Finite Group Databasebeta

← #30666408

Subgroups — #30666408

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#22553824C211yesmaximal, characteristic
2C2×#16862203C211yesmaximal, characteristic, complemented
3C2×#16862174C211yesmaximal, characteristic, complemented
4C2×#16862172C211yesmaximal, characteristic, complemented
5#22553275C211yesmaximal, characteristic, complemented
6#22553071C211yesmaximal, characteristic, complemented
7#339×C131C211yesmaximal, characteristic, complemented
8C2×#14523513C2211yescharacteristic
9C22×C4×D131C2211yescharacteristic
10C2×#14523497C2211yescharacteristic
11#16862230—21no—
12#16862226C412yescomplemented
13#16862221C412yescomplemented
14#16862216—21no—
15C4×D524—22no—
16#16862203—22no—
17#16862203—22no—
18C2×Dic131⋊C4—21no—
19C2×(C2×D131)⋊C4—21no—
20C2×C4×Dic131C2211yescharacteristic
21#16862174—22no—
22#16862172—22no—
23(C2×D131)⋊C8—22no—
24C2×M16×C131C2211yescharacteristic
25#117×C131C2211yescharacteristic
26C2×C42×C131C2211yescharacteristic
27C23×D131D411yescharacteristic
28#14523513—21no—
29#14523513—21no—
30#14523513—21no—
31#14523513—21no—
32C2×Dic262C2×C411yescharacteristic
33C2×D524—22no—
34C2×D524C2×C411yescharacteristic
35C2×C22⋊D131—21no—
36C22×Dic131—21no—
37C22×Dic131D411yescharacteristic
38C2×C4×D131—21no—
39C2×C4×D131—21no—
40C2×C4×D131—21no—
41C2×C4×D131—21no—
42C2×C4×D131—21no—
43C2×C4×D131C2×C412yes—
44C2×C4×D131D412yes—
45#14523497—21no—
46#14523497—21no—
47C2×C131⋊C8—21no—
48C4⋊Dic131—22no—
49Dic131⋊C4—24no—
50(C2×D131)⋊C4—24no—
51(C2×D131)⋊C4—22no—
52(C2×D131)⋊C4—22no—
53C4×Dic131—21no—
54C4×Dic131D412yescomplemented
55C22×C4×C131—21no—
56C22×C4×C131C2311yescharacteristic
57M16×C131—21no—
58M16×C131—21no—
59C2×C8×C131—21no—
60C4⋊C4×C131C2×C412yes—
61C22⋊C4×C131—21no—
62C42×C131—21no—
63C42×C131—21no—
64C42×C131D412yescomplemented
65C22×D131—21no—
66C22×D131—21no—
67C22×D131—21no—
68C22×D131—42no—
69C22×D131C22⋊C412yes—
70Dic262—21no—
71Dic262—21no—
72D524—21no—
73D524—21no—
74D524—44no—
75C22⋊D131—41no—
76C22⋊D131—41no—
77C2×Dic131—21no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—22no—
81C2×Dic131—42no—
82C2×Dic131C22⋊C412yes—
83C4×D131—22no—
84C4×D131—22no—
85C4×D131—22no—
86C4×D131—22no—
87C4×D131—41no—
88C4×D131—41no—
89C131⋊C8—41no—
90C23×C131C2×D411yescharacteristic
91C2×C4×C131—22no—
92C2×C4×C131—22no—
93C2×C4×C131—42no—
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×C131C2×C22⋊C411yescharacteristic
103C22×C131#12011yescharacteristic
104C22×C131#12311yescharacteristic
105C2×D131—22no—
106C2×D131—22no—
107C2×D131—22no—
108C2×D131—42no—
109C2×D131—42no—
110C2×D131—41no—
111C2×D131—42no—
112C4×C131—42no—
113C4×C131—42no—
114C4×C131C4×D411yescharacteristic
115C4×C131C4×D411yescharacteristic
116C4×C131#12111yescharacteristic
117C4×C131#12111yescharacteristic
118Dic131—22no—
119Dic131—42no—
120Dic131—41no—
121C2×C131—21no—
122C2×C131—21no—
123C2×C131#38611yescharacteristic
124C2×C131#42112yes—
125D131—42no—
126D131—81no—
127C131#189711yescharacteristic, complemented, Sylow
128#1897—1311nomaximal, Sylow
129#525—1311no—
130C2×#117—1311no—
131C2×#104—1311no—
132C2×#104—1311no—
133#407—1311no—
134#339—1311no—
135#339D13111yescharacteristic, complemented
136C2×#41—1311no—
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#117C2×D13111yescharacteristic
148C2×C4⋊C4—2621no—
149C2×C22⋊C4—2621no—
150C2×C42—1311no—
151C2×C42C2×D13111yescharacteristic
152#104—2622no—
153#104—2622no—
154C22⋊C8—2622no—
155C24—1311no—
156#41—2621no—
157#41—2621no—
158#41—2621no—
159#41—2621no—
160C2×Q8—1311no—
161C2×D4—2622no—
162C2×D4—2621no—
163C2×D4—1311no—
164C22×C4—21no—
165C22×C4—2621no—
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⋊C4C4×D13112yes—
184C22⋊C4—21no—
185C22⋊C4—2624no—
186C22⋊C4—2622no—
187C22⋊C4—2622no—
188C42—21no—
189C42—21no—
190C42—2621no—
191C42—1312no—
192C42D52412yescomplemented
193C23—2621no—
194C23—2621no—
195C23—2621no—
196C23—5242no—
197C23—1312no—
198C23D4×D13111yescharacteristic
199Q8—2621no—
200Q8—2621no—
201D4—2621no—
202D4—2621no—
203D4—5244no—
204D4—5241no—
205D4—5241no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—2621no—
210C2×C4—2621no—
211C2×C4—2622no—
212C2×C4—2622no—
213C2×C4—2622no—
214C2×C4—2622no—
215C2×C4—2622no—
216C2×C4—2621no—
217C2×C4—5242no—
218C2×C4—5241no—
219C2×C4—5241no—
220C2×C4—1312no—
221C2×C4C2×C4×D13111yescharacteristic
222C2×C4D4×D13111yescharacteristic
223C2×C4D4×D13111yescharacteristic
224C2×C4C2×D52411yescharacteristic
225C2×C4#1452351311yescharacteristic
226C2×C4#1452351411yescharacteristic
227C8—41no—
228C8—5241no—
229C22—22no—
230C22—2622no—
231C22—2622no—
232C22—2622no—
233C22—5242no—
234C22—5242no—
235C22—5241no—
236C22—5242no—
237C22C22⋊C4×D13111yescharacteristic
238C22#1686221611yescharacteristic
239C22#1686223111yescharacteristic
240C4—42no—
241C4—42no—
242C4—2622no—
243C4—5242no—
244C4—5241no—
245C4#1686220811yescharacteristic
246C4C4×D52411yescharacteristic
247C4#1686221911yescharacteristic
248C4#1686222111yescharacteristic
249C2—21no—
250C2—21no—
251C2—5242no—
252C2—10481no—
253C2#2255319011yescharacteristic
254C2#2255332712yes—
subgroupG/H
centerC2×C4D4×D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#339×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553824C2
cosocleC22×C4×C131C23
hypercenter#339D131
hypocenterC131#1897
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#16862203C2semidirect
3. C2×#16862174C2semidirect
4. C2×#16862172C2semidirect
5. #22553275C2semidirect
6. #22553071C2semidirect
7. #339×C131C2semidirect
12. #16862226C4semidirect
13. #16862221C4semidirect
54. C4×Dic131D4semidirect
64. C42×C131D4semidirect
127. C131#1897semidirect
135. #339D131semidirect
192. C42D524semidirect

Maximal subgroups

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

subgroupindex
#225538242
C2×#168622032
C2×#168621742
C2×#168621722
#225532752
#225530712
#339×C1312
#1897131