Finite Group Databasebeta

← #34183158

Subgroups — #34183158

Order 20224.

530 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771968C211yesmaximal, characteristic, complemented
2#23761851C211yesmaximal, characteristic, complemented
3#23761799C211yesmaximal, characteristic, complemented
4#23761782C211yesmaximal, characteristic, complemented
5#23761297C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C79×#3035C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098616C2211yescharacteristic
10#19098590—22no—
11#19098590—21no—
12#19098587—21no—
13#19098587C2211yescharacteristic
14C2×C4⋊(C79⋊C8)—21no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C4×C79⋊C8C2211yescharacteristic
17Q8⋊(C79⋊C8)—21no—
18D4⋊(C79⋊C8)—21no—
19D4⋊(C79⋊C8)—21no—
20D4⋊(C79⋊C8)—22no—
21D4⋊(C79⋊C8)—21no—
22C2×#135×C79—21no—
23#532×C79C2211yescharacteristic
24C4×#41×C79C2211yescharacteristic
25#464×C79—21no—
26#463×C79—21no—
27#462×C79C411yescharacteristic
28#461×C79C411yescharacteristic
29#460×C79C411yescharacteristic
30#459×C79C411yescharacteristic
31C2×C4⋊C8×C79C2211yescharacteristic
32#416×C79—21no—
33#14579168—42no—
34#14579168—42no—
35C2×#11929608—22no—
36C2×#11929608—21no—
37C22×C79⋊C8—22no—
38C22×C79⋊C8—21no—
39C22×C79⋊C8—21no—
40C4⋊(C79⋊C8)—22no—
41C4⋊(C79⋊C8)—22no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)D411yescharacteristic
46C4⋊(C79⋊C8)D411yescharacteristic
47C4⋊(C79⋊C8)D412yes—
48C22⋊(C79⋊C8)—21no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C2×#41×C79—21no—
53C2×#41×C79D411yescharacteristic
54#135×C79—42no—
55#135×C79—42no—
56C2×Q16×C79—21no—
57C2×SD16×C79—21no—
58C2×SD16×C79—21no—
59C2×D8×C79—21no—
60C22×C8×C79—21no—
61C4⋊Q8×C79C2×C411yescharacteristic
62#127×C79C2×C411yescharacteristic
63#124×C79—21no—
64#121×C79—21no—
65#121×C79—21no—
66C4×Q8×C79C2×C411yescharacteristic
67C4×Q8×C79D411yescharacteristic
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70C4×D4×C79—21no—
71C4×D4×C79—21no—
72C4×D4×C79C2×C411yescharacteristic
73C4×D4×C79D411yescharacteristic
74#117×C79—21no—
75#117×C79D411yescharacteristic
76C2×C42×C79—21no—
77C2×C42×C79C2311yescharacteristic
78C4⋊C8×C79—21no—
79C4⋊C8×C79C2×C411yescharacteristic
80C4⋊C8×C79C2×C411yescharacteristic
81#103×C79—21no—
82#103×C79—21no—
83#102×C79—21no—
84#102×C79—21no—
85#11929608—44no—
86#11929608—41no—
87#11929608—41no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—44no—
94C2×C79⋊C8—41no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98#41×C79—21no—
99#41×C79—21no—
100#41×C79—41no—
101#41×C79—41no—
102#41×C79—41no—
103#41×C79—41no—
104#41×C79D812yes—
105#41×C79SD1612yes—
106C2×Q8×C79—21no—
107C2×Q8×C79C22⋊C411yescharacteristic
108C2×D4×C79—21no—
109C2×D4×C79—21no—
110C2×D4×C79—21no—
111C2×D4×C79—21no—
112C2×D4×C79—41no—
113C2×D4×C79C22⋊C411yescharacteristic
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—21no—
117C22×C4×C79—41no—
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79#4111yescharacteristic
120C22×C4×C79#4111yescharacteristic
121Q16×C79—42no—
122SD16×C79—42no—
123SD16×C79—42no—
124D8×C79—42no—
125C2×C8×C79—21no—
126C2×C8×C79—21no—
127C2×C8×C79—41no—
128C2×C8×C79—41no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79—21no—
133C4⋊C4×C79C22⋊C411yescharacteristic
134C4⋊C4×C79C22⋊C411yescharacteristic
135C22⋊C4×C79—21no—
136C22⋊C4×C79—21no—
137C22⋊C4×C79—21no—
138C22⋊C4×C79—41no—
139C22⋊C4×C79—41no—
140C42×C79—21no—
141C42×C79—21no—
142C42×C79—21no—
143C42×C79C22×C411yescharacteristic
144C42×C79C2×D411yescharacteristic
145C42×C79C2×D411yescharacteristic
146C42×C79C2×D411yescharacteristic
147C23×C79—21no—
148C23×C79—41no—
149C23×C79#12311yescharacteristic
150Q8×C79—21no—
151Q8×C79—41no—
152Q8×C79—41no—
153Q8×C79#10212yes—
154D4×C79—22no—
155D4×C79—22no—
156D4×C79—21no—
157D4×C79—42no—
158D4×C79—41no—
159D4×C79—41no—
160D4×C79—41no—
161D4×C79—41no—
162D4×C79—81no—
163D4×C79—81no—
164D4×C79#10212yes—
165C2×C4×C79—21no—
166C2×C4×C79—21no—
167C2×C4×C79—21no—
168C2×C4×C79—22no—
169C2×C4×C79—22no—
170C2×C4×C79—21no—
171C2×C4×C79—21no—
172C2×C4×C79—41no—
173C2×C4×C79—41no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—81no—
177C2×C4×C79—81no—
178C2×C4×C79—81no—
179C2×C4×C79C2×C22⋊C411yescharacteristic
180C2×C4×C79C4×D411yescharacteristic
181C2×C4×C79C4×D411yescharacteristic
182C2×C4×C79#12011yescharacteristic
183C2×C4×C79#12111yescharacteristic
184C2×C4×C79#12111yescharacteristic
185C2×C4×C79C2×D811yescharacteristic
186C2×C4×C79C2×SD1611yescharacteristic
187C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C79#13711yescharacteristic
189C8×C79—41no—
190C8×C79—41no—
191C79⋊C8—42no—
192C79⋊C8—41no—
193C79⋊C8—42no—
194C79⋊C8—41no—
195C79⋊C8—81no—
196C22×C79—21no—
197C22×C79—21no—
198C22×C79—22no—
199C22×C79—41no—
200C22×C79—41no—
201C22×C79—81no—
202C22×C79—81no—
203C22×C79—81no—
204C22×C79#38611yescharacteristic
205C22×C79#48011yescharacteristic
206C22×C79#48111yescharacteristic
207C4×C79—22no—
208C4×C79—21no—
209C4×C79—21no—
210C4×C79—21no—
211C4×C79—41no—
212C4×C79—41no—
213C4×C79—81no—
214C4×C79C2×#10211yescharacteristic
215C4×C79#41711yescharacteristic
216C4×C79#44311yescharacteristic
217C4×C79#44411yescharacteristic
218C4×C79#44711yescharacteristic
219C4×C79#44811yescharacteristic
220#8995—791nomaximal, Sylow
221C2×C79—21no—
222C2×C79—21no—
223C2×C79—42no—
224C2×C79—161no—
225C2×C79#187411yescharacteristic
226C2×C79#187611yescharacteristic
227C2×C79#188111yescharacteristic
228#3035D7911yescharacteristic
229#2906—791no—
230#1492—791no—
231#1480—791no—
232#1476—791no—
233#1301—791no—
234#1277—791no—
235C79#899511yescharacteristic, complemented, Sylow
236C2×#135—21no—
237C2×#131—1581no—
238#532C2×D7911yescharacteristic
239C4×#41C2×D7911yescharacteristic
240#464—21no—
241#463—21no—
242#462Dic7911yescharacteristic
243#461Dic7911yescharacteristic
244#460Dic7911yescharacteristic
245#459Dic7911yescharacteristic
246#431—791no—
247#424—1582no—
248#424—1581no—
249#423—791no—
250#423—1581no—
251C2×C4⋊C8—791no—
252C2×C4⋊C8—1581no—
253C2×C4⋊C8C2×D7911yescharacteristic
254#416—21no—
255C2×C4×C8—791no—
256Q8⋊C8—1581no—
257D4⋊C8—1581no—
258D4⋊C8—1581no—
259D4⋊C8—1582no—
260D4⋊C8—1581no—
261C2×#41—21no—
262C2×#41C22⋊D7911yescharacteristic
263#135—42no—
264#135—42no—
265C2×Q16—21no—
266C2×SD16—21no—
267C2×SD16—21no—
268C2×D8—21no—
269#131—3162no—
270#131—3162no—
271C2×M16—1582no—
272C2×M16—1581no—
273C22×C8—21no—
274C22×C8—1582no—
275C22×C8—1581no—
276C22×C8—1581no—
277C4⋊Q8C2×Dic7911yescharacteristic
278#127C2×Dic7911yescharacteristic
279#124—21no—
280#121—21no—
281#121—21no—
282C4×Q8C2×Dic7911yescharacteristic
283C4×Q8C22⋊D7911yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4C2×Dic7911yescharacteristic
289C4×D4C22⋊D7911yescharacteristic
290#117—21no—
291#117C22⋊D7911yescharacteristic
292C2×C42—21no—
293C2×C42C22×D7911yescharacteristic
294C4⋊C8—21no—
295C4⋊C8—791no—
296C4⋊C8—791no—
297C4⋊C8—792no—
298C4⋊C8—1582no—
299C4⋊C8—1582no—
300C4⋊C8—1581no—
301C4⋊C8—1581no—
302C4⋊C8—1581no—
303C4⋊C8C2×Dic7911yescharacteristic
304C4⋊C8C2×Dic7911yescharacteristic
305#103—21no—
306#103—21no—
307#102—21no—
308#102—21no—
309C22⋊C8—1581no—
310C4×C8—1581no—
311C4×C8—1581no—
312C4×C8—1581no—
313#41—21no—
314#41—21no—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41#1192960912yes—
320#41#1192961212yes—
321C2×Q8—21no—
322C2×Q8C22⋊Dic7911yescharacteristic
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—41no—
328C2×D4C22⋊Dic7911yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4C2×C22⋊D7911yescharacteristic
334C22×C4#1192962311yescharacteristic
335C22×C4#1192962311yescharacteristic
336Q16—42no—
337SD16—42no—
338SD16—42no—
339D8—42no—
340M16—3164no—
341M16—3161no—
342M16—3161no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—3161no—
348C2×C8—3164no—
349C2×C8—3161no—
350C2×C8—3161no—
351C2×C8—3161no—
352C2×C8—3161no—
353C2×C8—1582no—
354C2×C8—1581no—
355C2×C8—1581no—
356C2×C8—1582no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4C22⋊Dic7911yescharacteristic
362C4⋊C4C22⋊Dic7911yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42C22×Dic7911yescharacteristic
372C42D4×D7911yescharacteristic
373C42D4×D7911yescharacteristic
374C42C2×C22⋊D7911yescharacteristic
375C23—21no—
376C23—41no—
377C23#1457914211yescharacteristic
378Q8—21no—
379Q8—41no—
380Q8—41no—
381Q8#1457908212yes—
382D4—22no—
383D4—22no—
384D4—21no—
385D4—42no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—81no—
391D4—81no—
392D4#1457908212yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4C2×C22⋊Dic7911yescharacteristic
408C2×C4D4×Dic7911yescharacteristic
409C2×C4D4×Dic7911yescharacteristic
410C2×C4#1457912911yescharacteristic
411C2×C4#1457913311yescharacteristic
412C2×C4#1457913311yescharacteristic
413C2×C4C2×#1192960911yescharacteristic
414C2×C4C2×#1192961211yescharacteristic
415C2×C4#1457918611yescharacteristic
416C2×C4#1457919111yescharacteristic
417C8—41no—
418C8—41no—
419C8—3162no—
420C8—3161no—
421C8—3162no—
422C8—3161no—
423C8—6321no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22#1909844311yescharacteristic
433C22#1909887411yescharacteristic
434C22#1909887911yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4C2×#1457908211yescharacteristic
443C4#1909856411yescharacteristic
444C4#1909868011yescharacteristic
445C4#1909868511yescharacteristic
446C4#1909869611yescharacteristic
447C4#1909870111yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—161no—
452C2#2376371811yescharacteristic
453C2#2376372911yescharacteristic
454C2#2376376411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3035C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3035D79
hypocenterC79#8995
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #23771968C2semidirect
2. #23761851C2semidirect
3. #23761799C2semidirect
4. #23761782C2semidirect
5. #23761297C2semidirect
6. #23761234C2semidirect
235. C79#8995semidirect

Maximal subgroups

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

subgroupindex
#237719682
#237618512
#237617992
#237617822
#237612972
#237612342
C79×#30352
#899579