Finite Group Databasebeta

← #34182160

Subgroups — #34182160

Order 20224.

446 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772508C211yesmaximal, characteristic
2#23763517C211yesmaximal, characteristic, complemented
3C4×#14579089C211yesmaximal, characteristic
4#23762153C211yesmaximal, characteristic, complemented
5#23761788C211yesmaximal, characteristic
6#23761249C211yesmaximal, characteristic, complemented
7C79×#1276C211yesmaximal, characteristic, complemented
8C4×#11929624C2211yescharacteristic
9C8×Dic158—21no—
10C8×D316—21no—
11(C22⋊D79)⋊C8—22no—
12#19098615—21no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×C4⋊(C79⋊C8)C2211yescharacteristic
15C4⋊C8×D79—21no—
16C2×#14579089C411yescharacteristic
17C2×#14579089C411yescharacteristic
18C4×#11929608C2211yescharacteristic
19#19098323C411yescharacteristic
20#19098323C411yescharacteristic
21#19098318—22no—
22#19098315—21no—
23#19098313—21no—
24#19098295—21no—
25#19098292—21no—
26#431×C79C2211yescharacteristic
27#431×C79C2211yescharacteristic
28C43×C79C2211yescharacteristic
29C2×#11929624C2×C411yescharacteristic
30C2×#11929608C2×C411yescharacteristic
31C2×#11929608C2×C411yescharacteristic
32C22×C79⋊C8—21no—
33C22×C79⋊C8D412yes—
34C2×C8×D79—22no—
35C4×Dic158D411yescharacteristic
36C4×Dic158Q811yescharacteristic
37C4×D316D411yescharacteristic
38C4×D316Q811yescharacteristic
39C4×C22⋊D79—21no—
40C4×C22⋊D79—21no—
41#14579120C2×C411yescharacteristic
42#14579118—21no—
43C42×D79—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)—21no—
48Dic79⋊C8—22no—
49#14579089—22no—
50#14579089C814yescomplemented
51(C2×D79)⋊C8—22no—
52#14579072—21no—
53C4×C79⋊C8—21no—
54C8×Dic79—22no—
55C2×C42×C79—21no—
56C2×C42×C79—21no—
57C2×C42×C79C2×C411yescharacteristic
58C2×C42×C79C2×C411yescharacteristic
59C2×C42×C79C2311yescharacteristic
60C4⋊C8×C79—21no—
61C4⋊C8×C79D411yescharacteristic
62C4⋊C8×C79D411yescharacteristic
63C22⋊C8×C79—21no—
64C22⋊C8×C79—21no—
65C4×C8×C79—21no—
66C4×C8×C79—21no—
67#11929624—21no—
68#11929624—21no—
69#11929624C2×C812yes—
70#11929624M1612yes—
71C2×Dic158C4⋊C411yescharacteristic
72C2×D316C4⋊C411yescharacteristic
73C2×C22⋊D79—21no—
74C2×C4×D79—21no—
75C2×C4×D79—21no—
76C2×C4×D79—21no—
77#11929608—21no—
78#11929608—21no—
79#11929608—21no—
80#11929608C2×C812yescomplemented
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—42no—
85C2×C79⋊C8—42no—
86C2×C79⋊C8—41no—
87C8×D79—82no—
88C4⋊Dic79C4⋊C411yescharacteristic
89C4⋊Dic79C4⋊C411yescharacteristic
90Dic79⋊C4—21no—
91Dic79⋊C4—21no—
92C22⋊Dic79—21no—
93(C2×D79)⋊C4—21no—
94(C2×D79)⋊C4—21no—
95C4×Dic79—21no—
96C4×Dic79—21no—
97C4×Dic79—21no—
98C22×C4×C79—21no—
99C22×C4×C79—21no—
100C22×C4×C79C22×C411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79#4111yescharacteristic
103C2×C8×C79—21no—
104C2×C8×C79—21no—
105C2×C8×C79—41no—
106C42×C79—21no—
107C42×C79—21no—
108C42×C79—21no—
109C42×C79—41no—
110C42×C79—41no—
111C42×C79—42no—
112C42×C79C2×C812yescomplemented
113C42×C79C2×D411yescharacteristic
114C42×C79C2×D411yescharacteristic
115C42×C79C2×Q811yescharacteristic
116C42×C79#4111yescharacteristic
117C23×C79C4×D411yescharacteristic
118C22×D79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—22no—
122C2×C4×C79—22no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79C2×C4⋊C411yescharacteristic
129C2×C4×C79C4×D411yescharacteristic
130C2×C4×C79#12011yescharacteristic
131C2×C4×C79C22⋊Q811yescharacteristic
132C2×C4×C79C22⋊Q811yescharacteristic
133C2×C4×C79#12311yescharacteristic
134C2×C4×C79C22×C811yescharacteristic
135C2×C4×C79C2×M1611yescharacteristic
136C2×C4×C79#13111yescharacteristic
137C2×C4×C79#13111yescharacteristic
138C22⋊D79—22no—
139C22⋊D79—22no—
140C2×Dic79—21no—
141C2×Dic79—21no—
142C2×Dic79—21no—
143D316—21no—
144D316C4⋊C812yes—
145C4×D79—22no—
146C4×D79—22no—
147C4×D79—41no—
148C4×D79—41no—
149C4×D79—81no—
150Dic158—21no—
151Dic158C4⋊C812yes—
152C8×C79—41no—
153C8×C79—41no—
154C8×C79—81no—
155C79⋊C8—22no—
156C79⋊C8—41no—
157C79⋊C8—42no—
158C79⋊C8—81no—
159C22×C79—21no—
160C22×C79—21no—
161C22×C79#38511yescharacteristic
162C22×C79C8×D411yescharacteristic
163C22×C79#43511yescharacteristic
164C2×D79—22no—
165C2×D79—41no—
166C2×D79—41no—
167C4×C79—21no—
168C4×C79—21no—
169C4×C79—21no—
170C4×C79—42no—
171C4×C79—42no—
172C4×C79C2×C4⋊C811yescharacteristic
173C4×C79#42411yescharacteristic
174C4×C79C8×D411yescharacteristic
175C4×C79#43511yescharacteristic
176C4×C79#45411yescharacteristic
177C4×C79#45511yescharacteristic
178Dic79—22no—
179Dic79—41no—
180Dic79—41no—
181#8840—791nomaximal, Sylow
182C2×C79—21no—
183C2×C79—21no—
184C2×C79#185011yescharacteristic
185C2×C79#186011yescharacteristic
186C2×C79#186111yescharacteristic
187D79—42no—
188#2988—791no—
189#1831—791no—
190C4×#104—791no—
191#1565—791no—
192#1477—791no—
193#1283—791no—
194#1276D7911yescharacteristic, complemented
195C79#884011yescharacteristic, complemented, Sylow
196C4×#41—791no—
197C8×Q8—1581no—
198C8×D4—1581no—
199C8×D4—1582no—
200#431—1581no—
201#431C2×D7911yescharacteristic
202#431C2×D7911yescharacteristic
203C2×C4⋊C8—791no—
204C2×C4⋊C8—791no—
205C2×C4⋊C8—1581no—
206C2×#104—791no—
207C2×#104—791no—
208C4×M16—791no—
209C43C2×D7911yescharacteristic
210#339—791no—
211#339—791no—
212#336—1582no—
213#335—1581no—
214#334—1581no—
215Q8⋊C8—1581no—
216D4⋊C8—1581no—
217C2×#41—791no—
218C2×M16—791no—
219C2×M16—791no—
220C22×C8—792no—
221C22×C8—1582no—
222C22×C8—1581no—
223C4×Q8—791no—
224C4×Q8—791no—
225C4×D4—791no—
226C4×D4—791no—
227C4×D4—1581no—
228C4×D4—1581no—
229#117—791no—
230#117—1581no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—1581no—
234C2×C42C4×D7911yescharacteristic
235C2×C42C4×D7911yescharacteristic
236C2×C42C22×D7911yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—1582no—
239C4⋊C8—1581no—
240C4⋊C8—1581no—
241C4⋊C8—1581no—
242C4⋊C8—1581no—
243C4⋊C8D31611yescharacteristic
244C4⋊C8D31611yescharacteristic
245#104—794no—
246#104—1582no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—1582no—
250#97—1581no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—1582no—
254C4×C8—1581no—
255#41—792no—
256#41—792no—
257#41—1581no—
258#41—1581no—
259C2×Q8—791no—
260C2×D4—791no—
261C2×D4—1581no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—1581no—
265C22×C4—1581no—
266C22×C4—1581no—
267C22×C4C2×C4×D7911yescharacteristic
268C22×C4D4×D7911yescharacteristic
269C22×C4#1192962311yescharacteristic
270M16—792no—
271M16—1581no—
272M16—1581no—
273M16—1581no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—3162no—
278C2×C8—3162no—
279C2×C8—3161no—
280C2×C8—6322no—
281C2×C8—1581no—
282C2×C8—1582no—
283C2×C8—1581no—
284C4⋊C4—791no—
285C4⋊C4—791no—
286C4⋊C4—1581no—
287C4⋊C4—1581no—
288C22⋊C4—1581no—
289C22⋊C4—1581no—
290C22⋊C4—1581no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—41no—
295C42—41no—
296C42—42no—
297C42—1581no—
298C42—1581no—
299C42—1581no—
300C42C8×D7912yescomplemented
301C42D4×D7911yescharacteristic
302C42C2×D31611yescharacteristic
303C42Q8×D7911yescharacteristic
304C42#1192962411yescharacteristic
305C23—1581no—
306C23(C22⋊D79)⋊C411yescharacteristic
307Q8—792no—
308Q8—1581no—
309D4—792no—
310D4—1582no—
311D4—1582no—
312D4—1581no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—3161no—
323C2×C4—3161no—
324C2×C4—6321no—
325C2×C4—1581no—
326C2×C4—1581no—
327C2×C4—1581no—
328C2×C4—1582no—
329C2×C4—1582no—
330C2×C4C4⋊C4×D7911yescharacteristic
331C2×C4C4×D31611yescharacteristic
332C2×C4#1457913111yescharacteristic
333C2×C4#1457913911yescharacteristic
334C2×C4#1457914111yescharacteristic
335C2×C4#1457914311yescharacteristic
336C2×C4C2×C8×D7911yescharacteristic
337C2×C4M16×D7911yescharacteristic
338C2×C4#1457916711yescharacteristic
339C2×C4#1457916911yescharacteristic
340C8—41no—
341C8—41no—
342C8—81no—
343C8—3161no—
344C8—3162no—
345C8—6321no—
346C8—1582no—
347C22—21no—
348C22—21no—
349C22—3161no—
350C22—3161no—
351C22—1582no—
352C22#1909843711yescharacteristic
353C22(C22⋊D79)⋊C811yescharacteristic
354C22#1909863711yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—3161no—
361C4—3161no—
362C4—1582no—
363C4C4⋊C8×D7911yescharacteristic
364C4#1909858811yescharacteristic
365C4C8×D31611yescharacteristic
366C4#1909863611yescharacteristic
367C4#1909873711yescharacteristic
368C4#1909874211yescharacteristic
369C2—21no—
370C2—21no—
371C2—3162no—
372C2#2376357911yescharacteristic
373C2#2376363411yescharacteristic
374C2#2376364111yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC2×C4×C79C22×C8
FrattiniC2×C42C22×D79
FittingC79×#1276C2
socleC22×C79#385
prime radicalC2×#11929624C2×C4
cosocleC2×C42×C79C23
hypercenter#1276D79
hypocenterC79#8840
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #23763517C2semidirect
4. #23762153C2semidirect
6. #23761249C2semidirect
7. C79×#1276C2semidirect
50. #14579089C8semidirect
80. #11929608C2×C8semidirect
112. C42×C79C2×C8semidirect
194. #1276D79semidirect
195. C79#8840semidirect
300. C42C8×D79semidirect

Maximal subgroups

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

subgroupindex
#237725082
#237635172
C4×#145790892
#237621532
#237617882
#237612492
C79×#12762
#884079