Finite Group Databasebeta

← #31792658

Subgroups — #31792658

Order 17536.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578111C211yesmaximal, characteristic
2Q8×C22⋊D137C211yesmaximal, characteristic
3#22577752C211yesmaximal, characteristic, complemented
4#22577739C211yesmaximal, characteristic, complemented
5#22577737C211yesmaximal, characteristic, complemented
6#22577696C211yesmaximal, characteristic, complemented
7#22577642C211yesmaximal, characteristic, complemented
8#22577641C211yesmaximal, characteristic, complemented
9#22577611C211yesmaximal, characteristic, complemented
10#22577557C211yesmaximal, characteristic, complemented
11Q16×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic
13C2×#17984012C211yesmaximal, characteristic, complemented
14#103×D137C211yesmaximal, characteristic, complemented
15#451×C137C211yesmaximal, characteristic, complemented
16C2×Q8×D137C2211yescharacteristic
17C2×#14555500C2211yescharacteristic
18C22×#11764040C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984087C2211yescharacteristic
21#17984086—21no—
22#17984086C2211yescharacteristic
23#17984078—21no—
24#17984068—21no—
25#17984067—21no—
26#17984067C2211yescharacteristic
27#17984065—21no—
28#17984064—21no—
29#17984064C2211yescharacteristic
30#17984064C2211yescharacteristic
31Q8×Dic137C2211yescharacteristic
32Q8×Dic137C2211yescharacteristic
33C4×C22⋊D137—21no—
34C4×C22⋊D137C2211yescharacteristic
35#17984048—21no—
36C2×C4⋊Dic137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38#17984022C2211yescharacteristic
39#17984022C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984012C2211yescharacteristic
48#17984012C2211yescharacteristic
49#17984011C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C22×Q8×C137C2211yescharacteristic
53C2×Q16×C137C2211yescharacteristic
54C2×Q16×C137C2211yescharacteristic
55C22⋊Q8×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59C2×Dic274—21no—
60C2×Dic274—21no—
61C2×Dic274C2311yescharacteristic
62Q8×D137—22no—
63Q8×D137—41no—
64C2×C22⋊D137D411yescharacteristic
65C22×Dic137—21no—
66C2×C4×D137—21no—
67C2×C4×D137—21no—
68C2×C4×D137C2311yescharacteristic
69#14555500—41no—
70C2×#11764040—41no—
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C8×D137—41no—
74C4⋊Dic137—21no—
75C4⋊Dic137—21no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489D411yescharacteristic
88C22⋊Dic137—21no—
89C22⋊Dic137D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487D411yescharacteristic
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137C2311yescharacteristic
96C2×Q8×C137—21no—
97C2×Q8×C137—22no—
98C2×Q8×C137—41no—
99C2×Q8×C137C2311yescharacteristic
100C2×Q8×C137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C22×C4×C137—21no—
103C22×C4×C137C2311yescharacteristic
104Q16×C137—22no—
105Q16×C137—41no—
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137—21no—
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114Dic274—42no—
115Dic274—81no—
116C22⋊D137Q1614yescomplemented
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—81no—
122C2×Dic137C2×D411yescharacteristic
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C4×D137—21no—
126C4×D137—21no—
127C4×D137—42no—
128C4×D137—81no—
129#11764040—21no—
130#11764040—21no—
131C23×C137C2×D411yescharacteristic
132Q8×C137—21no—
133Q8×C137—21no—
134Q8×C137—21no—
135Q8×C137—21no—
136Q8×C137—21no—
137Q8×C137—42no—
138Q8×C137#4112yes—
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—42no—
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2411yescharacteristic
148C8×C137—21no—
149C8×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137C2×Q1612yes—
153C22×C137C22×D411yescharacteristic
154C2×D137—21no—
155C2×D137—21no—
156C2×D137C2×Q1612yes—
157C4×C137—21no—
158C4×C137—22no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137C2×#4111yescharacteristic
163C4×C137H(5,𝔽2)11yescharacteristic
164Dic137—21no—
165Dic137—21no—
166Dic137—41no—
167Dic137—41no—
168Dic137—41no—
169Dic137—41no—
170Dic137C2×Q1612yes—
171C2×C137—22no—
172C2×C137#54611yescharacteristic
173C2×C137C22×Q1611yescharacteristic
174C2×C137#57711yescharacteristic
175D137—22no—
176C137#329911yescharacteristic, complemented, Sylow
177#3299—1371nomaximal, Sylow
178#554—1371no—
179D4×Q8—1371no—
180#487—1371no—
181#484—1371no—
182#484—1371no—
183#477—1371no—
184#467—1371no—
185#467—1371no—
186#462—1371no—
187#451—1371no—
188#451D13711yescharacteristic, complemented
189C4×Q16—1371no—
190C8×D4—1371no—
191C2×#103—1371no—
192C2×#103—1371no—
193C22×Q8—1371no—
194C22×Q8C2×D13711yescharacteristic
195C2×Q16—1371no—
196C2×Q16C2×D13711yescharacteristic
197C2×Q16C2×D13711yescharacteristic
198C22×C8—1371no—
199C22×C8—1371no—
200C4⋊Q8—2741no—
201C4⋊Q8—1371no—
202C4⋊Q8—1371no—
203#125—2741no—
204C22⋊Q8—2741no—
205C22⋊Q8—2741no—
206C22⋊Q8—2741no—
207C22⋊Q8—2741no—
208C22⋊Q8—1371no—
209C22⋊Q8—1371no—
210C22⋊Q8—1371no—
211C22⋊Q8C2×D13711yescharacteristic
212C4×Q8—1371no—
213C4×Q8—1371no—
214C4×D4—2741no—
215C4×D4—2741no—
216C4×D4—1371no—
217C2×C4⋊C4—1371no—
218C2×C4⋊C4—1371no—
219#107—1371no—
220#107—1371no—
221#107—1371no—
222C4⋊C8—1371no—
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103—1371no—
231#103C2×D13711yescharacteristic
232#103C2×D13711yescharacteristic
233C22⋊C8—1371no—
234C22⋊C8C2×D13711yescharacteristic
235C4×C8—1371no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—2741no—
240C2×Q8—2741no—
241C2×Q8—2742no—
242C2×Q8—5481no—
243C2×Q8—1371no—
244C2×Q8C22×D13711yescharacteristic
245C2×Q8C22×D13711yescharacteristic
246C2×Q8C22×D13711yescharacteristic
247C2×D4—1371no—
248C22×C4—21no—
249C22×C4—2741no—
250C22×C4—2741no—
251C22×C4—2741no—
252C22×C4—1371no—
253C22×C4C22×D13711yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—5481no—
257C2×C8—5481no—
258C2×C8—5481no—
259C2×C8—1371no—
260C2×C8—1371no—
261C2×C8C22×D13711yescharacteristic
262C2×C8C22×D13711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—2741no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—2741no—
273C4⋊C4—2741no—
274C4⋊C4—1371no—
275C4⋊C4—1371no—
276C4⋊C4—1371no—
277C4⋊C4—1371no—
278C4⋊C4C22×D13711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—1371no—
284C22⋊C4—1371no—
285C42—2741no—
286C42—2741no—
287C42—1371no—
288C23—1371no—
289C23D4×D13711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—2741no—
297Q8—2741no—
298Q8—5482no—
299Q8—10961no—
300Q8#1455550912yes—
301D4—1374no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—5482no—
316C2×C4—10961no—
317C2×C4—10961no—
318C2×C4—1371no—
319C2×C4—1371no—
320C2×C4—1371no—
321C2×C4D4×D13711yescharacteristic
322C2×C4C23×D13711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2741no—
326C8—2741no—
327C22—21no—
328C22—21no—
329C22—2741no—
330C22—2741no—
331C22—1372no—
332C22Q16×D13712yes—
333C22C2×D4×D13711yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—2741no—
340C4—2741no—
341C4—5481no—
342C4—5481no—
343C4—5481no—
344C4—5481no—
345C4—1372no—
346C4C2×#1455550911yescharacteristic
347C4#1798413311yescharacteristic
348C2—22no—
349C2—2742no—
350C2#2257804011yescharacteristic
351C2C2×Q16×D13711yescharacteristic
352C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#451×C137C2
socleC22×C137C22×D4
prime radicalC2×C22⋊D137D4
hypercenter#451D137
hypocenterC137#3299
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #22577752C2semidirect
4. #22577739C2semidirect
5. #22577737C2semidirect
6. #22577696C2semidirect
7. #22577642C2semidirect
8. #22577641C2semidirect
9. #22577611C2semidirect
10. #22577557C2semidirect
11. Q16×Dic137C2semidirect
13. C2×#17984012C2semidirect
14. #103×D137C2semidirect
15. #451×C137C2semidirect
116. C22⋊D137Q16semidirect
176. C137#3299semidirect
188. #451D137semidirect

Maximal subgroups

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

subgroupindex
#225781112
Q8×C22⋊D1372
#225777522
#225777392
#225777372
#225776962
#225776422
#225776412
#225776112
#225775572
Q16×Dic1372
#225774662
C2×#179840122
#103×D1372
#451×C1372
#3299137