Finite Group Databasebeta

← #30664882

Subgroups — #30664882

Order 16768.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#22553937C211yesmaximal, characteristic, complemented
2#22553463C211yesmaximal, characteristic, complemented
3#22553220C211yesmaximal, characteristic
4#22553050C211yesmaximal, characteristic, complemented
5#22553049C211yesmaximal, characteristic, complemented
6#22553035C211yesmaximal, characteristic, complemented
7#450×C131C211yesmaximal, characteristic, complemented
8C22×C22⋊D131—21no—
9C2×#14523501—21no—
10#16862226—21no—
11#16862225C2211yescharacteristic
12#16862224—21no—
13#16862224—21no—
14#16862223—21no—
15#16862222—21no—
16C2×C22⋊Dic131—21no—
17C2×(C2×D131)⋊C4C2211yescharacteristic
18C22⋊C4×D131—21no—
19#16862167—21no—
20C22⋊(C131⋊C8)C2211yescharacteristic
21C131⋊#95C2211yescharacteristic
22C22×D4×C131C2211yescharacteristic
23C2×SD16×C131—21no—
24C22⋊Q8×C131C2211yescharacteristic
25#102×C131—21no—
26C22⋊C8×C131C2211yescharacteristic
27C23×D131D411yescharacteristic
28C2×Dic262—21no—
29C2×Dic262D411yescharacteristic, complemented
30C2×C22⋊D131—42no—
31C2×C22⋊D131—42no—
32C2×C22⋊D131—42no—
33C22×Dic131—21no—
34C22×Dic131D411yescharacteristic
35C2×C4×D131—21no—
36C2×C4×D131—21no—
37#14523501—42no—
38C2×C131⋊C8—21no—
39C4⋊Dic131—21no—
40C4⋊Dic131D411yescharacteristic, complemented
41Dic131⋊C4—21no—
42Dic131⋊C4—21no—
43Dic131⋊C4—21no—
44C22⋊Dic131—21no—
45C22⋊Dic131—42no—
46(C2×D131)⋊C4—21no—
47(C2×D131)⋊C4—21no—
48(C2×D131)⋊C4—21no—
49(C2×D131)⋊C4—21no—
50C24×C131—21no—
51C2×Q8×C131D411yescharacteristic, complemented
52C2×D4×C131—21no—
53C2×D4×C131—21no—
54C2×D4×C131—42no—
55C22×C4×C131C2311yescharacteristic
56SD16×C131—42no—
57C2×C8×C131—21no—
58C4⋊C4×C131—21no—
59C4⋊C4×C131D411yescharacteristic, complemented
60C22⋊C4×C131—21no—
61C22×D131—21no—
62C22×D131—21no—
63C22×D131—41no—
64C22×D131—41no—
65C22×D131SD1612yescomplemented
66Dic262—41no—
67Dic262—81no—
68C22⋊D131—48no—
69C2×Dic131—22no—
70C2×Dic131—21no—
71C2×Dic131—21no—
72C2×Dic131—21no—
73C2×Dic131—41no—
74C2×Dic131—41no—
75C2×Dic131—41no—
76C4×D131—81no—
77C4×D131—81no—
78C131⋊C8—41no—
79C23×C131—22no—
80C23×C131—41no—
81C23×C131—41no—
82C23×C131—41no—
83C23×C131—41no—
84C23×C131—41no—
85C23×C131—41no—
86C23×C131C2×D411yescharacteristic
87Q8×C131—41no—
88D4×C131—22no—
89D4×C131—22no—
90D4×C131—41no—
91D4×C131—41no—
92C2×C4×C131—21no—
93C2×C4×C131—21no—
94C2×C4×C131—41no—
95C2×C4×C131C2×D411yescharacteristic
96C2×C4×C131C2×D411yescharacteristic
97C8×C131—41no—
98C22×C131—21no—
99C22×C131—21no—
100C22×C131—42no—
101C22×C131—42no—
102C22×C131—42no—
103C22×C131—81no—
104C22×C131—81no—
105C22×C131—81no—
106C22×C131—81no—
107C22×C131#12011yescharacteristic
108C22×C131C2×SD1611yescharacteristic
109C22×C131#13711yescharacteristic
110C2×D131—22no—
111C2×D131—41no—
112C2×D131—41no—
113C2×D131—42no—
114C2×D131—81no—
115C4×C131—21no—
116C4×C131—21no—
117C4×C131—41no—
118C4×C131—41no—
119Dic131—42no—
120Dic131—41no—
121Dic131—41no—
122Dic131—41no—
123C2×C131—21no—
124C2×C131—21no—
125C2×C131—82no—
126C2×C131#44811yescharacteristic
127C2×C131#45311yescharacteristic
128C2×C131B(4,𝔽2)11yescharacteristic
129D131—42no—
130C131#159611yescharacteristic, complemented, Sylow
131#1596—1311nomaximal, Sylow
132#543—1311no—
133#450—1311no—
134#450D13111yescharacteristic, complemented
135#392—1311no—
136#328—1311no—
137#328—1311no—
138#323—1311no—
139C22×D4—2621no—
140C22×D4C2×D13111yescharacteristic
141C2×SD16—21no—
142C2×SD16—2621no—
143C22⋊Q8—2621no—
144C22⋊Q8—2621no—
145C22⋊Q8—2621no—
146C22⋊Q8—2621no—
147C22⋊Q8—2621no—
148C22⋊Q8—1311no—
149C22⋊Q8C2×D13111yescharacteristic
150C2×C22⋊C4—2621no—
151C2×C22⋊C4—2621no—
152C2×C22⋊C4—1311no—
153#102—21no—
154#102—2621no—
155C22⋊C8—1311no—
156C22⋊C8C2×D13111yescharacteristic
157#95—1311no—
158C24—21no—
159C24—1311no—
160C2×Q8—2621no—
161C2×Q8—1311no—
162C2×Q8C22⋊D13111yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—5242no—
167C2×D4—5242no—
168C2×D4—5242no—
169C22×C4—2621no—
170C22×C4—2621no—
171C22×C4—2621no—
172C22×C4—1311no—
173C22×C4C22×D13111yescharacteristic
174SD16—42no—
175SD16—5242no—
176C2×C8—21no—
177C2×C8—2621no—
178C4⋊C4—21no—
179C4⋊C4—2621no—
180C4⋊C4—2621no—
181C4⋊C4—2621no—
182C4⋊C4—2621no—
183C4⋊C4—1311no—
184C4⋊C4C22⋊D13111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—2621no—
187C22⋊C4—2621no—
188C22⋊C4—2621no—
189C22⋊C4—2621no—
190C22⋊C4—2621no—
191C22⋊C4—5242no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—2621no—
200C23—2621no—
201C23—5241no—
202C23—5241no—
203C23—1312no—
204C23D4×D13111yescharacteristic
205Q8—41no—
206Q8—5241no—
207Q8—10481no—
208D4—22no—
209D4—22no—
210D4—41no—
211D4—41no—
212D4—5248no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—2622no—
217C2×C4—2621no—
218C2×C4—2621no—
219C2×C4—2621no—
220C2×C4—5241no—
221C2×C4—5241no—
222C2×C4—5241no—
223C2×C4—10481no—
224C2×C4—10481no—
225C2×C4D4×D13111yescharacteristic
226C2×C4C2×C22⋊D13111yescharacteristic
227C8—41no—
228C8—5241no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—2622no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—5242no—
240C22—5241no—
241C22—5241no—
242C22—10481no—
243C22#1686221411yescharacteristic
244C22SD16×D13111yescharacteristic
245C22#1686227511yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—5242no—
251C4—5241no—
252C4—5241no—
253C4—5241no—
254C2—21no—
255C2—21no—
256C2—82no—
257C2—5242no—
258C2#2255344911yescharacteristic
259C2#2255347811yescharacteristic
260C2#2255349811yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#450×C131C2
socleC22×C131#120
prime radical#22553220C2
hypercenter#450D131
hypocenterC131#1596
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22553937C2semidirect
2. #22553463C2semidirect
4. #22553050C2semidirect
5. #22553049C2semidirect
6. #22553035C2semidirect
7. #450×C131C2semidirect
29. C2×Dic262D4semidirect
40. C4⋊Dic131D4semidirect
51. C2×Q8×C131D4semidirect
59. C4⋊C4×C131D4semidirect
65. C22×D131SD16semidirect
130. C131#1596semidirect
134. #450D131semidirect
162. C2×Q8C22⋊D131semidirect
184. C4⋊C4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
#225539372
#225534632
#225532202
#225530502
#225530492
#225530352
#450×C1312
#1596131