Finite Group Databasebeta

← #34184814

Subgroups — #34184814

Order 20224.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774315C211yesmaximal, characteristic
2#23764240C211yesmaximal, characteristic, complemented
3#23762077C211yesmaximal, characteristic, complemented
4C2×#19098306C211yesmaximal, characteristic, complemented
5#23761297C211yesmaximal, characteristic, complemented
6#23761251C211yesmaximal, characteristic, complemented
7C79×#2935C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9#19098939—21no—
10#19098911—21no—
11#19098758—22no—
12#19098696C414yes—
13#19098616C2211yescharacteristic
14#19098587C2211yescharacteristic
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×#14579077—21no—
17C2×#14579077—21no—
18#19098309—21no—
19#19098307—21no—
20#19098306—21no—
21#19098306—21no—
22C2×#131×C79—21no—
23#532×C79C2211yescharacteristic
24#436×C79—21no—
25#435×C79—22no—
26C8×D4×C79—21no—
27C2×C4⋊C8×C79C2211yescharacteristic
28#408×C79—21no—
29C4×M16×C79C2211yescharacteristic
30C22×D316—21no—
31C22×D316—21no—
32C22×D316C2×C412yes—
33C2×#11929609—24no—
34C2×#11929608—21no—
35C22×C79⋊C8—21no—
36#14579158—21no—
37#14579158—21no—
38#14579158—21no—
39#14579158—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45#14579081—24no—
46#14579077—42no—
47#14579077—42no—
48C22⋊(C79⋊C8)C2×C412yes—
49C4×C79⋊C8—21no—
50C2×#41×C79—21no—
51#131×C79—42no—
52#131×C79—42no—
53C2×M16×C79—22no—
54C2×M16×C79D411yescharacteristic
55C2×M16×C79D411yescharacteristic
56C22×C8×C79—22no—
57C22×C8×C79—21no—
58C4⋊Q8×C79D411yescharacteristic
59#127×C79D411yescharacteristic
60#124×C79—21no—
61#121×C79—21no—
62#121×C79C2×C412yes—
63C4×D4×C79—21no—
64C4×D4×C79D412yes—
65C2×C42×C79C2311yescharacteristic
66C4⋊C8×C79—21no—
67C4⋊C8×C79D411yescharacteristic
68C4⋊C8×C79D411yescharacteristic
69C22⋊C8×C79—22no—
70C22⋊C8×C79—22no—
71#97×C79—21no—
72C4×C8×C79—21no—
73C23×D79—22no—
74C2×D316—22no—
75C2×D316—22no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—41no—
82C2×D316—41no—
83C2×D316—41no—
84C2×D316—81no—
85C2×D316—81no—
86C2×D316—82no—
87#11929609—84no—
88#11929608—41no—
89#11929608—41no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—81no—
94#41×C79—41no—
95#41×C79—41no—
96#41×C79—41no—
97#41×C79—41no—
98C2×Q8×C79—21no—
99C2×D4×C79—22no—
100C2×D4×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79C22⋊C412yes—
103C22×C4×C79—22no—
104C22×C4×C79C22×C411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79#4111yescharacteristic
107M16×C79—21no—
108M16×C79—21no—
109M16×C79—21no—
110M16×C79—44no—
111M16×C79D812yescomplemented
112C2×C8×C79—21no—
113C2×C8×C79—21no—
114C2×C8×C79—21no—
115C2×C8×C79—21no—
116C2×C8×C79—44no—
117C2×C8×C79—41no—
118C2×C8×C79—41no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79C22⋊C412yes—
121C22⋊C4×C79—22no—
122C22⋊C4×C79—22no—
123C42×C79C2×D411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79#4111yescharacteristic
127C23×C79—22no—
128C23×C79C4×D411yescharacteristic
129C22×D79—42no—
130C22×D79—42no—
131C22×D79—42no—
132C22×D79—82no—
133C22×D79—82no—
134Q8×C79—41no—
135Q8×C79—41no—
136D4×C79—42no—
137D4×C79—42no—
138D4×C79—42no—
139D4×C79—41no—
140D4×C79—41no—
141D4×C79—41no—
142D4×C79—41no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—22no—
146C2×C4×C79—41no—
147C2×C4×C79—42no—
148C2×C4×C79—42no—
149C2×C4×C79C2×C22⋊C411yescharacteristic
150C2×C4×C79C4×D411yescharacteristic
151C2×C4×C79#12011yescharacteristic
152C2×C4×C79#12111yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79#12311yescharacteristic
155C2×C4×C79C2×D811yescharacteristic
156C2×C4×C79#13511yescharacteristic
157C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
159D316—44no—
160D316—82no—
161D316—82no—
162D316—81no—
163D316—81no—
164D316—81no—
165D316—81no—
166D316—81no—
167D316—81no—
168C8×C79—22no—
169C8×C79—41no—
170C8×C79—41no—
171C8×C79—41no—
172C79⋊C8—42no—
173C79⋊C8—81no—
174C79⋊C8—81no—
175C22×C79—21no—
176C22×C79—21no—
177C22×C79—42no—
178C22×C79—42no—
179C22×C79—42no—
180C22×C79#38611yescharacteristic
181C22×C79C4×D811yescharacteristic
182C22×C79#44211yescharacteristic
183C2×D79—82no—
184C2×D79—82no—
185C2×D79—82no—
186C2×D79—162no—
187C2×D79—162no—
188C4×C79—21no—
189C4×C79—21no—
190C4×C79—21no—
191C4×C79—42no—
192C4×C79#44711yescharacteristic
193C4×C79#44911yescharacteristic
194C4×C79#45911yescharacteristic
195C4×C79#46311yescharacteristic
196C4×C79#47111yescharacteristic
197C4×C79#47211yescharacteristic
198#9273—791nomaximal, Sylow
199C2×C79—21no—
200C2×C79—21no—
201C2×C79—82no—
202C2×C79#187911yescharacteristic
203C2×C79#190711yescharacteristic
204C2×C79#191411yescharacteristic
205D79—162no—
206#3205—791no—
207#2935D7911yescharacteristic, complemented
208#1965—791no—
209#1547—791no—
210C2×#331—791no—
211#1301—791no—
212#1283—791no—
213C79#927311yescharacteristic, complemented, Sylow
214C2×#131—21no—
215#532C2×D7911yescharacteristic
216C2×#127—791no—
217#493—1581no—
218#486—1581no—
219#459—1582no—
220#447—794no—
221#436—21no—
222#435—22no—
223C8×D4—21no—
224#431—791no—
225#423—791no—
226C2×C4⋊C8—791no—
227C2×C4⋊C8C2×D7911yescharacteristic
228C2×#100—1581no—
229C2×#100—1581no—
230#408—21no—
231C4×M16C2×D7911yescharacteristic
232#332—1581no—
233#331—1581no—
234#331—1581no—
235#331—1581no—
236C2×#41—21no—
237C22×D4—792no—
238C22×D4—1581no—
239C22×D4—1581no—
240C2×D8—1584no—
241#131—42no—
242#131—42no—
243C2×M16—22no—
244C2×M16—1581no—
245C2×M16C22⋊D7911yescharacteristic
246C2×M16C22⋊D7911yescharacteristic
247C22×C8—22no—
248C22×C8—21no—
249C22×C8—1581no—
250C4⋊Q8D31611yescharacteristic
251#127—1581no—
252#127—1581no—
253#127—1581no—
254#127—1581no—
255#127D31611yescharacteristic
256#124—21no—
257#121—21no—
258#121C4×D7912yes—
259C4×D4—21no—
260C4×D4C22⋊D7912yes—
261C2×C42C22×D7911yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—1581no—
264C4⋊C8—1581no—
265C4⋊C8—1581no—
266C4⋊C8—1581no—
267C4⋊C8—1581no—
268C4⋊C8C22⋊D7911yescharacteristic
269C4⋊C8C22⋊D7911yescharacteristic
270#102—1584no—
271#100—3162no—
272#100—3162no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—792no—
276#97—21no—
277C4×C8—21no—
278C4×C8—1581no—
279C24—1582no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—3161no—
289C2×D4—3161no—
290C2×D4—3161no—
291C2×D4—3161no—
292C2×D4—3161no—
293C2×D4—3161no—
294C2×D4—3161no—
295C2×D4—3161no—
296C2×D4—6321no—
297C2×D4—6321no—
298C2×D4—6322no—
299C2×D4—1582no—
300C2×D4—1582no—
301C2×D4(C2×D79)⋊C412yes—
302C22×C4—22no—
303C22×C4C2×C4×D7911yescharacteristic
304C22×C4C2×C22⋊D7911yescharacteristic
305C22×C4#1192962411yescharacteristic
306D8—6324no—
307M16—21no—
308M16—21no—
309M16—21no—
310M16—44no—
311M16—3161no—
312M16—3161no—
313M16#1192960912yescomplemented
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—44no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—3161no—
322C2×C8—3161no—
323C2×C8—6321no—
324C2×C8—1582no—
325C4⋊C4—21no—
326C4⋊C4(C2×D79)⋊C412yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C42C2×C22⋊D7911yescharacteristic
330C42C2×C22⋊D7911yescharacteristic
331C42C2×D31611yescharacteristic
332C42#1192962411yescharacteristic
333C23—22no—
334C23—3162no—
335C23—3162no—
336C23—3162no—
337C23—6322no—
338C23—6322no—
339C23C4×C22⋊D7911yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—3164no—
350D4—6322no—
351D4—6322no—
352D4—6321no—
353D4—6321no—
354D4—6321no—
355D4—6321no—
356D4—6321no—
357D4—6321no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4C2×(C2×D79)⋊C411yescharacteristic
365C2×C4C4×C22⋊D7911yescharacteristic
366C2×C4#1457913011yescharacteristic
367C2×C4#1457913511yescharacteristic
368C2×C4#1457913511yescharacteristic
369C2×C4#1457914411yescharacteristic
370C2×C4C2×#1192960911yescharacteristic
371C2×C4#1457918111yescharacteristic
372C2×C4#1457918611yescharacteristic
373C2×C4#1457918711yescharacteristic
374C8—22no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—3162no—
379C8—6321no—
380C8—6321no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—6322no—
387C22—6322no—
388C22—6322no—
389C22—12642no—
390C22—12642no—
391C22#1909844211yescharacteristic
392C22C4×#1192960911yescharacteristic
393C22#1909867311yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4#1909869711yescharacteristic
399C4#1909870811yescharacteristic
400C4#1909875811yescharacteristic
401C4#1909877811yescharacteristic
402C4#1909882611yescharacteristic
403C4#1909883311yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—12642no—
408C2#2376374811yescharacteristic
409C2#2376390811yescharacteristic
410C2#2376394711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2935C2
socleC22×C79#386
prime radical#23774315C2
cosocleC2×C42×C79C23
hypercenter#2935D79
hypocenterC79#9273
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #23764240C2semidirect
3. #23762077C2semidirect
4. C2×#19098306C2semidirect
5. #23761297C2semidirect
6. #23761251C2semidirect
7. C79×#2935C2semidirect
111. M16×C79D8semidirect
207. #2935D79semidirect
213. C79#9273semidirect
313. M16#11929609semidirect

Maximal subgroups

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

subgroupindex
#237743152
#237642402
#237620772
C2×#190983062
#237612972
#237612512
C79×#29352
#927379