Finite Group Databasebeta

← #34182576

Subgroups — #34182576

Order 20224.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23763420C211yesmaximal, characteristic
2#23762170C211yesmaximal, characteristic, complemented
3#23762158C211yesmaximal, characteristic, complemented
4#23761782C211yesmaximal, characteristic
5#23761301C211yesmaximal, characteristic, complemented
6#23761239C211yesmaximal, characteristic, complemented
7C79×#2925C211yesmaximal, characteristic
8#19098616C2211yescharacteristic
9#19098587C2211yescharacteristic
10C2×#14579088—21no—
11C2×C4×C79⋊C8C2211yescharacteristic
12C4×C4⋊Dic79C2211yescharacteristic
13#19098322—21no—
14#19098315—22no—
15#19098313—21no—
16#19098313—21no—
17Q8⋊(C79⋊C8)C411yescharacteristic, complemented
18Q8⋊(C79⋊C8)C411yescharacteristic, complemented
19D4⋊(C79⋊C8)C411yescharacteristic, complemented
20D4⋊(C79⋊C8)C411yescharacteristic, complemented
21C4×#41×C79C2211yescharacteristic
22#433×C79—22no—
23#432×C79—21no—
24#431×C79—21no—
25#423×C79C2211yescharacteristic
26C2×C4⋊C8×C79C2211yescharacteristic
27#408×C79—21no—
28#405×C79—21no—
29C2×#11929608—21no—
30C22×C79⋊C8—21no—
31C2×C4⋊Dic79—21no—
32C2×C4×Dic79—21no—
33C4⋊(C79⋊C8)—21no—
34C4⋊(C79⋊C8)—21no—
35C4⋊(C79⋊C8)C2×C411yescharacteristic
36C4⋊(C79⋊C8)C2×C411yescharacteristic
37#14579088—42no—
38C22⋊(C79⋊C8)—21no—
39C4×C79⋊C8—21no—
40C4×C79⋊C8—21no—
41C4×C79⋊C8C2×C411yescharacteristic
42C4×C79⋊C8C2×C411yescharacteristic
43C79⋊#95—21no—
44C2×#41×C79D411yescharacteristic
45C2×M16×C79—21no—
46C22×C8×C79—21no—
47C4×Q8×C79C2×C411yescharacteristic
48C4×Q8×C79Q811yescharacteristic
49C4×D4×C79—21no—
50C4×D4×C79—21no—
51C4×D4×C79C2×C411yescharacteristic
52C4×D4×C79Q811yescharacteristic
53#117×C79—21no—
54#117×C79D411yescharacteristic
55C2×C42×C79—21no—
56C2×C42×C79C2311yescharacteristic
57C4⋊C8×C79—21no—
58C4⋊C8×C79—21no—
59C4⋊C8×C79D411yescharacteristic
60C4⋊C8×C79D411yescharacteristic
61C4⋊C8×C79D412yes—
62C22⋊C8×C79—22no—
63C22⋊C8×C79—22no—
64#97×C79—22no—
65C4×C8×C79—22no—
66C22×Dic79—41no—
67#11929608—41no—
68#11929608—41no—
69C2×C79⋊C8—21no—
70C2×C79⋊C8—21no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—42no—
74C2×C79⋊C8—41no—
75C4⋊Dic79—22no—
76C4⋊Dic79—41no—
77C4×Dic79—41no—
78C4×Dic79—41no—
79#41×C79—21no—
80#41×C79—21no—
81#41×C79—21no—
82#41×C79—21no—
83C2×Q8×C79C4⋊C411yescharacteristic
84C2×D4×C79—21no—
85C2×D4×C79C4⋊C411yescharacteristic
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79#4111yescharacteristic
91C22×C4×C79#4111yescharacteristic
92M16×C79—41no—
93M16×C79—41no—
94C2×C8×C79—21no—
95C2×C8×C79—22no—
96C2×C8×C79—21no—
97C2×C8×C79—41no—
98C2×C8×C79—41no—
99C4⋊C4×C79—21no—
100C4⋊C4×C79—21no—
101C4⋊C4×C79C4⋊C411yescharacteristic
102C4⋊C4×C79C4⋊C411yescharacteristic
103C22⋊C4×C79—21no—
104C22⋊C4×C79—21no—
105C22⋊C4×C79—21no—
106C42×C79—21no—
107C42×C79—21no—
108C42×C79—21no—
109C42×C79C22×C411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C42×C79C2×D411yescharacteristic
112C42×C79C2×Q811yescharacteristic
113C23×C79—21no—
114C23×C79#12311yescharacteristic
115Q8×C79—21no—
116Q8×C79—21no—
117D4×C79—21no—
118D4×C79—21no—
119D4×C79—41no—
120D4×C79—41no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—81no—
132C2×C4×C79C2×C4⋊C411yescharacteristic
133C2×C4×C79C4×D411yescharacteristic
134C2×C4×C79C4×D411yescharacteristic
135C2×C4×C79#12011yescharacteristic
136C2×C4×C79C22⋊Q811yescharacteristic
137C2×C4×C79C22⋊Q811yescharacteristic
138C2×C4×C79#13511yescharacteristic
139C2×C4×C79#13511yescharacteristic
140C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C79#13711yescharacteristic
142C2×Dic79—42no—
143C2×Dic79—81no—
144C2×Dic79—81no—
145C8×C79—42no—
146C8×C79—41no—
147C8×C79—41no—
148C79⋊C8—22no—
149C79⋊C8—41no—
150C79⋊C8—41no—
151C79⋊C8—41no—
152C79⋊C8—81no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—41no—
158C22×C79#38511yescharacteristic
159C22×C79#48211yescharacteristic
160C22×C79#48511yescharacteristic
161C4×C79—21no—
162C4×C79—21no—
163C4×C79—21no—
164C4×C79—41no—
165C4×C79—41no—
166C4×C79—41no—
167C4×C79#42711yescharacteristic
168C4×C79#42811yescharacteristic
169C4×C79#44311yescharacteristic
170C4×C79#44411yescharacteristic
171C4×C79#44911yescharacteristic
172C4×C79#45211yescharacteristic
173Dic79—82no—
174#8903—791nomaximal, Sylow
175C2×C79—21no—
176C2×C79—21no—
177C2×C79—81no—
178C2×C79#185411yescharacteristic
179C2×C79#186111yescharacteristic
180C2×C79#186611yescharacteristic
181#2925D7911yescharacteristic
182#1806—791no—
183#1570—791no—
184#1567—791no—
185#1476—791no—
186#1303—791no—
187#1279—791no—
188C79#890311yescharacteristic, complemented, Sylow
189C4×#41C2×D7911yescharacteristic
190#433—22no—
191#432—21no—
192#431—21no—
193#431—791no—
194#423—791no—
195#423C2×D7911yescharacteristic
196C2×C4⋊C8C2×D7911yescharacteristic
197C2×#104—1581no—
198#408—21no—
199#405—21no—
200C2×C4×C8—791no—
201C4×C4⋊C4—791no—
202#339—1581no—
203#335—1582no—
204#334—1581no—
205#334—1581no—
206Q8⋊C8—791no—
207Q8⋊C8—791no—
208D4⋊C8—791no—
209D4⋊C8—791no—
210C2×#41C22⋊D7911yescharacteristic
211C2×M16—21no—
212C2×M16—1581no—
213C22×C8—21no—
214C22×C8—1581no—
215C4×Q8Dic15811yescharacteristic
216C4×Q8C4×D7911yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic15811yescharacteristic
220C4×D4C4×D7911yescharacteristic
221#117—21no—
222#117C22⋊D7911yescharacteristic
223C2×C4⋊C4—1581no—
224C2×C42—21no—
225C2×C42—1581no—
226C2×C42C22×D7911yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—791no—
230C4⋊C8—791no—
231C4⋊C8—1581no—
232C4⋊C8—1581no—
233C4⋊C8C22⋊D7911yescharacteristic
234C4⋊C8C22⋊D7911yescharacteristic
235C4⋊C8C22⋊D7912yes—
236#104—3162no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—1581no—
240#97—22no—
241C4×C8—22no—
242C4×C8—791no—
243C4×C8—791no—
244C4×C8—1581no—
245C4×C8—1581no—
246#95—1581no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8Dic79⋊C411yescharacteristic
252C2×D4—21no—
253C2×D4Dic79⋊C411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—3161no—
258C22×C4C2×C22⋊D7911yescharacteristic
259C22×C4#1192962411yescharacteristic
260C22×C4#1192962411yescharacteristic
261M16—41no—
262M16—41no—
263M16—3161no—
264M16—3161no—
265C2×C8—21no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—3162no—
271C2×C8—3161no—
272C2×C8—1581no—
273C2×C8—1581no—
274C2×C8—1581no—
275C2×C8—1581no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3161no—
279C4⋊C4—1582no—
280C4⋊C4Dic79⋊C411yescharacteristic
281C4⋊C4Dic79⋊C411yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—3161no—
289C42—3161no—
290C42C2×C4×D7911yescharacteristic
291C42C2×C22⋊D7911yescharacteristic
292C42C2×C22⋊D7911yescharacteristic
293C42C2×Dic15811yescharacteristic
294C23—21no—
295C23#1457914411yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—3162no—
314C2×C4—6321no—
315C2×C4—6321no—
316C2×C4C2×Dic79⋊C411yescharacteristic
317C2×C4C4×C22⋊D7911yescharacteristic
318C2×C4C4×C22⋊D7911yescharacteristic
319C2×C4#1457913011yescharacteristic
320C2×C4#1457914011yescharacteristic
321C2×C4#1457914011yescharacteristic
322C2×C4#1457918111yescharacteristic
323C2×C4#1457918111yescharacteristic
324C2×C4#1457918611yescharacteristic
325C2×C4#1457919111yescharacteristic
326C8—42no—
327C8—41no—
328C8—41no—
329C8—3161no—
330C8—3161no—
331C8—3161no—
332C8—6321no—
333C8—1582no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1909843611yescharacteristic
340C22#1909888711yescharacteristic
341C22#1909890411yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—6322no—
349C4#1909859911yescharacteristic
350C4#1909860611yescharacteristic
351C4#1909867811yescharacteristic
352C4#1909868311yescharacteristic
353C4#1909870811yescharacteristic
354C4#1909872511yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2376359911yescharacteristic
359C2#2376364011yescharacteristic
360C2#2376367511yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2925C2
socleC22×C79#385
prime radicalC2×#41×C79D4
cosocleC2×C42×C79C23
hypercenter#2925D79
hypocenterC79#8903
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #23762170C2semidirect
3. #23762158C2semidirect
5. #23761301C2semidirect
6. #23761239C2semidirect
17. Q8⋊(C79⋊C8)C4semidirect
18. Q8⋊(C79⋊C8)C4semidirect
19. D4⋊(C79⋊C8)C4semidirect
20. D4⋊(C79⋊C8)C4semidirect
188. C79#8903semidirect

Maximal subgroups

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

subgroupindex
#237634202
#237621702
#237621582
#237617822
#237613012
#237612392
C79×#29252
#890379