Finite Group Databasebeta

← #34182167

Subgroups — #34182167

Order 20224.

440 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772513C211yesmaximal, characteristic
2#23763552C211yesmaximal, characteristic, complemented
3C4×#14579089C211yesmaximal, characteristic
4#23762153C211yesmaximal, characteristic, complemented
5#23761788C211yesmaximal, characteristic
6#23761249C211yesmaximal, characteristic, complemented
7C79×#1276C211yesmaximal, characteristic, complemented
8C4×#11929624C2211yescharacteristic
9#19098693—21no—
10#19098642—21no—
11#19098631—22no—
12#19098619—21no—
13#19098585—21no—
14C2×C4⋊(C79⋊C8)—22no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17C2×#14579089C411yescharacteristic
18C2×#14579089C411yescharacteristic
19C4×#11929608C2211yescharacteristic
20#19098323C411yescharacteristic
21#19098323C411yescharacteristic
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
32C2×#11929607—22no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8D412yes—
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—
48C4⋊(C79⋊C8)—44no—
49Dic79⋊C8—22no—
50#14579089—22no—
51#14579089—22no—
52(C2×D79)⋊C8—22no—
53#14579072—21no—
54#14579071—22no—
55C4×C79⋊C8—21no—
56C2×C42×C79—21no—
57C2×C42×C79—21no—
58C2×C42×C79C2×C411yescharacteristic
59C2×C42×C79C2×C411yescharacteristic
60C2×C42×C79C2311yescharacteristic
61C4⋊C8×C79—21no—
62C4⋊C8×C79D411yescharacteristic
63C4⋊C8×C79D411yescharacteristic
64C22⋊C8×C79—21no—
65C22⋊C8×C79—21no—
66C4×C8×C79—21no—
67C4×C8×C79—21no—
68#11929624—21no—
69#11929624—21no—
70#11929624—21no—
71#11929624—21no—
72C2×Dic158C4⋊C411yescharacteristic
73C2×D316C4⋊C411yescharacteristic
74C2×C22⋊D79—21no—
75C2×C4×D79—21no—
76C2×C4×D79—21no—
77C2×C4×D79—21no—
78#11929608—21no—
79#11929608—21no—
80#11929608—21no—
81#11929608M1612yescomplemented
82#11929607—82no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—42no—
87C2×C79⋊C8—42no—
88C2×C79⋊C8—41no—
89C4⋊Dic79C4⋊C411yescharacteristic
90C4⋊Dic79C4⋊C411yescharacteristic
91Dic79⋊C4—21no—
92Dic79⋊C4—21no—
93C22⋊Dic79—21no—
94(C2×D79)⋊C4—21no—
95(C2×D79)⋊C4—21no—
96C4×Dic79—21no—
97C4×Dic79—21no—
98C4×Dic79—21no—
99C22×C4×C79—21no—
100C22×C4×C79—21no—
101C22×C4×C79C22×C411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79#4111yescharacteristic
104C2×C8×C79—21no—
105C2×C8×C79—21no—
106C2×C8×C79—41no—
107C42×C79—22no—
108C42×C79—22no—
109C42×C79—21no—
110C42×C79—21no—
111C42×C79—21no—
112C42×C79—42no—
113C42×C79M1612yescomplemented
114C42×C79C2×D411yescharacteristic
115C42×C79C2×D411yescharacteristic
116C42×C79C2×Q811yescharacteristic
117C42×C79#4111yescharacteristic
118C23×C79C4×D411yescharacteristic
119C22×D79—21no—
120C2×C4×C79—22no—
121C2×C4×C79—22no—
122C2×C4×C79—22no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—22no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79C2×C4⋊C411yescharacteristic
130C2×C4×C79C4×D411yescharacteristic
131C2×C4×C79#12011yescharacteristic
132C2×C4×C79C22⋊Q811yescharacteristic
133C2×C4×C79C22⋊Q811yescharacteristic
134C2×C4×C79#12311yescharacteristic
135C2×C4×C79C2×M1611yescharacteristic
136C2×C4×C79C2×M1611yescharacteristic
137C2×C4×C79#13111yescharacteristic
138C2×C4×C79#13111yescharacteristic
139C22⋊D79—41no—
140C22⋊D79—41no—
141C2×Dic79—21no—
142C2×Dic79—21no—
143C2×Dic79—21no—
144D316—21no—
145D316—21no—
146C4×D79—41no—
147C4×D79—41no—
148C4×D79—41no—
149C4×D79—41no—
150C4×D79—81no—
151Dic158—21no—
152Dic158—21no—
153C8×C79—41no—
154C8×C79—41no—
155C8×C79—81no—
156C79⋊C8—22no—
157C79⋊C8—41no—
158C79⋊C8—42no—
159C79⋊C8—81no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79#38511yescharacteristic
163C22×C79#43511yescharacteristic
164C22×C79#43611yescharacteristic
165C2×D79—41no—
166C2×D79—41no—
167C2×D79—41no—
168C4×C79—24no—
169C4×C79—21no—
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—42no—
173C4×C79#42311yescharacteristic
174C4×C79#42411yescharacteristic
175C4×C79#43511yescharacteristic
176C4×C79#43611yescharacteristic
177C4×C79#45311yescharacteristic
178C4×C79#45611yescharacteristic
179Dic79—41no—
180Dic79—41no—
181Dic79—41no—
182#8841—791nomaximal, Sylow
183C2×C79—21no—
184C2×C79—21no—
185C2×C79#185111yescharacteristic
186C2×C79#185911yescharacteristic
187C2×C79#186111yescharacteristic
188D79—81no—
189#2989—791no—
190#1842—791no—
191C4×#104—791no—
192#1565—791no—
193#1477—791no—
194#1283—791no—
195#1276D7911yescharacteristic, complemented
196C79#884111yescharacteristic, complemented, Sylow
197C4×#41—791no—
198#446—1581no—
199#436—1581no—
200#435—1582no—
201#432—1581no—
202#431C2×D7911yescharacteristic
203#431C2×D7911yescharacteristic
204#423—1581no—
205C2×C4⋊C8—791no—
206C2×C4⋊C8—791no—
207C2×C4⋊C8—1582no—
208C2×#104—791no—
209C2×#104—791no—
210C4×M16—791no—
211C43C2×D7911yescharacteristic
212#339—791no—
213#339—791no—
214#335—1581no—
215#334—1581no—
216Q8⋊C8—1581no—
217D4⋊C8—1581no—
218C2×#41—791no—
219C2×M16—791no—
220C2×M16—791no—
221C2×M16—1582no—
222C22×C8—792no—
223C22×C8—1581no—
224C4×Q8—791no—
225C4×Q8—791no—
226C4×D4—791no—
227C4×D4—791no—
228C4×D4—1581no—
229C4×D4—1581no—
230#117—791no—
231#117—1581no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—1581no—
235C2×C42C4×D7911yescharacteristic
236C2×C42C4×D7911yescharacteristic
237C2×C42C22×D7911yescharacteristic
238C4⋊C8—21no—
239C4⋊C8—3164no—
240C4⋊C8—1582no—
241C4⋊C8—1581no—
242C4⋊C8—1581no—
243C4⋊C8—1581no—
244C4⋊C8—1581no—
245C4⋊C8D31611yescharacteristic
246C4⋊C8D31611yescharacteristic
247#104—1582no—
248#104—1582no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—1582no—
252#97—1582no—
253#97—1581no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—1581no—
257#41—1581no—
258#41—1581no—
259#41—1581no—
260#41—1581no—
261C2×Q8—791no—
262C2×D4—791no—
263C2×D4—1581no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—1581no—
267C22×C4—1581no—
268C22×C4—1581no—
269C22×C4C2×C4×D7911yescharacteristic
270C22×C4D4×D7911yescharacteristic
271C22×C4#1192962311yescharacteristic
272M16—792no—
273M16—6322no—
274M16—1581no—
275M16—1581no—
276M16—1581no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—3162no—
281C2×C8—3162no—
282C2×C8—3161no—
283C2×C8—1581no—
284C2×C8—1582no—
285C2×C8—1581no—
286C4⋊C4—791no—
287C4⋊C4—791no—
288C4⋊C4—1581no—
289C4⋊C4—1581no—
290C22⋊C4—1581no—
291C22⋊C4—1581no—
292C22⋊C4—1581no—
293C42—22no—
294C42—22no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—42no—
299C42—1581no—
300C42—1581no—
301C42—1581no—
302C42#1192960712yescomplemented
303C42D4×D7911yescharacteristic
304C42C2×D31611yescharacteristic
305C42Q8×D7911yescharacteristic
306C42#1192962411yescharacteristic
307C23—1581no—
308C23(C22⋊D79)⋊C411yescharacteristic
309Q8—1581no—
310Q8—1581no—
311D4—3161no—
312D4—3161no—
313D4—1581no—
314D4—1581no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—3161no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—6321no—
329C2×C4—1581no—
330C2×C4—1581no—
331C2×C4—1581no—
332C2×C4C4⋊C4×D7911yescharacteristic
333C2×C4C4×D31611yescharacteristic
334C2×C4#1457913111yescharacteristic
335C2×C4#1457913911yescharacteristic
336C2×C4#1457914111yescharacteristic
337C2×C4#1457914311yescharacteristic
338C2×C4M16×D7911yescharacteristic
339C2×C4C2×#1192960711yescharacteristic
340C2×C4#1457916711yescharacteristic
341C2×C4#1457916911yescharacteristic
342C8—41no—
343C8—41no—
344C8—81no—
345C8—3161no—
346C8—3162no—
347C8—6321no—
348C8—1582no—
349C22—21no—
350C22—21no—
351C22—3161no—
352C22—3161no—
353C22—3161no—
354C22#1909843711yescharacteristic
355C22#1909863111yescharacteristic
356C22#1909863811yescharacteristic
357C4—24no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—3161no—
363C4—3161no—
364C4—3161no—
365C4#1909858511yescharacteristic
366C4#1909858811yescharacteristic
367C4#1909863611yescharacteristic
368C4#1909864211yescharacteristic
369C4#1909873211yescharacteristic
370C4#1909874711yescharacteristic
371C2—21no—
372C2—21no—
373C2—6321no—
374C2#2376358411yescharacteristic
375C2#2376362711yescharacteristic
376C2#2376364111yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1276C2
socleC22×C79#385
prime radicalC2×#11929624C2×C4
cosocleC2×C42×C79C23
hypercenter#1276D79
hypocenterC79#8841
LCSG ▷ C22×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. #23763552C2semidirect
4. #23762153C2semidirect
6. #23761249C2semidirect
7. C79×#1276C2semidirect
81. #11929608M16semidirect
113. C42×C79M16semidirect
195. #1276D79semidirect
196. C79#8841semidirect
302. C42#11929607semidirect

Maximal subgroups

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

subgroupindex
#237725132
#237635522
C4×#145790892
#237621532
#237617882
#237612492
C79×#12762
#884179