Finite Group Databasebeta

← #34183504

Subgroups — #34183504

Order 20224.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773955C211yesmaximal, characteristic
2#23764344C211yesmaximal, characteristic, complemented
3#23761895C211yesmaximal, characteristic
4#23761802C211yesmaximal, characteristic, complemented
5#23761239C211yesmaximal, characteristic, complemented
6#23761238C211yesmaximal, characteristic, complemented
7C79×#2935C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098959C411yescharacteristic, complemented
10#19098935C411yescharacteristic, complemented
11#19098915C411yescharacteristic, complemented
12#19098909C411yescharacteristic, complemented
13#19098790—22no—
14C4×#11929612—22no—
15#19098616C2211yescharacteristic
16C2×C4×C79⋊C8C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098324—21no—
19#19098324—21no—
20#19098297—22no—
21#19098293—21no—
22#19098293—21no—
23C2×#131×C79—21no—
24#532×C79C2211yescharacteristic
25#436×C79—21no—
26#435×C79—22no—
27C8×D4×C79—21no—
28C2×C4⋊C8×C79C2211yescharacteristic
29#408×C79—21no—
30C4×M16×C79C2211yescharacteristic
31C2×#11929612—22no—
32C22×C79⋊C8—21no—
33#14579161C2×C411yescharacteristic
34#14579161C2×C411yescharacteristic
35#14579153—21no—
36#14579140—21no—
37#14579140—21no—
38C4×Dic158—21no—
39C4×Dic158—21no—
40#14579120—21no—
41C2×C4⋊Dic79—21no—
42C2×C4×Dic79—21no—
43#14579092—22no—
44C4⋊(C79⋊C8)—21no—
45#14579084—22no—
46#14579082—22no—
47C22⋊(C79⋊C8)—21no—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8C2×C411yescharacteristic
51C4×C79⋊C8C2×C411yescharacteristic
52C79⋊#95—21no—
53C2×#41×C79—21no—
54#131×C79—42no—
55#131×C79—42no—
56C2×M16×C79—22no—
57C2×M16×C79D411yescharacteristic
58C2×M16×C79D411yescharacteristic
59C22×C8×C79—22no—
60C22×C8×C79—21no—
61C4⋊Q8×C79C2×C411yescharacteristic
62#127×C79C2×C411yescharacteristic
63#124×C79—21no—
64#121×C79—21no—
65#121×C79D412yes—
66C4×D4×C79—21no—
67C4×D4×C79D412yes—
68C2×C42×C79C2311yescharacteristic
69C4⋊C8×C79—21no—
70C4⋊C8×C79D411yescharacteristic
71C4⋊C8×C79D411yescharacteristic
72C22⋊C8×C79—22no—
73C22⋊C8×C79—22no—
74#97×C79—21no—
75C4×C8×C79—21no—
76C2×Dic158—21no—
77C2×Dic158—21no—
78C22×Dic79—41no—
79#11929612—44no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—42no—
83C2×C79⋊C8—41no—
84C4⋊Dic79—22no—
85C4⋊Dic79—21no—
86C4⋊Dic79—21no—
87C4⋊Dic79—21no—
88C4⋊Dic79—21no—
89C4⋊Dic79—41no—
90Dic79⋊C4—41no—
91Dic79⋊C4—41no—
92C22⋊Dic79—41no—
93C4×Dic79—41no—
94C4×Dic79—41no—
95C4×Dic79—41no—
96#41×C79—41no—
97#41×C79—41no—
98#41×C79—41no—
99#41×C79—41no—
100C2×Q8×C79—21no—
101C2×D4×C79—22no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79C22⋊C412yes—
105C22×C4×C79—22no—
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79#4111yescharacteristic
109M16×C79—21no—
110M16×C79—21no—
111M16×C79—21no—
112M16×C79—44no—
113M16×C79SD1612yescomplemented
114C2×C8×C79—21no—
115C2×C8×C79—21no—
116C2×C8×C79—21no—
117C2×C8×C79—21no—
118C2×C8×C79—44no—
119C2×C8×C79—41no—
120C2×C8×C79—41no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79C22⋊C412yes—
123C22⋊C4×C79—22no—
124C22⋊C4×C79—22no—
125C42×C79C22×C411yescharacteristic
126C42×C79C2×D411yescharacteristic
127C42×C79C2×D411yescharacteristic
128C42×C79#4111yescharacteristic
129C23×C79—22no—
130C23×C79#12111yescharacteristic
131Q8×C79—41no—
132Q8×C79—41no—
133D4×C79—42no—
134D4×C79—42no—
135D4×C79—42no—
136D4×C79—41no—
137D4×C79—41no—
138D4×C79—41no—
139D4×C79—41no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—22no—
143C2×C4×C79—41no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79C2×C22⋊C411yescharacteristic
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79C4×D411yescharacteristic
149C2×C4×C79#12011yescharacteristic
150C2×C4×C79#12111yescharacteristic
151C2×C4×C79#12311yescharacteristic
152C2×C4×C79C2×SD1611yescharacteristic
153C2×C4×C79#13511yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C79#13711yescharacteristic
156C2×Dic79—42no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159C2×Dic79—81no—
160C2×Dic79—81no—
161Dic158—42no—
162Dic158—41no—
163Dic158—41no—
164C8×C79—22no—
165C8×C79—41no—
166C8×C79—41no—
167C8×C79—41no—
168C79⋊C8—22no—
169C79⋊C8—41no—
170C79⋊C8—81no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—42no—
174C22×C79—42no—
175C22×C79—42no—
176C22×C79#38611yescharacteristic
177C22×C79#46111yescharacteristic
178C22×C79#46311yescharacteristic
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—42no—
183C4×C79C4×SD1611yescharacteristic
184C4×C79#44011yescharacteristic
185C4×C79#44311yescharacteristic
186C4×C79#44411yescharacteristic
187C4×C79#45011yescharacteristic
188C4×C79#45211yescharacteristic
189Dic79—82no—
190Dic79—81no—
191Dic79—81no—
192#9055—791nomaximal, Sylow
193C2×C79—21no—
194C2×C79—21no—
195C2×C79—82no—
196C2×C79#187611yescharacteristic
197C2×C79#187711yescharacteristic
198C2×C79#188411yescharacteristic
199#3167—791no—
200#2935D7911yescharacteristic
201#1986—791no—
202#1502—791no—
203#1481—791no—
204#1279—791no—
205#1278—791no—
206C79#905511yescharacteristic, complemented, Sylow
207C2×#131—21no—
208#533—791no—
209#532C2×D7911yescharacteristic
210#498—791no—
211#492—791no—
212#487—791no—
213#486—791no—
214#465—1582no—
215C4×SD16—1582no—
216#436—21no—
217#435—22no—
218C8×D4—21no—
219#431—791no—
220C2×C4⋊C8C2×D7911yescharacteristic
221#408—21no—
222C4×M16C2×D7911yescharacteristic
223C2×C4×C8—791no—
224C4×C4⋊C4—791no—
225#339—1581no—
226#339—1581no—
227#327—1582no—
228Q8⋊C8—1581no—
229Q8⋊C8—1581no—
230C2×#41—21no—
231C2×SD16—1582no—
232#131—42no—
233#131—42no—
234C2×M16—22no—
235C2×M16C22⋊D7911yescharacteristic
236C2×M16C22⋊D7911yescharacteristic
237C22×C8—22no—
238C22×C8—21no—
239C22×C8—1581no—
240C4⋊Q8—791no—
241C4⋊Q8—791no—
242C4⋊Q8C4×D7911yescharacteristic
243#127C4×D7911yescharacteristic
244#125—1581no—
245#124—21no—
246C22⋊Q8—1581no—
247C22⋊Q8—1581no—
248#121—21no—
249#121D31612yes—
250C4×Q8—1581no—
251C4×Q8—1581no—
252C4×D4—21no—
253C4×D4C22⋊D7912yes—
254#117—1581no—
255C2×C4⋊C4—1581no—
256C2×C42—1581no—
257C2×C42C22×D7911yescharacteristic
258#106—1582no—
259C4⋊C8—21no—
260C4⋊C8—1581no—
261C4⋊C8C22⋊D7911yescharacteristic
262C4⋊C8C22⋊D7911yescharacteristic
263#103—1582no—
264#102—1582no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—1581no—
268#97—21no—
269C4×C8—21no—
270C4×C8—791no—
271C4×C8—791no—
272C4×C8—1581no—
273C4×C8—1581no—
274#95—1581no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—1581no—
281C2×Q8—1581no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4(C2×D79)⋊C412yes—
286C22×C4—22no—
287C22×C4—3161no—
288C22×C4C2×C22⋊D7911yescharacteristic
289C22×C4C2×D31611yescharacteristic
290C22×C4#1192962411yescharacteristic
291SD16—3164no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—44no—
296M16#1192961212yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—44no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—3162no—
305C2×C8—3161no—
306C2×C8—1581no—
307C2×C8—1581no—
308C4⋊C4—21no—
309C4⋊C4—3161no—
310C4⋊C4—3161no—
311C4⋊C4—3161no—
312C4⋊C4—1582no—
313C4⋊C4—1581no—
314C4⋊C4—1581no—
315C4⋊C4—1581no—
316C4⋊C4—1581no—
317C4⋊C4(C2×D79)⋊C412yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—3161no—
321C42—3161no—
322C42—3161no—
323C42—3161no—
324C42C2×C4×D7911yescharacteristic
325C42C2×C22⋊D7911yescharacteristic
326C42C2×C22⋊D7911yescharacteristic
327C42#1192962411yescharacteristic
328C23—22no—
329C23#1457913511yescharacteristic
330Q8—41no—
331Q8—41no—
332Q8—3162no—
333Q8—3161no—
334Q8—3161no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—3162no—
349C2×C4—3161no—
350C2×C4—3161no—
351C2×C4—6321no—
352C2×C4—6321no—
353C2×C4C2×(C2×D79)⋊C411yescharacteristic
354C2×C4C4×C22⋊D7911yescharacteristic
355C2×C4C4×C22⋊D7911yescharacteristic
356C2×C4#1457913011yescharacteristic
357C2×C4#1457913511yescharacteristic
358C2×C4#1457914411yescharacteristic
359C2×C4C2×#1192961211yescharacteristic
360C2×C4#1457918111yescharacteristic
361C2×C4#1457918711yescharacteristic
362C2×C4#1457919111yescharacteristic
363C8—22no—
364C8—41no—
365C8—41no—
366C8—41no—
367C8—3161no—
368C8—6321no—
369C8—1582no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1909844211yescharacteristic
376C22#1909876811yescharacteristic
377C22#1909877811yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—6322no—
383C4—6321no—
384C4—6321no—
385C4C4×#1192961211yescharacteristic
386C4#1909866311yescharacteristic
387C4#1909867811yescharacteristic
388C4#1909868311yescharacteristic
389C4#1909871411yescharacteristic
390C4#1909872511yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2376372711yescharacteristic
395C2#2376373411yescharacteristic
396C2#2376378311yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2935C2
socleC22×C79#386
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#2935D79
hypocenterC79#9055
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #23764344C2semidirect
4. #23761802C2semidirect
5. #23761239C2semidirect
6. #23761238C2semidirect
9. #19098959C4semidirect
10. #19098935C4semidirect
11. #19098915C4semidirect
12. #19098909C4semidirect
113. M16×C79SD16semidirect
206. C79#9055semidirect
296. M16#11929612semidirect

Maximal subgroups

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

subgroupindex
#237739552
#237643442
#237618952
#237618022
#237612392
#237612382
C79×#29352
#905579