Finite Group Databasebeta

← #34185003

Subgroups — #34185003

Order 20224.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774521C211yesmaximal, characteristic
2#23764232C211yesmaximal, characteristic
3#23762072C211yesmaximal, characteristic, complemented
4#23761959C211yesmaximal, characteristic
5#23761258C211yesmaximal, characteristic, complemented
6#23761229C211yesmaximal, characteristic, complemented
7C79×#2327C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098966C412yescomplemented
10#19098911C412yescomplemented
11#19098876—22no—
12#19098840—22no—
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098587C2211yescharacteristic
16C2×#14579089—21no—
17C2×#14579089—21no—
18#19098309—22no—
19#19098296—22no—
20C2×C4⋊Q8×C79C2211yescharacteristic
21C2×C4×Q8×C79—21no—
22#391×C79—21no—
23#387×C79—22no—
24#387×C79—22no—
25#382×C79C2211yescharacteristic
26#376×C79C2211yescharacteristic
27C2×#11929624—21no—
28C2×#11929608—21no—
29#14579161C2×C411yescharacteristic
30#14579158C2×C411yescharacteristic
31#14579149—21no—
32#14579135—21no—
33#14579135—21no—
34C4×D316—21no—
35C4×D316—21no—
36#14579094—24no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41#14579089—42no—
42#14579089—42no—
43#14579081—24no—
44C22⋊(C79⋊C8)—21no—
45C22⋊(C79⋊C8)—21no—
46C4×C79⋊C8—21no—
47C4×C79⋊C8C2×C412yes—
48C22×Q8×C79—21no—
49C4⋊Q8×C79—21no—
50C4⋊Q8×C79—21no—
51C4⋊Q8×C79C2×C412yes—
52C4⋊Q8×C79D412yes—
53C4×Q8×C79—42no—
54C4×Q8×C79—42no—
55C2×C4⋊C4×C79—21no—
56C2×C4⋊C4×C79—22no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79D412yes—
59C2×C42×C79—22no—
60C2×C42×C79D411yescharacteristic
61C2×C42×C79D411yescharacteristic
62C2×C42×C79C2311yescharacteristic
63#95×C79—24no—
64#95×C79—21no—
65#95×C79—21no—
66#95×C79—21no—
67#95×C79D412yes—
68#11929624—41no—
69#11929624—41no—
70#11929624—41no—
71#11929624—41no—
72C2×Dic158—21no—
73C2×D316—21no—
74C2×D316—21no—
75C2×D316—21no—
76C2×C22⋊D79—41no—
77C2×C4×D79—41no—
78#11929608—41no—
79#11929608—41no—
80C2×C79⋊C8—22no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—41no—
83C4⋊Dic79—21no—
84C4⋊Dic79—21no—
85(C2×D79)⋊C4—41no—
86(C2×D79)⋊C4—41no—
87C2×Q8×C79—22no—
88C2×Q8×C79—22no—
89C2×Q8×C79—41no—
90C2×Q8×C79—41no—
91C2×Q8×C79—41no—
92C2×Q8×C79—41no—
93C22×C4×C79—21no—
94C22×C4×C79—22no—
95C22×C4×C79—21no—
96C22×C4×C79—22no—
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79C2×D411yescharacteristic
99C22×C4×C79C2×D411yescharacteristic
100C4⋊C4×C79—22no—
101C4⋊C4×C79—22no—
102C4⋊C4×C79—22no—
103C4⋊C4×C79—44no—
104C4⋊C4×C79—41no—
105C4⋊C4×C79—41no—
106C4⋊C4×C79C22⋊C414yes—
107C42×C79—21no—
108C42×C79—21no—
109C42×C79—21no—
110C42×C79—44no—
111C42×C79D812yes—
112C42×C79C22×C411yescharacteristic
113C42×C79C2×D411yescharacteristic
114C42×C79#4111yescharacteristic
115C42×C79#4111yescharacteristic
116C23×C79#12011yescharacteristic
117C22×D79—41no—
118Q8×C79—42no—
119Q8×C79—42no—
120Q8×C79—42no—
121Q8×C79—42no—
122C2×C4×C79—24no—
123C2×C4×C79—22no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—42no—
129C2×C4×C79—42no—
130C2×C4×C79—41no—
131C2×C4×C79—42no—
132C2×C4×C79—41no—
133C2×C4×C79—42no—
134C2×C4×C79—42no—
135C2×C4×C79—41no—
136C2×C4×C79C2×C22⋊C411yescharacteristic
137C2×C4×C79C4×D411yescharacteristic
138C2×C4×C79C4×D411yescharacteristic
139C2×C4×C79#12111yescharacteristic
140C2×C4×C79#12111yescharacteristic
141C2×C4×C79#12311yescharacteristic
142C2×C4×C79C2×D811yescharacteristic
143C2×C4×C79#13511yescharacteristic
144C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
146C22⋊D79—81no—
147C22⋊D79—81no—
148C2×Dic79—41no—
149D316—42no—
150D316—41no—
151D316—41no—
152D316—81no—
153C4×D79—81no—
154C4×D79—81no—
155Dic158—41no—
156Dic158—41no—
157C79⋊C8—42no—
158C79⋊C8—81no—
159C79⋊C8—81no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79#38611yescharacteristic
163C22×C79#44711yescharacteristic
164C22×C79#44911yescharacteristic
165C2×D79—81no—
166C2×D79—81no—
167C2×D79—81no—
168C4×C79—21no—
169C4×C79—21no—
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—42no—
173C4×C79—44no—
174C4×C79—82no—
175C4×C79—81no—
176C4×C79C4×D811yescharacteristic
177C4×C79#44211yescharacteristic
178C4×C79#45911yescharacteristic
179C4×C79#46311yescharacteristic
180Dic79—81no—
181#9300—791nomaximal, Sylow
182C2×C79—21no—
183C2×C79—21no—
184C2×C79#187911yescharacteristic
185C2×C79#190511yescharacteristic
186C2×C79#191211yescharacteristic
187D79—161no—
188#3226—791no—
189#2327D7911yescharacteristic, complemented
190#1963—791no—
191#1546—791no—
192#1517—791no—
193#1286—791no—
194#1274—791no—
195C79#930011yescharacteristic, complemented, Sylow
196#532—791no—
197C2×C4⋊Q8C2×D7911yescharacteristic
198C2×C4×Q8—21no—
199#500—792no—
200#486—792no—
201#480—1582no—
202#474—1582no—
203#431—791no—
204#431—791no—
205#423—791no—
206C2×#104—1581no—
207C2×#104—1581no—
208#391—21no—
209#387—22no—
210#387—22no—
211#382C2×D7911yescharacteristic
212#376C2×D7911yescharacteristic
213#332—1582no—
214#327—1582no—
215C2×#41—1581no—
216C22×Q8—21no—
217C2×M16—1581no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—791no—
221C4⋊Q8C4×D7912yes—
222C4⋊Q8D31612yes—
223#127—791no—
224#124—1581no—
225#121—1581no—
226#121—1581no—
227C4×Q8—42no—
228C4×Q8—42no—
229C4×D4—1581no—
230C4×D4—1581no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22⋊D7912yes—
235C2×C42—22no—
236C2×C42C22⋊D7911yescharacteristic
237C2×C42C22⋊D7911yescharacteristic
238C2×C42C22×D7911yescharacteristic
239#107—1584no—
240C4⋊C8—1581no—
241C4⋊C8—1581no—
242C4⋊C8—1581no—
243C4⋊C8—1581no—
244#104—3162no—
245#104—3162no—
246#102—1584no—
247C22⋊C8—1581no—
248C22⋊C8—1581no—
249C4×C8—792no—
250C4×C8—1581no—
251#95—24no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95C22⋊D7912yes—
256#41—3161no—
257#41—3161no—
258#41—3161no—
259#41—3161no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—1581no—
267C2×D4—3161no—
268C2×D4—1581no—
269C2×D4—1581no—
270C2×D4—1581no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—3161no—
276C22×C4C2×C22⋊D7911yescharacteristic
277C22×C4C2×C22⋊D7911yescharacteristic
278C22×C4C2×C22⋊D7911yescharacteristic
279M16—3161no—
280M16—3161no—
281C2×C8—3161no—
282C2×C8—3161no—
283C2×C8—1582no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—1581no—
291C4⋊C4—1581no—
292C4⋊C4(C2×D79)⋊C414yes—
293C22⋊C4—3161no—
294C22⋊C4—3161no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—44no—
299C42#1192960912yes—
300C42C2×C4×D7911yescharacteristic
301C42C2×D31611yescharacteristic
302C42#1192962411yescharacteristic
303C42#1192962411yescharacteristic
304C23—3161no—
305C23#1457913011yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—42no—
310Q8—3161no—
311Q8—3161no—
312D4—3162no—
313D4—3161no—
314D4—3161no—
315D4—6321no—
316D4—6321no—
317D4—6321no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—3161no—
333C2×C4—6321no—
334C2×C4—6321no—
335C2×C4C2×(C2×D79)⋊C411yescharacteristic
336C2×C4C4×C22⋊D7911yescharacteristic
337C2×C4C4×C22⋊D7911yescharacteristic
338C2×C4#1457913511yescharacteristic
339C2×C4#1457913511yescharacteristic
340C2×C4#1457914411yescharacteristic
341C2×C4C2×#1192960911yescharacteristic
342C2×C4#1457918111yescharacteristic
343C2×C4#1457918611yescharacteristic
344C2×C4#1457918711yescharacteristic
345C8—3162no—
346C8—6321no—
347C8—6321no—
348C22—21no—
349C22—21no—
350C22—6321no—
351C22—6321no—
352C22—6321no—
353C22#1909844211yescharacteristic
354C22#1909869711yescharacteristic
355C22#1909870811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—44no—
362C4—82no—
363C4—81no—
364C4—6321no—
365C4C4×#1192960911yescharacteristic
366C4#1909867311yescharacteristic
367C4#1909875811yescharacteristic
368C4#1909877811yescharacteristic
369C2—21no—
370C2—21no—
371C2—12641no—
372C2#2376374811yescharacteristic
373C2#2376389411yescharacteristic
374C2#2376393711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2327C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#2327D79
hypocenterC79#9300
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
3. #23762072C2semidirect
5. #23761258C2semidirect
6. #23761229C2semidirect
7. C79×#2327C2semidirect
9. #19098966C4semidirect
10. #19098911C4semidirect
189. #2327D79semidirect
195. C79#9300semidirect

Maximal subgroups

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

subgroupindex
#237745212
#237642322
#237620722
#237619592
#237612582
#237612292
C79×#23272
#930079