Finite Group Databasebeta

← #30666406

Subgroups — #30666406

Order 16768.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#22553819C211yesmaximal, characteristic
2C2×#16862203C211yesmaximal, characteristic, complemented
3C2×#16862173C211yesmaximal, characteristic, complemented
4#22553275C211yesmaximal, characteristic, complemented
5#22553073C211yesmaximal, characteristic, complemented
6#22553072C211yesmaximal, characteristic, complemented
7C2×#104×C131C211yesmaximal, characteristic, complemented
8C22×C4×D131C2211yescharacteristic
9C2×#14523497C2211yescharacteristic
10#16862227—21no—
11#16862222C412yescomplemented
12#16862217C412yescomplemented
13#16862214—21no—
14C4×C22⋊D131—22no—
15#16862205C2211yescharacteristic
16#16862203—22no—
17#16862203—22no—
18C2×Dic131⋊C4—21no—
19C2×(C2×D131)⋊C4—21no—
20C2×C4×Dic131C2211yescharacteristic
21#16862173—22no—
22(C2×D131)⋊C8—22no—
23C2×#41×C131C2211yescharacteristic
24C2×M16×C131C2211yescharacteristic
25C2×C42×C131C2211yescharacteristic
26#104×C131—22no—
27C23×D131D411yescharacteristic
28C2×C22⋊D131—22no—
29C22×Dic131—21no—
30C22×Dic131D411yescharacteristic
31C2×C4×D131—21no—
32C2×C4×D131—21no—
33C2×C4×D131—21no—
34C2×C4×D131—21no—
35C2×C4×D131C2×C412yes—
36C2×C4×D131D412yes—
37#14523497—21no—
38#14523497—21no—
39C2×C131⋊C8—21no—
40C4⋊Dic131C2×C412yes—
41Dic131⋊C4—24no—
42Dic131⋊C4—22no—
43C22⋊Dic131—22no—
44C22⋊Dic131—21no—
45(C2×D131)⋊C4—24no—
46(C2×D131)⋊C4—22no—
47C4×Dic131—21no—
48C4×Dic131—21no—
49C4×Dic131D412yescomplemented
50#41×C131—21no—
51#41×C131—21no—
52#41×C131—21no—
53#41×C131—21no—
54C2×Q8×C131C2×C411yescharacteristic
55C2×D4×C131—21no—
56C2×D4×C131C2×C411yescharacteristic
57C22×C4×C131—21no—
58C22×C4×C131—21no—
59C22×C4×C131C2311yescharacteristic
60M16×C131—21no—
61M16×C131—21no—
62C2×C8×C131—21no—
63C42×C131—21no—
64C42×C131D412yescomplemented
65C22×D131—21no—
66C22×D131—21no—
67C22×D131—42no—
68C22×D131C22⋊C412yes—
69C22⋊D131—44no—
70C2×Dic131—21no—
71C2×Dic131—21no—
72C2×Dic131—22no—
73C2×Dic131—22no—
74C2×Dic131—42no—
75C2×Dic131C22⋊C412yes—
76C4×D131—22no—
77C4×D131—22no—
78C4×D131—22no—
79C4×D131—22no—
80C131⋊C8—41no—
81C23×C131—21no—
82C23×C131C2×D411yescharacteristic
83Q8×C131—21no—
84Q8×C131—21no—
85D4×C131—21no—
86D4×C131—21no—
87D4×C131—41no—
88D4×C131—41no—
89C2×C4×C131—21no—
90C2×C4×C131—22no—
91C2×C4×C131—42no—
92C2×C4×C131—41no—
93C2×C4×C131—41no—
94C2×C4×C131C22×C411yescharacteristic
95C2×C4×C131C2×D411yescharacteristic
96C2×C4×C131C2×D411yescharacteristic
97C2×C4×C131C2×D411yescharacteristic
98C2×C4×C131#4111yescharacteristic
99C2×C4×C131#4111yescharacteristic
100C8×C131—41no—
101C22×C131—22no—
102C22×C131—41no—
103C22×C131—42no—
104C22×C131C2×C22⋊C411yescharacteristic
105C22×C131#12011yescharacteristic
106C22×C131#12311yescharacteristic
107C2×D131—22no—
108C2×D131—22no—
109C2×D131—22no—
110C2×D131—42no—
111C2×D131—42no—
112C4×C131—42no—
113C4×C131—41no—
114C4×C131C4×D411yescharacteristic
115C4×C131C4×D411yescharacteristic
116C4×C131#12111yescharacteristic
117C4×C131#12111yescharacteristic
118Dic131—22no—
119Dic131—42no—
120Dic131—42no—
121C2×C131—21no—
122C2×C131—21no—
123C2×C131—81no—
124C2×C131#38611yescharacteristic
125C2×C131#42112yes—
126D131—42no—
127C131#189711yescharacteristic, complemented, Sylow
128#1897—1311nomaximal, Sylow
129#525—1311no—
130C2×#117—1311no—
131C2×#104—1311no—
132C2×#104D13111yescharacteristic, complemented
133#407—1311no—
134#339—1311no—
135#339—1311no—
136C2×#41C2×D13111yescharacteristic
137C23×C4—1311no—
138C2×M16—1311no—
139C2×M16C2×D13111yescharacteristic
140#123—2621no—
141C22⋊Q8—1312no—
142#121—1312no—
143#120—2621no—
144C4×D4—2622no—
145#117—2622no—
146#117—2622no—
147#117—1311no—
148C2×C4⋊C4—2621no—
149C2×C22⋊C4—2621no—
150C2×C42—1311no—
151C2×C42C2×D13111yescharacteristic
152#104—22no—
153#104—2622no—
154C22⋊C8—2622no—
155C24—1311no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160C2×Q8C4×D13111yescharacteristic
161C2×D4—21no—
162C2×D4—2622no—
163C2×D4C4×D13111yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2621no—
167C22×C4—2621no—
168C22×C4—2621no—
169C22×C4—2621no—
170C22×C4—2621no—
171C22×C4—1311no—
172C22×C4—1312no—
173C22×C4—1312no—
174C22×C4C22×D13111yescharacteristic
175M16—21no—
176M16—21no—
177M16—2621no—
178M16—2621no—
179C2×C8—21no—
180C2×C8—2621no—
181C4⋊C4—2624no—
182C4⋊C4—2622no—
183C4⋊C4—1312no—
184C22⋊C4—2624no—
185C22⋊C4—2622no—
186C22⋊C4—2621no—
187C22⋊C4—2622no—
188C42—21no—
189C42—2621no—
190C42—2621no—
191C42—1312no—
192C42C22⋊D13112yescomplemented
193C23—21no—
194C23—2621no—
195C23—2621no—
196C23—5242no—
197C23—1312no—
198C23D4×D13111yescharacteristic
199Q8—21no—
200Q8—21no—
201D4—21no—
202D4—21no—
203D4—41no—
204D4—41no—
205D4—5244no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—2621no—
212C2×C4—2621no—
213C2×C4—2622no—
214C2×C4—2622no—
215C2×C4—2622no—
216C2×C4—2622no—
217C2×C4—2622no—
218C2×C4—2622no—
219C2×C4—5242no—
220C2×C4—1312no—
221C2×C4C2×C4×D13111yescharacteristic
222C2×C4D4×D13111yescharacteristic
223C2×C4D4×D13111yescharacteristic
224C2×C4C2×C22⋊D13111yescharacteristic
225C2×C4#1452351211yescharacteristic
226C2×C4#1452351311yescharacteristic
227C8—41no—
228C8—5241no—
229C22—22no—
230C22—41no—
231C22—42no—
232C22—2622no—
233C22—2622no—
234C22—2622no—
235C22—5242no—
236C22—5242no—
237C22C22⋊C4×D13111yescharacteristic
238C22#1686221411yescharacteristic
239C22#1686222811yescharacteristic
240C4—42no—
241C4—41no—
242C4—2622no—
243C4—5242no—
244C4—5242no—
245C4(C22⋊D131)⋊C411yescharacteristic
246C4C4×C22⋊D13111yescharacteristic
247C4#1686221711yescharacteristic
248C4#1686221911yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—5242no—
253C2#2255318811yescharacteristic
254C2#2255332712yes—
subgroupG/H
centerC2×C4D4×D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
FittingC2×#104×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553819C2
cosocleC22×C4×C131C23
hypercenterC2×#104D131
hypocenterC131#1897
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16862203C2semidirect
3. C2×#16862173C2semidirect
4. #22553275C2semidirect
5. #22553073C2semidirect
6. #22553072C2semidirect
7. C2×#104×C131C2semidirect
11. #16862222C4semidirect
12. #16862217C4semidirect
49. C4×Dic131D4semidirect
64. C42×C131D4semidirect
127. C131#1897semidirect
132. C2×#104D131semidirect
192. C42C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
#225538192
C2×#168622032
C2×#168621732
#225532752
#225530732
#225530722
C2×#104×C1312
#1897131