Finite Group Databasebeta

← #34258107

Subgroups — #34258107

Order 20224.

662 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772483C211yesmaximal, characteristic, complemented
2#23772440C211yesmaximal, characteristic
3C2×C4×#11929613C211yesmaximal, characteristic
4#23762304C214yesmaximal, complemented
5#23762292C214yesmaximal, complemented
6#23762148C211yesmaximal, characteristic, complemented
7#23761796C211yesmaximal, characteristic, complemented
8#23761792C211yesmaximal, characteristic
9C79×#1450C211yesmaximal, characteristic, complemented
10C22×#11929613C411yescharacteristic
11C2×C4×Dic158C2211yescharacteristic
12C2×C4×D316C2211yescharacteristic
13#19098693C2214yescomplemented
14#19098690—22no—
15C4×#11929613C412yes—
16C4×#11929613C412yes—
17C4×#11929613C2212yes—
18C4×#11929613C2212yes—
19#19098642C2214yes—
20#19098640—22no—
21#19098617—22no—
22#19098616C2211yescharacteristic
23#19098616C2211yescharacteristic
24C2×#14579093C411yescharacteristic
25C2×#14579086C411yescharacteristic
26C2×#14579083C411yescharacteristic
27#19098525—22no—
28#19098316C2212yescomplemented
29#19098314C2212yescomplemented
30#19098295C2214yescomplemented
31#19098292C2214yes—
32#431×C79C2211yescharacteristic
33#431×C79C2211yescharacteristic
34C2×C4×C8×C79C2211yescharacteristic
35#322×C79C2212yescomplemented
36C82×C79C2212yescomplemented
37C22×Dic158C2×C411yescharacteristic
38C22×D316C2×C411yescharacteristic
39C22×C4×D79—21no—
40C2×#11929613—24no—
41C2×#11929613C2×C412yes—
42C2×#11929613C2×C412yes—
43C2×#11929607—24no—
44C4×Dic158—21no—
45C4×Dic158—21no—
46C4×Dic158C2×C412yes—
47C4×Dic158C2312yes—
48C4×D316—21no—
49C4×D316—21no—
50C4×D316C2×C412yes—
51C4×D316C2312yes—
52C2×C4⋊Dic79C2×C411yescharacteristic
53C2×C4⋊Dic79C2×C411yescharacteristic
54C2×Dic79⋊C4—21no—
55C2×(C2×D79)⋊C4—21no—
56C2×C4×Dic79—21no—
57#14579093C2×C412yes—
58#14579093C2×C412yes—
59C4⋊(C79⋊C8)C2312yes—
60C4⋊(C79⋊C8)C2312yes—
61Dic79⋊C8—24no—
62#14579086C2×C412yes—
63#14579086C2×C412yes—
64#14579083C2×C412yes—
65#14579083C2×C412yes—
66C22⋊(C79⋊C8)—21no—
67C22⋊(C79⋊C8)—21no—
68(C2×D79)⋊C8—24no—
69#14579071—24no—
70C4×C79⋊C8—21no—
71C4×C79⋊C8—21no—
72C22×C8×C79C2×C411yescharacteristic
73C22×C8×C79C2×C411yescharacteristic
74C2×C42×C79C2311yescharacteristic
75C4⋊C8×C79—21no—
76C4⋊C8×C79D412yes—
77C22⋊C8×C79—21no—
78C22⋊C8×C79D412yes—
79C4×C8×C79C2×C411yescharacteristic
80C4×C8×C79C2×C411yescharacteristic
81C4×C8×C79C2312yes—
82C4×C8×C79C2311yescharacteristic
83C4×C8×C79C2311yescharacteristic
84C4×C8×C79C2312yes—
85C23×D79—21no—
86C2×Dic158—21no—
87C2×Dic158—22no—
88C2×Dic158—22no—
89C2×Dic158C22×C412yes—
90C2×D316—21no—
91C2×D316—22no—
92C2×D316—22no—
93C2×D316C22×C412yes—
94C22×Dic79—21no—
95C22×Dic79—21no—
96C22×Dic79—21no—
97C2×C4×D79—22no—
98C2×C4×D79—22no—
99C2×C4×D79—82no—
100#11929613—24no—
101#11929613—24no—
102#11929607—84no—
103C2×C79⋊C8—22no—
104C2×C79⋊C8—22no—
105C4⋊Dic79—21no—
106C4⋊Dic79—21no—
107C4⋊Dic79C22×C412yes—
108C4⋊Dic79C22×C412yes—
109Dic79⋊C4—22no—
110Dic79⋊C4—22no—
111(C2×D79)⋊C4—22no—
112(C2×D79)⋊C4—22no—
113C4×Dic79—22no—
114C4×Dic79—22no—
115C22×C4×C79C22×C411yescharacteristic
116C22×C4×C79C2×D411yescharacteristic
117C22×C4×C79#4111yescharacteristic
118C2×C8×C79—22no—
119C2×C8×C79—22no—
120C2×C8×C79M1614yescomplemented
121C2×C8×C79C22×C411yescharacteristic
122C2×C8×C79C22×C411yescharacteristic
123C2×C8×C79C22×C411yescharacteristic
124C2×C8×C79C22×C411yescharacteristic
125C2×C8×C79C2×D414yes—
126C42×C79C22×C411yescharacteristic
127C42×C79C2×D411yescharacteristic
128C42×C79#4111yescharacteristic
129C42×C79C2411yescharacteristic
130C23×C79C4×D411yescharacteristic
131C22×D79—22no—
132C22×D79—42no—
133C22×D79—81no—
134C22×D79—81no—
135C2×C4×C79—22no—
136C2×C4×C79—41no—
137C2×C4×C79C4×D411yescharacteristic
138C2×C4×C79C4×D411yescharacteristic
139C2×C4×C79C4×D411yescharacteristic
140C2×C4×C79C2×M1612yes—
141C2×C4×C79#13112yes—
142C2×C4×C79C23×C411yescharacteristic
143C2×C4×C79C22×D411yescharacteristic
144C2×C4×C79C2×#4111yescharacteristic
145C2×Dic79—22no—
146C2×Dic79—22no—
147C2×Dic79—22no—
148C2×Dic79—42no—
149C2×Dic79—81no—
150C2×Dic79—81no—
151D316—22no—
152D316—22no—
153D316—42no—
154C4×D79—44no—
155C4×D79—82no—
156Dic158—22no—
157Dic158—22no—
158Dic158—42no—
159C8×C79—21no—
160C8×C79—21no—
161C8×C79—24no—
162C8×C79—42no—
163C8×C79C2×M1612yes—
164C8×C79C2×M1612yes—
165C79⋊C8—42no—
166C79⋊C8—42no—
167C22×C79—21no—
168C22×C79—21no—
169C22×C79#43612yes—
170C22×C79C2×C4×D411yescharacteristic
171C2×D79—42no—
172C2×D79—42no—
173C2×D79—42no—
174C2×D79—84no—
175C4×C79—21no—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79#43612yes—
179C4×C79C22×M1611yescharacteristic
180C4×C79C2×#13111yescharacteristic
181C4×C79#57712yes—
182Dic79—42no—
183Dic79—42no—
184Dic79—42no—
185#17896—791nomaximal, Sylow
186C2×C79—22no—
187C2×C79C2×#43611yescharacteristic
188C2×C79#294911yescharacteristic
189C2×C79#295711yescharacteristic
190D79—82no—
191#2983—791no—
192#2975—791no—
193C2×C4×SD16—791no—
194#1592—794no—
195#1590—794no—
196#1564—791no—
197#1479—791no—
198#1478—791no—
199#1450D7911yescharacteristic, complemented
200C79#1789611yescharacteristic, complemented, Sylow
201C22×SD16—791no—
202C2×C4×Q8—791no—
203C2×C4×D4—791no—
204#446—794no—
205#446—1582no—
206C4×SD16—792no—
207C4×SD16—792no—
208C4×SD16—792no—
209C4×SD16—792no—
210#436—794no—
211#436—1582no—
212#432—1582no—
213#431—791no—
214#431—791no—
215#431C2×D7911yescharacteristic
216#431C2×D7911yescharacteristic
217C2×#106—791no—
218C2×#103—791no—
219C2×#102—791no—
220#407—1582no—
221C2×C4×C8C2×D7911yescharacteristic
222#335—792no—
223#334—792no—
224Q8⋊C8—794no—
225D4⋊C8—794no—
226#322C2×D7912yescomplemented
227C82C2×D7912yescomplemented
228C22×Q8—791no—
229C22×D4—791no—
230C23×C4—1581no—
231C2×SD16—792no—
232C2×SD16—792no—
233C2×SD16—1584no—
234C2×M16—1584no—
235C22×C8C4×D7911yescharacteristic
236C22×C8C4×D7911yescharacteristic
237C4×Q8—792no—
238C4×Q8—792no—
239C4×Q8—1581no—
240C4×Q8—1581no—
241C4×D4—792no—
242C4×D4—792no—
243C4×D4—1581no—
244C4×D4—1581no—
245C2×C4⋊C4—791no—
246C2×C4⋊C4—791no—
247C2×C4⋊C4—1581no—
248C2×C22⋊C4—1581no—
249C2×C42—1581no—
250C2×C42C22×D7911yescharacteristic
251#106—792no—
252#106—792no—
253C4⋊C8—21no—
254C4⋊C8—792no—
255C4⋊C8—792no—
256C4⋊C8—1584no—
257C4⋊C8D31612yes—
258#103—792no—
259#103—792no—
260#102—792no—
261#102—792no—
262C22⋊C8—21no—
263C22⋊C8—1581no—
264C22⋊C8—1581no—
265C22⋊C8—1584no—
266C22⋊C8D31612yes—
267#97—1584no—
268C4×C8—1581no—
269C4×C8—1581no—
270C4×C8C4×D7911yescharacteristic
271C4×C8C4×D7911yescharacteristic
272C4×C8C22×D7912yes—
273C4×C8C22×D7911yescharacteristic
274C4×C8C22×D7911yescharacteristic
275C4×C8C22×D7912yes—
276C24—1581no—
277C2×Q8—792no—
278C2×Q8—1582no—
279C2×Q8—1582no—
280C2×Q8—1581no—
281C2×D4—792no—
282C2×D4—1581no—
283C2×D4—1582no—
284C2×D4—1582no—
285C22×C4—6322no—
286C22×C4—1581no—
287C22×C4—1582no—
288C22×C4—1581no—
289C22×C4—1581no—
290C22×C4—1582no—
291C22×C4C2×C4×D7911yescharacteristic
292C22×C4C2×D31611yescharacteristic
293C22×C4#1192962411yescharacteristic
294SD16—1584no—
295SD16—1584no—
296M16—6324no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—1582no—
300C2×C8—1582no—
301C2×C8#1192960714yescomplemented
302C2×C8C2×C4×D7911yescharacteristic
303C2×C8C2×C4×D7911yescharacteristic
304C2×C8C2×C4×D7911yescharacteristic
305C2×C8C2×C4×D7911yescharacteristic
306C2×C8C2×D31614yes—
307C4⋊C4—792no—
308C4⋊C4—792no—
309C4⋊C4—1582no—
310C4⋊C4—1582no—
311C4⋊C4—1581no—
312C4⋊C4—1581no—
313C22⋊C4—1582no—
314C22⋊C4—1582no—
315C42—1582no—
316C42—1582no—
317C42C2×C4×D7911yescharacteristic
318C42C2×D31611yescharacteristic
319C42#1192962411yescharacteristic
320C42C23×D7911yescharacteristic
321C23—3162no—
322C23—6321no—
323C23—6321no—
324C23—1582no—
325C23C4×D31611yescharacteristic
326Q8—3162no—
327Q8—1582no—
328Q8—1582no—
329D4—3162no—
330D4—1582no—
331D4—1582no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—3162no—
335C2×C4—3164no—
336C2×C4—6322no—
337C2×C4—6321no—
338C2×C4—6321no—
339C2×C4—1582no—
340C2×C4—1582no—
341C2×C4—1582no—
342C2×C4C4×D31611yescharacteristic
343C2×C4C4×D31611yescharacteristic
344C2×C4C4×D31611yescharacteristic
345C2×C4C2×#1192960712yes—
346C2×C4#1457916712yes—
347C2×C4C22×C4×D7911yescharacteristic
348C2×C4C22×D31611yescharacteristic
349C2×C4C2×#1192962411yescharacteristic
350C8—21no—
351C8—21no—
352C8—24no—
353C8—42no—
354C8—3162no—
355C8—3162no—
356C8C2×#1192960712yes—
357C8C2×#1192960712yes—
358C22—21no—
359C22—21no—
360C22—3162no—
361C22—3162no—
362C22—3162no—
363C22—6324no—
364C22#1909864212yes—
365C22C2×C4×D31611yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—3162no—
370C4—3162no—
371C4—3162no—
372C4#1909864212yes—
373C4C22×#1192960711yescharacteristic
374C4C2×#1457916711yescharacteristic
375C4#1909937512yes—
376C2—22no—
377C2—6322no—
378C2C2×#1909864211yescharacteristic
379C2#2377226111yescharacteristic
380C2#2377231811yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1450C2
socleC22×C79C2×C4×D4
prime radicalC22×D316C2×C4
cosocleC42×C79C24
hypercenter#1450D79
hypocenterC79#17896
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
1. #23772483C2semidirect
4. #23762304C2semidirect
5. #23762292C2semidirect
6. #23762148C2semidirect
7. #23761796C2semidirect
9. C79×#1450C2semidirect
13. #19098693C22semidirect
28. #19098316C22semidirect
29. #19098314C22semidirect
30. #19098295C22semidirect
35. #322×C79C22semidirect
36. C82×C79C22semidirect
120. C2×C8×C79M16semidirect
199. #1450D79semidirect
200. C79#17896semidirect
226. #322C2×D79semidirect
227. C82C2×D79semidirect
301. C2×C8#11929607semidirect

Maximal subgroups

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

subgroupindex
#237724832
#237724402
C2×C4×#119296132
#237623042
#237622922
#237621482
#237617962
#237617922
C79×#14502
#1789679