Finite Group Databasebeta

← #34182475

Subgroups — #34182475

Order 20224.

546 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766351C211yesmaximal, characteristic
2#23764156C211yesmaximal, characteristic, complemented
3#23763397C211yesmaximal, characteristic
4C2×#19098292C211yesmaximal, characteristic
5#23761283C211yesmaximal, characteristic, complemented
6#23761240C211yesmaximal, characteristic, complemented
7C79×#1844C211yesmaximal, characteristic, complemented
8C2×C4×D316C2211yescharacteristic
9C2×#14579119—21no—
10C2×#14579094—21no—
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×#14579081—21no—
14#19098464—21no—
15#19098453—22no—
16#19098420—21no—
17C4⋊C4×Dic79—21no—
18C4×C4⋊Dic79C2211yescharacteristic
19#19098325—21no—
20#19098325—21no—
21#19098292C414yescomplemented
22C2×#106×C79—22no—
23C2×C4×C8×C79C2211yescharacteristic
24#384×C79C2211yescharacteristic
25#384×C79C2211yescharacteristic
26C22×D316D411yescharacteristic
27C22×C4×D79—21no—
28C22×C79⋊C8—21no—
29C22×C79⋊C8—21no—
30C4×D316—21no—
31C4×D316—21no—
32C4×D316C2×C412yes—
33C4×D316Q812yes—
34#14579119—42no—
35#14579119—42no—
36C2×C4⋊Dic79D411yescharacteristic
37C2×C4⋊Dic79D411yescharacteristic
38C2×C4⋊Dic79D411yescharacteristic
39C2×(C2×D79)⋊C4—21no—
40C2×(C2×D79)⋊C4—22no—
41C2×(C2×D79)⋊C4—22no—
42C2×C4×Dic79—22no—
43C2×C4×Dic79—21no—
44#14579094—42no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)C2×C412yes—
49#14579081—42no—
50C79⋊#95—22no—
51C79⋊#95—21no—
52C22×C8×C79—21no—
53C2×C4⋊C4×C79—21no—
54C2×C4⋊C4×C79—21no—
55C2×C4⋊C4×C79—21no—
56C2×C4⋊C4×C79—21no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79D412yes—
59C2×C42×C79C2311yescharacteristic
60#106×C79—44no—
61C4×C8×C79—21no—
62C4×C8×C79C2×C412yes—
63C23×D79—21no—
64C2×D316—21no—
65C2×D316—21no—
66C2×D316—21no—
67C2×D316—21no—
68C2×D316—21no—
69C2×D316C4⋊C412yes—
70C22×Dic79—21no—
71C22×Dic79—21no—
72C22×Dic79—21no—
73C2×C4×D79—22no—
74C2×C4×D79—22no—
75C2×C4×D79—41no—
76C2×C4×D79—41no—
77C2×C4×D79—41no—
78C2×C4×D79—41no—
79C2×C79⋊C8—22no—
80C2×C79⋊C8—41no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C4⋊Dic79—21no—
86C4⋊Dic79—21no—
87C4⋊Dic79—21no—
88C4⋊Dic79—21no—
89C4⋊Dic79C4⋊C412yes—
90C4⋊Dic79SD1612yes—
91(C2×D79)⋊C4—22no—
92(C2×D79)⋊C4—22no—
93(C2×D79)⋊C4—44no—
94(C2×D79)⋊C4—82no—
95C4×Dic79—44no—
96C4×Dic79—41no—
97C4×Dic79—41no—
98C22×C4×C79—22no—
99C22×C4×C79—41no—
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C2×C8×C79—22no—
104C2×C8×C79—22no—
105C2×C8×C79—22no—
106C4⋊C4×C79—42no—
107C4⋊C4×C79—42no—
108C4⋊C4×C79—41no—
109C4⋊C4×C79—41no—
110C4⋊C4×C79—41no—
111C4⋊C4×C79—41no—
112C4⋊C4×C79—41no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79—41no—
115C4⋊C4×C79—41no—
116C42×C79C22×C411yescharacteristic
117C42×C79C2×Q811yescharacteristic
118C42×C79#4111yescharacteristic
119C42×C79#4111yescharacteristic
120C23×C79#12011yescharacteristic
121C22×D79—22no—
122C22×D79—41no—
123C22×D79—41no—
124C22×D79—41no—
125C22×D79—41no—
126C22×D79—41no—
127C22×D79—41no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—42no—
131C2×C4×C79—42no—
132C2×C4×C79—42no—
133C2×C4×C79—81no—
134C2×C4×C79—81no—
135C2×C4×C79—81no—
136C2×C4×C79C2×C4⋊C411yescharacteristic
137C2×C4×C79C4×D411yescharacteristic
138C2×C4×C79C4×D411yescharacteristic
139C2×C4×C79C22⋊Q811yescharacteristic
140C2×C4×C79C22⋊Q811yescharacteristic
141C2×C4×C79#12311yescharacteristic
142C2×C4×C79C2×SD1611yescharacteristic
143C2×C4×C79#13511yescharacteristic
144C2×C4×C79#13511yescharacteristic
145C2×C4×C79#13511yescharacteristic
146C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C79#13711yescharacteristic
150C2×Dic79—22no—
151C2×Dic79—22no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159D316—22no—
160D316—22no—
161D316—41no—
162D316—41no—
163C4×D79—42no—
164C4×D79—42no—
165C4×D79—42no—
166C4×D79—42no—
167C8×C79—24no—
168C79⋊C8—42no—
169C79⋊C8—81no—
170C22×C79#38511yescharacteristic
171C22×C79#44911yescharacteristic
172C22×C79#44911yescharacteristic
173C22×C79#45011yescharacteristic
174C22×C79#45211yescharacteristic
175C22×C79#45311yescharacteristic
176C22×C79#45611yescharacteristic
177C2×D79—42no—
178C2×D79—42no—
179C2×D79—42no—
180C2×D79—81no—
181C2×D79—81no—
182C2×D79—81no—
183C2×D79—81no—
184C4×C79—21no—
185C4×C79—21no—
186C4×C79—82no—
187C4×C79—161no—
188C4×C79#42711yescharacteristic
189C4×C79#42811yescharacteristic
190C4×C79C4×SD1611yescharacteristic
191C4×C79#44211yescharacteristic
192C4×C79#47611yescharacteristic
193C4×C79#47811yescharacteristic
194C4×C79#48211yescharacteristic
195C4×C79#48511yescharacteristic
196Dic79—42no—
197Dic79—42no—
198Dic79—81no—
199#8889—791nomaximal, Sylow
200C2×C79#162411yescharacteristic
201C2×C79#163111yescharacteristic
202C2×C79#165211yescharacteristic
203C2×C79#165411yescharacteristic
204C2×C79#185411yescharacteristic
205C2×C79#185911yescharacteristic
206C2×C79#186411yescharacteristic
207D79—82no—
208#2354—791no—
209#1947—791no—
210#1844D7911yescharacteristic, complemented
211#1802—791no—
212C2×D4⋊C8—791no—
213#1296—791no—
214#1279—791no—
215C79#888911yescharacteristic, complemented, Sylow
216C2×C4×D4—791no—
217C2×#117—1581no—
218C2×#107—1581no—
219C2×#106—22no—
220C2×C4⋊C8—791no—
221C2×C4⋊C8—791no—
222C2×#102—1581no—
223C2×C4×C8C2×D7911yescharacteristic
224#390—1581no—
225#388—1582no—
226#384C2×D7911yescharacteristic
227#384C2×D7911yescharacteristic
228#380—1581no—
229C4×C4⋊C4—791no—
230C4×C4⋊C4—1581no—
231#340—1581no—
232#340—1581no—
233D4⋊C8—794no—
234C22×D4—791no—
235C23×C4—1581no—
236C22×C8—21no—
237C22×C8—1581no—
238C22×C8—1581no—
239C4×D4—792no—
240C4×D4—792no—
241C4×D4—1581no—
242C4×D4—1581no—
243#117—3162no—
244#117—3162no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—791no—
251C2×C4⋊C4—791no—
252C2×C4⋊C4—791no—
253C2×C4⋊C4C22⋊D7912yes—
254C2×C22⋊C4—1582no—
255C2×C22⋊C4—1582no—
256C2×C22⋊C4—1581no—
257C2×C42—1582no—
258C2×C42—1581no—
259C2×C42C22×D7911yescharacteristic
260#107—3162no—
261#106—44no—
262C4⋊C8—792no—
263C4⋊C8—1581no—
264C4⋊C8—1581no—
265C4⋊C8—1581no—
266#102—3162no—
267C4×C8—21no—
268C4×C8C4×D7912yes—
269#95—1582no—
270#95—1581no—
271C24—1581no—
272C2×D4—792no—
273C2×D4—1581no—
274C2×D4—1581no—
275C2×D4—1581no—
276C2×D4—1581no—
277C2×D4—1581no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—3161no—
281C22×C4—3161no—
282C22×C4—3161no—
283C22×C4—3161no—
284C22×C4—1581no—
285C22×C4—1581no—
286C22×C4—1581no—
287C22×C4—1582no—
288C22×C4—1582no—
289C22×C4D4×D7911yescharacteristic
290C22×C4D4×D7911yescharacteristic
291C22×C4C2×C22⋊D7911yescharacteristic
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—3161no—
296C2×C8—3161no—
297C2×C8—3161no—
298C2×C8—3161no—
299C2×C8—3161no—
300C2×C8—1582no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—792no—
312C4⋊C4—792no—
313C4⋊C4—1581no—
314C4⋊C4—1581no—
315C4⋊C4—1581no—
316C4⋊C4—1581no—
317C22⋊C4—3164no—
318C22⋊C4—6322no—
319C22⋊C4—1582no—
320C22⋊C4—1582no—
321C42—3164no—
322C42—3161no—
323C42—3161no—
324C42C2×C4×D7911yescharacteristic
325C42Q8×D7911yescharacteristic
326C42#1192962411yescharacteristic
327C42#1192962511yescharacteristic
328C23—3161no—
329C23—3161no—
330C23—3161no—
331C23—3161no—
332C23—3161no—
333C23—3161no—
334C23—1582no—
335C23#1457912911yescharacteristic
336D4—3161no—
337D4—3161no—
338D4—1582no—
339D4—1582no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—3161no—
349C2×C4—3161no—
350C2×C4—3161no—
351C2×C4—3161no—
352C2×C4—3162no—
353C2×C4—3162no—
354C2×C4—3162no—
355C2×C4—3162no—
356C2×C4—3161no—
357C2×C4—3161no—
358C2×C4—3161no—
359C2×C4—1582no—
360C2×C4—1582no—
361C2×C4C4⋊C4×D7911yescharacteristic
362C2×C4C4×C22⋊D7911yescharacteristic
363C2×C4#1457912311yescharacteristic
364C2×C4#1457913711yescharacteristic
365C2×C4#1457913911yescharacteristic
366C2×C4#1457914611yescharacteristic
367C2×C4SD16×D7911yescharacteristic
368C2×C4#1457918211yescharacteristic
369C2×C4#1457918211yescharacteristic
370C2×C4#1457918211yescharacteristic
371C2×C4#1457918811yescharacteristic
372C2×C4#1457918811yescharacteristic
373C2×C4#1457918811yescharacteristic
374C2×C4#1457919011yescharacteristic
375C8—24no—
376C8—3162no—
377C8—6321no—
378C22—3162no—
379C22—3162no—
380C22—3162no—
381C22—6321no—
382C22—6321no—
383C22—6321no—
384C22—6321no—
385C22#1909843411yescharacteristic
386C22#1909870711yescharacteristic
387C22#1909870711yescharacteristic
388C22#1909871211yescharacteristic
389C22#1909872211yescharacteristic
390C22#1909873011yescharacteristic
391C22#1909874511yescharacteristic
392C4—21no—
393C4—21no—
394C4—82no—
395C4—161no—
396C4—3162no—
397C4—3162no—
398C4—6321no—
399C4#1909859811yescharacteristic
400C4#1909860211yescharacteristic
401C4#1909865011yescharacteristic
402C4#1909867411yescharacteristic
403C4#1909885211yescharacteristic
404C4#1909886211yescharacteristic
405C4#1909888911yescharacteristic
406C4#1909890711yescharacteristic
407C2—6322no—
408C2#2376248111yescharacteristic
409C2#2376252411yescharacteristic
410C2#2376266711yescharacteristic
411C2#2376268111yescharacteristic
412C2#2376359811yescharacteristic
413C2#2376362511yescharacteristic
414C2#2376365811yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1844C2
socleC23×C79#120
prime radicalC22×D316D4
cosocleC2×C42×C79C23
hypercenter#1844D79
hypocenterC79#8889
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
2. #23764156C2semidirect
5. #23761283C2semidirect
6. #23761240C2semidirect
7. C79×#1844C2semidirect
21. #19098292C4semidirect
210. #1844D79semidirect
215. C79#8889semidirect

Maximal subgroups

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

subgroupindex
#237663512
#237641562
#237633972
C2×#190982922
#237612832
#237612402
C79×#18442
#888979