Finite Group Databasebeta

← #30671133

Subgroups — #30671133

Order 16768.

564 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16862245C211yesmaximal, characteristic, complemented
2C2×#16862208C211yesmaximal, characteristic
3#22553225C214yesmaximal, complemented
4#22553214C211yesmaximal, characteristic
5#22553213C211yesmaximal, characteristic
6#22553177C211yesmaximal, characteristic, complemented
7#22553171C214yesmaximal, complemented
8C4×Dic131⋊C4C211yesmaximal, characteristic, complemented
9#382×C131C211yesmaximal, characteristic, complemented
10C22×Dic262C2212yes—
11C22×D524C2211yescharacteristic
12C22×C4×D131C2212yes—
13#16862245—24no—
14#16862208—24no—
15C2×C4⋊Dic131C2211yescharacteristic
16C2×Dic131⋊C4C2212yes—
17C2×Dic131⋊C4C2212yes—
18C2×(C2×D131)⋊C4C2212yes—
19C2×(C2×D131)⋊C4C2214yes—
20C2×(C2×D131)⋊C4C2214yes—
21C2×C4×Dic131C2211yescharacteristic
22C2×C4×Dic131C2211yescharacteristic
23C131⋊#95C2212yes—
24C131⋊#95C2214yes—
25C2×C4⋊C4×C131C2211yescharacteristic
26C2×C4⋊C4×C131C2211yescharacteristic
27C2×C42×C131C2211yescharacteristic
28#95×C131C2214yes—
29C23×D131C2312yes—
30C2×Dic262—24no—
31C2×Dic262—44no—
32C2×D524—22no—
33C2×D524—22no—
34C2×D524Q814yescomplemented
35C22×Dic131C2312yes—
36C22×Dic131C2312yes—
37C22×Dic131C2311yescharacteristic
38C22×Dic131C2311yescharacteristic
39C2×C4×D131—24no—
40C2×C4×D131—28no—
41C4⋊Dic131—22no—
42Dic131⋊C4—24no—
43Dic131⋊C4—24no—
44(C2×D131)⋊C4—24no—
45(C2×D131)⋊C4—44no—
46(C2×D131)⋊C4—44no—
47C4×Dic131—22no—
48C4×Dic131D414yes—
49C22×C4×C131C2312yes—
50C22×C4×C131C2312yes—
51C22×C4×C131C2312yes—
52C22×C4×C131C2311yescharacteristic
53C4⋊C4×C131—22no—
54C4⋊C4×C131—22no—
55C42×C131—22no—
56C22×D131—22no—
57C22×D131—24no—
58C22×D131—22no—
59C22×D131—24no—
60C22×D131C2×Q814yes—
61Dic262—48no—
62D524—28no—
63C2×Dic131—22no—
64C2×Dic131—22no—
65C2×Dic131—22no—
66C2×Dic131—22no—
67C2×Dic131—22no—
68C2×Dic131—24no—
69C2×Dic131C2×D412yes—
70C2×Dic131C2×D412yes—
71C2×Dic131#4114yes—
72C4×D131—48no—
73C23×C131C2411yescharacteristic
74C2×C4×C131—22no—
75C2×C4×C131—22no—
76C2×C4×C131—22no—
77C2×C4×C131—22no—
78C2×C4×C131—24no—
79C2×C4×C131—22no—
80C2×C4×C131—24no—
81C2×C4×C131C2×D412yes—
82C2×C4×C131C2×Q812yes—
83C2×C4×C131#4112yes—
84C22×C131C22×D411yescharacteristic
85C22×C131C22×Q811yescharacteristic
86C22×C131C2×#4112yes—
87C22×C131C2×#4111yescharacteristic
88C22×C131H(5,𝔽2)12yes—
89C2×D131—24no—
90C2×D131—28no—
91C2×D131—42no—
92C2×D131—42no—
93C2×D131—44no—
94C4×C131—42no—
95C4×C131—42no—
96C4×C131—42no—
97C4×C131C22⋊Q814yes—
98Dic131—22no—
99Dic131—42no—
100Dic131—42no—
101Dic131#12414yes—
102C2×C131C2×C22⋊Q811yescharacteristic
103C2×C131C2×#12411yescharacteristic
104C2×C131#53511yescharacteristic
105C2×C131#55412yes—
106C2×C131#55512yes—
107D131—44no—
108C131#256311yescharacteristic, complemented, Sylow
109#2563—1311nomaximal, Sylow
110C2×C4⋊Q8—1311no—
111C2×C4×D4—1311no—
112#393—1314no—
113#390—1311no—
114#390—1311no—
115#382—1311no—
116#382D13111yescharacteristic, complemented
117#380—1314no—
118C4×C4⋊C4—1311no—
119C22×Q8—1312no—
120C22×D4—1311no—
121C23×C4—1312no—
122C4⋊Q8—2624no—
123C4×D4—2624no—
124C2×C4⋊C4—1312no—
125C2×C4⋊C4—1311no—
126C2×C4⋊C4—1312no—
127C2×C4⋊C4C2×D13111yescharacteristic
128C2×C4⋊C4C2×D13111yescharacteristic
129C2×C22⋊C4—1312no—
130C2×C22⋊C4—1314no—
131C2×C22⋊C4—1314no—
132C2×C42—1311no—
133C2×C42—1311no—
134C2×C42C2×D13111yescharacteristic
135#95—1312no—
136#95—1314no—
137#95C2×D13114yes—
138C24—1312no—
139C2×Q8—2624no—
140C2×Q8—5244no—
141C2×D4—2622no—
142C2×D4—2622no—
143C2×D4—1314no—
144C22×C4—2624no—
145C22×C4—2628no—
146C22×C4—1312no—
147C22×C4—1312no—
148C22×C4—1311no—
149C22×C4—1311no—
150C22×C4C22×D13112yes—
151C22×C4C22×D13112yes—
152C22×C4C22×D13112yes—
153C22×C4C22×D13111yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—2624no—
157C4⋊C4—2624no—
158C4⋊C4—2622no—
159C22⋊C4—2624no—
160C22⋊C4—5244no—
161C22⋊C4—5244no—
162C42—22no—
163C42—2622no—
164C42—1314no—
165C23—2622no—
166C23—2624no—
167C23—2622no—
168C23—2624no—
169C23—1314no—
170C23C23×D13111yescharacteristic
171Q8—5248no—
172D4—2628no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—2622no—
181C2×C4—2622no—
182C2×C4—2622no—
183C2×C4—2622no—
184C2×C4—2622no—
185C2×C4—2624no—
186C2×C4—5248no—
187C2×C4—1312no—
188C2×C4—1312no—
189C2×C4—1314no—
190C2×C4D4×D13112yes—
191C2×C4Q8×D13112yes—
192C2×C4#1452351312yes—
193C22—2628no—
194C22—2624no—
195C22—5242no—
196C22—5244no—
197C22—5242no—
198C22C2×D4×D13111yescharacteristic
199C22C2×Q8×D13111yescharacteristic
200C22#41×D13112yes—
201C22C2×#1452351311yescharacteristic
202C22#1686229212yes—
203C4—42no—
204C4—42no—
205C4—42no—
206C4—2622no—
207C4—5242no—
208C4—5242no—
209C4—1314no—
210C4#1686222414yes—
211C2—5244no—
212C2C2×#1686222411yescharacteristic
213C2#124×D13111yescharacteristic
214C2#2255388011yescharacteristic
215C2#2255401812yes—
216C2#2255402512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#382×C131C2
prime radicalC22×D524C22
hypercenter#382D131
hypocenterC131#2563
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#16862245C2semidirect
3. #22553225C2semidirect
6. #22553177C2semidirect
7. #22553171C2semidirect
8. C4×Dic131⋊C4C2semidirect
9. #382×C131C2semidirect
34. C2×D524Q8semidirect
108. C131#2563semidirect
116. #382D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622452
C2×#168622082
#225532252
#225532142
#225532132
#225531772
#225531712
C4×Dic131⋊C42
#382×C1312
#2563131