Finite Group Databasebeta

← #30664939

Subgroups — #30664939

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#22553900C211yesmaximal, characteristic
2#22553468C211yesmaximal, characteristic, complemented
3#22553451C211yesmaximal, characteristic, complemented
4#22553048C211yesmaximal, characteristic, complemented
5#22553045C211yesmaximal, characteristic, complemented
6#22553036C211yesmaximal, characteristic, complemented
7#393×C131C211yesmaximal, characteristic, complemented
8C2×#14523503—21no—
9C2×#14523500—21no—
10#16862239—21no—
11#16862232—21no—
12#16862228—21no—
13#16862225C2211yescharacteristic
14#16862220C2211yescharacteristic
15#16862219—21no—
16#16862215—21no—
17C4×C22⋊D131—21no—
18#16862170—21no—
19#16862170—21no—
20C22⋊(C131⋊C8)C2211yescharacteristic
21C22⋊(C131⋊C8)C2211yescharacteristic
22C22×Q8×C131C2211yescharacteristic
23C2×C22⋊C4×C131—21no—
24C2×C22⋊C4×C131C2211yescharacteristic
25#95×C131—21no—
26#95×C131C2211yescharacteristic
27C2×Dic262D411yescharacteristic
28C2×D524D411yescharacteristic
29C2×C22⋊D131—21no—
30C2×C22⋊D131—21no—
31C2×C22⋊D131—21no—
32C2×C4×D131—21no—
33#14523503—42no—
34#14523500—42no—
35C2×C131⋊C8—21no—
36C2×C131⋊C8—21no—
37C4⋊Dic131D411yescharacteristic
38C4⋊Dic131D411yescharacteristic
39Dic131⋊C4—21no—
40Dic131⋊C4—21no—
41C22⋊Dic131—21no—
42C22⋊Dic131—21no—
43C22⋊Dic131—21no—
44(C2×D131)⋊C4—21no—
45(C2×D131)⋊C4—21no—
46C4×Dic131—21no—
47C24×C131D411yescharacteristic
48C2×Q8×C131—21no—
49C2×Q8×C131—21no—
50C2×Q8×C131—42no—
51C22×C4×C131—21no—
52C22×C4×C131—21no—
53C22×C4×C131D411yescharacteristic
54C22×C4×C131C2311yescharacteristic
55C22⋊C4×C131—21no—
56C22⋊C4×C131—21no—
57C22⋊C4×C131—42no—
58C22×D131—21no—
59Dic262—41no—
60D524—41no—
61C22⋊D131—41no—
62C22⋊D131—41no—
63C22⋊D131—81no—
64C22⋊D131—81no—
65C2×Dic131—21no—
66C2×Dic131—21no—
67C2×Dic131—21no—
68C4×D131—81no—
69C131⋊C8—41no—
70C131⋊C8—41no—
71C23×C131—21no—
72C23×C131—21no—
73C23×C131—21no—
74C23×C131—41no—
75C23×C131—41no—
76C23×C131C2×D411yescharacteristic
77Q8×C131—22no—
78Q8×C131—22no—
79Q8×C131—41no—
80Q8×C131—41no—
81C2×C4×C131—22no—
82C2×C4×C131—21no—
83C2×C4×C131—41no—
84C2×C4×C131—41no—
85C2×C4×C131—41no—
86C2×C4×C131—41no—
87C2×C4×C131—41no—
88C2×C4×C131—41no—
89C2×C4×C131—41no—
90C2×C4×C131C2×D411yescharacteristic
91C2×C4×C131C2×D411yescharacteristic
92C22×C131—21no—
93C22×C131—21no—
94C22×C131—42no—
95C22×C131—41no—
96C22×C131—41no—
97C22×C131—41no—
98C22×C131—81no—
99C22×C131#12011yescharacteristic
100C22×C131#13511yescharacteristic
101C22×C131#13711yescharacteristic
102C2×D131—41no—
103C2×D131—41no—
104C2×D131—41no—
105C4×C131—21no—
106C4×C131—21no—
107C4×C131—41no—
108C4×C131—42no—
109C4×C131—81no—
110Dic131—41no—
111Dic131—41no—
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#393D13111yescharacteristic, complemented
126#328—1311no—
127#327—1311no—
128#323—1311no—
129C22×Q8C2×D13111yescharacteristic
130C2×Q16—2621no—
131C2×SD16—2621no—
132#126—2621no—
133#124—2621no—
134#123—2621no—
135C22⋊Q8—1311no—
136#121—2621no—
137#121—1311no—
138#120—2621no—
139C4×D4—2621no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4C2×D13111yescharacteristic
142#103—2621no—
143#103—2621no—
144C22⋊C8—1311no—
145C22⋊C8—1311no—
146#95—21no—
147#95C2×D13111yescharacteristic
148C24C22⋊D13111yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—1311no—
153C2×D4—2621no—
154C2×D4—2621no—
155C2×D4—2621no—
156C2×D4—1311no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—2621no—
160C22×C4C22⋊D13111yescharacteristic
161C22×C4C22×D13111yescharacteristic
162Q16—5242no—
163SD16—5242no—
164C2×C8—2621no—
165C2×C8—2621no—
166C4⋊C4—2621no—
167C4⋊C4—2621no—
168C4⋊C4—1311no—
169C4⋊C4—1311no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—42no—
173C22⋊C4—2621no—
174C22⋊C4—2621no—
175C22⋊C4—2621no—
176C22⋊C4—2621no—
177C22⋊C4—2621no—
178C42—2621no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—2621no—
185C23C2×C22⋊D13111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—5241no—
191D4—5241no—
192D4—5241no—
193D4—5241no—
194D4—10481no—
195D4—10481no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—2621no—
206C2×C4—2621no—
207C2×C4—2621no—
208C2×C4—10481no—
209C2×C4D4×D13111yescharacteristic
210C2×C4D4×D13111yescharacteristic
211C8—5241no—
212C8—5241no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—5241no—
221C22—5241no—
222C22—5241no—
223C22#1686221411yescharacteristic
224C22#1686226611yescharacteristic
225C22#1686227711yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—81no—
231C4—5241no—
232C4—5241no—
233C4—5241no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—10481no—
238C2#2255347211yescharacteristic
239C2#2255348911yescharacteristic
240C2#2255349811yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#393×C131C2
socleC22×C131#120
prime radical#22553900C2
hypercenter#393D131
hypocenterC131#1606
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #22553468C2semidirect
3. #22553451C2semidirect
4. #22553048C2semidirect
5. #22553045C2semidirect
6. #22553036C2semidirect
7. #393×C131C2semidirect
120. C131#1606semidirect
125. #393D131semidirect

Maximal subgroups

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

subgroupindex
#225539002
#225534682
#225534512
#225530482
#225530452
#225530362
#393×C1312
#1606131