Finite Group Databasebeta

← #323×D137 #31779094

Subgroups — #323×D137 #31779094

Order 17536.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×C22⋊C4×D137C211yesmaximal, characteristic
2C22⋊C8×D137C212yesmaximal, complemented
3#22577127C211yesmaximal, characteristic, complemented
4#22577126C212yesmaximal, complemented
5#323×C137C211yesmaximal, characteristic, complemented
6C24×D137C411yescharacteristic
7C22×C4×D137C411yescharacteristic
8C22×C4×D137C2211yescharacteristic
9C2×C8×D137—22no—
10C2×C22⋊Dic137C411yescharacteristic
11C2×#14555487C411yescharacteristic
12C2×#14555487C2211yescharacteristic
13C22⋊C4×D137—22no—
14C22⋊C4×D137—22no—
15C22⋊#11764040C2212yescomplemented
16#17984000—22no—
17C2×C22⋊C4×C137C2211yescharacteristic
18C22⋊C8×C137C2212yescomplemented
19C23×D137—21no—
20C23×D137—21no—
21C23×D137—22no—
22C23×D137—42no—
23C23×D137—42no—
24C23×D137C814yescomplemented
25C23×D137C2×C411yescharacteristic
26C23×D137C2×C411yescharacteristic
27C22×Dic137C2×C411yescharacteristic
28C22×Dic137C2×C411yescharacteristic
29C2×C4×D137—21no—
30C2×C4×D137—21no—
31C2×C4×D137—42no—
32C2×C4×D137—42no—
33C2×C4×D137D412yes—
34C2×C4×D137D412yes—
35C2×#11764040—22no—
36C8×D137—44no—
37C22⋊Dic137—22no—
38#14555487—22no—
39#14555487—22no—
40C24×C137C2×C411yescharacteristic
41C22×C4×C137C2×C411yescharacteristic
42C22×C4×C137C2311yescharacteristic
43C2×C8×C137—22no—
44C22⋊C4×C137—22no—
45C22×D137—22no—
46C22×D137—22no—
47C22×D137—21no—
48C22×D137—21no—
49C22×D137—24no—
50C22×D137—24no—
51C22×D137—42no—
52C22×D137—42no—
53C22×D137—48no—
54C22×D137—48no—
55C22×D137—42no—
56C22×D137—42no—
57C22×D137—41no—
58C22×D137—41no—
59C22×D137C22⋊C411yescharacteristic
60C22×D137C22⋊C411yescharacteristic
61C22×D137C2×C812yes—
62C22×D137C2×C812yescomplemented
63C22×D137M1612yes—
64C2×Dic137—21no—
65C2×Dic137—41no—
66C2×Dic137—41no—
67C2×Dic137C22⋊C412yes—
68C4×D137—24no—
69C4×D137—42no—
70C4×D137—42no—
71C4×D137—42no—
72#11764040—42no—
73C23×C137—21no—
74C23×C137—21no—
75C23×C137—41no—
76C23×C137—41no—
77C23×C137C2×C812yescomplemented
78C23×C137C22×C411yescharacteristic
79C2×C4×C137—21no—
80C2×C4×C137—41no—
81C2×C4×C137—41no—
82C2×C4×C137C2×D412yes—
83C8×C137—42no—
84C22×C137—21no—
85C22×C137—21no—
86C22×C137—22no—
87C22×C137—42no—
88C22×C137—42no—
89C22×C137—41no—
90C22×C137—41no—
91C22×C137C2×C22⋊C411yescharacteristic
92C22×C137C22×C811yescharacteristic
93C22×C137C2×M1611yescharacteristic
94C2×D137—21no—
95C2×D137—21no—
96C2×D137—24no—
97C2×D137—24no—
98C2×D137—22no—
99C2×D137—48no—
100C2×D137—42no—
101C2×D137—41no—
102C2×D137—41no—
103C2×D137—48no—
104C2×D137—42no—
105C2×D137C22⋊C812yes—
106C2×D137C22⋊C812yes—
107C2×D137#9912yes—
108C2×D137#10012yes—
109C4×C137—22no—
110C4×C137—41no—
111Dic137—22no—
112Dic137—41no—
113C2×C137—21no—
114C2×C137—21no—
115C2×C137—42no—
116C2×C137C2×C22⋊C811yescharacteristic
117C2×C137C2×#9911yescharacteristic
118C2×C137C2×#10011yescharacteristic
119D137—22no—
120D137—42no—
121D137#32314yesdirect product factor
122C137C2×#32311yescharacteristic, complemented, Sylow
123C2×#323—1371nomaximal, Sylow
124C22×C22⋊C4—1371no—
125C2×C22⋊C8—1372no—
126#323—1371no—
127#323—1372no—
128#323D13711yescharacteristic, direct product factor
129C25—1371no—
130C23×C4—1371no—
131C23×C4—1371no—
132C22×C8—2742no—
133C2×C22⋊C4—2742no—
134C2×C22⋊C4—2742no—
135C2×C22⋊C4—1371no—
136C2×C22⋊C4—1371no—
137C2×C22⋊C4—1371no—
138C2×C22⋊C4C2×D13711yescharacteristic
139C22⋊C8—2742no—
140C22⋊C8—1372no—
141C22⋊C8C2×D13712yescomplemented
142C24—2741no—
143C24—2741no—
144C24—2742no—
145C24—5482no—
146C24—5482no—
147C24—1371no—
148C24—1374no—
149C24—1371no—
150C24C4×D13711yescharacteristic
151C22×C4—2741no—
152C22×C4—2741no—
153C22×C4—5482no—
154C22×C4—5482no—
155C22×C4—1372no—
156C22×C4—1371no—
157C22×C4—1371no—
158C22×C4—1372no—
159C22×C4C4×D13711yescharacteristic
160C22×C4C22×D13711yescharacteristic
161C2×C8—22no—
162C2×C8—2742no—
163C2×C8—5484no—
164C22⋊C4—22no—
165C22⋊C4—2742no—
166C22⋊C4—2742no—
167C22⋊C4—2742no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—2742no—
173C23—2742no—
174C23—2744no—
175C23—2744no—
176C23—2741no—
177C23—2741no—
178C23—5481no—
179C23—5481no—
180C23—5488no—
181C23—5488no—
182C23—5482no—
183C23—5482no—
184C23—5482no—
185C23—5482no—
186C23—1371no—
187C23—1372no—
188C23—1372no—
189C23—1372no—
190C23—1371no—
191C23C8×D13712yescomplemented
192C23C2×C4×D13711yescharacteristic
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—2744no—
197C2×C4—2741no—
198C2×C4—5482no—
199C2×C4—5482no—
200C2×C4—5482no—
201C2×C4—5481no—
202C2×C4—5481no—
203C2×C4—1372no—
204C2×C4D4×D13712yes—
205C8—42no—
206C8—5482no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—2744no—
215C22—2744no—
216C22—2741no—
217C22—2741no—
218C22—2742no—
219C22—5482no—
220C22—5482no—
221C22—5488no—
222C22—5488no—
223C22—5481no—
224C22—5481no—
225C22—1372no—
226C22—1372no—
227C22—1372no—
228C22—1372no—
229C22C22⋊C4×D13711yescharacteristic
230C22C2×C8×D13711yescharacteristic
231C22M16×D13711yescharacteristic
232C4—22no—
233C4—41no—
234C4—2742no—
235C4—5481no—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—2742no—
240C2—5482no—
241C2—1374no—
242C2C22⋊C8×D13711yescharacteristic
243C2#99×D13711yescharacteristic
244C2#100×D13711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC22×C137C22×C8
FrattiniC22×C4C22×D137
Fitting#323×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC24×D137C4
cosocleC22×C4×C137C23
hypercenter#323D137
hypocenterC137C2×#323
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D137C2semidirect
3. #22577127C2semidirect
4. #22577126C2semidirect
5. #323×C137C2semidirect
15. C22⋊#11764040C22semidirect
18. C22⋊C8×C137C22semidirect
24. C23×D137C8semidirect
62. C22×D137C2×C8semidirect
77. C23×C137C2×C8semidirect
121. D137#323direct (N × H)
122. C137C2×#323semidirect
128. #323D137direct (N × H)
141. C22⋊C8C2×D137semidirect
191. C23C8×D137semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1372
C22⋊C8×D1372
#225771272
#225771262
#323×C1372
C2×#323137