Finite Group Databasebeta

← #34184108

Subgroups — #34184108

Order 20224.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772435C211yesmaximal, characteristic
2#23763225C211yesmaximal, characteristic
3#23762015C211yesmaximal, characteristic, complemented
4#23761899C211yesmaximal, characteristic, complemented
5#23761298C211yesmaximal, characteristic, complemented
6#23761257C211yesmaximal, characteristic, complemented
7C79×#3110C211yesmaximal, characteristic
8#19098641—21no—
9#19098635—22no—
10D4×C79⋊C8—21no—
11#19098620C2211yescharacteristic
12#19098616C2211yescharacteristic
13C2×#14579082—21no—
14#19098527—21no—
15C2×C22⋊(C79⋊C8)—21no—
16C2×#14579072C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098325—21no—
19#19098305—22no—
20#19098300—21no—
21#19098300—21no—
22#533×C79C2211yescharacteristic
23C2×C4×D4×C79C2211yescharacteristic
24#484×C79—21no—
25#474×C79C411yescharacteristic, complemented
26#474×C79C411yescharacteristic, complemented
27#461×C79C411yescharacteristic, complemented
28#461×C79C411yescharacteristic, complemented
29#450×C79—21no—
30#440×C79—21no—
31#440×C79—21no—
32#432×C79C2211yescharacteristic
33C2×#11929608—22no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—22no—
36C2×C4⋊Dic79—21no—
37C2×C4×Dic79—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40#14579082—24no—
41C22⋊(C79⋊C8)—22no—
42C22⋊(C79⋊C8)—22no—
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)D411yescharacteristic
45C22⋊(C79⋊C8)D411yescharacteristic
46#14579072—21no—
47#14579072—21no—
48#14579072D412yes—
49C4×C79⋊C8—21no—
50C79⋊#95—21no—
51C22×D4×C79D411yescharacteristic
52C23×C4×C79—21no—
53C2×SD16×C79—21no—
54C2×SD16×C79—21no—
55C4⋊Q8×C79C2×C411yescharacteristic
56C4⋊Q8×C79C2×C411yescharacteristic
57#125×C79—21no—
58C22⋊Q8×C79—21no—
59C22⋊Q8×C79—21no—
60C4×Q8×C79—21no—
61C4×Q8×C79—21no—
62C4×D4×C79—21no—
63C4×D4×C79—21no—
64C4×D4×C79C2×C411yescharacteristic
65C4×D4×C79C2×C411yescharacteristic
66C4×D4×C79D411yescharacteristic
67C4×D4×C79D411yescharacteristic
68#117×C79—21no—
69C2×C4⋊C4×C79D411yescharacteristic
70C2×C22⋊C4×C79—21no—
71C2×C42×C79C2311yescharacteristic
72#107×C79—21no—
73#107×C79—21no—
74C4⋊C8×C79C2×C411yescharacteristic
75C4⋊C8×C79C2×C411yescharacteristic
76#103×C79—21no—
77#103×C79—21no—
78#102×C79—21no—
79#102×C79—21no—
80C22⋊C8×C79—21no—
81#97×C79—21no—
82C22×Dic79—41no—
83#11929608—82no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—82no—
91C4⋊Dic79—22no—
92C4⋊Dic79—41no—
93C4×Dic79—81no—
94C24×C79—21no—
95C2×Q8×C79—21no—
96C2×Q8×C79—21no—
97C2×D4×C79—22no—
98C2×D4×C79—21no—
99C2×D4×C79—41no—
100C2×D4×C79C22⋊C411yescharacteristic
101C2×D4×C79C22⋊C411yescharacteristic
102C22×C4×C79—21no—
103C22×C4×C79—21no—
104C22×C4×C79—21no—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—82no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79#4111yescharacteristic
111SD16×C79—42no—
112SD16×C79—42no—
113C2×C8×C79—21no—
114C2×C8×C79—21no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—41no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79C22⋊C411yescharacteristic
123C4⋊C4×C79C22⋊C411yescharacteristic
124C22⋊C4×C79—21no—
125C22⋊C4×C79—21no—
126C22⋊C4×C79—21no—
127C22⋊C4×C79—21no—
128C22⋊C4×C79—41no—
129C42×C79—41no—
130C42×C79C22×C411yescharacteristic
131C42×C79C2×D411yescharacteristic
132C42×C79C2×D411yescharacteristic
133C42×C79#4111yescharacteristic
134C23×C79—21no—
135C23×C79—21no—
136C23×C79—42no—
137C23×C79—41no—
138C23×C79—41no—
139C23×C79—81no—
140C23×C79#12111yescharacteristic
141Q8×C79—41no—
142Q8×C79—41no—
143Q8×C79—81no—
144D4×C79—21no—
145D4×C79—21no—
146D4×C79—21no—
147D4×C79—42no—
148D4×C79#10412yes—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—21no—
155C2×C4×C79—41no—
156C2×C4×C79—42no—
157C2×C4×C79—42no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—81no—
161C2×C4×C79—81no—
162C2×C4×C79—81no—
163C2×C4×C79C2×C22⋊C411yescharacteristic
164C2×C4×C79C4×D411yescharacteristic
165C2×C4×C79C4×D411yescharacteristic
166C2×C4×C79#12011yescharacteristic
167C2×C4×C79#12111yescharacteristic
168C2×C4×C79#12311yescharacteristic
169C2×Dic79—42no—
170C2×Dic79—81no—
171C2×Dic79—81no—
172C8×C79—41no—
173C8×C79—41no—
174C79⋊C8—42no—
175C79⋊C8—41no—
176C79⋊C8—42no—
177C79⋊C8—41no—
178C22×C79—21no—
179C22×C79—21no—
180C22×C79—22no—
181C22×C79—41no—
182C22×C79—41no—
183C22×C79—41no—
184C22×C79—41no—
185C22×C79—41no—
186C22×C79—82no—
187C22×C79—82no—
188C22×C79#38611yescharacteristic
189C22×C79#47111yescharacteristic
190C22×C79#47311yescharacteristic
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—21no—
195C4×C79—21no—
196C4×C79—41no—
197C4×C79—41no—
198C4×C79—81no—
199C4×C79—81no—
200C4×C79C2×#10411yescharacteristic
201C4×C79#42111yescharacteristic
202Dic79—82no—
203#9165—791nomaximal, Sylow
204C2×C79—21no—
205C2×C79—21no—
206C2×C79—42no—
207C2×C79—81no—
208C2×C79#187111yescharacteristic
209C2×C79#190411yescharacteristic
210C2×C79#190811yescharacteristic
211#3110D7911yescharacteristic
212#2974—791no—
213#1762—791no—
214#1531—791no—
215#1503—791no—
216#1302—791no—
217#1285—791no—
218C79#916511yescharacteristic, complemented, Sylow
219#533C2×D7911yescharacteristic
220C2×C4×D4C2×D7911yescharacteristic
221#484—21no—
222#474Dic7911yescharacteristic, complemented
223#474Dic7911yescharacteristic, complemented
224#461Dic7911yescharacteristic, complemented
225#461Dic7911yescharacteristic, complemented
226#450—21no—
227#440—21no—
228#440—21no—
229#436—1581no—
230#435—1582no—
231C8×D4—1581no—
232#432—791no—
233#432C2×D7911yescharacteristic
234#431—791no—
235C2×#102—1581no—
236#407—1581no—
237C2×C22⋊C8—1581no—
238C2×#97—791no—
239C4×C4⋊C4—791no—
240#340—1581no—
241#330—1582no—
242#328—1581no—
243#328—1581no—
244C22×D4C22⋊D7911yescharacteristic
245C23×C4—21no—
246C2×SD16—21no—
247C2×SD16—21no—
248C2×M16—1582no—
249C22×C8—1581no—
250C22×C8—1582no—
251C4⋊Q8C2×Dic7911yescharacteristic
252C4⋊Q8C2×Dic7911yescharacteristic
253#125—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C2×Dic7911yescharacteristic
261C4×D4C2×Dic7911yescharacteristic
262C4×D4C22⋊D7911yescharacteristic
263C4×D4C22⋊D7911yescharacteristic
264#117—21no—
265C2×C4⋊C4—1581no—
266C2×C4⋊C4C22⋊D7911yescharacteristic
267C2×C22⋊C4—21no—
268C2×C42—1581no—
269C2×C42C22×D7911yescharacteristic
270#107—21no—
271#107—21no—
272C4⋊C8—1581no—
273C4⋊C8—1581no—
274C4⋊C8C2×Dic7911yescharacteristic
275C4⋊C8C2×Dic7911yescharacteristic
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280#102—1584no—
281C22⋊C8—21no—
282C22⋊C8—791no—
283C22⋊C8—791no—
284C22⋊C8—1582no—
285C22⋊C8—1582no—
286C22⋊C8—1581no—
287#97—21no—
288#97—792no—
289#97—1581no—
290#97—1581no—
291C4×C8—1581no—
292#95—1581no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4C22⋊Dic7911yescharacteristic
300C2×D4C22⋊Dic7911yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—82no—
307C22×C4—3161no—
308C22×C4D4×D7911yescharacteristic
309C22×C4C2×C22⋊D7911yescharacteristic
310C22×C4#1192962311yescharacteristic
311SD16—42no—
312SD16—42no—
313M16—6322no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—3161no—
317C2×C8—3161no—
318C2×C8—6322no—
319C2×C8—1582no—
320C2×C8—1581no—
321C2×C8—1581no—
322C2×C8—1582no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—3161no—
331C4⋊C4—1582no—
332C4⋊C4C22⋊Dic7911yescharacteristic
333C4⋊C4C22⋊Dic7911yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—41no—
339C42—41no—
340C42—6321no—
341C42C22×Dic7911yescharacteristic
342C42D4×D7911yescharacteristic
343C42C2×C22⋊D7911yescharacteristic
344C42#1192962311yescharacteristic
345C23—21no—
346C23—21no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—81no—
351C23#1457913311yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—81no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4#1457908812yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—3162no—
375C2×C4—6321no—
376C2×C4—6321no—
377C2×C4C2×C22⋊Dic7911yescharacteristic
378C2×C4D4×Dic7911yescharacteristic
379C2×C4D4×Dic7911yescharacteristic
380C2×C4#1457912911yescharacteristic
381C2×C4#1457913311yescharacteristic
382C2×C4#1457914211yescharacteristic
383C8—41no—
384C8—41no—
385C8—3162no—
386C8—3161no—
387C8—3162no—
388C8—3161no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—82no—
398C22—82no—
399C22#1909844311yescharacteristic
400C22#1909882811yescharacteristic
401C22#1909883611yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C4—6322no—
412C4C2×#1457908811yescharacteristic
413C4#1909858011yescharacteristic
414C2—21no—
415C2—21no—
416C2—42no—
417C2—81no—
418C2#2376370311yescharacteristic
419C2#2376388911yescharacteristic
420C2#2376391711yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3110C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenter#3110D79
hypocenterC79#9165
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
3. #23762015C2semidirect
4. #23761899C2semidirect
5. #23761298C2semidirect
6. #23761257C2semidirect
25. #474×C79C4semidirect
26. #474×C79C4semidirect
27. #461×C79C4semidirect
28. #461×C79C4semidirect
218. C79#9165semidirect
222. #474Dic79semidirect
223. #474Dic79semidirect
224. #461Dic79semidirect
225. #461Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724352
#237632252
#237620152
#237618992
#237612982
#237612572
C79×#31102
#916579