Finite Group Databasebeta

← #34443774

Subgroups — #34443774

Order 20224.

474 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773937C211yesmaximal, characteristic
2#23773933C211yesmaximal, characteristic
3#23765686C211yesmaximal, characteristic, complemented
4#23765685C211yesmaximal, characteristic, complemented
5#23765684C211yesmaximal, characteristic, complemented
6#23765678C211yesmaximal, characteristic, complemented
7#23765676C211yesmaximal, characteristic, complemented
8#23765674C211yesmaximal, characteristic, complemented
9#23765673C211yesmaximal, characteristic, complemented
10#23765187C211yesmaximal, characteristic
11#23765185C211yesmaximal, characteristic
12#23765183C211yesmaximal, characteristic
13#366×D79C211yesmaximal, characteristic
14#23764998C211yesmaximal, characteristic
15C79×#2255C211yesmaximal, characteristic, complemented
16#19099193—21no—
17#19099190—21no—
18C2×#14579100C2211yescharacteristic
19C2×C16×D79C2211yescharacteristic
20#19098973—21no—
21#19098971—21no—
22#19098970—21no—
23#19098969—21no—
24#19098968C2211yescharacteristic
25#19098967C2211yescharacteristic
26#19098967C2211yescharacteristic
27#19098965C2211yescharacteristic
28#19098964C2211yescharacteristic
29#19098964C2211yescharacteristic
30#19098604—21no—
31#19098602—21no—
32#19098600C2211yescharacteristic
33#19098600C2211yescharacteristic
34#107×D79C2211yescharacteristic
35#107×D79C2211yescharacteristic
36#19098517C2211yescharacteristic
37#19098391C2211yescharacteristic
38#19098391C2211yescharacteristic
39#19098390C2211yescharacteristic
40#19098390C2211yescharacteristic
41#19098389C2211yescharacteristic
42#19098389C2211yescharacteristic
43#19098388C2211yescharacteristic
44#19098388C2211yescharacteristic
45#19098388C2211yescharacteristic
46#19098388C2211yescharacteristic
47#19098388C2211yescharacteristic
48#19098388C2211yescharacteristic
49#19098338C2211yescharacteristic
50C4×C79⋊C16C2211yescharacteristic
51C16×Dic79C2211yescharacteristic
52#500×C79C2211yescharacteristic
53#499×C79C2211yescharacteristic
54#367×C79C2211yescharacteristic
55#367×C79C2211yescharacteristic
56#366×C79C2211yescharacteristic
57#366×C79C2211yescharacteristic
58#346×C79C2211yescharacteristic
59C2×#11929607D411yescharacteristic
60C2×C8×D79C2311yescharacteristic
61#14579161—21no—
62#14579160—21no—
63#14579160—21no—
64#14579160—41no—
65#14579159—41no—
66#14579153—21no—
67#14579152—21no—
68#14579152—21no—
69#14579151—41no—
70#14579151—41no—
71#14579141—41no—
72#14579139—41no—
73#14579139—41no—
74#14579137—41no—
75#14579119—21no—
76#14579119—21no—
77#14579118D411yescharacteristic
78C4⋊C4×D79—21no—
79C4⋊C4×D79—21no—
80#14579100Q814yescomplemented
81C2×C79⋊C16C2311yescharacteristic
82C2×C79⋊C16C2311yescharacteristic
83C16×D79Q814yescomplemented
84#14579095—21no—
85#14579095C2311yescharacteristic
86#14579095C2311yescharacteristic
87#14579095C2311yescharacteristic
88#14579095C2311yescharacteristic
89#14579094—21no—
90#14579094—21no—
91#14579094—21no—
92#14579094—21no—
93#14579093—21no—
94#14579092—21no—
95#14579092—21no—
96#14579092—21no—
97#14579092—21no—
98#14579072D411yescharacteristic
99#14579071D411yescharacteristic
100C8×Dic79C2311yescharacteristic
101C4⋊Q8×C79—21no—
102#125×C79—21no—
103C2×C16×C79C2311yescharacteristic
104C2×C16×C79C2311yescharacteristic
105#107×C79—21no—
106#107×C79C2311yescharacteristic
107#107×C79C2311yescharacteristic
108#107×C79C2311yescharacteristic
109#107×C79C2311yescharacteristic
110#106×C79—21no—
111C4×C8×C79C2311yescharacteristic
112C2×Dic158—41no—
113C2×Dic158—41no—
114C2×Dic158—41no—
115C2×C4×D79—41no—
116C2×C4×D79—41no—
117C2×C4×D79C2×D411yescharacteristic
118#11929607Q1614yes—
119C2×C79⋊C8C2×D411yescharacteristic
120C2×C79⋊C8C2×D411yescharacteristic
121C8×D79C2×Q812yes—
122C8×D79C2×Q812yes—
123C4⋊Dic79—21no—
124C4⋊Dic79—21no—
125C4⋊Dic79—21no—
126C4⋊Dic79—21no—
127C4⋊Dic79—41no—
128C4⋊Dic79—41no—
129C4⋊Dic79—41no—
130Dic79⋊C4—41no—
131Dic79⋊C4—41no—
132Dic79⋊C4—41no—
133Dic79⋊C4—41no—
134Dic79⋊C4—41no—
135Dic79⋊C4—41no—
136Dic79⋊C4—41no—
137Dic79⋊C4—41no—
138Dic79⋊C4D811yescharacteristic
139Dic79⋊C4D811yescharacteristic
140(C2×D79)⋊C4—41no—
141(C2×D79)⋊C4—41no—
142(C2×D79)⋊C4D811yescharacteristic
143(C2×D79)⋊C4D811yescharacteristic
144C4×Dic79—41no—
145C4×Dic79—41no—
146C4×Dic79C2×D411yescharacteristic
147C79⋊C16C2×Q812yes—
148C79⋊C16C2×Q812yes—
149C2×Q8×C79—41no—
150C2×C8×C79C2×D411yescharacteristic
151C2×C8×C79C2411yescharacteristic
152C4⋊C4×C79—21no—
153C4⋊C4×C79—21no—
154C4⋊C4×C79—21no—
155C4⋊C4×C79—21no—
156C4⋊C4×C79—41no—
157C4⋊C4×C79—41no—
158C4⋊C4×C79—41no—
159C42×C79C2×D411yescharacteristic
160C16×C79C2×Q812yes—
161C16×C79C2×Q812yes—
162C22×D79C2×D811yescharacteristic
163Q8×C79—161no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79C2×D811yescharacteristic
169C2×C4×C79C2×D811yescharacteristic
170C2×C4×C79C22×D411yescharacteristic
171C2×Dic79—41no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79C2×D811yescharacteristic
176C2×Dic79C2×D811yescharacteristic
177C2×Dic79C2×D811yescharacteristic
178C4×D79—161no—
179C4×D79—161no—
180C4×D79C4⋊Q812yes—
181C4×D79C2×Q1612yes—
182Dic158—161no—
183Dic158—161no—
184Dic158—161no—
185C8×C79C2×Q1612yes—
186C8×C79C22×Q811yescharacteristic
187C8×C79C22×Q811yescharacteristic
188C79⋊C8C4⋊Q812yes—
189C79⋊C8C2×Q1612yes—
190C22×C79C22×D811yescharacteristic
191C2×D79—21no—
192C2×D79—21no—
193C2×D79#50012yes—
194C4×C79—21no—
195C4×C79—21no—
196C4×C79—81no—
197C4×C79—81no—
198C4×C79—81no—
199C4×C79—81no—
200C4×C79C2×C4⋊Q811yescharacteristic
201C4×C79C22×Q1611yescharacteristic
202Dic79—21no—
203Dic79—21no—
204Dic79—81no—
205Dic79—81no—
206Dic79—81no—
207Dic79—81no—
208Dic79#50012yes—
209#33729—791nomaximal, Sylow
210C2×C79C2×#50011yescharacteristic
211C2×C79#341511yescharacteristic
212C2×C79#341711yescharacteristic
213D79—22no—
214#3164—791no—
215#3164—791no—
216#2255—791no—
217#2255—791no—
218#2255—791no—
219#2255D7911yescharacteristic, complemented
220#2253—791no—
221#2253—791no—
222#2252—791no—
223#2252—791no—
224#2159—791no—
225#2159—791no—
226#2158—791no—
227C2×#366—791no—
228#2108—791no—
229C79#3372911yescharacteristic, complemented, Sylow
230#544—1581no—
231#544—1581no—
232C2×M32—791no—
233C22×C16—791no—
234#501—1581no—
235#501—1581no—
236#501—1581no—
237#501—1581no—
238#500—791no—
239#500—791no—
240#500—791no—
241#500C2×D7911yescharacteristic
242#499—791no—
243#499—791no—
244#499—791no—
245#499C2×D7911yescharacteristic
246#428—1581no—
247#428—1581no—
248#427—791no—
249#427—791no—
250C2×#107—791no—
251C2×#107—791no—
252#405—791no—
253#367—791no—
254#367—791no—
255#367—791no—
256#367—791no—
257#367C2×D7911yescharacteristic
258#367C2×D7911yescharacteristic
259#366—791no—
260#366—791no—
261#366—791no—
262#366—791no—
263#366—791no—
264#366—791no—
265#366—791no—
266#366—791no—
267#366C2×D7911yescharacteristic
268#366C2×D7911yescharacteristic
269#346—791no—
270#346C2×D7911yescharacteristic
271C4×C16—791no—
272C4×C16—791no—
273C2×M16—791no—
274C22×C8—791no—
275C4⋊Q8—21no—
276C4⋊Q8—3161no—
277C4⋊Q8—3161no—
278C4⋊Q8—1581no—
279C4⋊Q8—1581no—
280C4⋊Q8—1581no—
281#125—21no—
282#125—3161no—
283#125—3161no—
284#125—1581no—
285#125—1581no—
286#125—1581no—
287C22⋊Q8—3161no—
288C22⋊Q8—3161no—
289C22⋊Q8—3161no—
290C22⋊Q8—3161no—
291#117—791no—
292#117—1581no—
293#117—1581no—
294C2×C4⋊C4—1581no—
295C2×C4⋊C4—1581no—
296M32—794no—
297C2×C16—791no—
298C2×C16—791no—
299C2×C16—794no—
300C2×C16C22×D7911yescharacteristic
301C2×C16C22×D7911yescharacteristic
302#107—21no—
303#107—791no—
304#107—791no—
305#107—791no—
306#107—791no—
307#107—1581no—
308#107—1581no—
309#107—1581no—
310#107—1581no—
311#107—1581no—
312#107C22×D7911yescharacteristic
313#107C22×D7911yescharacteristic
314#107C22×D7911yescharacteristic
315#107C22×D7911yescharacteristic
316#106—21no—
317#106—1581no—
318#106—1581no—
319#106—1581no—
320#106—1581no—
321#106—1581no—
322#97—791no—
323#97—791no—
324C4×C8—791no—
325C4×C8C22×D7911yescharacteristic
326C2×Q8—41no—
327C2×Q8—3161no—
328C2×Q8—3161no—
329C2×Q8—3161no—
330C22×C4—3161no—
331C22×C4—3161no—
332C22×C4—791no—
333M16—794no—
334C2×C8—791no—
335C2×C8—791no—
336C2×C8—792no—
337C2×C8—792no—
338C2×C8D4×D7911yescharacteristic
339C2×C8C23×D7911yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—3161no—
348C4⋊C4—3161no—
349C4⋊C4—3161no—
350C4⋊C4—3161no—
351C4⋊C4—3161no—
352C4⋊C4—3161no—
353C4⋊C4—3161no—
354C4⋊C4—3161no—
355C4⋊C4—3161no—
356C4⋊C4—3161no—
357C4⋊C4—3161no—
358C4⋊C4—791no—
359C4⋊C4—791no—
360C4⋊C4—1581no—
361C4⋊C4—1581no—
362C4⋊C4—1581no—
363C4⋊C4—1581no—
364C22⋊C4—3161no—
365C22⋊C4—3161no—
366C22⋊C4—791no—
367C22⋊C4—791no—
368C42—3161no—
369C42—3161no—
370C42—791no—
371C42D4×D7911yescharacteristic
372C16—792no—
373C16—792no—
374C16Q8×D7912yes—
375C16Q8×D7912yes—
376C23—791no—
377Q8—161no—
378Q8—12641no—
379Q8—12641no—
380Q8—12641no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—3161no—
386C2×C4—3161no—
387C2×C4—3161no—
388C2×C4—3161no—
389C2×C4—791no—
390C2×C4—792no—
391C2×C4—792no—
392C2×C4—791no—
393C2×C4—791no—
394C2×C4—12641no—
395C2×C4—12641no—
396C2×C4D8×D7911yescharacteristic
397C2×C4D8×D7911yescharacteristic
398C2×C4C2×D4×D7911yescharacteristic
399C8—792no—
400C8—792no—
401C8Q16×D7912yes—
402C8C2×Q8×D7911yescharacteristic
403C8C2×Q8×D7911yescharacteristic
404C22—792no—
405C22—1581no—
406C22—1581no—
407C22C2×D8×D7911yescharacteristic
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—792no—
415C4—6321no—
416C4—6321no—
417C4—6321no—
418C4—6321no—
419C4—1581no—
420C4—1581no—
421C4C4⋊Q8×D7911yescharacteristic
422C4C2×Q16×D7911yescharacteristic
423C2—1582no—
424C2#500×D7911yescharacteristic
425C2#2377654011yescharacteristic
426C2#2377655211yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#2255C2
socleC22×C79C22×D8
prime radicalC22×D79C2×D8
hypercenter#2255D79
hypocenterC79#33729
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #23765686C2semidirect
4. #23765685C2semidirect
5. #23765684C2semidirect
6. #23765678C2semidirect
7. #23765676C2semidirect
8. #23765674C2semidirect
9. #23765673C2semidirect
15. C79×#2255C2semidirect
80. #14579100Q8semidirect
83. C16×D79Q8semidirect
219. #2255D79semidirect
229. C79#33729semidirect

Maximal subgroups

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

subgroupindex
#237739372
#237739332
#237656862
#237656852
#237656842
#237656782
#237656762
#237656742
#237656732
#237651872
#237651852
#237651832
#366×D792
#237649982
C79×#22552
#3372979