435—1582no—
224C8×D4—1582no—
225C8×D4C2×D7912yes—
226#433—1582no—
227#432—792no—
228#405—1582no—
229C2×C4×C8—791no—
230C2×C4×C8—791no—
231C2×C4×C8—791no—
232#335—791no—
233#335C2×D7911yescharacteristic
234#334—792no—
235Q8⋊C8—792no—
236Q8⋊C8—792no—
237D4⋊C8—792no—
238D4⋊C8C2×D7912yes—
239C82—791no—
240C82—791no—
241C82—791no—
242C82C2×D7911yescharacteristic
243C2×Q16—791no—
244C2×SD16—792no—
245C2×D8C4×D7911yescharacteristic
246C2×M16—1582no—
247C22×C8—22no—
248C22×C8—791no—
249C22×C8—791no—
250C22×C8—791no—
251C22×C8—791no—
252C22×C8—1581no—
253#126—1582no—
254#125—792no—
255#124—792no—
256#123—1582no—
257C22⋊Q8—792no—
258#121—792no—
259C4×Q8—792no—
260C4×D4—1582no—
261C4×D4C22×D7912yes—
262#117—1582no—
263C2×C42—791no—
264#107—791no—
265#107C4×D7911yescharacteristic
266#106—792no—
267C4⋊C8—792no—
268C4⋊C8—1582no—
269C4⋊C8—1582no—
270C4⋊C8C22×D7912yes—
271#103—792no—
272#103—792no—
273#102—792no—
274#102C4×D7912yes—
275C22⋊C8—22no—
276C22⋊C8—1582no—
277C22⋊C8—1582no—
278#97—1582no—
279#97—1582no—
280C4×C8—791no—
281C4×C8—791no—
282C4×C8—791no—
283C4×C8—791no—
284C4×C8—791no—
285C4×C8—791no—
286C4×C8—791no—
287C4×C8—1581no—
288C4×C8—1582no—
289C4×C8C22×D7911yescharacteristic
290C4×C8C22×D7911yescharacteristic
291C4×C8C22×D7911yescharacteristic
292C2×Q8—792no—
293C2×D4—1582no—
294C2×D4C2×C4×D7912yes—
295C22×C4—22no—
296C22×C4—791no—
297C22×C4—791no—
298C22×C4—791no—
299Q16—1582no—
300SD16—1584no—
301D8—22no—
302M16—3164no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—44no—
306C2×C8—791no—
307C2×C8—791no—
308C2×C8—791no—
309C2×C8—791no—
310C2×C8—1582no—
311C2×C8—1582no—
312C2×C8—1584no—
313C2×C8—1582no—
314C2×C8—1581no—
315C2×C8—1582no—
316C2×C8—1582no—
317C2×C8C2×C4×D7911yescharacteristic
318C2×C8C2×C4×D7911yescharacteristic
319C2×C8D4×D7911yescharacteristic
320C2×C8D4×D7911yescharacteristic
321C4⋊C4—792no—
322C4⋊C4—1582no—
323C4⋊C4—1582no—
324C4⋊C4—1582no—
325C4⋊C4—1582no—
326C4⋊C4C2×C4×D7912yes—
327C22⋊C4—22no—
328C22⋊C4—1582no—
329C22⋊C4—1582no—
330C22⋊C4—1582no—
331C22⋊C4—1582no—
332C42—791no—
333C42—791no—
334C42—791no—
335C42—1582no—
336C42C23×D7911yescharacteristic
337C23—22no—
338C23—791no—
339Q8—1582no—
340Q8—1582no—
341D4—22no—
342D4—22no—
343D4—3164no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—791no—
348C2×C4—791no—
349C2×C4—791no—
350C2×C4—1582no—
351C2×C4—1582no—
352C2×C4—1582no—
353C2×C4—1581no—
354C2×C4—1581no—
355C2×C4—1581no—
356C2×C4—1581no—
357C2×C4C22×C4×D7911yescharacteristic
358C2×C4C2×D4×D7911yescharacteristic
359C2×C4#41×D7911yescharacteristic
360C8—22no—
361C8—21no—
362C8—21no—
363C8—42no—
364C8—3162no—
365C8—792no—
366C8—792no—
367C8—1582no—
368C8—1581no—
369C8—1581no—
370C8#1457916712yes—
371C8#1457918012yes—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—792no—
376C22—1581no—
377C22—1581no—
378C22C4×D4×D7911yescharacteristic
379C4—21no—
380C4—42no—
381C4—3162no—
382C4—3162no—
383C4—1581no—
384C4—1581no—
385C4—1581no—
386C4#131×D7911yescharacteristic
387C4C2×#1457916711yescharacteristic
388C4#135×D7911yescharacteristic
389C4C2×#1457918011yescharacteristic
390C2—82no—
391C2—1582no—
392C2#2377210711yescharacteristic
393C2C4×#1457918011yescharacteristic
394C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C8D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×D8×C79C2
socleC22×C79C2×C4×D4
prime radical#19098796C4
cosocleC42×C79C24
hypercenterC8×D8D79
hypocenterC79#17959
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. #23772372C2semidirect
3. #23762296C2semidirect
4. #23762293C2semidirect
5. #23762285C2semidirect
6. C8×Dic316C2semidirect
7. C8×#11929612C2semidirect
8. #23762162C2semidirect
9. #23761857C2semidirect
10. C82×D79C2semidirect
11. C8×D8×C79C2semidirect
205. C8×D8D79semidirect
209. C79#17959semidirect

Maximal subgroups

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

subgroupindex
#237739682
#237723722
#237622962
#237622932
#237622852
C8×Dic3162
C8×#119296122
#237621622
#237618572
C82×D792
C8×D8×C792
#1795979