Finite Group Databasebeta

← #47985855

Subgroups — #47985855

Order 35456.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22579853C211yesmaximal, characteristic
2C2×#22579840C211yesmaximal, characteristic, complemented
3C2×#22579838C211yesmaximal, characteristic, complemented
4#31799904C211yesmaximal, characteristic, complemented
5#31799902C211yesmaximal, characteristic, complemented
6#31799902C211yesmaximal, characteristic, complemented
7#31799899C211yesmaximal, characteristic, complemented
8#31799892C211yesmaximal, characteristic, complemented
9#31799880C211yesmaximal, characteristic, complemented
10#31799879C211yesmaximal, characteristic, complemented
11#31799877C211yesmaximal, characteristic, complemented
12#31799876C211yesmaximal, characteristic, complemented
13#31799861C211yesmaximal, characteristic, complemented
14C4×#17984542C211yesmaximal, characteristic, complemented
15#388×C277C211yesmaximal, characteristic, complemented
16C22×D1108C2211yescharacteristic
17C22×C22⋊D277C2211yescharacteristic
18C22×C22⋊D277C2211yescharacteristic
19C23×Dic277C2211yescharacteristic
20C22×C4×D277C2211yescharacteristic
21C22×C4×D277C2211yescharacteristic
22#22579853C418yescomplemented
23#22579840—24no—
24#22579838—24no—
25C2×C4⋊Dic277C2211yescharacteristic
26C2×C4⋊Dic277C2211yescharacteristic
27C2×#17984542C2211yescharacteristic
28C2×#17984542C2211yescharacteristic
29C2×#17984542C2211yescharacteristic
30C2×#17984542C2211yescharacteristic
31C2×C22⋊Dic277C2211yescharacteristic
32C2×#17984540C2211yescharacteristic
33C2×#17984540C2211yescharacteristic
34C2×#17984540C2211yescharacteristic
35C2×#17984540C2211yescharacteristic
36C2×#17984540C2211yescharacteristic
37C2×#17984540C2211yescharacteristic
38C2×#17984540C2211yescharacteristic
39C2×#17984540C2211yescharacteristic
40C2×C4×Dic277C2211yescharacteristic
41C2×C4×Dic277C2211yescharacteristic
42C2×C4×Dic277C2211yescharacteristic
43#22579787C2211yescharacteristic
44#22579787C2211yescharacteristic
45#22579787C2211yescharacteristic
46#22579787C2211yescharacteristic
47C2×C4⋊C4×C277C2211yescharacteristic
48C2×C22⋊C4×C277C2211yescharacteristic
49C2×C22⋊C4×C277C2211yescharacteristic
50C2×C22⋊C4×C277C2211yescharacteristic
51C2×C42×C277C2211yescharacteristic
52#95×C277C2211yescharacteristic
53#95×C277C2211yescharacteristic
54C23×D277C2311yescharacteristic
55C23×D277C2311yescharacteristic
56C2×D1108—41no—
57C2×D1108—41no—
58C2×D1108C2×C414yescomplemented
59C2×C22⋊D277—22no—
60C2×C22⋊D277—22no—
61C2×C22⋊D277—41no—
62C2×C22⋊D277—41no—
63C2×C22⋊D277C2×C414yes—
64C2×C22⋊D277C2×C414yescomplemented
65C22×Dic277—22no—
66C22×Dic277—41no—
67C22×Dic277—41no—
68C22×Dic277C2311yescharacteristic
69C22×Dic277C2311yescharacteristic
70C22×Dic277C2311yescharacteristic
71C22×Dic277C2311yescharacteristic
72C22×Dic277C2311yescharacteristic
73C22×Dic277C2311yescharacteristic
74C2×C4×D277—22no—
75C2×C4×D277—22no—
76C2×C4×D277—22no—
77C2×C4×D277—41no—
78C2×C4×D277—41no—
79C2×C4×D277C2×C414yescomplemented
80C4⋊Dic277—22no—
81C4⋊Dic277—41no—
82#17984542—22no—
83#17984542—22no—
84#17984542—41no—
85#17984542C2×C414yescomplemented
86C22⋊Dic277—41no—
87#17984540—22no—
88#17984540—22no—
89#17984540—22no—
90#17984540—41no—
91#17984540—41no—
92#17984540—41no—
93#17984540—41no—
94#17984540C2×C414yescomplemented
95C4×Dic277—22no—
96C4×Dic277—41no—
97C4×Dic277D414yescomplemented
98C24×C277C2311yescharacteristic
99C22×C4×C277C2311yescharacteristic
100C22×C4×C277C2311yescharacteristic
101C22×C4×C277C2311yescharacteristic
102C22×C4×C277C2311yescharacteristic
103C22×C4×C277C2311yescharacteristic
104C22×C4×C277C2311yescharacteristic
105C4⋊C4×C277—22no—
106C22⋊C4×C277—22no—
107C22⋊C4×C277—41no—
108C22⋊C4×C277C2×C414yescomplemented
109C42×C277—41no—
110C22×D277—21no—
111C22×D277—21no—
112C22×D277—21no—
113C22×D277—21no—
114C22×D277—21no—
115C22×D277—21no—
116C22×D277—21no—
117C22×D277—21no—
118C22×D277—21no—
119C22×D277—21no—
120C22×D277—21no—
121C22×D277—21no—
122C22×D277C22×C412yes—
123C22×D277C22×C412yes—
124D1108—44no—
125C22⋊D277—28no—
126C22⋊D277—44no—
127C2×Dic277—21no—
128C2×Dic277—21no—
129C2×Dic277—21no—
130C2×Dic277—21no—
131C2×Dic277—21no—
132C2×Dic277—21no—
133C2×Dic277—21no—
134C2×Dic277—21no—
135C2×Dic277—21no—
136C2×Dic277—21no—
137C2×Dic277—21no—
138C2×Dic277—21no—
139C2×Dic277—21no—
140C2×Dic277—44no—
141C2×Dic277C22×C412yes—
142C2×Dic277C22×C412yes—
143C2×Dic277C2×D412yes—
144C2×Dic277C2×D412yes—
145C2×Dic277#4112yes—
146C4×D277—28no—
147C4×D277—44no—
148C23×C277—21no—
149C23×C277—21no—
150C23×C277—21no—
151C23×C277—21no—
152C23×C277—21no—
153C23×C277—21no—
154C23×C277C22×C412yes—
155C23×C277C2411yescharacteristic
156C2×C4×C277—21no—
157C2×C4×C277—21no—
158C2×C4×C277—21no—
159C2×C4×C277—21no—
160C2×C4×C277—21no—
161C2×C4×C277—21no—
162C2×C4×C277—21no—
163C2×C4×C277—21no—
164C2×C4×C277—21no—
165C2×C4×C277—21no—
166C2×C4×C277—21no—
167C2×C4×C277—21no—
168C2×C4×C277—21no—
169C2×C4×C277—21no—
170C2×C4×C277—21no—
171C2×C4×C277C22×C412yes—
172C2×C4×C277C22×C412yes—
173C2×C4×C277C2×D412yes—
174C22×C277—22no—
175C22×C277—22no—
176C22×C277—22no—
177C22×C277—41no—
178C22×C277—41no—
179C22×C277—41no—
180C22×C277—41no—
181C22×C277C23×C411yescharacteristic
182C22×C277C22×D411yescharacteristic
183C22×C277C2×#4111yescharacteristic
184C22×C277H(5,𝔽2)11yescharacteristic
185C22×C277H(5,𝔽2)11yescharacteristic
186C22×C277H(5,𝔽2)11yescharacteristic
187C22×C2772−1+411yescharacteristic
188C2×D277—22no—
189C2×D277—22no—
190C2×D277—22no—
191C2×D277—22no—
192C2×D277—22no—
193C2×D277—22no—
194C2×D277—41no—
195C2×D277—41no—
196C2×D277—41no—
197C2×D277—41no—
198C2×D277—41no—
199C2×D277—41no—
200C2×D277—41no—
201C2×D277—41no—
202C4×C277—22no—
203C4×C277—22no—
204C4×C277—22no—
205C4×C277—41no—
206C4×C277—41no—
207C4×C277—41no—
208Dic277—22no—
209Dic277—22no—
210Dic277—41no—
211Dic277—41no—
212Dic277—41no—
213Dic277C4×D414yes—
214C2×C277—42no—
215C2×C277C2×C4×D411yescharacteristic
216C2×C277#51811yescharacteristic
217C2×C277#52011yescharacteristic
218C2×C277#53511yescharacteristic
219C2×C277#53711yescharacteristic
220C2×C277#54011yescharacteristic
221C2×C277#54211yescharacteristic
222D277—42no—
223D277—42no—
224C277#232411yescharacteristic, complemented, Sylow
225#2324—2771nomaximal, Sylow
226C2×#121—2771no—
227C2×C4×D4—2771no—
228C2×#117—2771no—
229#390—2771no—
230#390—2771no—
231#390—2771no—
232#389—2771no—
233#388—2771no—
234#388D27711yescharacteristic, complemented
235#386—2771no—
236#386—2771no—
237#385—2771no—
238#385—2771no—
239#380—2771no—
240C4×C4⋊C4—2771no—
241C22×D4—2771no—
242C22×D4—2771no—
243C22×D4—2771no—
244C23×C4—2771no—
245C23×C4—2771no—
246C23×C4—2771no—
247#121—2778no—
248C4×D4—5544no—
249#117—5544no—
250C2×C4⋊C4—2771no—
251C2×C4⋊C4—2771no—
252C2×C4⋊C4—2771no—
253C2×C4⋊C4—2771no—
254C2×C4⋊C4—2771no—
255C2×C4⋊C4—2771no—
256C2×C4⋊C4C2×D27711yescharacteristic
257C2×C22⋊C4—2771no—
258C2×C22⋊C4—2771no—
259C2×C22⋊C4—2771no—
260C2×C22⋊C4—2771no—
261C2×C22⋊C4—2771no—
262C2×C22⋊C4—2771no—
263C2×C22⋊C4—2771no—
264C2×C22⋊C4—2771no—
265C2×C22⋊C4—2771no—
266C2×C22⋊C4C2×D27711yescharacteristic
267C2×C22⋊C4C2×D27711yescharacteristic
268C2×C22⋊C4C2×D27711yescharacteristic
269C2×C42—2771no—
270C2×C42—2771no—
271C2×C42—2771no—
272C2×C42C2×D27711yescharacteristic
273#95—2771no—
274#95—2771no—
275#95—2771no—
276#95—2771no—
277#95C2×D27711yescharacteristic
278#95C2×D27711yescharacteristic
279C24—2771no—
280C24—2771no—
281C24C22×D27711yescharacteristic
282C2×D4—2774no—
283C2×D4—2774no—
284C2×D4—2774no—
285C2×D4—5542no—
286C2×D4—5542no—
287C2×D4—11081no—
288C2×D4—11081no—
289C2×D4—11081no—
290C2×D4—11081no—
291C22×C4—2771no—
292C22×C4—2771no—
293C22×C4—2771no—
294C22×C4—2774no—
295C22×C4—2771no—
296C22×C4—2771no—
297C22×C4—2771no—
298C22×C4—5542no—
299C22×C4—5542no—
300C22×C4—5542no—
301C22×C4—5542no—
302C22×C4—11081no—
303C22×C4—11081no—
304C22×C4—11081no—
305C22×C4—11081no—
306C22×C4C22×D27711yescharacteristic
307C22×C4C22×D27711yescharacteristic
308C22×C4C22×D27711yescharacteristic
309C22×C4C22×D27711yescharacteristic
310C22×C4C22×D27711yescharacteristic
311C22×C4C22×D27711yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—2774no—
314C4⋊C4—5542no—
315C4⋊C4—5542no—
316C4⋊C4—5542no—
317C4⋊C4—11081no—
318C4⋊C4—11081no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—2774no—
322C22⋊C4—5542no—
323C22⋊C4—5542no—
324C22⋊C4—5542no—
325C22⋊C4—11081no—
326C22⋊C4—11081no—
327C22⋊C4—11081no—
328C22⋊C4—11081no—
329C22⋊C4—11081no—
330C22⋊C4C4×D27714yescomplemented
331C42—41no—
332C42—2774no—
333C42—5542no—
334C42—11081no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2772no—
342C23—2772no—
343C23—5541no—
344C23—5541no—
345C23—5541no—
346C23—5541no—
347C23—5541no—
348C23—5541no—
349C23—5541no—
350C23—5541no—
351C23—5541no—
352C23—5541no—
353C23—5541no—
354C23—5541no—
355C23C2×C4×D27712yes—
356C23C23×D27711yescharacteristic
357D4—5548no—
358D4—11084no—
359D4—11084no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—2772no—
376C2×C4—2772no—
377C2×C4—2772no—
378C2×C4—2772no—
379C2×C4—2772no—
380C2×C4—5548no—
381C2×C4—5541no—
382C2×C4—5541no—
383C2×C4—5541no—
384C2×C4—5541no—
385C2×C4—5541no—
386C2×C4—5541no—
387C2×C4—5541no—
388C2×C4—5541no—
389C2×C4—5541no—
390C2×C4—5541no—
391C2×C4—5541no—
392C2×C4—5541no—
393C2×C4—5541no—
394C2×C4—11084no—
395C2×C4—11084no—
396C2×C4C2×C4×D27712yes—
397C2×C4C2×C4×D27712yes—
398C2×C4D4×D27712yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—5542no—
407C22—5542no—
408C22—5542no—
409C22—5542no—
410C22—5542no—
411C22—5542no—
412C22—11081no—
413C22—11081no—
414C22—11081no—
415C22—11081no—
416C22—11081no—
417C22—11081no—
418C22—11081no—
419C22—11081no—
420C22C22×C4×D27711yescharacteristic
421C22C2×D4×D27711yescharacteristic
422C22#41×D27711yescharacteristic
423C22#2257992511yescharacteristic
424C22#2257992611yescharacteristic
425C22#2257992611yescharacteristic
426C22#2257992711yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—2774no—
434C4—5542no—
435C4—5542no—
436C4—11081no—
437C4—11081no—
438C4—11081no—
439C2—42no—
440C2—11082no—
441C2—11082no—
442C2C4×D4×D27711yescharacteristic
443C2#3180047411yescharacteristic
444C2#3180048211yescharacteristic
445C2#3180057011yescharacteristic
446C2#3180057911yescharacteristic
447C2#3180060411yescharacteristic
448C2#3180061811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D277
commutatorC22×C277C23×C4
Fitting#388×C277C2
socle, cosocleC23×C277C24
prime radicalC2×#22579853C2
hypercenter#388D277
hypocenterC277#2324
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#22579840C2semidirect
3. C2×#22579838C2semidirect
4. #31799904C2semidirect
5. #31799902C2semidirect
6. #31799902C2semidirect
7. #31799899C2semidirect
8. #31799892C2semidirect
9. #31799880C2semidirect
10. #31799879C2semidirect
11. #31799877C2semidirect
12. #31799876C2semidirect
13. #31799861C2semidirect
14. C4×#17984542C2semidirect
15. #388×C277C2semidirect
22. #22579853C4semidirect
58. C2×D1108C2×C4semidirect
64. C2×C22⋊D277C2×C4semidirect
79. C2×C4×D277C2×C4semidirect
85. #17984542C2×C4semidirect
94. #17984540C2×C4semidirect
97. C4×Dic277D4semidirect
108. C22⋊C4×C277C2×C4semidirect
224. C277#2324semidirect
234. #388D277semidirect
330. C22⋊C4C4×D277semidirect

Maximal subgroups

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

subgroupindex
C2×#225798532
C2×#225798402
C2×#225798382
#317999042
#317999022
#317998992
#317998922
#317998802
#317998792
#317998772
#317998762
#317998612
C4×#179845422
#388×C2772
#2324277