Finite Group Databasebeta

← #30670413

Subgroups — #30670413

Order 16768.

484 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#16862222C211yesmaximal, characteristic
2C2×#16862204C211yesmaximal, characteristic
3#22553241C214yesmaximal
4#22553219C211yesmaximal, characteristic, complemented
5#22553194C214yesmaximal, complemented
6#22553181C211yesmaximal, characteristic, complemented
7#22553178C211yesmaximal, characteristic, complemented
8C4×(C2×D131)⋊C4C211yesmaximal, characteristic
9#391×C131C211yesmaximal, characteristic, complemented
10C22×C4×D131C2211yescharacteristic
11#16862222—24no—
12#16862204—24no—
13C2×C4⋊Dic131C2211yescharacteristic
14C2×C4⋊Dic131C2211yescharacteristic
15C2×Dic131⋊C4C2214yes—
16C2×Dic131⋊C4C2212yes—
17C2×(C2×D131)⋊C4C2212yes—
18C2×(C2×D131)⋊C4C2212yes—
19C2×(C2×D131)⋊C4C2212yes—
20C2×C4×Dic131C2211yescharacteristic
21C2×C4×Dic131C2212yes—
22C131⋊#95C2212yes—
23C131⋊#95C2214yes—
24C131⋊#95C2214yes—
25C22×Q8×C131C2211yescharacteristic
26C2×C4⋊C4×C131C2211yescharacteristic
27C2×C42×C131C2211yescharacteristic
28#95×C131C2214yes—
29C23×D131C2311yescharacteristic
30C22×Dic131C2312yes—
31C22×Dic131C2312yes—
32C22×Dic131C2311yescharacteristic
33C22×Dic131C2312yes—
34C2×C4×D131—22no—
35C2×C4×D131—22no—
36C2×C4×D131D414yes—
37C4⋊Dic131—22no—
38C4⋊Dic131—22no—
39Dic131⋊C4—24no—
40Dic131⋊C4—28no—
41(C2×D131)⋊C4—24no—
42(C2×D131)⋊C4—24no—
43(C2×D131)⋊C4—24no—
44C4×Dic131—24no—
45C4×Dic131—22no—
46C2×Q8×C131—22no—
47C2×Q8×C131—22no—
48C2×Q8×C131—22no—
49C22×C4×C131C2312yes—
50C22×C4×C131C2312yes—
51C22×C4×C131C2312yes—
52C22×C4×C131C2311yescharacteristic
53C4⋊C4×C131—22no—
54C42×C131—22no—
55C22×D131—22no—
56C22×D131—21no—
57C22×D131—21no—
58C22×D131—22no—
59C22×D131C2×D412yes—
60C2×Dic131—22no—
61C2×Dic131—21no—
62C2×Dic131—21no—
63C2×Dic131—22no—
64C2×Dic131—22no—
65C2×Dic131—22no—
66C2×Dic131—24no—
67C2×Dic131—24no—
68C2×Dic131C2×D412yes—
69C2×Dic131#4114yes—
70C4×D131—28no—
71C23×C131C2411yescharacteristic
72Q8×C131—44no—
73C2×C4×C131—22no—
74C2×C4×C131—22no—
75C2×C4×C131—22no—
76C2×C4×C131—22no—
77C2×C4×C131—22no—
78C2×C4×C131—22no—
79C2×C4×C131—24no—
80C2×C4×C131—22no—
81C2×C4×C131C2×D412yes—
82C2×C4×C131#4112yes—
83C2×C4×C131#4112yes—
84C22×C131C22×D411yescharacteristic
85C22×C131C2×#4112yes—
86C22×C131C2×#4111yescharacteristic
87C22×C131C2×#4111yescharacteristic
88C22×C1312−1+412yes—
89C2×D131—22no—
90C2×D131—22no—
91C2×D131—22no—
92C2×D131—42no—
93C2×D131—42no—
94C4×C131—42no—
95C4×C131—42no—
96C4×C131—42no—
97C4×C131#12314yes—
98Dic131—22no—
99Dic131—24no—
100Dic131—42no—
101Dic131—42no—
102C2×C131C2×#12311yescharacteristic
103C2×C131#53211yescharacteristic
104C2×C131#53611yescharacteristic
105C2×C131#55212yes—
106C2×C131#55612yes—
107D131—42no—
108C131#249911yescharacteristic, complemented, Sylow
109#2499—1311nomaximal, Sylow
110C2×C22⋊Q8—1311no—
111C2×#117—1311no—
112#397—1314no—
113#391—1311no—
114#391D13111yescharacteristic, complemented
115#387—1314no—
116#384—1311no—
117#382—1311no—
118C4×C22⋊C4—1311no—
119C22×Q8C2×D13111yescharacteristic
120C23×C4—1311no—
121C22⋊Q8—2624no—
122#117—2624no—
123C2×C4⋊C4—1311no—
124C2×C4⋊C4—1311no—
125C2×C4⋊C4—1312no—
126C2×C4⋊C4—1314no—
127C2×C4⋊C4C2×D13111yescharacteristic
128C2×C22⋊C4—1312no—
129C2×C22⋊C4—1312no—
130C2×C22⋊C4—1312no—
131C2×C42—1312no—
132C2×C42—1311no—
133C2×C42C2×D13111yescharacteristic
134#95—1312no—
135#95—1314no—
136#95—1314no—
137#95C2×D13114yes—
138C24—1311no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C22×C4—2622no—
143C22×C4—2622no—
144C22×C4—1312no—
145C22×C4—1312no—
146C22×C4—1312no—
147C22×C4—1311no—
148C22×C4—1314no—
149C22×C4C22×D13112yes—
150C22×C4C22×D13112yes—
151C22×C4C22×D13112yes—
152C22×C4C22×D13111yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—2628no—
155C4⋊C4—2624no—
156C4⋊C4—2622no—
157C4⋊C4—2622no—
158C22⋊C4—2624no—
159C22⋊C4—2624no—
160C22⋊C4—2624no—
161C42—22no—
162C42—2624no—
163C42—2622no—
164C23—2622no—
165C23—2621no—
166C23—2621no—
167C23—2622no—
168C23—1312no—
169C23C23×D13111yescharacteristic
170Q8—44no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—2621no—
180C2×C4—2622no—
181C2×C4—2621no—
182C2×C4—2622no—
183C2×C4—2628no—
184C2×C4—2622no—
185C2×C4—2622no—
186C2×C4—2624no—
187C2×C4—2624no—
188C2×C4—1312no—
189C2×C4—1314no—
190C2×C4D4×D13112yes—
191C2×C4#1452351212yes—
192C2×C4#1452351312yes—
193C22—2622no—
194C22—2622no—
195C22—2622no—
196C22—5242no—
197C22—5242no—
198C22C2×D4×D13111yescharacteristic
199C22#41×D13112yes—
200C22C2×#1452351211yescharacteristic
201C22C2×#1452351311yescharacteristic
202C22#1686229412yes—
203C4—42no—
204C4—42no—
205C4—42no—
206C4—2624no—
207C4—2622no—
208C4—5242no—
209C4—5242no—
210C4#1686222814yes—
211C2—5242no—
212C2C2×#1686222811yescharacteristic
213C2#2255386111yescharacteristic
214C2#2255388711yescharacteristic
215C2#2255400312yes—
216C2#2255403312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#391×C131C2
prime radicalC23×D131C23
hypercenter#391D131
hypocenterC131#2499
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #22553219C2semidirect
5. #22553194C2semidirect
6. #22553181C2semidirect
7. #22553178C2semidirect
9. #391×C131C2semidirect
108. C131#2499semidirect
114. #391D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622222
C2×#168622042
#225532412
#225532192
#225531942
#225531812
#225531782
C4×(C2×D131)⋊C42
#391×C1312
#2499131