Finite Group Databasebeta

← #34184178

Subgroups — #34184178

Order 20224.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766310C211yesmaximal, characteristic
2#23761964C211yesmaximal, characteristic, complemented
3#23761899C211yesmaximal, characteristic, complemented
4#23761791C211yesmaximal, characteristic
5#23761301C211yesmaximal, characteristic, complemented
6#23761260C211yesmaximal, characteristic, complemented
7C79×#3077C211yesmaximal, characteristic
8C2×D4×Dic79—21no—
9#19098616C2211yescharacteristic
10#19098616C2211yescharacteristic
11#19098462—21no—
12#19098438—22no—
13#19098429C2211yescharacteristic
14#19098423—21no—
15C4×C4⋊Dic79C2211yescharacteristic
16C2×C79⋊#95—21no—
17#19098300—22no—
18#19098300—21no—
19#19098300—21no—
20D4⋊(C79⋊C8)—22no—
21#533×C79C2211yescharacteristic
22C2×C4×D4×C79C2211yescharacteristic
23#478×C79—22no—
24#476×C79C412yescomplemented
25#474×C79C412yescomplemented
26#428×C79—21no—
27#423×C79C2211yescharacteristic
28#418×C79—21no—
29C23×Dic79—22no—
30D4×Dic79—42no—
31D4×Dic79—42no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79—22no—
35C2×C22⋊Dic79—22no—
36C2×C4×Dic79—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C4×C79⋊C8—21no—
42C4×C79⋊C8—21no—
43C79⋊#95—22no—
44C79⋊#95—21no—
45C79⋊#95D411yescharacteristic
46C79⋊#95D411yescharacteristic
47C79⋊#95D412yes—
48C22×D4×C79D411yescharacteristic
49C23×C4×C79—21no—
50C2×M16×C79—21no—
51C4⋊Q8×C79C2×C411yescharacteristic, complemented
52C4⋊Q8×C79C2×C411yescharacteristic, complemented
53#125×C79—21no—
54C22⋊Q8×C79—21no—
55C22⋊Q8×C79—21no—
56C4×Q8×C79—21no—
57C4×Q8×C79—21no—
58C4×D4×C79—21no—
59C4×D4×C79—21no—
60C4×D4×C79C2×C412yes—
61C4×D4×C79D412yes—
62#117×C79—21no—
63C2×C4⋊C4×C79D411yescharacteristic
64C2×C22⋊C4×C79—21no—
65C2×C42×C79C2311yescharacteristic
66#107×C79—22no—
67#106×C79—22no—
68C4⋊C8×C79—21no—
69C4⋊C8×C79C2×C412yescomplemented
70#102×C79—22no—
71#102×C79—22no—
72C22×Dic79—21no—
73C22×Dic79—22no—
74C22×Dic79—21no—
75C22×Dic79—44no—
76C22×Dic79—44no—
77C22×Dic79—82no—
78C2×C79⋊C8—41no—
79C2×C79⋊C8—41no—
80C4⋊Dic79—42no—
81C4⋊Dic79—41no—
82C4⋊Dic79—41no—
83C4⋊Dic79—82no—
84C22⋊Dic79—44no—
85C4×Dic79—41no—
86C4×Dic79—41no—
87C24×C79—21no—
88C2×Q8×C79—21no—
89C2×Q8×C79—21no—
90C2×D4×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79—21no—
93C2×D4×C79C22⋊C412yes—
94C2×D4×C79SD1612yes—
95C2×D4×C79Q1612yes—
96C22×C4×C79—21no—
97C22×C4×C79—22no—
98C22×C4×C79—22no—
99C22×C4×C79—82no—
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103M16×C79—41no—
104M16×C79—41no—
105C2×C8×C79—22no—
106C4⋊C4×C79—21no—
107C4⋊C4×C79—21no—
108C4⋊C4×C79—21no—
109C4⋊C4×C79—21no—
110C4⋊C4×C79—21no—
111C4⋊C4×C79—41no—
112C4⋊C4×C79—41no—
113C4⋊C4×C79C22⋊C412yes—
114C22⋊C4×C79—22no—
115C22⋊C4×C79—22no—
116C22⋊C4×C79—41no—
117C42×C79—41no—
118C42×C79C22×C411yescharacteristic
119C42×C79C2×D411yescharacteristic
120C42×C79#4111yescharacteristic
121C42×C79#4111yescharacteristic
122C23×C79—22no—
123C23×C79—22no—
124C23×C79—22no—
125C23×C79—42no—
126C23×C79—42no—
127C23×C79#12011yescharacteristic
128Q8×C79—81no—
129Q8×C79—81no—
130D4×C79—22no—
131D4×C79—22no—
132D4×C79—22no—
133D4×C79#10314yes—
134C2×C4×C79—22no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—42no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—82no—
141C2×C4×C79—82no—
142C2×C4×C79C2×C22⋊C411yescharacteristic
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79C4×D411yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79#12111yescharacteristic
147C2×C4×C79#12311yescharacteristic
148C2×C4×C79C2×SD1611yescharacteristic
149C2×C4×C79C2×Q1611yescharacteristic
150C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C79#13711yescharacteristic
152C2×Dic79—22no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—42no—
158C2×Dic79—42no—
159C2×Dic79—41no—
160C2×Dic79—42no—
161C2×Dic79—88no—
162C8×C79—42no—
163C79⋊C8—81no—
164C79⋊C8—81no—
165C22×C79—21no—
166C22×C79—21no—
167C22×C79—24no—
168C22×C79—44no—
169C22×C79—44no—
170C22×C79—42no—
171C22×C79—42no—
172C22×C79#38611yescharacteristic
173C22×C79#45011yescharacteristic
174C22×C79#45111yescharacteristic
175C4×C79—21no—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—42no—
180C4×C79—81no—
181C4×C79—81no—
182C4×C79C2×#10311yescharacteristic
183C4×C79#41711yescharacteristic
184C4×C79#48311yescharacteristic
185C4×C79#48411yescharacteristic
186Dic79—42no—
187Dic79—81no—
188Dic79—82no—
189#9175—791nomaximal, Sylow
190C2×C79—21no—
191C2×C79—21no—
192C2×C79—44no—
193C2×C79#187211yescharacteristic
194C2×C79#190411yescharacteristic
195C2×C79#190611yescharacteristic
196#3077D7911yescharacteristic
197#2349—791no—
198#1518—791no—
199#1503—791no—
200#1478—791no—
201#1303—791no—
202#1286—791no—
203C79#917511yescharacteristic, complemented, Sylow
204#533C2×D7911yescharacteristic
205C2×C4×D4—1581no—
206C2×C4×D4C2×D7911yescharacteristic
207#478—22no—
208#476Dic7912yescomplemented
209#474Dic7912yescomplemented
210#431—791no—
211#431—791no—
212#428—21no—
213#423C2×D7911yescharacteristic
214#418—21no—
215#389—1581no—
216#385—1582no—
217#382—791no—
218#380—1581no—
219C4×C4⋊C4—791no—
220C2×#95—1581no—
221#328—1581no—
222#328—1581no—
223#328—1582no—
224D4⋊C8—1582no—
225C22×D4C22⋊D7911yescharacteristic
226C23×C4—21no—
227C23×C4—1582no—
228C2×M16—21no—
229C4⋊Q8C2×Dic7911yescharacteristic, complemented
230C4⋊Q8C2×Dic7911yescharacteristic, complemented
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—3162no—
239C4×D4—3162no—
240C4×D4C2×Dic7912yes—
241C4×D4C22⋊D7912yes—
242#117—21no—
243C2×C4⋊C4—1581no—
244C2×C4⋊C4—1581no—
245C2×C4⋊C4—1582no—
246C2×C4⋊C4C22⋊D7911yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1582no—
249C2×C42—1581no—
250C2×C42C22×D7911yescharacteristic
251#107—22no—
252#106—22no—
253C4⋊C8—21no—
254C4⋊C8—1581no—
255C4⋊C8—1581no—
256C4⋊C8C2×Dic7912yescomplemented
257#102—22no—
258#102—22no—
259C22⋊C8—1581no—
260C22⋊C8—1581no—
261C4×C8—1581no—
262C4×C8—1581no—
263#95—791no—
264#95—791no—
265#95—792no—
266#95—1582no—
267#95—1581no—
268C24—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4C22⋊Dic7912yes—
275C2×D4#1192961112yes—
276C2×D4#1192961412yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—3164no—
282C22×C4—3164no—
283C22×C4—6322no—
284C22×C4—1582no—
285C22×C4—1581no—
286C22×C4—1581no—
287C22×C4D4×D7911yescharacteristic
288C22×C4D4×D7911yescharacteristic
289C22×C4C2×C22⋊D7911yescharacteristic
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—3161no—
294C2×C8—3161no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—3162no—
303C4⋊C4—3161no—
304C4⋊C4—3161no—
305C4⋊C4—6322no—
306C4⋊C4C22⋊Dic7912yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—3164no—
311C42—41no—
312C42—3161no—
313C42—3161no—
314C42C22×Dic7911yescharacteristic
315C42C2×C22⋊D7911yescharacteristic
316C42#1192962311yescharacteristic
317C42#1192962311yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1457912911yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1457908514yes—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—3161no—
339C2×C4—3161no—
340C2×C4—3161no—
341C2×C4—3161no—
342C2×C4—3162no—
343C2×C4—3162no—
344C2×C4—3161no—
345C2×C4—3162no—
346C2×C4—6328no—
347C2×C4—1582no—
348C2×C4C2×C22⋊Dic7911yescharacteristic
349C2×C4D4×Dic7911yescharacteristic
350C2×C4D4×Dic7911yescharacteristic
351C2×C4#1457913311yescharacteristic
352C2×C4#1457913311yescharacteristic
353C2×C4#1457914211yescharacteristic
354C2×C4C2×#1192961111yescharacteristic
355C2×C4C2×#1192961411yescharacteristic
356C2×C4#1457918611yescharacteristic
357C2×C4#1457919111yescharacteristic
358C8—42no—
359C8—6321no—
360C8—6321no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1909844311yescharacteristic
369C22#1909871311yescharacteristic
370C22#1909871811yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—3162no—
379C4—6322no—
380C4—6321no—
381C4C2×#1457908511yescharacteristic
382C4#1909856411yescharacteristic
383C4#1909889111yescharacteristic
384C4#1909889611yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2376370811yescharacteristic
389C2#2376388911yescharacteristic
390C2#2376390311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3077C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenter#3077D79
hypocenterC79#9175
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #23761964C2semidirect
3. #23761899C2semidirect
5. #23761301C2semidirect
6. #23761260C2semidirect
24. #476×C79C4semidirect
25. #474×C79C4semidirect
51. C4⋊Q8×C79C2×C4semidirect
52. C4⋊Q8×C79C2×C4semidirect
69. C4⋊C8×C79C2×C4semidirect
203. C79#9175semidirect
208. #476Dic79semidirect
209. #474Dic79semidirect
229. C4⋊Q8C2×Dic79semidirect
230. C4⋊Q8C2×Dic79semidirect
256. C4⋊C8C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237663102
#237619642
#237618992
#237617912
#237613012
#237612602
C79×#30772
#917579