Finite Group Databasebeta

← #31781330

Subgroups — #31781330

Order 17536.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577965C211yesmaximal, characteristic
2#22577876C211yesmaximal, characteristic, complemented
3C2×#17984002C211yesmaximal, characteristic, complemented
4C2×#17984002C211yesmaximal, characteristic, complemented
5C2×#17984002C211yesmaximal, characteristic, complemented
6#22577169C211yesmaximal, characteristic, complemented
7#342×C137C211yesmaximal, characteristic, complemented
8C22×C22⋊D137C2211yescharacteristic
9C22×C22⋊D137C2211yescharacteristic
10#17984072—21no—
11#17984072C412yescomplemented
12#17984062—22no—
13#17984058—21no—
14#17984058C412yescomplemented
15C4×C22⋊D137—22no—
16#17984048—22no—
17#17984044—21no—
18C2×C22⋊Dic137C2211yescharacteristic
19C2×C22⋊Dic137C2211yescharacteristic
20C22⋊C4×D137—21no—
21#17984002—22no—
22#17984002—22no—
23#17984002—22no—
24C2×C22⋊C4×C137C2211yescharacteristic
25C2×C22⋊C4×C137C2211yescharacteristic
26C2×C22⋊C4×C137C2211yescharacteristic
27C23×D137C2×C411yescharacteristic, complemented
28C23×D137D411yescharacteristic
29C2×D548—22no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C2×C22⋊D137—21no—
33C2×C22⋊D137—21no—
34C2×C22⋊D137—21no—
35C2×C22⋊D137—21no—
36C2×C22⋊D137—21no—
37C2×C22⋊D137—21no—
38C2×C22⋊D137—21no—
39C2×C22⋊D137—21no—
40C2×C22⋊D137C2×C412yes—
41C2×C22⋊D137D412yes—
42C22×Dic137—21no—
43C22×Dic137—21no—
44C22×Dic137C2×C411yescharacteristic, complemented
45C22×Dic137D411yescharacteristic
46C2×C4×D137—22no—
47C4⋊Dic137—22no—
48#14555489—22no—
49C22⋊Dic137—21no—
50C22⋊Dic137—21no—
51C22⋊Dic137—21no—
52C22⋊Dic137D412yescomplemented
53#14555487—22no—
54#14555487—22no—
55#14555487—22no—
56C4×Dic137—22no—
57C24×C137C2311yescharacteristic
58C22×C4×C137—21no—
59C22×C4×C137—21no—
60C22×C4×C137—21no—
61C22⋊C4×C137—21no—
62C22⋊C4×C137—21no—
63C22⋊C4×C137—21no—
64C22⋊C4×C137—21no—
65C22⋊C4×C137C2×C412yescomplemented
66C22⋊C4×C137D412yescomplemented
67C22×D137—21no—
68C22×D137—21no—
69C22×D137—21no—
70C22×D137—21no—
71C22×D137—21no—
72C22×D137—81no—
73C22×D137—81no—
74C22×D137C22⋊C412yes—
75D548—82no—
76C22⋊D137—22no—
77C22⋊D137—22no—
78C22⋊D137—22no—
79C22⋊D137—22no—
80C22⋊D137—41no—
81C22⋊D137—41no—
82C22⋊D137—41no—
83C22⋊D137—41no—
84C2×Dic137—21no—
85C2×Dic137—21no—
86C2×Dic137—21no—
87C2×Dic137—21no—
88C2×Dic137—21no—
89C2×Dic137—22no—
90C2×Dic137—41no—
91C2×Dic137—81no—
92C2×Dic137—81no—
93C2×Dic137C22⋊C412yes—
94C4×D137—82no—
95C23×C137—81no—
96C23×C137C22×C411yescharacteristic
97C23×C137C2×D411yescharacteristic
98C23×C137C2×D411yescharacteristic
99C23×C137C2×D411yescharacteristic
100C23×C137C2×D411yescharacteristic
101C23×C137#4111yescharacteristic
102C23×C137#4111yescharacteristic
103C2×C4×C137—22no—
104C2×C4×C137—22no—
105C2×C4×C137—41no—
106C2×C4×C137—81no—
107C2×C4×C137—81no—
108C2×C4×C137—81no—
109C22×C137—41no—
110C22×C137—41no—
111C22×C137—41no—
112C22×C137—41no—
113C22×C137—41no—
114C22×C137—41no—
115C22×C137—41no—
116C22×C137C2×C22⋊C411yescharacteristic
117C22×C137C4×D411yescharacteristic
118C22×C137C4×D411yescharacteristic
119C22×C137#12011yescharacteristic
120C22×C137#12111yescharacteristic
121C22×C137#12111yescharacteristic
122C22×C137#12311yescharacteristic
123C2×D137—22no—
124C2×D137—42no—
125C2×D137—42no—
126C2×D137—42no—
127C2×D137—41no—
128C2×D137—81no—
129C2×D137—81no—
130C2×D137—81no—
131C4×C137—42no—
132C4×C137—42no—
133C4×C137—81no—
134Dic137—22no—
135Dic137—42no—
136Dic137—41no—
137Dic137—81no—
138C2×C137—21no—
139C2×C137—21no—
140C2×C137—21no—
141C2×C137—21no—
142C2×C137—21no—
143C2×C137—21no—
144C2×C137—21no—
145C2×C137#38611yescharacteristic
146D137—42no—
147D137—81no—
148C137#191311yescharacteristic, complemented, Sylow
149#1913—1371nomaximal, Sylow
150#534—1371no—
151#518—1371no—
152C2×#99—1371no—
153C2×#99—1371no—
154C2×#99—1371no—
155#342—1371no—
156#342D13711yescharacteristic, complemented
157C22×D4—1371no—
158C22×D4—1371no—
159#123—2741no—
160#123—1372no—
161#121—2742no—
162#120—2741no—
163#120—1372no—
164C4×D4—2742no—
165C4×D4—2742no—
166#117—2741no—
167C2×C22⋊C4—2741no—
168C2×C22⋊C4—1371no—
169C2×C22⋊C4—1371no—
170C2×C22⋊C4C2×D13711yescharacteristic
171C2×C22⋊C4C2×D13711yescharacteristic
172C2×C22⋊C4C2×D13711yescharacteristic
173#99—2742no—
174#99—2742no—
175#99—2742no—
176C24—1371no—
177C24—1371no—
178C24C22×D13711yescharacteristic
179C2×D4—2742no—
180C2×D4—2741no—
181C2×D4—2741no—
182C2×D4—2741no—
183C2×D4—2741no—
184C2×D4—2741no—
185C2×D4—2741no—
186C2×D4—2741no—
187C2×D4—2741no—
188C2×D4—2741no—
189C2×D4—2741no—
190C2×D4—1372no—
191C2×D4—1372no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—2742no—
196C22×C4—2741no—
197C22×C4—2741no—
198C22×C4—1371no—
199C22×C4—1371no—
200C4⋊C4—2742no—
201C4⋊C4—2742no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—2742no—
207C22⋊C4—2742no—
208C22⋊C4—2742no—
209C22⋊C4—2741no—
210C22⋊C4—2741no—
211C22⋊C4—2741no—
212C22⋊C4—1372no—
213C22⋊C4C4×D13712yescomplemented
214C22⋊C4D54812yescomplemented
215C42—2742no—
216C23—81no—
217C23—2741no—
218C23—2741no—
219C23—2741no—
220C23—2741no—
221C23—2741no—
222C23—10961no—
223C23—10961no—
224C23—1372no—
225C23C2×C4×D13711yescharacteristic
226C23D4×D13711yescharacteristic
227C23D4×D13711yescharacteristic
228C23D4×D13711yescharacteristic
229C23C2×D54811yescharacteristic
230C23#1455551011yescharacteristic
231C23#1455551111yescharacteristic
232D4—2742no—
233D4—2742no—
234D4—2742no—
235D4—2742no—
236D4—5481no—
237D4—5481no—
238D4—5481no—
239D4—5481no—
240D4—10962no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—2741no—
248C2×C4—2741no—
249C2×C4—2741no—
250C2×C4—2741no—
251C2×C4—2741no—
252C2×C4—2742no—
253C2×C4—5481no—
254C2×C4—10962no—
255C2×C4—10961no—
256C2×C4—10961no—
257C2×C4—1372no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—2742no—
266C22—5482no—
267C22—5482no—
268C22—5482no—
269C22—5481no—
270C22—10961no—
271C22—10961no—
272C22—10961no—
273C22C22⋊C4×D13711yescharacteristic
274C22#1798405011yescharacteristic
275C22C4×D54811yescharacteristic
276C22#1798405811yescharacteristic
277C22#1798406111yescharacteristic
278C22#1798406311yescharacteristic
279C22#1798407311yescharacteristic
280C4—42no—
281C4—42no—
282C4—81no—
283C4—2742no—
284C4—5482no—
285C4—5481no—
286C4—10961no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—5482no—
295C2—10961no—
296C2#2257728111yescharacteristic
subgroupG/H
centerC2#22577281
commutatorC23×C137C22×C4
FrattiniC24C22×D137
Fitting#342×C137C2
socleC2×C137#386
prime radical#22577965C2
cosocleC24×C137C23
hypercenter#342D137
hypocenterC137#1913
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #22577876C2semidirect
3. C2×#17984002C2semidirect
4. C2×#17984002C2semidirect
5. C2×#17984002C2semidirect
6. #22577169C2semidirect
7. #342×C137C2semidirect
11. #17984072C4semidirect
14. #17984058C4semidirect
27. C23×D137C2×C4semidirect
44. C22×Dic137C2×C4semidirect
52. C22⋊Dic137D4semidirect
65. C22⋊C4×C137C2×C4semidirect
66. C22⋊C4×C137D4semidirect
148. C137#1913semidirect
156. #342D137semidirect
213. C22⋊C4C4×D137semidirect
214. C22⋊C4D548semidirect

Maximal subgroups

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

subgroupindex
#225779652
#225778762
C2×#179840022
#225771692
#342×C1372
#1913137