Finite Group Databasebeta

← #34184681

Subgroups — #34184681

Order 20224.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098763C211yesmaximal, characteristic
2#23764240C211yesmaximal, characteristic
3#23762069C211yesmaximal, characteristic
4#23761829C211yesmaximal, characteristic, complemented
5#23761274C211yesmaximal, characteristic, complemented
6#23761251C211yesmaximal, characteristic, complemented
7Q8×M16×C79C211yesmaximal, characteristic, complemented
8C22×#11929611C412yes—
9C2×#14579158C2211yescharacteristic
10#19098763—21no—
11#19098763—21no—
12#19098763—21no—
13#19098763—21no—
14#19098616C2211yescharacteristic
15#19098587C2211yescharacteristic
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17C2×#14579081C412yes—
18C2×#14579077—21no—
19C2×#14579077—21no—
20#19098306—22no—
21Q8⋊(C79⋊C8)—21no—
22Q8⋊(C79⋊C8)—21no—
23C2×C4×Q8×C79C2211yescharacteristic
24#446×C79—21no—
25#446×C79—22no—
26C8×Q8×C79—21no—
27#423×C79—21no—
28#423×C79C2211yescharacteristic
29C4×M16×C79—21no—
30C4×M16×C79C2211yescharacteristic
31C22×D316—21no—
32C22×D316—21no—
33C22×D316C2×C412yes—
34C2×#11929611—22no—
35C2×#11929611—22no—
36C2×#11929611—44no—
37C2×#11929608—21no—
38C22×C79⋊C8C2×C412yes—
39#14579158—21no—
40#14579158—21no—
41#14579158—21no—
42#14579158—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)—21no—
48#14579081—22no—
49#14579081—22no—
50#14579077—42no—
51#14579077—42no—
52C22⋊(C79⋊C8)—21no—
53C4×C79⋊C8—21no—
54C22×Q8×C79C2×C411yescharacteristic
55C2×M16×C79—21no—
56C2×M16×C79D411yescharacteristic
57C2×M16×C79D411yescharacteristic
58C4×Q8×C79—21no—
59C4×Q8×C79—21no—
60C4×Q8×C79D411yescharacteristic
61C4×Q8×C79D411yescharacteristic
62C4×Q8×C79D411yescharacteristic
63C4×Q8×C79D411yescharacteristic
64C2×C4⋊C4×C79—21no—
65C2×C4⋊C4×C79C2×C411yescharacteristic
66C2×C42×C79—21no—
67C2×C42×C79C2311yescharacteristic
68C4⋊C8×C79—22no—
69C4⋊C8×C79—22no—
70C4⋊C8×C79—21no—
71C4⋊C8×C79D412yes—
72#97×C79—22no—
73#97×C79—21no—
74C4×C8×C79—22no—
75C4×C8×C79—21no—
76C23×D79—22no—
77C2×D316—22no—
78C2×D316—22no—
79C2×D316—42no—
80C2×D316—42no—
81C2×D316—41no—
82C2×D316—41no—
83C2×D316—41no—
84C2×D316—41no—
85C2×D316—41no—
86C2×D316—41no—
87C2×D316—41no—
88C2×D316—41no—
89C2×D316—82no—
90#11929611—44no—
91#11929611—44no—
92#11929608—41no—
93#11929608—41no—
94C2×C79⋊C8—22no—
95C2×C79⋊C8—42no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×Q8×C79—21no—
99C2×Q8×C79—22no—
100C2×Q8×C79—41no—
101C2×Q8×C79C22⋊C411yescharacteristic
102C2×Q8×C79C22⋊C411yescharacteristic
103C22×C4×C79—21no—
104C22×C4×C79—21no—
105C22×C4×C79C22×C411yescharacteristic
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79#4111yescharacteristic
108M16×C79—21no—
109M16×C79—21no—
110M16×C79—21no—
111M16×C79—41no—
112M16×C79—41no—
113M16×C79SD1612yescomplemented
114C2×C8×C79—21no—
115C2×C8×C79—21no—
116C2×C8×C79—22no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79C22⋊C411yescharacteristic
123C4⋊C4×C79C22⋊C411yescharacteristic
124C42×C79—21no—
125C42×C79—21no—
126C42×C79—21no—
127C42×C79—21no—
128C42×C79C2×D411yescharacteristic
129C42×C79C2×D411yescharacteristic
130C42×C79C2×D411yescharacteristic
131C42×C79#4111yescharacteristic
132C23×C79C4×D411yescharacteristic
133C22×D79—42no—
134C22×D79—42no—
135C22×D79—42no—
136C22×D79—82no—
137C22×D79—82no—
138Q8×C79—21no—
139Q8×C79—21no—
140Q8×C79—21no—
141Q8×C79—42no—
142Q8×C79#10012yescomplemented
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—41no—
150C2×C4×C79—42no—
151C2×C4×C79—81no—
152C2×C4×C79C2×C22⋊C411yescharacteristic
153C2×C4×C79C4×D411yescharacteristic
154C2×C4×C79#12011yescharacteristic
155C2×C4×C79#12111yescharacteristic
156C2×C4×C79#12111yescharacteristic
157C2×C4×C79#12311yescharacteristic
158C2×C4×C79C2×SD1611yescharacteristic
159C2×C4×C79#13511yescharacteristic
160C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C79#13711yescharacteristic
162D316—42no—
163D316—42no—
164D316—82no—
165D316—82no—
166D316—82no—
167D316—81no—
168D316—81no—
169D316—81no—
170D316—81no—
171C8×C79—22no—
172C8×C79—41no—
173C8×C79—42no—
174C79⋊C8—42no—
175C79⋊C8—81no—
176C79⋊C8—81no—
177C22×C79—21no—
178C22×C79—21no—
179C22×C79#38611yescharacteristic
180C22×C79C4×SD1611yescharacteristic
181C22×C79#44011yescharacteristic
182C2×D79—82no—
183C2×D79—82no—
184C2×D79—82no—
185C2×D79—162no—
186C2×D79—162no—
187C4×C79—21no—
188C4×C79—22no—
189C4×C79—21no—
190C4×C79—21no—
191C4×C79—41no—
192C4×C79—41no—
193C4×C79—41no—
194C4×C79C2×#10011yescharacteristic
195C4×C79#41311yescharacteristic
196C4×C79#44811yescharacteristic
197C4×C79#44911yescharacteristic
198C4×C79#46011yescharacteristic
199C4×C79#46411yescharacteristic
200#9254—791nomaximal, Sylow
201C2×C79—21no—
202C2×C79—21no—
203C2×C79#188011yescharacteristic
204C2×C79#190011yescharacteristic
205C2×C79#190311yescharacteristic
206D79—162no—
207C2×#460—791no—
208Q8×M16D7911yescharacteristic, complemented
209#1965—791no—
210#1545—791no—
211#1487—791no—
212#1292—791no—
213#1283—791no—
214C79#925411yescharacteristic, complemented, Sylow
215C22×SD16—792no—
216C2×#127—791no—
217C2×C4×Q8C2×D7911yescharacteristic
218#460—1581no—
219#460—1581no—
220#460—1581no—
221#460—1581no—
222#446—21no—
223#446—22no—
224C8×Q8—21no—
225#431—791no—
226#423—21no—
227#423—791no—
228#423C2×D7911yescharacteristic
229C2×C4⋊C8—791no—
230C2×#102—792no—
231C2×#100—1581no—
232C2×#100—1581no—
233C4×M16—21no—
234C4×M16C2×D7911yescharacteristic
235#331—1582no—
236Q8⋊C8—1581no—
237Q8⋊C8—1581no—
238C22×Q8C4×D7911yescharacteristic
239C22×D4—792no—
240C22×D4—1581no—
241C22×D4—1581no—
242C2×SD16—3164no—
243C2×SD16—1582no—
244C2×SD16—1582no—
245C2×M16—21no—
246C2×M16—1581no—
247C2×M16C22⋊D7911yescharacteristic
248C2×M16C22⋊D7911yescharacteristic
249C22×C8—792no—
250#127—1581no—
251#127—1581no—
252#127—1581no—
253#127—1581no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8D31611yescharacteristic
257C4×Q8D31611yescharacteristic
258C4×Q8C22⋊D7911yescharacteristic
259C4×Q8C22⋊D7911yescharacteristic
260C2×C4⋊C4—21no—
261C2×C4⋊C4C4×D7911yescharacteristic
262C2×C42—21no—
263C2×C42C22×D7911yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—1581no—
268C4⋊C8—1581no—
269C4⋊C8—1581no—
270C4⋊C8—1581no—
271C4⋊C8—1581no—
272C4⋊C8C22⋊D7912yes—
273#102—1582no—
274#102—1582no—
275#100—3162no—
276#100—3162no—
277C22⋊C8—1581no—
278#97—22no—
279#97—21no—
280C4×C8—22no—
281C4×C8—21no—
282C4×C8—1581no—
283C24—1582no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8(C2×D79)⋊C411yescharacteristic
288C2×Q8(C2×D79)⋊C411yescharacteristic
289C2×D4—3162no—
290C2×D4—3162no—
291C2×D4—3161no—
292C2×D4—3161no—
293C2×D4—3161no—
294C2×D4—3161no—
295C2×D4—3161no—
296C2×D4—3161no—
297C2×D4—3161no—
298C2×D4—3161no—
299C2×D4—6322no—
300C2×D4—1582no—
301C2×D4—1582no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C2×C4×D7911yescharacteristic
305C22×C4C2×C22⋊D7911yescharacteristic
306C22×C4#1192962411yescharacteristic
307SD16—3164no—
308SD16—3164no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16—3161no—
315M16—3161no—
316M16#1192961112yescomplemented
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—3162no—
321C2×C8—3161no—
322C2×C8—3161no—
323C2×C8—1582no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4(C2×D79)⋊C411yescharacteristic
330C4⋊C4(C2×D79)⋊C411yescharacteristic
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×C22⋊D7911yescharacteristic
336C42C2×C22⋊D7911yescharacteristic
337C42C2×D31611yescharacteristic
338C42#1192962411yescharacteristic
339C23—3162no—
340C23—3162no—
341C23—3162no—
342C23—6322no—
343C23—6322no—
344C23C4×C22⋊D7911yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8#1457907712yescomplemented
350D4—3162no—
351D4—3162no—
352D4—6322no—
353D4—6322no—
354D4—6322no—
355D4—6321no—
356D4—6321no—
357D4—6321no—
358D4—6321no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4C2×(C2×D79)⋊C411yescharacteristic
369C2×C4C4×C22⋊D7911yescharacteristic
370C2×C4#1457913011yescharacteristic
371C2×C4#1457913511yescharacteristic
372C2×C4#1457913511yescharacteristic
373C2×C4#1457914411yescharacteristic
374C2×C4C2×#1192961111yescharacteristic
375C2×C4#1457918111yescharacteristic
376C2×C4#1457918611yescharacteristic
377C2×C4#1457919211yescharacteristic
378C8—22no—
379C8—41no—
380C8—42no—
381C8—3162no—
382C8—6321no—
383C8—6321no—
384C22—21no—
385C22—21no—
386C22—6322no—
387C22—6322no—
388C22—6322no—
389C22—12642no—
390C22—12642no—
391C22#1909844211yescharacteristic
392C22C4×#1192961111yescharacteristic
393C22#1909866111yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4C2×#1457907711yescharacteristic
402C4#1909854511yescharacteristic
403C4#1909870211yescharacteristic
404C4#1909870811yescharacteristic
405C4#1909876311yescharacteristic
406C4#1909878511yescharacteristic
407C2—21no—
408C2—21no—
409C2—12642no—
410C2#2376375511yescharacteristic
411C2#2376386311yescharacteristic
412C2#2376388011yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingQ8×M16×C79C2
socleC22×C79#386
prime radicalC2×#14579158C22
cosocleC2×C42×C79C23
hypercenterQ8×M16D79
hypocenterC79#9254
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
4. #23761829C2semidirect
5. #23761274C2semidirect
6. #23761251C2semidirect
7. Q8×M16×C79C2semidirect
113. M16×C79SD16semidirect
142. Q8×C79#100semidirect
208. Q8×M16D79semidirect
214. C79#9254semidirect
316. M16#11929611semidirect
349. Q8#14579077semidirect

Maximal subgroups

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

subgroupindex
C2×#190987632
#237642402
#237620692
#237618292
#237612742
#237612512
Q8×M16×C792
#925479