Finite Group Databasebeta

← #34187654

Subgroups — #34187654

Order 20224.

408 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773725C211yesmaximal, characteristic
2#23763562C211yesmaximal, characteristic, complemented
3#23761964C211yesmaximal, characteristic, complemented
4#23761961C211yesmaximal, characteristic
5#23761959C211yesmaximal, characteristic
6#23761693C211yesmaximal, characteristic, complemented
7C79×#1518C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098920—21no—
10#19098920—21no—
11#19098915C411yescharacteristic, complemented
12#19098915C411yescharacteristic, complemented
13#19098911C411yescharacteristic, complemented
14#19098911C411yescharacteristic, complemented
15#19098616C2211yescharacteristic
16C2×#14579094—21no—
17C2×#14579092—21no—
18#19098557—21no—
19#19098557—21no—
20C2×C4×C79⋊C8C2211yescharacteristic
21#19098429C2211yescharacteristic
22#19098300—22no—
23#19098298—22no—
24#19098296—22no—
25#19098285—21no—
26#19098284—21no—
27#533×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29#382×C79C2211yescharacteristic
30#328×C79—22no—
31C2×#11929624—21no—
32C22×C79⋊C8D411yescharacteristic
33C22×C79⋊C8D411yescharacteristic
34#14579161C2×C411yescharacteristic
35#14579158C2×C411yescharacteristic
36#14579149—21no—
37#14579135—21no—
38#14579135—21no—
39C4×D316—21no—
40C4×D316—21no—
41C2×C4⋊Dic79—21no—
42#14579094—42no—
43#14579092—42no—
44C4⋊(C79⋊C8)—21no—
45#14579084—21no—
46#14579084—21no—
47#14579084—21no—
48#14579084—21no—
49#14579081—21no—
50#14579081—21no—
51#14579081—21no—
52#14579081—21no—
53C22⋊(C79⋊C8)D412yes—
54C4×C79⋊C8—21no—
55C4×C79⋊C8C2×C411yescharacteristic
56C4×C79⋊C8C2×C411yescharacteristic
57C4×C79⋊C8D411yescharacteristic
58C4×C79⋊C8D411yescharacteristic
59C79⋊#95—21no—
60C79⋊#95—21no—
61C4⋊Q8×C79C2×C411yescharacteristic
62C4⋊Q8×C79C2×C411yescharacteristic
63#125×C79—21no—
64C22⋊Q8×C79—21no—
65C22⋊Q8×C79—21no—
66C4×Q8×C79—21no—
67C4×Q8×C79—21no—
68#117×C79—21no—
69C2×C4⋊C4×C79—21no—
70C2×C42×C79C2311yescharacteristic
71C4⋊C8×C79—21no—
72C22⋊C8×C79D412yes—
73C4×C8×C79—21no—
74#95×C79—21no—
75#95×C79—21no—
76#11929624—41no—
77#11929624—41no—
78#11929624—41no—
79#11929624—41no—
80C2×Dic158—21no—
81C2×D316—21no—
82C2×D316—21no—
83C2×D316—21no—
84C2×C22⋊D79—41no—
85C22×Dic79—41no—
86C2×C4×D79—41no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—21no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8C22⋊C411yescharacteristic
93C2×C79⋊C8C22⋊C411yescharacteristic
94C2×C79⋊C8C22⋊C411yescharacteristic
95C2×C79⋊C8C22⋊C411yescharacteristic
96C4⋊Dic79—21no—
97C4⋊Dic79—21no—
98C4⋊Dic79—41no—
99C4⋊Dic79—41no—
100(C2×D79)⋊C4—41no—
101(C2×D79)⋊C4—41no—
102C2×Q8×C79—21no—
103C2×Q8×C79—21no—
104C22×C4×C79—41no—
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C2×C8×C79—22no—
109C4⋊C4×C79—21no—
110C4⋊C4×C79—21no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79—41no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C22⋊C4×C79—41no—
118C42×C79—41no—
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79#4111yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79#12711yescharacteristic
124C22×D79—41no—
125Q8×C79—81no—
126Q8×C79—81no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—81no—
132C2×C4×C79—81no—
133C2×C4×C79—81no—
134C2×C4×C79C2×C22⋊C411yescharacteristic
135C2×C4×C79C4×D411yescharacteristic
136C2×C4×C79C4×D411yescharacteristic
137C2×C4×C79#12111yescharacteristic
138C2×C4×C79#12111yescharacteristic
139C2×C4×C79#12411yescharacteristic
140C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C79#13711yescharacteristic
143C2×C4×C79#13711yescharacteristic
144C22⋊D79—81no—
145C22⋊D79—81no—
146C2×Dic79—41no—
147C2×Dic79—81no—
148C2×Dic79—81no—
149C2×Dic79—81no—
150D316—41no—
151D316—41no—
152D316—81no—
153D316—81no—
154C4×D79—81no—
155C4×D79—81no—
156Dic158—41no—
157Dic158—41no—
158C8×C79—42no—
159C79⋊C8—21no—
160C79⋊C8—21no—
161C79⋊C8—21no—
162C79⋊C8—21no—
163C79⋊C8—42no—
164C22×C79—21no—
165C22×C79—21no—
166C22×C79#39011yescharacteristic
167C22×C79#49611yescharacteristic
168C22×C79#49711yescharacteristic
169C2×D79—81no—
170C2×D79—81no—
171C2×D79—81no—
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—21no—
175C4×C79—21no—
176C4×C79—81no—
177C4×C79—81no—
178C4×C79—161no—
179C4×C79#41711yescharacteristic
180C4×C79#41711yescharacteristic
181C4×C79#48911yescharacteristic
182C4×C79#48911yescharacteristic
183Dic79—81no—
184Dic79—161no—
185#9758—791nomaximal, Sylow
186C2×C79—21no—
187C2×C79—21no—
188C2×C79#195911yescharacteristic
189C2×C79#197611yescharacteristic
190C2×C79#197611yescharacteristic
191D79—161no—
192#3131—791no—
193#1846—791no—
194#1518—791no—
195#1518D7911yescharacteristic, complemented
196#1517—791no—
197#1517—791no—
198#1452—791no—
199C79#975811yescharacteristic, complemented, Sylow
200#533C2×D7911yescharacteristic
201#532—791no—
202#488—1581no—
203#488—1581no—
204#487—791no—
205#487—791no—
206#486—791no—
207#486—791no—
208#431—791no—
209#431C2×D7911yescharacteristic
210C2×#107—1581no—
211C2×#106—1581no—
212#416—1581no—
213#416—1581no—
214C2×C4×C8—791no—
215#382—791no—
216#382C2×D7911yescharacteristic
217#328—22no—
218#328—1582no—
219#327—1582no—
220#327—1582no—
221#322—1581no—
222#322—1581no—
223C2×#41—1581no—
224C22×C8—791no—
225C22×C8—791no—
226C4⋊Q8—791no—
227C4⋊Q8C4×D7911yescharacteristic
228C4⋊Q8C4×D7911yescharacteristic
229#127—791no—
230#125—21no—
231#124—1581no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234#121—1581no—
235#121—1581no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×D4—1581no—
239C4×D4—1581no—
240#117—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—1581no—
243C2×C42C22×D7911yescharacteristic
244#107—3162no—
245#106—3162no—
246C4⋊C8—21no—
247C4⋊C8—1581no—
248#103—1581no—
249#103—1581no—
250#103—1581no—
251#103—1581no—
252#102—1581no—
253#102—1581no—
254#102—1581no—
255#102—1581no—
256C22⋊C8—792no—
257C22⋊C8C22⋊D7912yes—
258C4×C8—21no—
259C4×C8—791no—
260C4×C8—791no—
261C4×C8—791no—
262C4×C8—791no—
263C4×C8—1581no—
264#95—21no—
265#95—21no—
266#95—1581no—
267#95—1581no—
268#41—3161no—
269#41—3161no—
270#41—3161no—
271#41—3161no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—1581no—
275C2×D4—3161no—
276C2×D4—1581no—
277C2×D4—1581no—
278C2×D4—1581no—
279C22×C4—41no—
280C22×C4—3161no—
281C22×C4—3161no—
282C22×C4D4×D7911yescharacteristic
283C22×C4D4×D7911yescharacteristic
284C22×C4C2×C22⋊D7911yescharacteristic
285C2×C8—22no—
286C2×C8—791no—
287C2×C8—791no—
288C2×C8—791no—
289C2×C8—791no—
290C2×C8—1582no—
291C2×C8—1581no—
292C2×C8—1581no—
293C2×C8—1581no—
294C2×C8—1581no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—3161no—
304C4⋊C4—3161no—
305C4⋊C4—1581no—
306C4⋊C4—1581no—
307C22⋊C4—41no—
308C22⋊C4—3161no—
309C22⋊C4—3161no—
310C42—41no—
311C42C2×C4×D7911yescharacteristic
312C42D4×D7911yescharacteristic
313C42#1192962311yescharacteristic
314C42#1192962411yescharacteristic
315C23—3161no—
316C23#1457915711yescharacteristic
317Q8—81no—
318Q8—81no—
319Q8—3161no—
320Q8—3161no—
321D4—3161no—
322D4—3161no—
323D4—6321no—
324D4—6321no—
325D4—6321no—
326D4—6321no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—3161no—
335C2×C4—6321no—
336C2×C4—6321no—
337C2×C4—6321no—
338C2×C4—6321no—
339C2×C4—6321no—
340C2×C4C22⋊C4×D7911yescharacteristic
341C2×C4(C22⋊D79)⋊C411yescharacteristic
342C2×C4C4×C22⋊D7911yescharacteristic
343C2×C4#1457913311yescharacteristic
344C2×C4#1457913411yescharacteristic
345C2×C4#1457914711yescharacteristic
346C2×C4#1457918811yescharacteristic
347C2×C4#1457918811yescharacteristic
348C2×C4#1457919311yescharacteristic
349C2×C4#1457919311yescharacteristic
350C8—42no—
351C8—3162no—
352C8—1581no—
353C8—1581no—
354C8—1581no—
355C8—1581no—
356C22—21no—
357C22—21no—
358C22—6321no—
359C22—6321no—
360C22—6321no—
361C22#1909846311yescharacteristic
362C22#1909894911yescharacteristic
363C22#1909895611yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—161no—
371C4—6321no—
372C4—12641no—
373C4#1909856311yescharacteristic
374C4#1909856311yescharacteristic
375C4#1909892411yescharacteristic
376C4#1909892411yescharacteristic
377C2—21no—
378C2—21no—
379C2—12641no—
380C2#2376421311yescharacteristic
381C2#2376429511yescharacteristic
382C2#2376429711yescharacteristic
subgroupG/H
centerC22#19098463
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1518C2
socleC22×C79#390
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1518D79
hypocenterC79#9758
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. #23763562C2semidirect
3. #23761964C2semidirect
6. #23761693C2semidirect
7. C79×#1518C2semidirect
11. #19098915C4semidirect
12. #19098915C4semidirect
13. #19098911C4semidirect
14. #19098911C4semidirect
195. #1518D79semidirect
199. C79#9758semidirect

Maximal subgroups

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

subgroupindex
#237737252
#237635622
#237619642
#237619612
#237619592
#237616932
C79×#15182
#975879