Finite Group Databasebeta

← #30664942

Subgroups — #30664942

Order 16768.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#22553899C211yesmaximal, characteristic
2#22553470C211yesmaximal, characteristic, complemented
3#22553450C211yesmaximal, characteristic, complemented
4#22553225C211yesmaximal, characteristic, complemented
5#22553048C211yesmaximal, characteristic, complemented
6#22553035C211yesmaximal, characteristic, complemented
7#327×C131C211yesmaximal, characteristic, complemented
8C22×Dic262C2211yescharacteristic
9C2×Dic524—21no—
10C2×#14523501—21no—
11#16862241—21no—
12#16862232—21no—
13#16862230—21no—
14#16862225C2211yescharacteristic
15#16862218—21no—
16#16862214—21no—
17D4×Dic131—21no—
18C2×(C2×D131)⋊C4—21no—
19C2×(C2×D131)⋊C4C2211yescharacteristic
20#16862171—21no—
21#16862169—21no—
22C22⋊(C131⋊C8)C2211yescharacteristic
23C131⋊#95—21no—
24#121×C131C2211yescharacteristic
25C22⋊C8×C131C2211yescharacteristic
26#95×C131C2211yescharacteristic
27C23×D131D411yescharacteristic
28C2×Dic262—21no—
29C2×Dic262—21no—
30C2×Dic262—42no—
31C2×Dic262D411yescharacteristic
32C2×C22⋊D131—21no—
33C2×C22⋊D131—21no—
34C22×Dic131—21no—
35C22×Dic131—21no—
36C22×Dic131D411yescharacteristic
37Dic524—42no—
38#14523501—42no—
39C2×C131⋊C8—21no—
40C4⋊Dic131—21no—
41C4⋊Dic131D411yescharacteristic
42Dic131⋊C4—21no—
43C22⋊Dic131—21no—
44C22⋊Dic131—21no—
45(C2×D131)⋊C4—21no—
46(C2×D131)⋊C4—21no—
47(C2×D131)⋊C4—21no—
48(C2×D131)⋊C4—21no—
49(C2×D131)⋊C4—42no—
50C4×Dic131—21no—
51C2×D4×C131—21no—
52C2×D4×C131D411yescharacteristic
53C22×C4×C131—21no—
54C22×C4×C131C2311yescharacteristic
55C2×C8×C131—21no—
56C4⋊C4×C131D411yescharacteristic
57C22⋊C4×C131—21no—
58C22×D131—21no—
59C22×D131—21no—
60C22×D131—21no—
61C22×D131—41no—
62C22×D131—41no—
63Dic262—22no—
64Dic262—22no—
65Dic262—41no—
66Dic262—41no—
67Dic262—41no—
68C22⋊D131—81no—
69C22⋊D131—81no—
70C2×Dic131—22no—
71C2×Dic131—21no—
72C2×Dic131—21no—
73C2×Dic131—21no—
74C2×Dic131—41no—
75C2×Dic131—41no—
76C2×Dic131—41no—
77C2×Dic131—81no—
78C131⋊C8—41no—
79C23×C131—21no—
80C23×C131C2×D411yescharacteristic
81D4×C131—41no—
82D4×C131—41no—
83D4×C131—41no—
84C2×C4×C131—21no—
85C2×C4×C131—41no—
86C2×C4×C131—41no—
87C2×C4×C131—41no—
88C2×C4×C131—41no—
89C2×C4×C131C2×D411yescharacteristic
90C2×C4×C131C2×D411yescharacteristic
91C8×C131—41no—
92C22×C131—21no—
93C22×C131—21no—
94C22×C131—41no—
95C22×C131—41no—
96C22×C131—41no—
97C22×C131#12011yescharacteristic
98C22×C131#13511yescharacteristic
99C22×C131#13711yescharacteristic
100C2×D131—42no—
101C2×D131—41no—
102C2×D131—41no—
103C2×D131—41no—
104C2×D131—81no—
105C4×C131—21no—
106C4×C131—21no—
107C4×C131—41no—
108C4×C131—81no—
109Dic131—41no—
110Dic131—41no—
111Dic131—42no—
112Dic131—41no—
113C2×C131—21no—
114C2×C131—21no—
115C2×C131—81no—
116C2×C131#45211yescharacteristic
117C2×C131#45511yescharacteristic
118C2×C131B(4,𝔽2)11yescharacteristic
119D131—81no—
120C131#160611yescharacteristic, complemented, Sylow
121#1606—1311nomaximal, Sylow
122#538—1311no—
123#451—1311no—
124#448—1311no—
125#393—1311no—
126#328—1311no—
127#327D13111yescharacteristic, complemented
128#323—1311no—
129C22×Q8—1311no—
130C2×Q16—2621no—
131C2×SD16—2621no—
132#126—2621no—
133#124—2621no—
134#123—2621no—
135C22⋊Q8—1311no—
136#121—2621no—
137#121C2×D13111yescharacteristic
138#120—2621no—
139C4×D4—2621no—
140C2×C22⋊C4—2621no—
141C2×C22⋊C4—1311no—
142#103—2621no—
143#103—2621no—
144C22⋊C8—1311no—
145C22⋊C8C2×D13111yescharacteristic
146#95—2621no—
147#95C2×D13111yescharacteristic
148C24—1311no—
149C2×Q8—2621no—
150C2×Q8—2621no—
151C2×Q8—5242no—
152C2×Q8—1311no—
153C2×D4—21no—
154C2×D4—2621no—
155C2×D4—2621no—
156C2×D4D52411yescharacteristic
157C22×C4—21no—
158C22×C4—2621no—
159C22×C4—2621no—
160C22×C4—1311no—
161C22×C4C22×D13111yescharacteristic
162Q16—5242no—
163SD16—5242no—
164C2×C8—21no—
165C2×C8—2621no—
166C4⋊C4—2621no—
167C4⋊C4—2621no—
168C4⋊C4—1311no—
169C4⋊C4D52411yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—2621no—
172C22⋊C4—2621no—
173C22⋊C4—2621no—
174C22⋊C4—2621no—
175C22⋊C4—2621no—
176C22⋊C4—2621no—
177C22⋊C4—5242no—
178C42—2621no—
179C23—21no—
180C23—2621no—
181C23—2621no—
182C23—2621no—
183C23—5241no—
184C23—5241no—
185C23D4×D13111yescharacteristic
186Q8—2622no—
187Q8—2622no—
188Q8—5241no—
189Q8—5241no—
190Q8—5241no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—10481no—
195D4—10481no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—2622no—
202C2×C4—2621no—
203C2×C4—2621no—
204C2×C4—2621no—
205C2×C4—5241no—
206C2×C4—5241no—
207C2×C4—5241no—
208C2×C4—10481no—
209C2×C4D4×D13111yescharacteristic
210C2×C4C2×D52411yescharacteristic
211C8—41no—
212C8—5241no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—5242no—
219C22—5241no—
220C22—5241no—
221C22—5241no—
222C22—10481no—
223C22#1686221611yescharacteristic
224C22#1686226511yescharacteristic
225C22#1686227511yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—81no—
230C4—5241no—
231C4—5241no—
232C4—5242no—
233C4—5241no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—10481no—
238C2#2255347511yescharacteristic
239C2#2255349111yescharacteristic
240C2#2255350011yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#327×C131C2
socleC22×C131#120
prime radical#22553899C2
hypercenter#327D131
hypocenterC131#1606
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22553470C2semidirect
3. #22553450C2semidirect
4. #22553225C2semidirect
5. #22553048C2semidirect
6. #22553035C2semidirect
7. #327×C131C2semidirect
120. C131#1606semidirect
127. #327D131semidirect

Maximal subgroups

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

subgroupindex
#225538992
#225534702
#225534502
#225532252
#225530482
#225530352
#327×C1312
#1606131