Finite Group Databasebeta

← #34446026

Subgroups — #34446026

Order 20224.

494 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776268C211yesmaximal, characteristic, complemented
2#23765641C211yesmaximal, characteristic, complemented
3#23765636C211yesmaximal, characteristic, complemented
4#23765631C211yesmaximal, characteristic, complemented
5#23765564C211yesmaximal, characteristic, complemented
6#23765557C211yesmaximal, characteristic, complemented
7#23765537C211yesmaximal, characteristic, complemented
8#23765452C211yesmaximal, characteristic, complemented
9#23765452C211yesmaximal, characteristic, complemented
10#23765432C211yesmaximal, characteristic
11#23765309C211yesmaximal, characteristic, complemented
12#23765274C211yesmaximal, characteristic, complemented
13#23765262C211yesmaximal, characteristic, complemented
14C4×#14579102C211yesmaximal, characteristic, complemented
15C79×#3396C211yesmaximal, characteristic, complemented
16#19099273—21no—
17C2×#14579104C2211yescharacteristic
18C2×#14579102C2211yescharacteristic
19C2×#14579102C2211yescharacteristic
20#19098968C2211yescharacteristic
21#19098965C2211yescharacteristic
22#19098965C2211yescharacteristic
23#19098864—21no—
24#19098864—21no—
25#19098854—21no—
26C4×D632C2211yescharacteristic
27C4×D632C2211yescharacteristic
28C4×D632C2211yescharacteristic
29#19098390C2211yescharacteristic
30#19098388C2211yescharacteristic
31#19098388C2211yescharacteristic
32C4⋊(C79⋊C16)C2211yescharacteristic
33C4⋊(C79⋊C16)C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098364C2211yescharacteristic
39#19098364C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098363C2211yescharacteristic
45#19098339C2211yescharacteristic
46#19098339C2211yescharacteristic
47C4×C79⋊C16C2211yescharacteristic
48#559×C79—21no—
49#495×C79C2211yescharacteristic
50#495×C79C2211yescharacteristic
51#493×C79C2211yescharacteristic
52#463×C79—21no—
53#463×C79—21no—
54#460×C79—21no—
55C8×Q8×C79C2211yescharacteristic
56C4×Q16×C79C2211yescharacteristic
57C4×D8×C79C2211yescharacteristic
58C4×D8×C79C2211yescharacteristic
59C2×D632C2311yescharacteristic
60#14579161—21no—
61#14579153—21no—
62#14579153—21no—
63#14579139—41no—
64#14579139—41no—
65#14579137—41no—
66C4×D316—21no—
67C4×D316—21no—
68C4×D316—21no—
69C4⋊C4×D79—41no—
70#14579104—22no—
71#14579102—22no—
72#14579102—22no—
73C2×C79⋊C16C2311yescharacteristic
74C2×C79⋊C16C2311yescharacteristic
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77#14579095—21no—
78#14579095C2311yescharacteristic
79#14579095C2311yescharacteristic
80#14579095C2311yescharacteristic
81#14579093—21no—
82#14579093—21no—
83#14579083—21no—
84#14579083—21no—
85#14579083—21no—
86C2×Q16×C79C2311yescharacteristic
87C2×SD16×C79—21no—
88C2×SD16×C79—21no—
89C2×D8×C79—21no—
90C2×D8×C79C2311yescharacteristic
91C2×D8×C79C2311yescharacteristic
92#127×C79—21no—
93#124×C79—21no—
94#124×C79—21no—
95#124×C79—41no—
96#123×C79—41no—
97#123×C79—41no—
98C4×Q8×C79—21no—
99C4×Q8×C79D411yescharacteristic
100C4×D4×C79—21no—
101C4×D4×C79—21no—
102C4×D4×C79—41no—
103#107×C79C2311yescharacteristic
104C4⋊C8×C79D411yescharacteristic
105C4⋊C8×C79D411yescharacteristic
106C4⋊C8×C79D411yescharacteristic
107#103×C79—21no—
108#102×C79—21no—
109#102×C79—21no—
110C4×C8×C79C2311yescharacteristic
111C4×C8×C79C2311yescharacteristic
112C4×C8×C79C2311yescharacteristic
113C2×Dic158—41no—
114C2×D316—21no—
115C2×C4×D79—41no—
116C2×C4×D79—41no—
117C2×C4×D79—41no—
118D632—21no—
119D632#4112yes—
120C4⋊Dic79—21no—
121C4⋊Dic79—21no—
122C4⋊Dic79—21no—
123C4⋊Dic79—41no—
124Dic79⋊C4—41no—
125Dic79⋊C4—41no—
126Dic79⋊C4—41no—
127Dic79⋊C4—41no—
128(C2×D79)⋊C4—41no—
129(C2×D79)⋊C4—41no—
130(C2×D79)⋊C4—41no—
131C79⋊C16—21no—
132C79⋊C16—21no—
133C79⋊C16—21no—
134C79⋊C16—21no—
135C2×Q8×C79—21no—
136C2×Q8×C79D811yescharacteristic
137C2×D4×C79—21no—
138C2×D4×C79—21no—
139C2×D4×C79—41no—
140C2×D4×C79—41no—
141C22×C4×C79—41no—
142C22×C4×C79—41no—
143Q16×C79—21no—
144Q16×C79—21no—
145SD16×C79—81no—
146SD16×C79—81no—
147D8×C79—21no—
148D8×C79—21no—
149D8×C79—21no—
150D8×C79—21no—
151D8×C79—81no—
152C2×C8×C79C2×D411yescharacteristic
153C2×C8×C79C2×D411yescharacteristic
154C2×C8×C79C2×D411yescharacteristic
155C2×C8×C79C2411yescharacteristic
156C4⋊C4×C79—21no—
157C4⋊C4×C79—41no—
158C4⋊C4×C79D811yescharacteristic
159C4⋊C4×C79D811yescharacteristic
160C4⋊C4×C79D811yescharacteristic
161C22⋊C4×C79—41no—
162C22⋊C4×C79—41no—
163C22⋊C4×C79—41no—
164C22⋊C4×C79—41no—
165C22⋊C4×C79—41no—
166C22⋊C4×C79—41no—
167C42×C79—41no—
168C42×C79C2×D411yescharacteristic
169C42×C79C2×D411yescharacteristic
170C42×C79C2×D411yescharacteristic
171C23×C79—41no—
172C23×C79—41no—
173C22×D79—41no—
174Q8×C79—41no—
175Q8×C79—41no—
176Q8×C79—41no—
177D4×C79—41no—
178D4×C79—41no—
179D4×C79—41no—
180D4×C79—41no—
181D4×C79—82no—
182D4×C79—161no—
183C2×C4×C79—41no—
184C2×C4×C79—41no—
185C2×C4×C79—161no—
186C2×C4×C79—161no—
187C2×C4×C79C2×D811yescharacteristic
188C2×C4×C79C2×D811yescharacteristic
189C2×C4×C79C2×D811yescharacteristic
190C2×C4×C79C2×D811yescharacteristic
191C2×C4×C79C2×D811yescharacteristic
192C2×C4×C79C2×D811yescharacteristic
193C2×C4×C79C22×D411yescharacteristic
194C2×Dic79—41no—
195C2×Dic79—41no—
196C2×Dic79—41no—
197D316—22no—
198D316—41no—
199C4×D79—82no—
200C4×D79—161no—
201C4×D79—161no—
202Dic158—161no—
203C8×C79—21no—
204C8×C79—21no—
205C8×C79—21no—
206C8×C79C2×#4111yescharacteristic
207C8×C792−1+411yescharacteristic
208C22×C79—81no—
209C22×C79—81no—
210C22×C79—81no—
211C22×C79—81no—
212C22×C79—81no—
213C22×C79—81no—
214C22×C79C22×D811yescharacteristic
215C2×D79—42no—
216C2×D79—81no—
217C2×D79—81no—
218C4×C79—21no—
219C4×C79—21no—
220C4×C79—21no—
221C4×C79—21no—
222C4×C79—21no—
223C4×C79—21no—
224C4×C79—81no—
225C4×C79—81no—
226C4×C79#54811yescharacteristic
227C4×C79#57711yescharacteristic
228Dic79—81no—
229Dic79—81no—
230Dic79—81no—
231#33937—791nomaximal, Sylow
232C2×C79—161no—
233C2×C79—161no—
234C2×C79#335811yescharacteristic
235C2×C79#341411yescharacteristic
236C2×C79#341511yescharacteristic
237D79—82no—
238#3396D7911yescharacteristic, complemented
239#3380—791no—
240#2244—791no—
241#2243—791no—
242#2242—791no—
243#2229—791no—
244#2228—791no—
245#2224—791no—
246#2210—791no—
247#2210—791no—
248#2206—791no—
249#2183—791no—
250#2177—791no—
251#2175—791no—
252C4×D16—791no—
253C79#3393711yescharacteristic, complemented, Sylow
254#559—21no—
255#554—1581no—
256C2×SD32—791no—
257C2×D16—791no—
258C2×D16—791no—
259#500—791no—
260#499—791no—
261#499—791no—
262#495C2×D7911yescharacteristic
263#495C2×D7911yescharacteristic
264#493C2×D7911yescharacteristic
265#478—1581no—
266#478—1581no—
267#476—1581no—
268#463—21no—
269#463—21no—
270#460—21no—
271C8×Q8C2×D7911yescharacteristic
272C4×Q16C2×D7911yescharacteristic
273C4×D8—791no—
274C4×D8—791no—
275C4×D8—791no—
276C4×D8C2×D7911yescharacteristic
277C4×D8C2×D7911yescharacteristic
278#367—791no—
279#366—791no—
280#366—791no—
281C4⋊C16—791no—
282C4⋊C16—791no—
283C4⋊C16—791no—
284#358—791no—
285#358—791no—
286#358—791no—
287#357—791no—
288#357—791no—
289#357—791no—
290#357—791no—
291#357—791no—
292#357—791no—
293#357—791no—
294#346—791no—
295#346—791no—
296C4×C16—791no—
297C2×Q16C22×D7911yescharacteristic
298C2×SD16—21no—
299C2×SD16—21no—
300C2×D8—21no—
301C2×D8—791no—
302C2×D8C22×D7911yescharacteristic
303C2×D8C22×D7911yescharacteristic
304C4⋊Q8—1581no—
305#127—21no—
306#125—1581no—
307#125—1581no—
308#124—21no—
309#124—21no—
310#124—41no—
311#123—41no—
312#123—41no—
313C22⋊Q8—3161no—
314C22⋊Q8—3161no—
315C22⋊Q8—3161no—
316C4×Q8—21no—
317C4×Q8C22⋊D7911yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—41no—
321C4×D4—1581no—
322C4×D4—1581no—
323C4×D4—1581no—
324C2×C4⋊C4—3161no—
325SD32—1582no—
326D16—1582no—
327D16—1582no—
328C2×C16—791no—
329C2×C16—791no—
330C2×C16—791no—
331C2×C16—791no—
332#107—791no—
333#107—791no—
334#107—791no—
335#107—1581no—
336#107C22×D7911yescharacteristic
337#106—1581no—
338#106—1581no—
339C4⋊C8C22⋊D7911yescharacteristic
340C4⋊C8C22⋊D7911yescharacteristic
341C4⋊C8C22⋊D7911yescharacteristic
342#103—21no—
343#102—21no—
344#102—21no—
345#102—1581no—
346#102—1581no—
347#102—1581no—
348C4×C8C22×D7911yescharacteristic
349C4×C8C22×D7911yescharacteristic
350C4×C8C22×D7911yescharacteristic
351C2×Q8—21no—
352C2×Q8—3161no—
353C2×Q8#1192960911yescharacteristic
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—1581no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—3161no—
362C22×C4—3161no—
363C22×C4—3161no—
364Q16—21no—
365Q16—21no—
366SD16—81no—
367SD16—81no—
368D8—21no—
369D8—21no—
370D8—21no—
371D8—21no—
372D8—81no—
373D8—792no—
374D8—1581no—
375C2×C8C2×C22⋊D7911yescharacteristic
376C2×C8C2×C22⋊D7911yescharacteristic
377C2×C8C2×C22⋊D7911yescharacteristic
378C2×C8C23×D7911yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—41no—
381C4⋊C4—3161no—
382C4⋊C4—3161no—
383C4⋊C4—3161no—
384C4⋊C4—3161no—
385C4⋊C4—3161no—
386C4⋊C4—1581no—
387C4⋊C4—1581no—
388C4⋊C4—1581no—
389C4⋊C4#1192960911yescharacteristic
390C4⋊C4#1192960911yescharacteristic
391C4⋊C4#1192960911yescharacteristic
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—3161no—
399C22⋊C4—3161no—
400C22⋊C4—3161no—
401C42—41no—
402C42C2×C22⋊D7911yescharacteristic
403C42C2×C22⋊D7911yescharacteristic
404C42C2×C22⋊D7911yescharacteristic
405C16—1581no—
406C16—1581no—
407C16—1581no—
408C16—1581no—
409C23—41no—
410C23—41no—
411C23—3161no—
412Q8—41no—
413Q8—41no—
414Q8—41no—
415Q8—12641no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—82no—
421D4—161no—
422D4—3161no—
423D4—1582no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—161no—
428C2×C4—3161no—
429C2×C4—3161no—
430C2×C4—3161no—
431C2×C4—6322no—
432C2×C4—12641no—
433C2×C4—12641no—
434C2×C4C2×#1192960911yescharacteristic
435C2×C4C2×#1192960911yescharacteristic
436C2×C4C2×#1192960911yescharacteristic
437C2×C4C2×#1192960911yescharacteristic
438C2×C4C2×#1192960911yescharacteristic
439C2×C4C2×#1192960911yescharacteristic
440C2×C4C22×C22⋊D7911yescharacteristic
441C8—21no—
442C8—21no—
443C8—21no—
444C8#41×D7911yescharacteristic
445C8#1457920811yescharacteristic
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—3162no—
453C22—6321no—
454C22—6321no—
455C22C22×#1192960911yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—81no—
463C4—81no—
464C4—6321no—
465C4—6321no—
466C4—6321no—
467C4#1909922011yescharacteristic
468C4#1909937411yescharacteristic
469C2—161no—
470C2—161no—
471C2—6322no—
472C2#2377603111yescharacteristic
473C2#2377653411yescharacteristic
474C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3396C2
socleC22×C79C22×D8
prime radical#23765432C2
hypercenter#3396D79
hypocenterC79#33937
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
1. #23776268C2semidirect
2. #23765641C2semidirect
3. #23765636C2semidirect
4. #23765631C2semidirect
5. #23765564C2semidirect
6. #23765557C2semidirect
7. #23765537C2semidirect
8. #23765452C2semidirect
9. #23765452C2semidirect
11. #23765309C2semidirect
12. #23765274C2semidirect
13. #23765262C2semidirect
14. C4×#14579102C2semidirect
15. C79×#3396C2semidirect
238. #3396D79semidirect
253. C79#33937semidirect

Maximal subgroups

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

subgroupindex
#237762682
#237656412
#237656362
#237656312
#237655642
#237655572
#237655372
#237654522
#237654322
#237653092
#237652742
#237652622
C4×#145791022
C79×#33962
#3393779