Finite Group Databasebeta

← #34443355

Subgroups — #34443355

Order 20224.

548 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773986C211yesmaximal, characteristic, complemented
2#23765608C212yesmaximal, complemented
3#23765552C211yesmaximal, characteristic, complemented
4#23765547C211yesmaximal, characteristic, complemented
5#23765450C211yesmaximal, characteristic, complemented
6#23765445C211yesmaximal, characteristic, complemented
7#23765340C212yesmaximal
8#23765269C212yesmaximal, complemented
9#23765264C212yesmaximal, complemented
10#23765203C211yesmaximal, characteristic, complemented
11C4×#135×C79C211yesmaximal, characteristic
12#19099120—21no—
13C2×#14579107C2212yes—
14C2×#14579105C2212yes—
15#19098968C2211yescharacteristic
16#19098943C2211yescharacteristic
17#19098914—21no—
18#19098804—21no—
19#19098804C2212yes—
20C4×Dic316—21no—
21C4×Dic316C2212yes—
22#19098390C2212yes—
23#19098388C2212yes—
24C4⋊(C79⋊C16)C2211yescharacteristic
25C4⋊(C79⋊C16)C2211yescharacteristic
26#19098367C2212yes—
27#19098366C2212yes—
28#19098366C2212yes—
29#19098364C2212yes—
30C22⋊(C79⋊C16)C2212yes—
31#19098339C2212yes—
32C2×#135×C79C2211yescharacteristic
33C4×#41×C79—21no—
34C4×Q16×C79C2211yescharacteristic
35C4×Q16×C79C2211yescharacteristic
36C4×SD16×C79—21no—
37C4×SD16×C79—21no—
38C4×D8×C79C2211yescharacteristic
39C4×D8×C79C2211yescharacteristic
40#427×C79C2211yescharacteristic
41#416×C79—21no—
42C2×C4×C8×C79C2211yescharacteristic
43C2×Dic316—21no—
44C2×Dic316C2312yes—
45#14579161—21no—
46#14579161—21no—
47#14579153—41no—
48#14579140—22no—
49#14579140—41no—
50C4×Dic158—22no—
51C4×Dic158—41no—
52#14579120—41no—
53#14579107—24no—
54#14579105—24no—
55C2×C79⋊C16C2312yes—
56C2×C79⋊C16C2312yes—
57#14579095—21no—
58#14579095C2312yes—
59#14579086—22no—
60#14579086—22no—
61C2×#41×C79—21no—
62#135×C79—22no—
63#135×C79—22no—
64C2×Q16×C79C2311yescharacteristic
65C2×SD16×C79—21no—
66C2×D8×C79C2311yescharacteristic
67C22×C8×C79D411yescharacteristic
68C22×C8×C79C2311yescharacteristic
69C4×Q8×C79—21no—
70C4×Q8×C79—21no—
71C4×D4×C79—21no—
72C4×D4×C79—21no—
73C4×D4×C79—41no—
74C4×D4×C79—41no—
75#117×C79—21no—
76#117×C79—41no—
77C2×C42×C79—41no—
78C2×C42×C79D411yescharacteristic
79#107×C79C2311yescharacteristic
80#107×C79C2311yescharacteristic
81#106×C79—21no—
82#103×C79—21no—
83#103×C79—21no—
84#102×C79—21no—
85#102×C79—21no—
86C4×C8×C79D411yescharacteristic
87C4×C8×C79D411yescharacteristic
88C4×C8×C79C2311yescharacteristic
89C4×C8×C79C2311yescharacteristic
90C2×Dic158—22no—
91C2×Dic158—41no—
92Dic316—22no—
93Dic316—22no—
94Dic316—42no—
95C4⋊Dic79—22no—
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98C4⋊Dic79—41no—
99Dic79⋊C4—42no—
100Dic79⋊C4—42no—
101C22⋊Dic79—42no—
102C4×Dic79—42no—
103C79⋊C16—22no—
104C79⋊C16—22no—
105#41×C79—24no—
106#41×C79—42no—
107C2×Q8×C79—21no—
108C2×D4×C79—21no—
109C2×D4×C79—41no—
110C22×C4×C79—41no—
111C22×C4×C79—41no—
112C22×C4×C79—41no—
113C22×C4×C79D811yescharacteristic
114C22×C4×C79D811yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116Q16×C79—21no—
117Q16×C79#4112yes—
118SD16×C79—24no—
119D8×C79—21no—
120D8×C79#4112yes—
121C2×C8×C79—21no—
122C2×C8×C79—21no—
123C2×C8×C79—41no—
124C2×C8×C79C2×D411yescharacteristic
125C2×C8×C79C2×D411yescharacteristic
126C2×C8×C79C2×D411yescharacteristic
127C2×C8×C79C2411yescharacteristic
128C4⋊C4×C79—21no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—41no—
131C4⋊C4×C79—41no—
132C22⋊C4×C79—41no—
133C22⋊C4×C79—41no—
134C22⋊C4×C79—41no—
135C42×C79—41no—
136C42×C79—41no—
137C42×C79—41no—
138C42×C79D811yescharacteristic
139C42×C79D811yescharacteristic
140C42×C79C2×D411yescharacteristic
141C42×C79C2×D411yescharacteristic
142C23×C79—41no—
143C23×C79C2×D811yescharacteristic
144Q8×C79—22no—
145Q8×C79—41no—
146D4×C79—22no—
147D4×C79—44no—
148D4×C79—41no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—22no—
152C2×C4×C79—41no—
153C2×C4×C79—44no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—82no—
158C2×C4×C79—82no—
159C2×C4×C79C2×D811yescharacteristic
160C2×C4×C79C2×D811yescharacteristic
161C2×C4×C79C2×D811yescharacteristic
162C2×C4×C79C2×D811yescharacteristic
163C2×C4×C79C2×D811yescharacteristic
164C2×C4×C79C22×D411yescharacteristic
165C2×Dic79—42no—
166C2×Dic79—42no—
167Dic158—42no—
168Dic158—42no—
169Dic158—82no—
170C8×C79—21no—
171C8×C79—21no—
172C8×C79—21no—
173C8×C79C2×#4111yescharacteristic
174C8×C79C2×#4111yescharacteristic
175C22×C79—21no—
176C22×C79—21no—
177C22×C79—21no—
178C22×C79—42no—
179C22×C79—81no—
180C22×C79—81no—
181C22×C79C22×D811yescharacteristic
182C4×C79—21no—
183C4×C79—21no—
184C4×C79—21no—
185C4×C79—21no—
186C4×C79—42no—
187C4×C79—81no—
188C4×C79—81no—
189C4×C79#51012yes—
190C4×C79#52511yescharacteristic
191C4×C79#57511yescharacteristic
192Dic79—82no—
193Dic79—82no—
194#33676—791nomaximal, Sylow
195C2×C79—41no—
196C2×C79—82no—
197C2×C79#310011yescharacteristic
198C2×C79C2×#51011yescharacteristic
199C2×C79#341411yescharacteristic
200#3170—791no—
201C4×#135D7911yescharacteristic
202#2237—792no—
203#2227—791no—
204#2226—791no—
205#2209—791no—
206#2208—791no—
207#2189—792no—
208#2176—792no—
209#2175—792no—
210#2163—791no—
211C79#3367611yescharacteristic, complemented, Sylow
212C2×#135C2×D7911yescharacteristic
213#533—1581no—
214C4×#41—21no—
215C2×Q32—792no—
216C2×SD32—792no—
217#500—791no—
218#494—791no—
219#487—1581no—
220#467—792no—
221#467—1581no—
222C4×Q16—792no—
223C4×Q16—1581no—
224C4×Q16C2×D7911yescharacteristic
225C4×Q16C2×D7911yescharacteristic
226C4×SD16—21no—
227C4×SD16—21no—
228C4×D8C2×D7911yescharacteristic
229C4×D8C2×D7911yescharacteristic
230#427C2×D7911yescharacteristic
231#416—21no—
232C2×C4×C8C2×D7911yescharacteristic
233#367—792no—
234#366—792no—
235C4⋊C16—791no—
236C4⋊C16—791no—
237#358—792no—
238#358—792no—
239#358—792no—
240#357—792no—
241C22⋊C16—792no—
242#346—792no—
243C2×#41—21no—
244#135—22no—
245#135—22no—
246C2×Q16—792no—
247C2×Q16—1581no—
248C2×Q16C22×D7911yescharacteristic
249C2×SD16—21no—
250C2×D8C22×D7911yescharacteristic
251C22×C8C22⋊D7911yescharacteristic
252C22×C8C22×D7911yescharacteristic
253C4⋊Q8—1581no—
254C4⋊Q8—1581no—
255#125—3161no—
256C22⋊Q8—3161no—
257C22⋊Q8—1582no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—3161no—
261C4×Q8—1582no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—41no—
265C4×D4—41no—
266#117—21no—
267#117—41no—
268#117—3161no—
269C2×C42—41no—
270C2×C42C22⋊D7911yescharacteristic
271Q32—1584no—
272SD32—1584no—
273C2×C16—792no—
274C2×C16—792no—
275#107—792no—
276#107—1581no—
277#107C22×D7911yescharacteristic
278#107C22×D7911yescharacteristic
279#106—21no—
280#103—21no—
281#103—21no—
282#103—1582no—
283#103—1582no—
284#102—21no—
285#102—21no—
286C4×C8C22⋊D7911yescharacteristic
287C4×C8C22⋊D7911yescharacteristic
288C4×C8C22×D7911yescharacteristic
289C4×C8C22×D7911yescharacteristic
290#41—24no—
291#41—42no—
292C2×Q8—21no—
293C2×Q8—3161no—
294C2×Q8—1582no—
295C2×D4—21no—
296C2×D4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4#1192960911yescharacteristic
301C22×C4#1192960911yescharacteristic
302C22×C4C2×C22⋊D7911yescharacteristic
303Q16—21no—
304Q16—3162no—
305Q16—1582no—
306Q16—1582no—
307Q16#1192962412yes—
308SD16—24no—
309D8—21no—
310D8#1192962412yes—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8C2×C22⋊D7911yescharacteristic
315C2×C8C2×C22⋊D7911yescharacteristic
316C2×C8C2×C22⋊D7911yescharacteristic
317C2×C8C23×D7911yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—3162no—
323C4⋊C4—3162no—
324C4⋊C4—3161no—
325C4⋊C4—3161no—
326C4⋊C4—3161no—
327C4⋊C4—1582no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—3162no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—3162no—
336C42#1192960911yescharacteristic
337C42#1192960911yescharacteristic
338C42C2×C22⋊D7911yescharacteristic
339C42C2×C22⋊D7911yescharacteristic
340C16—1582no—
341C16—1582no—
342C23—41no—
343C23C2×#1192960911yescharacteristic
344Q8—22no—
345Q8—41no—
346Q8—3162no—
347Q8—3162no—
348Q8—6322no—
349D4—22no—
350D4—44no—
351D4—41no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—3162no—
363C2×C4—3162no—
364C2×C4C2×#1192960911yescharacteristic
365C2×C4C2×#1192960911yescharacteristic
366C2×C4C2×#1192960911yescharacteristic
367C2×C4C2×#1192960911yescharacteristic
368C2×C4C2×#1192960911yescharacteristic
369C2×C4C22×C22⋊D7911yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8C2×#1192962411yescharacteristic
374C8C2×#1192962411yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22C22×#1192960911yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—42no—
387C4—81no—
388C4—81no—
389C4—6322no—
390C4—6322no—
391C4#1909900312yes—
392C4#1909907411yescharacteristic
393C4#1909936111yescharacteristic
394C2—41no—
395C2—82no—
396C2#2377340011yescharacteristic
397C2C2×#1909900311yescharacteristic
398C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC4×#135×C79C2
socleC22×C79C22×D8
prime radicalC2×#135×C79C22
hypercenterC4×#135D79
hypocenterC79#33676
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #23773986C2semidirect
2. #23765608C2semidirect
3. #23765552C2semidirect
4. #23765547C2semidirect
5. #23765450C2semidirect
6. #23765445C2semidirect
8. #23765269C2semidirect
9. #23765264C2semidirect
10. #23765203C2semidirect
211. C79#33676semidirect

Maximal subgroups

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

subgroupindex
#237739862
#237656082
#237655522
#237655472
#237654502
#237654452
#237653402
#237652692
#237652642
#237652032
C4×#135×C792
#3367679