Finite Group Databasebeta

← #31780887

Subgroups — #31780887

Order 17536.

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

normal subgroups only

This group has 306 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#117×D137C211yesmaximal, characteristic
2#22577366C211yesmaximal, characteristic
3C4×#14555487C211yesmaximal, characteristic
4#22577164C211yesmaximal, characteristic, complemented
5#22577163C211yesmaximal, characteristic, complemented
6#22577162C211yesmaximal, characteristic, complemented
7#339×C137C211yesmaximal, characteristic, complemented
8C22×C4×D137C2211yescharacteristic
9C2×#14555494C2211yescharacteristic
10#17984047C2211yescharacteristic
11#17984046C411yescharacteristic, complemented
12#17984046C411yescharacteristic, complemented
13#17984045—21no—
14#17984044—21no—
15C4⋊C4×D137C411yescharacteristic, complemented
16C4⋊C4×D137C411yescharacteristic, complemented
17C2×#14555487—21no—
18C22⋊C4×D137—21no—
19C2×C4×Dic137C2211yescharacteristic
20C42×D137—21no—
21#17984000—21no—
22#17984000—21no—
23#17983995—21no—
24C2×M16×C137C2211yescharacteristic
25#117×C137C2211yescharacteristic
26C2×C42×C137C2211yescharacteristic
27C23×D137D411yescharacteristic
28C22×Dic137—21no—
29C22×Dic137D411yescharacteristic
30C2×C4×D137—21no—
31C2×C4×D137—21no—
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137—21no—
35C2×C4×D137—21no—
36C2×C4×D137C2×C411yescharacteristic
37C2×C4×D137C2×C411yescharacteristic
38C2×C4×D137D411yescharacteristic
39C2×C4×D137D411yescharacteristic
40#14555494—21no—
41#14555494—21no—
42C2×#11764040—21no—
43C4⋊Dic137C2×C411yescharacteristic
44C4⋊Dic137C2×C411yescharacteristic
45#14555489—21no—
46#14555489—21no—
47C22⋊Dic137—21no—
48#14555487—24no—
49#14555487—21no—
50#14555487—21no—
51C4×Dic137—21no—
52C4×Dic137—21no—
53C4×Dic137—21no—
54C4×Dic137—21no—
55C4×Dic137—21no—
56C22×C4×C137—21no—
57C22×C4×C137C2311yescharacteristic
58M16×C137—21no—
59M16×C137—21no—
60C2×C8×C137—21no—
61C4⋊C4×C137C2×C411yescharacteristic
62C4⋊C4×C137C2×C411yescharacteristic
63C22⋊C4×C137—21no—
64C42×C137—21no—
65C42×C137—21no—
66C42×C137—21no—
67C22×D137—21no—
68C22×D137—21no—
69C22×D137—22no—
70C22×D137—41no—
71C22×D137C22⋊C411yescharacteristic
72C22×D137C22⋊C411yescharacteristic
73C2×Dic137—21no—
74C2×Dic137—21no—
75C2×Dic137—22no—
76C2×Dic137—21no—
77C2×Dic137—21no—
78C2×Dic137—41no—
79C2×Dic137—41no—
80C2×Dic137C22⋊C411yescharacteristic
81C2×Dic137C22⋊C411yescharacteristic
82C4×D137—21no—
83C4×D137—21no—
84C4×D137—21no—
85C4×D137—21no—
86C4×D137—21no—
87C4×D137—21no—
88C4×D137—21no—
89C4×D137—21no—
90C4×D137—42no—
91C4×D137—42no—
92#11764040—41no—
93C23×C137C2×D411yescharacteristic
94C2×C4×C137—22no—
95C2×C4×C137—21no—
96C2×C4×C137—21no—
97C2×C4×C137—41no—
98C2×C4×C137—41no—
99C2×C4×C137C22×C411yescharacteristic
100C2×C4×C137C2×D411yescharacteristic
101C2×C4×C137#4111yescharacteristic
102C2×C4×C137#4111yescharacteristic
103C2×C4×C137#4111yescharacteristic
104C2×C4×C137#4111yescharacteristic
105C8×C137—41no—
106C22×C137—21no—
107C22×C137—21no—
108C22×C137C2×C22⋊C411yescharacteristic
109C22×C137#12311yescharacteristic
110C22×C137#12311yescharacteristic
111C2×D137—21no—
112C2×D137—21no—
113C2×D137—21no—
114C2×D137—21no—
115C2×D137—21no—
116C2×D137—42no—
117C2×D137—42no—
118C2×D137#10412yes—
119C4×C137—42no—
120C4×C137—41no—
121C4×C137—41no—
122C4×C137#11711yescharacteristic
123C4×C137#11711yescharacteristic
124C4×C137#12311yescharacteristic
125C4×C137#12311yescharacteristic
126Dic137—21no—
127Dic137—21no—
128Dic137—42no—
129Dic137—41no—
130Dic137—41no—
131C2×C137—21no—
132C2×C137—21no—
133C2×C137#38111yescharacteristic
134C2×C137C2×#10411yescharacteristic
135C2×C137#42111yescharacteristic
136D137—22no—
137D137—41no—
138C137#182611yescharacteristic, complemented, Sylow
139#1826—1371nomaximal, Sylow
140C2×#117—1371no—
141#407—1371no—
142C4×C22⋊C4—1371no—
143#339—1371no—
144#339—1371no—
145#339—1371no—
146#339D13711yescharacteristic, complemented
147C23×C4—1371no—
148C2×M16—1371no—
149C2×M16C2×D13711yescharacteristic
150#117—2741no—
151#117—2741no—
152#117—1371no—
153#117—1371no—
154#117—1371no—
155#117C2×D13711yescharacteristic
156C2×C4⋊C4—1371no—
157C2×C4⋊C4—1371no—
158C2×C22⋊C4—2741no—
159C2×C22⋊C4—2741no—
160C2×C42—2741no—
161C2×C42—1371no—
162C2×C42C2×D13711yescharacteristic
163C22⋊C8—2741no—
164C22⋊C8—2741no—
165#95—2741no—
166C24—1371no—
167C22×C4—21no—
168C22×C4—2741no—
169C22×C4—2741no—
170C22×C4—2741no—
171C22×C4—2741no—
172C22×C4—2741no—
173C22×C4—2741no—
174C22×C4—2741no—
175C22×C4—1371no—
176C22×C4—1371no—
177C22×C4—1371no—
178C22×C4—1371no—
179C22×C4—1371no—
180C22×C4C22×D13711yescharacteristic
181M16—21no—
182M16—21no—
183M16—2741no—
184M16—2741no—
185C2×C8—21no—
186C2×C8—2741no—
187C4⋊C4—2741no—
188C4⋊C4—2741no—
189C4⋊C4—1371no—
190C4⋊C4—1371no—
191C4⋊C4C4×D13711yescharacteristic
192C4⋊C4C4×D13711yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—2744no—
195C22⋊C4—2741no—
196C22⋊C4—2741no—
197C22⋊C4—2741no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—2741no—
202C42—2741no—
203C42—2741no—
204C42—2741no—
205C42—2741no—
206C23—2742no—
207C23—2741no—
208C23—2741no—
209C23—5481no—
210C23—1371no—
211C23—1371no—
212C23D4×D13711yescharacteristic
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—2741no—
219C2×C4—2741no—
220C2×C4—2741no—
221C2×C4—2741no—
222C2×C4—2741no—
223C2×C4—2741no—
224C2×C4—2741no—
225C2×C4—2741no—
226C2×C4—2741no—
227C2×C4—2741no—
228C2×C4—2742no—
229C2×C4—2741no—
230C2×C4—2741no—
231C2×C4—5482no—
232C2×C4—5482no—
233C2×C4—5481no—
234C2×C4—5481no—
235C2×C4—1371no—
236C2×C4—1371no—
237C2×C4C2×C4×D13711yescharacteristic
238C2×C4D4×D13711yescharacteristic
239C2×C4#1455550911yescharacteristic
240C2×C4#1455551011yescharacteristic
241C2×C4#1455551011yescharacteristic
242C2×C4#1455551111yescharacteristic
243C8—41no—
244C8—5481no—
245C22—21no—
246C22—21no—
247C22—2741no—
248C22—2741no—
249C22—2741no—
250C22—2741no—
251C22—2741no—
252C22—5482no—
253C22—5482no—
254C22—1372no—
255C22C22⋊C4×D13711yescharacteristic
256C22#1798407011yescharacteristic
257C22#1798407311yescharacteristic
258C4—42no—
259C4—41no—
260C4—41no—
261C4—2741no—
262C4—2741no—
263C4—5482no—
264C4—5481no—
265C4—5481no—
266C4#1798404511yescharacteristic
267C4#1798404611yescharacteristic
268C4#1798407011yescharacteristic
269C4#1798407311yescharacteristic
270C2—21no—
271C2—21no—
272C2—2742no—
273C2—5481no—
274C2#2257726511yescharacteristic
275C2#104×D13711yescharacteristic
276C2#2257741811yescharacteristic
subgroupG/H
centerC2×C4D4×D137
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#339×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC22×C4×C137C23
hypercenter#339D137
hypocenterC137#1826
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
4. #22577164C2semidirect
5. #22577163C2semidirect
6. #22577162C2semidirect
7. #339×C137C2semidirect
11. #17984046C4semidirect
12. #17984046C4semidirect
15. C4⋊C4×D137C4semidirect
16. C4⋊C4×D137C4semidirect
138. C137#1826semidirect
146. #339D137semidirect

Maximal subgroups

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

subgroupindex
#117×D1372
#225773662
C4×#145554872
#225771642
#225771632
#225771622
#339×C1372
#1826137