Finite Group Databasebeta

← #31784064

Subgroups — #31784064

Order 17536.

528 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984053C211yesmaximal, characteristic, complemented
2C2×#17984044C211yesmaximal, characteristic
3#22577344C211yesmaximal, characteristic, complemented
4#22577339C211yesmaximal, characteristic
5#22577336C211yesmaximal, characteristic, complemented
6#22577333C211yesmaximal, characteristic, complemented
7#22577326C211yesmaximal, characteristic
8#22577323C211yesmaximal, characteristic, complemented
9#22577315C211yesmaximal, characteristic
10C4⋊C4×Dic137C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic137C211yesmaximal, characteristic, complemented
12#22577254C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic137C211yesmaximal, characteristic
14C22⋊C4×Dic137C211yesmaximal, characteristic
15#396×C137C211yesmaximal, characteristic
16C22×Dic274C2211yescharacteristic
17C23×Dic137C2211yescharacteristic
18#17984053—24no—
19#17984044—24no—
20C2×C4⋊Dic137C2211yescharacteristic
21C2×C4⋊Dic137C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×#14555489C2211yescharacteristic
26C2×#14555489C2211yescharacteristic
27C2×#14555489C2211yescharacteristic
28C2×C22⋊Dic137C2211yescharacteristic
29C2×C22⋊Dic137C2211yescharacteristic
30C2×C22⋊Dic137C2211yescharacteristic
31C2×C4×Dic137C2211yescharacteristic
32C2×C4×Dic137C2211yescharacteristic
33C2×C4×Dic137C2211yescharacteristic
34C2×C4×Dic137C2211yescharacteristic
35C2×C4×Dic137C2211yescharacteristic
36C2×C4×Dic137C2211yescharacteristic
37#17983995C2211yescharacteristic
38#17983995C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45#17983995C2211yescharacteristic
46C2×C4⋊C4×C137C2211yescharacteristic
47C2×C4⋊C4×C137C2211yescharacteristic
48C2×C4⋊C4×C137C2211yescharacteristic
49C2×C22⋊C4×C137C2211yescharacteristic
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C22⋊C4×C137C2211yescharacteristic
52#95×C137C2211yescharacteristic
53C2×Dic274—22no—
54C2×Dic274—22no—
55C2×Dic274—22no—
56C22×Dic137—22no—
57C22×Dic137—22no—
58C22×Dic137Q814yescomplemented
59C22×Dic137C2311yescharacteristic
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C22×Dic137C2311yescharacteristic
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C4⋊Dic137—22no—
68C4⋊Dic137—22no—
69#14555489—22no—
70#14555489—22no—
71#14555489—22no—
72#14555489—22no—
73#14555489—22no—
74#14555489—22no—
75C22⋊Dic137—22no—
76C22⋊Dic137—22no—
77C22⋊Dic137—22no—
78C4×Dic137—22no—
79C4×Dic137—22no—
80C4×Dic137—22no—
81C4×Dic137—22no—
82C4×Dic137—22no—
83C4×Dic137D414yes—
84C24×C137C2311yescharacteristic
85C22×C4×C137C2311yescharacteristic
86C22×C4×C137C2311yescharacteristic
87C22×C4×C137C2311yescharacteristic
88C22×C4×C137C2311yescharacteristic
89C22×C4×C137C2311yescharacteristic
90C22×C4×C137C2311yescharacteristic
91C4⋊C4×C137—22no—
92C4⋊C4×C137—22no—
93C4⋊C4×C137—22no—
94C22⋊C4×C137—22no—
95C22⋊C4×C137—22no—
96C22⋊C4×C137—22no—
97Dic274—28no—
98C2×Dic137—21no—
99C2×Dic137—21no—
100C2×Dic137—21no—
101C2×Dic137—21no—
102C2×Dic137—21no—
103C2×Dic137—21no—
104C2×Dic137—21no—
105C2×Dic137—21no—
106C2×Dic137—28no—
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×Dic137C2×D412yes—
117C2×Dic137C2×D412yes—
118C2×Dic137C2×Q812yes—
119C2×Dic137C2×Q812yes—
120C2×Dic137#4112yes—
121C2×Dic137#4112yes—
122C2×Dic137#4112yes—
123C23×C137—21no—
124C23×C137—21no—
125C23×C137—21no—
126C23×C137—21no—
127C23×C137—21no—
128C23×C137—21no—
129C23×C137C2×Q812yes—
130C23×C137C2411yescharacteristic
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×C137C2×D412yes—
147C2×C4×C137#4112yes—
148C2×C4×C137#4112yes—
149C22×C137—22no—
150C22×C137—22no—
151C22×C137—22no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137—41no—
156C22×C137C22×D411yescharacteristic
157C22×C137C22×Q811yescharacteristic
158C22×C137C2×#4111yescharacteristic
159C22×C137C2×#4111yescharacteristic
160C22×C137C2×#4111yescharacteristic
161C22×C137C2×#4111yescharacteristic
162C22×C137C2×#4111yescharacteristic
163C4×C137—22no—
164C4×C137—22no—
165C4×C137—22no—
166C4×C137—41no—
167C4×C137—41no—
168C4×C137—41no—
169Dic137—22no—
170Dic137—22no—
171Dic137—22no—
172Dic137—22no—
173Dic137—41no—
174Dic137—41no—
175Dic137—41no—
176Dic137C22⋊Q814yes—
177C2×C137—42no—
178C2×C137C2×C22⋊Q811yescharacteristic
179C2×C137#52511yescharacteristic
180C2×C137#52911yescharacteristic
181C2×C137#52911yescharacteristic
182C2×C137#53211yescharacteristic
183C2×C137#53311yescharacteristic
184C2×C137#53311yescharacteristic
185C137#239611yescharacteristic, complemented, Sylow
186#2396—1371nomaximal, Sylow
187C2×C4×Q8—1371no—
188C2×#117—1371no—
189#400—1371no—
190#399—1371no—
191#398—1371no—
192#398—1371no—
193#396—1371no—
194#396D13711yescharacteristic
195#395—1371no—
196#393—1371no—
197C4×C4⋊C4—1371no—
198C4×C4⋊C4—1371no—
199C4×C4⋊C4—1371no—
200C4×C22⋊C4—1371no—
201C4×C22⋊C4—1371no—
202C22×Q8—1371no—
203C23×C4—1371no—
204C4×Q8—2744no—
205#117—2744no—
206C2×C4⋊C4—1371no—
207C2×C4⋊C4—1371no—
208C2×C4⋊C4—1371no—
209C2×C4⋊C4—1371no—
210C2×C4⋊C4—1371no—
211C2×C4⋊C4—1371no—
212C2×C4⋊C4—1371no—
213C2×C4⋊C4—1371no—
214C2×C4⋊C4C2×D13711yescharacteristic
215C2×C4⋊C4C2×D13711yescharacteristic
216C2×C4⋊C4C2×D13711yescharacteristic
217C2×C22⋊C4—1371no—
218C2×C22⋊C4—1371no—
219C2×C22⋊C4—1371no—
220C2×C22⋊C4C2×D13711yescharacteristic
221C2×C22⋊C4C2×D13711yescharacteristic
222C2×C22⋊C4C2×D13711yescharacteristic
223C2×C42—1371no—
224C2×C42—1371no—
225C2×C42—1371no—
226C2×C42—1371no—
227C2×C42—1371no—
228C2×C42—1371no—
229#95—1371no—
230#95—1371no—
231#95—1371no—
232#95—1371no—
233#95—1371no—
234#95—1371no—
235#95—1371no—
236#95—1371no—
237#95—1371no—
238#95C2×D13711yescharacteristic
239C24C22×D13711yescharacteristic
240C2×Q8—2742no—
241C2×Q8—2742no—
242C2×Q8—2742no—
243C22×C4—2742no—
244C22×C4—2742no—
245C22×C4—1371no—
246C22×C4—1371no—
247C22×C4—1371no—
248C22×C4—1374no—
249C22×C4—1371no—
250C22×C4—1371no—
251C22×C4—1371no—
252C22×C4—1371no—
253C22×C4—1371no—
254C22×C4C22×D13711yescharacteristic
255C22×C4C22×D13711yescharacteristic
256C22×C4C22×D13711yescharacteristic
257C22×C4C22×D13711yescharacteristic
258C22×C4C22×D13711yescharacteristic
259C22×C4C22×D13711yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—2742no—
264C4⋊C4—2742no—
265C4⋊C4—2742no—
266C4⋊C4—2742no—
267C4⋊C4—2742no—
268C4⋊C4—2742no—
269C4⋊C4—2742no—
270C4⋊C4—2742no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—2742no—
275C22⋊C4—2742no—
276C22⋊C4—2742no—
277C42—2742no—
278C42—2742no—
279C42—2742no—
280C42—2742no—
281C42—2742no—
282C42—1374no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23Q8×D13712yes—
290C23C23×D13711yescharacteristic
291Q8—2748no—
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—2741no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2748no—
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—2741no—
325C2×C4—1372no—
326C2×C4—1372no—
327C2×C4—1372no—
328C2×C4—1372no—
329C2×C4—1372no—
330C2×C4—1372no—
331C2×C4—1372no—
332C2×C4D4×D13712yes—
333C2×C4#1455550912yes—
334C2×C4#1455550912yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C2×D4×D13711yescharacteristic
343C22C2×Q8×D13711yescharacteristic
344C22#41×D13711yescharacteristic
345C22#41×D13711yescharacteristic
346C22#41×D13711yescharacteristic
347C22C2×#1455550911yescharacteristic
348C22C2×#1455550911yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—2742no—
356C4—2742no—
357C4—2742no—
358C4—2742no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4—1374no—
363C2—42no—
364C2C22⋊Q8×D13711yescharacteristic
365C2#2257791311yescharacteristic
366C2#2257793411yescharacteristic
367C2#2257793611yescharacteristic
368C2#2257794811yescharacteristic
369C2#2257795511yescharacteristic
370C2#2257795511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#396×C137C2
prime radicalC24×C137C23
hypercenter#396D137
hypocenterC137#2396
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#17984053C2semidirect
3. #22577344C2semidirect
5. #22577336C2semidirect
6. #22577333C2semidirect
8. #22577323C2semidirect
10. C4⋊C4×Dic137C2semidirect
11. C4⋊C4×Dic137C2semidirect
12. #22577254C2semidirect
58. C22×Dic137Q8semidirect
185. C137#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#179840532
C2×#179840442
#225773442
#225773392
#225773362
#225773332
#225773262
#225773232
#225773152
C4⋊C4×Dic1372
#225772542
C22⋊C4×Dic1372
#396×C1372
#2396137