Finite Group Databasebeta

← #31779331

Subgroups — #31779331

Order 17536.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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#22577916C211yesmaximal, characteristic
2#22577365C211yesmaximal, characteristic, complemented
3#22577262C211yesmaximal, characteristic, complemented
4#22577140C211yesmaximal, characteristic, complemented
5#22577138C211yesmaximal, characteristic, complemented
6#22577136C211yesmaximal, characteristic, complemented
7#328×C137C211yesmaximal, characteristic, complemented
8C2×#14555511C411yescharacteristic, complemented
9C22×C4×D137C2211yescharacteristic
10C2×#14555493—21no—
11#17984073—21no—
12#17984064C411yescharacteristic, complemented
13#17984063C411yescharacteristic, complemented
14#17984062C2211yescharacteristic
15#17984058—21no—
16#17984050—21no—
17#17984048—21no—
18#17984046C411yescharacteristic, complemented
19C2×#14555489—21no—
20C2×#14555487—21no—
21C22⋊#11764040C2211yescharacteristic
22#17984000—21no—
23#17983995C2211yescharacteristic
24C22⋊Q8×C137C2211yescharacteristic
25C22⋊C8×C137C2211yescharacteristic
26#95×C137C2211yescharacteristic
27C23×D137D411yescharacteristic
28#14555511—42no—
29C2×D548—21no—
30C2×D548—21no—
31C2×D548C2×C411yescharacteristic
32C2×C22⋊D137—21no—
33C22×Dic137—21no—
34C22×Dic137D411yescharacteristic
35C2×C4×D137—21no—
36C2×C4×D137—42no—
37C2×C4×D137C2×C411yescharacteristic
38C2×C4×D137C2×C411yescharacteristic
39C2×C4×D137D411yescharacteristic
40C2×C4×D137D411yescharacteristic
41#14555493—42no—
42C2×#11764040—21no—
43C4⋊Dic137C2×C411yescharacteristic
44#14555489—21no—
45#14555489—42no—
46#14555487—21no—
47#14555487—21no—
48#14555487—21no—
49#14555487—21no—
50#14555487—42no—
51C4×Dic137—21no—
52C2×Q8×C137C2×C411yescharacteristic
53C22×C4×C137—21no—
54C22×C4×C137C2311yescharacteristic
55C2×C8×C137—21no—
56C4⋊C4×C137—21no—
57C4⋊C4×C137C2×C411yescharacteristic
58C22⋊C4×C137—21no—
59C22×D137—21no—
60C22×D137—21no—
61C22×D137—21no—
62C22×D137—41no—
63C22×D137—41no—
64C22×D137C22⋊C411yescharacteristic
65C22×D137C22⋊C411yescharacteristic
66D548—41no—
67D548—42no—
68D548—42no—
69C22⋊D137—41no—
70C22⋊D137—41no—
71C2×Dic137—22no—
72C2×Dic137—21no—
73C2×Dic137—41no—
74C2×Dic137—41no—
75C2×Dic137—41no—
76C2×Dic137C22⋊C411yescharacteristic
77C2×Dic137C22⋊C411yescharacteristic
78C4×D137—22no—
79C4×D137—22no—
80C4×D137—41no—
81C4×D137—41no—
82C4×D137—42no—
83#11764040—41no—
84C23×C137C2×D411yescharacteristic
85Q8×C137—41no—
86C2×C4×C137—21no—
87C2×C4×C137—21no—
88C2×C4×C137—41no—
89C2×C4×C137—41no—
90C2×C4×C137—41no—
91C2×C4×C137—41no—
92C2×C4×C137C22×C411yescharacteristic
93C2×C4×C137C2×D411yescharacteristic
94C8×C137—41no—
95C22×C137—21no—
96C22×C137—21no—
97C22×C137C2×C22⋊C411yescharacteristic
98C22×C137AGL(1,ℤ/8)11yescharacteristic
99C22×C137#13711yescharacteristic
100C2×D137—21no—
101C2×D137—21no—
102C2×D137—21no—
103C2×D137—21no—
104C2×D137—21no—
105C2×D137—21no—
106C2×D137—42no—
107C2×D137—42no—
108C2×D137—41no—
109C2×D137—41no—
110C2×D137—41no—
111C4×C137—21no—
112C4×C137—21no—
113C4×C137—41no—
114C4×C137—41no—
115C4×C137—81no—
116Dic137—21no—
117Dic137—41no—
118Dic137—81no—
119Dic137#9912yes—
120C2×C137—21no—
121C2×C137—21no—
122C2×C137C2×#9911yescharacteristic
123C2×C137#41711yescharacteristic
124C2×C137#42111yescharacteristic
125D137—41no—
126D137—41no—
127D137—81no—
128C137#151311yescharacteristic, complemented, Sylow
129#1513—1371nomaximal, Sylow
130#525—1371no—
131#407—1371no—
132#380—1371no—
133#328—1371no—
134#328D13711yescharacteristic, complemented
135#327—1371no—
136#327—1371no—
137C2×#41—1371no—
138C23×C4—1371no—
139C2×M16—2741no—
140#123—2741no—
141C22⋊Q8—1371no—
142C22⋊Q8C2×D13711yescharacteristic
143#121—1371no—
144#121—1371no—
145#120—2741no—
146C4×D4—2741no—
147C4×D4—2741no—
148#117—1371no—
149C2×C4⋊C4—2741no—
150C2×C22⋊C4—2741no—
151C22⋊C8—2741no—
152C22⋊C8—1371no—
153C22⋊C8C2×D13711yescharacteristic
154#95—1371no—
155#95C2×D13711yescharacteristic
156C24—1371no—
157#41—5482no—
158C2×Q8C4×D13711yescharacteristic
159C2×D4—2741no—
160C2×D4—2741no—
161C2×D4—2741no—
162C2×D4—1371no—
163C22×C4—21no—
164C22×C4—2741no—
165C22×C4—2741no—
166C22×C4—5482no—
167C22×C4—1371no—
168C22×C4—1371no—
169C22×C4—1371no—
170C22×C4—1371no—
171C22×C4—1371no—
172C22×C4C22×D13711yescharacteristic
173M16—5482no—
174C2×C8—21no—
175C2×C8—2741no—
176C4⋊C4—21no—
177C4⋊C4—2741no—
178C4⋊C4—5482no—
179C4⋊C4—1371no—
180C4⋊C4C4×D13711yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—2741no—
183C22⋊C4—2741no—
184C22⋊C4—2741no—
185C22⋊C4—2741no—
186C22⋊C4—5482no—
187C42—2741no—
188C23—2741no—
189C23—2741no—
190C23—2741no—
191C23—5481no—
192C23—5481no—
193C23—1371no—
194C23—1371no—
195C23D4×D13711yescharacteristic
196Q8—41no—
197D4—5481no—
198D4—5482no—
199D4—5482no—
200D4—5481no—
201D4—5481no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—2742no—
209C2×C4—2742no—
210C2×C4—2742no—
211C2×C4—2741no—
212C2×C4—5481no—
213C2×C4—5481no—
214C2×C4—5482no—
215C2×C4—5481no—
216C2×C4—5481no—
217C2×C4—5481no—
218C2×C4—1371no—
219C2×C4—1371no—
220C2×C4C2×C4×D13711yescharacteristic
221C2×C4D4×D13711yescharacteristic
222C8—41no—
223C8—5481no—
224C22—21no—
225C22—21no—
226C22—2741no—
227C22—2741no—
228C22—2741no—
229C22—2741no—
230C22—2741no—
231C22—2741no—
232C22—5482no—
233C22—5482no—
234C22—5481no—
235C22—5481no—
236C22—5481no—
237C22C22⋊C4×D13711yescharacteristic
238C22#1798411511yescharacteristic
239C22#1798412011yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—2741no—
246C4—5481no—
247C4—10961no—
248C4—1372no—
249C2—21no—
250C2—21no—
251C2—5481no—
252C2—5481no—
253C2—10961no—
254C2#99×D13711yescharacteristic
255C2#2257740311yescharacteristic
256C2#2257741811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#328×C137C2
socleC22×C137C2×C22⋊C4
prime radical#22577916C2
cosocleC22×C4×C137C23
hypercenter#328D137
hypocenterC137#1513
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #22577365C2semidirect
3. #22577262C2semidirect
4. #22577140C2semidirect
5. #22577138C2semidirect
6. #22577136C2semidirect
7. #328×C137C2semidirect
8. C2×#14555511C4semidirect
12. #17984064C4semidirect
13. #17984063C4semidirect
18. #17984046C4semidirect
128. C137#1513semidirect
134. #328D137semidirect

Maximal subgroups

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

subgroupindex
#225779162
#225773652
#225772622
#225771402
#225771382
#225771362
#328×C1372
#1513137