Finite Group Databasebeta

← #31783906

Subgroups — #31783906

Order 17536.

496 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984081C211yesmaximal, characteristic
2#22577299C211yesmaximal, characteristic, complemented
3#22577291C211yesmaximal, characteristic
4#22577291C211yesmaximal, characteristic
5#22577286C211yesmaximal, characteristic, complemented
6#22577285C211yesmaximal, characteristic, complemented
7#22577284C211yesmaximal, characteristic, complemented
8#22577275C211yesmaximal, characteristic
9C4×#14555489C211yesmaximal, characteristic, complemented
10#22577254C211yesmaximal, characteristic, complemented
11#22577254C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic137C211yesmaximal, characteristic
13C22⋊C4×Dic137C211yesmaximal, characteristic
14#22577249C211yesmaximal, characteristic, complemented
15#388×C137C211yesmaximal, characteristic
16C23×Dic137C2211yescharacteristic
17#17984081C418yescomplemented
18C2×C4⋊Dic137C2211yescharacteristic
19C2×#14555489C2211yescharacteristic
20C2×#14555489C2211yescharacteristic
21C2×#14555489C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×C22⋊Dic137C2211yescharacteristic
26C2×C22⋊Dic137C2211yescharacteristic
27C2×C22⋊Dic137C2211yescharacteristic
28C2×C4×Dic137C2211yescharacteristic
29C2×C4×Dic137C2211yescharacteristic
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×C22⋊C4×C137C2211yescharacteristic
47C2×C22⋊C4×C137C2211yescharacteristic
48C2×C22⋊C4×C137C2211yescharacteristic
49C2×C42×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52C22×Dic137—22no—
53C22×Dic137—22no—
54C22×Dic137—41no—
55C22×Dic137C2311yescharacteristic
56C22×Dic137C2311yescharacteristic
57C22×Dic137C2311yescharacteristic
58C22×Dic137C2311yescharacteristic
59C22×Dic137C2311yescharacteristic
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C22×Dic137C2311yescharacteristic
63C4⋊Dic137C2×C414yescomplemented
64#14555489—22no—
65#14555489—22no—
66#14555489—22no—
67#14555489—22no—
68#14555489C2×C414yescomplemented
69#14555489C2×C414yescomplemented
70C22⋊Dic137—22no—
71C22⋊Dic137C2×C414yes—
72C22⋊Dic137C2×C414yes—
73C4×Dic137—22no—
74C4×Dic137—22no—
75C4×Dic137—22no—
76C4×Dic137—41no—
77C4×Dic137—41no—
78C4×Dic137—41no—
79C4×Dic137C2×C414yescomplemented
80C24×C137C2311yescharacteristic
81C22×C4×C137C2311yescharacteristic
82C22×C4×C137C2311yescharacteristic
83C22×C4×C137C2311yescharacteristic
84C22×C4×C137C2311yescharacteristic
85C22×C4×C137C2311yescharacteristic
86C22×C4×C137C2311yescharacteristic
87C4⋊C4×C137—41no—
88C22⋊C4×C137—22no—
89C22⋊C4×C137—41no—
90C22⋊C4×C137C2×C414yes—
91C42×C137—22no—
92C2×Dic137—21no—
93C2×Dic137—21no—
94C2×Dic137—21no—
95C2×Dic137—21no—
96C2×Dic137—21no—
97C2×Dic137—21no—
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—21no—
107C2×Dic137—21no—
108C2×Dic137—21no—
109C2×Dic137—44no—
110C2×Dic137C22×C412yes—
111C2×Dic137C22×C412yes—
112C2×Dic137C22×C412yes—
113C2×Dic137C22×C412yes—
114C2×Dic137#4112yes—
115C2×Dic137#4112yes—
116C2×Dic137#4112yes—
117C23×C137—21no—
118C23×C137—21no—
119C23×C137—21no—
120C23×C137—21no—
121C23×C137—21no—
122C23×C137—21no—
123C23×C137C22×C412yes—
124C23×C137C2411yescharacteristic
125C2×C4×C137—21no—
126C2×C4×C137—21no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
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×C137C22×C412yes—
141C2×C4×C137C22×C412yes—
142C2×C4×C137#4112yes—
143C22×C137—22no—
144C22×C137—22no—
145C22×C137—22no—
146C22×C137—41no—
147C22×C137—41no—
148C22×C137—41no—
149C22×C137—41no—
150C22×C137C23×C411yescharacteristic
151C22×C137C2×#4111yescharacteristic
152C22×C137C2×#4111yescharacteristic
153C22×C137C2×#4111yescharacteristic
154C22×C137C2×#4111yescharacteristic
155C22×C137H(5,𝔽2)11yescharacteristic
156C22×C1372−1+411yescharacteristic
157C4×C137—22no—
158C4×C137—22no—
159C4×C137—22no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137—41no—
163Dic137—22no—
164Dic137—22no—
165Dic137—22no—
166Dic137—22no—
167Dic137—22no—
168Dic137—22no—
169Dic137—22no—
170Dic137—41no—
171C2×C137—42no—
172C2×C137C4×#4111yescharacteristic
173C2×C137C4×#4111yescharacteristic
174C2×C137#52011yescharacteristic
175C2×C137#55111yescharacteristic
176C2×C137#55211yescharacteristic
177C2×C137#55311yescharacteristic
178C2×C137#55611yescharacteristic
179C137#237311yescharacteristic, complemented, Sylow
180#2373—1371nomaximal, Sylow
181C2×#126—1371no—
182#389—1371no—
183#388—1371no—
184#388—1371no—
185#388D13711yescharacteristic
186#387—1371no—
187#387—1371no—
188#387—1371no—
189#385—1371no—
190C4×C4⋊C4—1371no—
191C4×C4⋊C4—1371no—
192C4×C4⋊C4—1371no—
193C4×C22⋊C4—1371no—
194C4×C22⋊C4—1371no—
195#376—1371no—
196C23×C4—1371no—
197#126—1378no—
198C2×C4⋊C4—1371no—
199C2×C4⋊C4—1371no—
200C2×C4⋊C4—1371no—
201C2×C4⋊C4—1371no—
202C2×C4⋊C4—1371no—
203C2×C4⋊C4—1371no—
204C2×C4⋊C4—1371no—
205C2×C4⋊C4C2×D13711yescharacteristic
206C2×C22⋊C4—1371no—
207C2×C22⋊C4—1371no—
208C2×C22⋊C4—1371no—
209C2×C22⋊C4C2×D13711yescharacteristic
210C2×C22⋊C4C2×D13711yescharacteristic
211C2×C22⋊C4C2×D13711yescharacteristic
212C2×C42—1371no—
213C2×C42—1371no—
214C2×C42—1371no—
215C2×C42—1371no—
216C2×C42—1371no—
217C2×C42—1371no—
218C2×C42—1371no—
219C2×C42C2×D13711yescharacteristic
220#95—1371no—
221#95—1371no—
222#95—1371no—
223#95—1371no—
224#95—1371no—
225#95—1371no—
226#95—1371no—
227#95—1371no—
228#95—1371no—
229#95—1371no—
230#95C2×D13711yescharacteristic
231#95C2×D13711yescharacteristic
232C24C22×D13711yescharacteristic
233C22×C4—2742no—
234C22×C4—2742no—
235C22×C4—5481no—
236C22×C4—1371no—
237C22×C4—1371no—
238C22×C4—1371no—
239C22×C4—1371no—
240C22×C4—1371no—
241C22×C4—1371no—
242C22×C4—1371no—
243C22×C4—1371no—
244C22×C4C22×D13711yescharacteristic
245C22×C4C22×D13711yescharacteristic
246C22×C4C22×D13711yescharacteristic
247C22×C4C22×D13711yescharacteristic
248C22×C4C22×D13711yescharacteristic
249C22×C4C22×D13711yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—2742no—
252C4⋊C4—2742no—
253C4⋊C4—2742no—
254C4⋊C4—2742no—
255C4⋊C4—1374no—
256C4⋊C4—1374no—
257C4⋊C4—1374no—
258C22⋊C4—22no—
259C22⋊C4—41no—
260C22⋊C4—2742no—
261C22⋊C4—1374no—
262C22⋊C4—1374no—
263C22⋊C4C4×D13714yes—
264C42—22no—
265C42—2742no—
266C42—2742no—
267C42—2742no—
268C42—5481no—
269C42—5481no—
270C42—5481no—
271C42—1374no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D13712yes—
279C23C23×D13711yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—2741no—
296C2×C4—2741no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2741no—
300C2×C4—2741no—
301C2×C4—2741no—
302C2×C4—2741no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—5484no—
313C2×C4—1372no—
314C2×C4—1372no—
315C2×C4—1372no—
316C2×C4—1372no—
317C2×C4—1372no—
318C2×C4—1372no—
319C2×C4—1372no—
320C2×C4C2×C4×D13712yes—
321C2×C4C2×C4×D13712yes—
322C2×C4#1455550912yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D13711yescharacteristic
331C22#41×D13711yescharacteristic
332C22#41×D13711yescharacteristic
333C22#41×D13711yescharacteristic
334C22C2×#1455550911yescharacteristic
335C22#1798413311yescharacteristic
336C22#1798413511yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—2742no—
344C4—2742no—
345C4—2742no—
346C4—2742no—
347C4—2742no—
348C4—2742no—
349C4—2742no—
350C4—5481no—
351C2—42no—
352C2C4×#1455550911yescharacteristic
353C2#2257787111yescharacteristic
354C2#2257788211yescharacteristic
355C2#2257808311yescharacteristic
356C2#2257809011yescharacteristic
357C2#2257810011yescharacteristic
358C2#2257812011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#388×C137C2
socle, cosocleC23×C137C24
prime radicalC24×C137C23
hypercenter#388D137
hypocenterC137#2373
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #22577299C2semidirect
5. #22577286C2semidirect
6. #22577285C2semidirect
7. #22577284C2semidirect
9. C4×#14555489C2semidirect
10. #22577254C2semidirect
11. #22577254C2semidirect
14. #22577249C2semidirect
17. #17984081C4semidirect
63. C4⋊Dic137C2×C4semidirect
68. #14555489C2×C4semidirect
69. #14555489C2×C4semidirect
79. C4×Dic137C2×C4semidirect
179. C137#2373semidirect

Maximal subgroups

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

subgroupindex
C2×#179840812
#225772992
#225772912
#225772862
#225772852
#225772842
#225772752
C4×#145554892
#225772542
C22⋊C4×Dic1372
#225772492
#388×C1372
#2373137