Finite Group Databasebeta

← #31783866

Subgroups — #31783866

Order 17536.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984066C211yesmaximal, characteristic
2#22577308C211yesmaximal, characteristic, complemented
3#22577307C211yesmaximal, characteristic, complemented
4#22577290C211yesmaximal, characteristic, complemented
5#22577285C211yesmaximal, characteristic, complemented
6#22577285C211yesmaximal, characteristic, complemented
7#22577277C211yesmaximal, characteristic
8#22577274C211yesmaximal, characteristic
9C4⋊C4×Dic137C211yesmaximal, characteristic, complemented
10C4×#14555489C211yesmaximal, characteristic, complemented
11C4×#14555487C211yesmaximal, characteristic
12#22577250C211yesmaximal, characteristic
13#22577250C211yesmaximal, characteristic
14#95×D137C211yesmaximal, characteristic
15#387×C137C211yesmaximal, characteristic, complemented
16C22×Dic274C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20#17984066C418yescomplemented
21C2×C4⋊Dic137C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×#14555489C2211yescharacteristic
26C2×#14555487C2211yescharacteristic
27C2×#14555487C2211yescharacteristic
28C2×#14555487C2211yescharacteristic
29C2×#14555487C2211yescharacteristic
30C2×C4×Dic137C2211yescharacteristic
31C2×C4×Dic137C2211yescharacteristic
32C2×C4×Dic137C2211yescharacteristic
33C2×C4×Dic137C2211yescharacteristic
34C2×C4×Dic137C2211yescharacteristic
35#17983995C2211yescharacteristic
36#17983995C2211yescharacteristic
37#17983995C2211yescharacteristic
38#17983995C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45C2×C4⋊C4×C137C2211yescharacteristic
46C2×C4⋊C4×C137C2211yescharacteristic
47C2×C42×C137C2211yescharacteristic
48#95×C137C2211yescharacteristic
49#95×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52C23×D137C2311yescharacteristic
53C2×Dic274—41no—
54C2×Dic274—41no—
55C2×Dic274C2×C414yescomplemented
56C22×Dic137C2311yescharacteristic
57C22×Dic137C2311yescharacteristic
58C22×Dic137C2311yescharacteristic
59C22×Dic137C2311yescharacteristic
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C22×Dic137C2311yescharacteristic
63C2×C4×D137—22no—
64C2×C4×D137—22no—
65C2×C4×D137—22no—
66C2×C4×D137—22no—
67C2×C4×D137—41no—
68C2×C4×D137—41no—
69C2×C4×D137—41no—
70C2×C4×D137—41no—
71C2×C4×D137C2×C414yes—
72C4⋊Dic137—22no—
73#14555489—22no—
74#14555489—41no—
75#14555489C2×C414yescomplemented
76#14555489C2×C414yescomplemented
77#14555487—22no—
78#14555487C2×C414yes—
79#14555487C2×C414yes—
80#14555487Q814yescomplemented
81C4×Dic137—22no—
82C4×Dic137—22no—
83C4×Dic137—41no—
84C4×Dic137—41no—
85C4×Dic137—41no—
86C22×C4×C137C2311yescharacteristic
87C22×C4×C137C2311yescharacteristic
88C22×C4×C137C2311yescharacteristic
89C22×C4×C137C2311yescharacteristic
90C22×C4×C137C2311yescharacteristic
91C22×C4×C137C2311yescharacteristic
92C22×C4×C137C2311yescharacteristic
93C4⋊C4×C137—41no—
94C4⋊C4×C137C2×C414yescomplemented
95C42×C137—41no—
96C22×D137—21no—
97C22×D137—21no—
98C22×D137—21no—
99C22×D137—21no—
100C22×D137C22×C412yes—
101C22×D137C2×Q812yes—
102C22×D137#4112yes—
103Dic274—44no—
104C2×Dic137—21no—
105C2×Dic137—21no—
106C2×Dic137—21no—
107C2×Dic137—21no—
108C2×Dic137—21no—
109C2×Dic137—21no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137—21no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137C22×C412yes—
120C2×Dic137C22×C412yes—
121C2×Dic137C22×C412yes—
122C2×Dic137C2×Q812yes—
123C2×Dic137#4112yes—
124C2×Dic137#4112yes—
125C4×D137—44no—
126C4×D137—44no—
127C4×D137—44no—
128C23×C137C2411yescharacteristic
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137C22×C412yes—
147C2×C4×C137C22×C412yes—
148C2×C4×C137C22×C412yes—
149C2×C4×C137C2×Q812yes—
150C22×C137C23×C411yescharacteristic
151C22×C137C22×Q811yescharacteristic
152C22×C137C2×#4111yescharacteristic
153C22×C137C2×#4111yescharacteristic
154C22×C137C2×#4111yescharacteristic
155C22×C137H(5,𝔽2)11yescharacteristic
156C22×C137H(5,𝔽2)11yescharacteristic
157C2×D137—22no—
158C2×D137—22no—
159C2×D137—22no—
160C2×D137—22no—
161C2×D137—22no—
162C2×D137—22no—
163C2×D137C4×Q814yes—
164C4×C137—22no—
165C4×C137—22no—
166C4×C137—22no—
167C4×C137—22no—
168C4×C137—41no—
169C4×C137—41no—
170C4×C137—41no—
171Dic137—22no—
172Dic137—22no—
173Dic137—22no—
174Dic137—22no—
175Dic137—22no—
176Dic137—22no—
177Dic137—41no—
178C2×C137C2×C4×Q811yescharacteristic
179C2×C137C4×#4111yescharacteristic
180C2×C137#51811yescharacteristic
181C2×C137#55111yescharacteristic
182C2×C137#55111yescharacteristic
183C2×C137#55411yescharacteristic
184C2×C137#55411yescharacteristic
185D137—24no—
186C137#236811yescharacteristic, complemented, Sylow
187#2368—1371nomaximal, Sylow
188C2×C22⋊Q8—1371no—
189#391—1371no—
190#391—1371no—
191#387—1371no—
192#387—1371no—
193#387—1371no—
194#387D13711yescharacteristic, complemented
195#385—1371no—
196#385—1371no—
197C4×C4⋊C4—1371no—
198C4×C4⋊C4—1371no—
199C4×C22⋊C4—1371no—
200C4×C22⋊C4—1371no—
201C4×C22⋊C4—1371no—
202C2×#95—1371no—
203C22×Q8—1371no—
204C23×C4—1371no—
205C23×C4—1371no—
206C23×C4—1371no—
207C22⋊Q8—1378no—
208C2×C4⋊C4—1371no—
209C2×C4⋊C4—1371no—
210C2×C4⋊C4—1371no—
211C2×C4⋊C4—1371no—
212C2×C4⋊C4—1371no—
213C2×C4⋊C4C2×D13711yescharacteristic
214C2×C4⋊C4C2×D13711yescharacteristic
215C2×C22⋊C4—1371no—
216C2×C22⋊C4—1371no—
217C2×C22⋊C4—1371no—
218C2×C22⋊C4—1371no—
219C2×C42—1371no—
220C2×C42—1371no—
221C2×C42—1371no—
222C2×C42—1371no—
223C2×C42—1371no—
224C2×C42C2×D13711yescharacteristic
225#95—1371no—
226#95—1371no—
227#95—1371no—
228#95—1371no—
229#95—1371no—
230#95—1371no—
231#95—1371no—
232#95—1371no—
233#95—1371no—
234#95—1371no—
235#95C2×D13711yescharacteristic
236#95C2×D13711yescharacteristic
237#95C2×D13711yescharacteristic
238#95C2×D13711yescharacteristic
239C24—1371no—
240C2×Q8—5481no—
241C2×Q8—5481no—
242C2×Q8—1374no—
243C22×C4—2742no—
244C22×C4—2742no—
245C22×C4—2742no—
246C22×C4—2742no—
247C22×C4—5481no—
248C22×C4—5481no—
249C22×C4—5481no—
250C22×C4—5481no—
251C22×C4—1371no—
252C22×C4—1371no—
253C22×C4—1371no—
254C22×C4—1371no—
255C22×C4—1371no—
256C22×C4—1371no—
257C22×C4—1371no—
258C22×C4—1374no—
259C22×C4C22×D13711yescharacteristic
260C22×C4C22×D13711yescharacteristic
261C22×C4C22×D13711yescharacteristic
262C22×C4C22×D13711yescharacteristic
263C22×C4C22×D13711yescharacteristic
264C22×C4C22×D13711yescharacteristic
265C22×C4C22×D13711yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—2742no—
268C4⋊C4—2742no—
269C4⋊C4—5481no—
270C4⋊C4—1374no—
271C4⋊C4—1374no—
272C4⋊C4C4×D13714yescomplemented
273C22⋊C4—2742no—
274C22⋊C4—1374no—
275C22⋊C4—1374no—
276C22⋊C4—1374no—
277C42—41no—
278C42—2742no—
279C42—2742no—
280C42—5481no—
281C42—5481no—
282C42—5481no—
283C23—2741no—
284C23—2741no—
285C23—2741no—
286C23—2741no—
287C23—1372no—
288C23—1372no—
289C23—1372no—
290C23C23×D13711yescharacteristic
291Q8—5484no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—2741no—
318C2×C4—2741no—
319C2×C4—2741no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—5484no—
325C2×C4—5484no—
326C2×C4—5484no—
327C2×C4—1372no—
328C2×C4—1372no—
329C2×C4—1372no—
330C2×C4—1372no—
331C2×C4—1372no—
332C2×C4—1372no—
333C2×C4C2×C4×D13712yes—
334C2×C4C2×C4×D13712yes—
335C2×C4C2×C4×D13712yes—
336C2×C4Q8×D13712yes—
337C22—2742no—
338C22—2742no—
339C22—2742no—
340C22—2742no—
341C22—2742no—
342C22—2742no—
343C22—1374no—
344C22C22×C4×D13711yescharacteristic
345C22C2×Q8×D13711yescharacteristic
346C22#41×D13711yescharacteristic
347C22#41×D13711yescharacteristic
348C22#41×D13711yescharacteristic
349C22#1798413311yescharacteristic
350C22#1798413411yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—2742no—
359C4—2742no—
360C4—2742no—
361C4—2742no—
362C4—2742no—
363C4—2742no—
364C4—5481no—
365C2—2744no—
366C2C4×Q8×D13711yescharacteristic
367C2#2257787111yescharacteristic
368C2#2257787511yescharacteristic
369C2#2257807811yescharacteristic
370C2#2257808111yescharacteristic
371C2#2257810711yescharacteristic
372C2#2257810911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#387×C137C2
socle, cosocleC23×C137C24
prime radicalC23×D137C23
hypercenter#387D137
hypocenterC137#2368
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #22577308C2semidirect
3. #22577307C2semidirect
4. #22577290C2semidirect
5. #22577285C2semidirect
6. #22577285C2semidirect
9. C4⋊C4×Dic137C2semidirect
10. C4×#14555489C2semidirect
15. #387×C137C2semidirect
20. #17984066C4semidirect
55. C2×Dic274C2×C4semidirect
75. #14555489C2×C4semidirect
76. #14555489C2×C4semidirect
80. #14555487Q8semidirect
94. C4⋊C4×C137C2×C4semidirect
186. C137#2368semidirect
194. #387D137semidirect
272. C4⋊C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840662
#225773082
#225773072
#225772902
#225772852
#225772772
#225772742
C4⋊C4×Dic1372
C4×#145554892
C4×#145554872
#225772502
#95×D1372
#387×C1372
#2368137