Finite Group Databasebeta

← #47986090

Subgroups — #47986090

Order 35456.

660 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×#22579840C211yesmaximal, characteristic
2C2×#22579837C211yesmaximal, characteristic, complemented
3C22×#17984542C211yesmaximal, characteristic, complemented
4#31799893C211yesmaximal, characteristic, complemented
5#31799891C211yesmaximal, characteristic, complemented
6#31799886C211yesmaximal, characteristic, complemented
7#31799884C211yesmaximal, characteristic, complemented
8#31799881C211yesmaximal, characteristic
9#31799879C211yesmaximal, characteristic
10#31799866C211yesmaximal, characteristic, complemented
11#31799861C211yesmaximal, characteristic, complemented
12#31799860C211yesmaximal, characteristic, complemented
13C4×#17984542C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic277C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C277C211yesmaximal, characteristic, complemented
16C22×C22⋊D277C2211yescharacteristic
17C23×Dic277C2211yescharacteristic
18C23×Dic277C2211yescharacteristic
19C22×C4×D277C2211yescharacteristic
20#22579840C418yescomplemented
21#22579837—24no—
22C2×#17984542—24no—
23C2×#17984542C2211yescharacteristic
24C2×#17984542C2211yescharacteristic
25C2×#17984542C2211yescharacteristic
26C2×#17984542C2211yescharacteristic
27C2×#17984542C2211yescharacteristic
28C2×#17984542C2211yescharacteristic
29C2×#17984542C2211yescharacteristic
30C2×C22⋊Dic277C2211yescharacteristic
31C2×#17984540C2211yescharacteristic
32C2×#17984540C2211yescharacteristic
33C2×#17984540C2211yescharacteristic
34C2×#17984540C2211yescharacteristic
35C2×#17984540C2211yescharacteristic
36C2×C4×Dic277C2211yescharacteristic
37C2×C4×Dic277C2211yescharacteristic
38C2×C4×Dic277C2211yescharacteristic
39#22579787C2211yescharacteristic
40#22579787C2211yescharacteristic
41#22579787C2211yescharacteristic
42#22579787C2211yescharacteristic
43#22579787C2211yescharacteristic
44#22579787C2211yescharacteristic
45#22579787C2211yescharacteristic
46#22579787C2211yescharacteristic
47C23×C4×C277C2211yescharacteristic
48C2×C22⋊C4×C277C2211yescharacteristic
49C2×C22⋊C4×C277C2211yescharacteristic
50C2×C42×C277C2211yescharacteristic
51C2×C42×C277C2211yescharacteristic
52#95×C277C2211yescharacteristic
53#95×C277C2211yescharacteristic
54C23×D277C2311yescharacteristic
55C2×C22⋊D277—22no—
56C2×C22⋊D277—22no—
57C2×C22⋊D277C2×C414yes—
58C22×Dic277—22no—
59C22×Dic277—22no—
60C22×Dic277—22no—
61C22×Dic277—22no—
62C22×Dic277—41no—
63C22×Dic277C2×C414yes—
64C22×Dic277C2311yescharacteristic
65C22×Dic277C2311yescharacteristic
66C22×Dic277C2311yescharacteristic
67C22×Dic277C2311yescharacteristic
68C22×Dic277C2311yescharacteristic
69C22×Dic277C2311yescharacteristic
70C22×Dic277C2311yescharacteristic
71C2×C4×D277—22no—
72C2×C4×D277—22no—
73C2×C4×D277C2×C414yescomplemented
74#17984542—22no—
75#17984542—22no—
76#17984542—22no—
77#17984542—22no—
78#17984542—22no—
79#17984542C2×C414yescomplemented
80#17984542D414yescomplemented
81C22⋊Dic277—22no—
82#17984540—22no—
83#17984540—22no—
84#17984540—41no—
85#17984540—41no—
86#17984540C2×C414yescomplemented
87C4×Dic277—22no—
88C4×Dic277—41no—
89C4×Dic277C2×C414yescomplemented
90C24×C277C2311yescharacteristic
91C22×C4×C277—22no—
92C22×C4×C277—22no—
93C22×C4×C277—41no—
94C22×C4×C277C2311yescharacteristic
95C22×C4×C277C2311yescharacteristic
96C22×C4×C277C2311yescharacteristic
97C22×C4×C277C2311yescharacteristic
98C22×C4×C277C2311yescharacteristic
99C22×C4×C277C2311yescharacteristic
100C22⋊C4×C277—22no—
101C22⋊C4×C277C2×C414yes—
102C42×C277—22no—
103C42×C277—41no—
104C22×D277—21no—
105C22×D277—21no—
106C22×D277—21no—
107C22×D277—21no—
108C22×D277—21no—
109C22×D277—21no—
110C22×D277C22×C412yes—
111C22⋊D277—28no—
112C2×Dic277—21no—
113C2×Dic277—21no—
114C2×Dic277—21no—
115C2×Dic277—21no—
116C2×Dic277—21no—
117C2×Dic277—21no—
118C2×Dic277—28no—
119C2×Dic277—21no—
120C2×Dic277—21no—
121C2×Dic277—21no—
122C2×Dic277—21no—
123C2×Dic277—21no—
124C2×Dic277—21no—
125C2×Dic277—21no—
126C2×Dic277—21no—
127C2×Dic277—21no—
128C2×Dic277—44no—
129C2×Dic277C22×C412yes—
130C2×Dic277C22×C412yes—
131C2×Dic277C22×C412yes—
132C2×Dic277C2×D412yes—
133C2×Dic277C2×D412yes—
134C2×Dic277#4112yes—
135C4×D277—28no—
136C23×C277—21no—
137C23×C277—21no—
138C23×C277—21no—
139C23×C277—21no—
140C23×C277C22×C412yes—
141C23×C277#4112yes—
142C23×C277#4112yes—
143C23×C277C2411yescharacteristic
144C2×C4×C277—21no—
145C2×C4×C277—21no—
146C2×C4×C277—21no—
147C2×C4×C277—21no—
148C2×C4×C277—21no—
149C2×C4×C277—21no—
150C2×C4×C277—21no—
151C2×C4×C277—21no—
152C2×C4×C277—21no—
153C2×C4×C277—21no—
154C2×C4×C277—21no—
155C2×C4×C277—21no—
156C2×C4×C277—21no—
157C2×C4×C277—21no—
158C2×C4×C277—21no—
159C2×C4×C277—44no—
160C2×C4×C277C22×C412yes—
161C2×C4×C277C22×C412yes—
162C2×C4×C277C2×D412yes—
163C22×C277—22no—
164C22×C277—22no—
165C22×C277—22no—
166C22×C277—22no—
167C22×C277—22no—
168C22×C277—22no—
169C22×C277#11714yescomplemented
170C22×C277C23×C411yescharacteristic
171C22×C277C22×D411yescharacteristic
172C22×C277C2×#4111yescharacteristic
173C22×C277C2×#4111yescharacteristic
174C22×C277C2×#4111yescharacteristic
175C22×C277H(5,𝔽2)11yescharacteristic
176C22×C2772−1+411yescharacteristic
177C2×D277—22no—
178C2×D277—22no—
179C2×D277—22no—
180C2×D277—41no—
181C2×D277—41no—
182C2×D277—41no—
183C2×D277—41no—
184C4×C277—22no—
185C4×C277—22no—
186C4×C277—22no—
187C4×C277—41no—
188C4×C277—41no—
189C4×C277—41no—
190Dic277—22no—
191Dic277—22no—
192Dic277—22no—
193Dic277—41no—
194Dic277—41no—
195Dic277—41no—
196Dic277C4×D414yes—
197C2×C277—24no—
198C2×C277C2×#11711yescharacteristic
199C2×C277C2×C4×D411yescharacteristic
200C2×C277#52011yescharacteristic
201C2×C277#54611yescharacteristic
202C2×C277#54811yescharacteristic
203C2×C277#55211yescharacteristic
204C2×C277#55311yescharacteristic
205D277—42no—
206C277#235211yescharacteristic, complemented, Sylow
207#2352—2771nomaximal, Sylow
208C2×C4×D4—2771no—
209C2×#117—2771no—
210C22×C4⋊C4—2771no—
211#388—2771no—
212#388—2771no—
213#387—2771no—
214#387—2771no—
215#386—2771no—
216#386—2771no—
217#382—2771no—
218#380—2771no—
219#380—2771no—
220C4×C4⋊C4—2771no—
221C4×C22⋊C4—2771no—
222C4×C22⋊C4D27711yescharacteristic, complemented
223C22×D4—2771no—
224C23×C4—2771no—
225C23×C4—2771no—
226C23×C4—2771no—
227C23×C4C2×D27711yescharacteristic
228C4×D4—2778no—
229#117—5544no—
230C2×C4⋊C4—2771no—
231C2×C4⋊C4—2771no—
232C2×C4⋊C4—2771no—
233C2×C4⋊C4—2771no—
234C2×C4⋊C4—2771no—
235C2×C4⋊C4—2771no—
236C2×C4⋊C4—2771no—
237C2×C4⋊C4—5544no—
238C2×C22⋊C4—2771no—
239C2×C22⋊C4—2771no—
240C2×C22⋊C4—2771no—
241C2×C22⋊C4—2771no—
242C2×C22⋊C4—2771no—
243C2×C22⋊C4—2771no—
244C2×C22⋊C4C2×D27711yescharacteristic
245C2×C22⋊C4C2×D27711yescharacteristic
246C2×C42—2771no—
247C2×C42—2771no—
248C2×C42—2771no—
249C2×C42C2×D27711yescharacteristic
250C2×C42C2×D27711yescharacteristic
251#95—2771no—
252#95—2771no—
253#95—2771no—
254#95—2771no—
255#95—2771no—
256#95—2771no—
257#95—2771no—
258#95—2771no—
259#95C2×D27711yescharacteristic
260#95C2×D27711yescharacteristic
261C24—2771no—
262C24C22×D27711yescharacteristic
263C2×D4—2774no—
264C2×D4—5542no—
265C2×D4—5542no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—2771no—
270C22×C4—2774no—
271C22×C4—2771no—
272C22×C4—2771no—
273C22×C4—2771no—
274C22×C4—2774no—
275C22×C4—2771no—
276C22×C4—2771no—
277C22×C4—2771no—
278C22×C4—5542no—
279C22×C4—5542no—
280C22×C4—5542no—
281C22×C4—5542no—
282C22×C4—5542no—
283C22×C4—5542no—
284C22×C4—11081no—
285C22×C4C22×D27711yescharacteristic
286C22×C4C22×D27711yescharacteristic
287C22×C4C22×D27711yescharacteristic
288C22×C4C22×D27711yescharacteristic
289C22×C4C22×D27711yescharacteristic
290C22×C4C22×D27711yescharacteristic
291C4⋊C4—2774no—
292C4⋊C4—2774no—
293C4⋊C4—5542no—
294C4⋊C4—5542no—
295C4⋊C4—5542no—
296C4⋊C4—5542no—
297C4⋊C4—5542no—
298C22⋊C4—22no—
299C22⋊C4—2774no—
300C22⋊C4—5542no—
301C22⋊C4—5542no—
302C22⋊C4—5542no—
303C22⋊C4—11081no—
304C22⋊C4—11081no—
305C22⋊C4C4×D27714yes—
306C42—22no—
307C42—41no—
308C42—2774no—
309C42—5542no—
310C42—11081no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—2772no—
316C23—5541no—
317C23—5541no—
318C23—5541no—
319C23—5541no—
320C23—5541no—
321C23—5541no—
322C23C2×C4×D27712yes—
323C23#1798456312yes—
324C23#1798456312yes—
325C23C23×D27711yescharacteristic
326D4—5548no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—2772no—
344C2×C4—2772no—
345C2×C4—2772no—
346C2×C4—2772no—
347C2×C4—2772no—
348C2×C4—2772no—
349C2×C4—5548no—
350C2×C4—5548no—
351C2×C4—5541no—
352C2×C4—5541no—
353C2×C4—5541no—
354C2×C4—5541no—
355C2×C4—5541no—
356C2×C4—5541no—
357C2×C4—5541no—
358C2×C4—5541no—
359C2×C4—5541no—
360C2×C4—5541no—
361C2×C4—5541no—
362C2×C4—5541no—
363C2×C4—5541no—
364C2×C4—5541no—
365C2×C4—5541no—
366C2×C4—11084no—
367C2×C4C2×C4×D27712yes—
368C2×C4C2×C4×D27712yes—
369C2×C4D4×D27712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—5542no—
377C22—5542no—
378C22—5542no—
379C22—11081no—
380C22—11081no—
381C22—11081no—
382C22—11081no—
383C22#2257983714yescomplemented
384C22C22×C4×D27711yescharacteristic
385C22C2×D4×D27711yescharacteristic
386C22#41×D27711yescharacteristic
387C22C2×#1798456311yescharacteristic
388C22C2×#1798456311yescharacteristic
389C22#2257992511yescharacteristic
390C22#2257992711yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—2774no—
398C4—5542no—
399C4—5542no—
400C4—5542no—
401C4—11081no—
402C4—11081no—
403C4—11081no—
404C2—24no—
405C2—11082no—
406C2C2×#2257983711yescharacteristic
407C2C4×D4×D27711yescharacteristic
408C2#3180048111yescharacteristic
409C2#3180064211yescharacteristic
410C2#3180066111yescharacteristic
411C2#3180068511yescharacteristic
412C2#3180070111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D277
commutatorC22×C277C23×C4
FittingC4×C22⋊C4×C277C2
socle, cosocleC23×C277C24
prime radicalC22×C22⋊D277C22
hypercenterC4×C22⋊C4D277
hypocenterC277#2352
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#22579837C2semidirect
3. C22×#17984542C2semidirect
4. #31799893C2semidirect
5. #31799891C2semidirect
6. #31799886C2semidirect
7. #31799884C2semidirect
10. #31799866C2semidirect
11. #31799861C2semidirect
12. #31799860C2semidirect
13. C4×#17984542C2semidirect
14. C22⋊C4×Dic277C2semidirect
15. C4×C22⋊C4×C277C2semidirect
20. #22579840C4semidirect
73. C2×C4×D277C2×C4semidirect
79. #17984542C2×C4semidirect
80. #17984542D4semidirect
86. #17984540C2×C4semidirect
89. C4×Dic277C2×C4semidirect
169. C22×C277#117semidirect
206. C277#2352semidirect
222. C4×C22⋊C4D277semidirect
383. C22#22579837semidirect

Maximal subgroups

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

subgroupindex
C2×#225798402
C2×#225798372
C22×#179845422
#317998932
#317998912
#317998862
#317998842
#317998812
#317998792
#317998662
#317998612
#317998602
C4×#179845422
C22⋊C4×Dic2772
C4×C22⋊C4×C2772
#2352277