Finite Group Databasebeta

← #392×D131 #30669155

Subgroups — #392×D131 #30669155

Order 16768.

1812 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 223 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
1C22×D4×D131C213yesmaximal, complemented
2C2×C22⋊C4×D131C213yesmaximal, complemented
3#22553222C211yesmaximal, characteristic, complemented
4#22553221C213yesmaximal, complemented
5#22553220C213yesmaximal, complemented
6#95×D131C211yesmaximal, characteristic, complemented
7#392×C131C211yesmaximal, characteristic, complemented
8C24×D131C2213yes—
9C22×D524C2213yescomplemented
10C22×C22⋊D131C2216yescomplemented
11C2×D4×D131—224no—
12C2×D4×D131—212no—
13C22×C4×D131C2211yescharacteristic
14C22×C4×D131C2213yes—
15C2×C22⋊Dic131C2213yescomplemented
16C2×(C2×D131)⋊C4C2213yescomplemented
17C2×(C2×D131)⋊C4C2213yescomplemented
18C22⋊C4×D131—212no—
19C131⋊#95C2213yescomplemented
20C22×D4×C131C2213yescomplemented
21C2×C22⋊C4×C131C2213yescomplemented
22#95×C131C2211yescharacteristic, complemented
23C23×D131—212no—
24C23×D131—212no—
25C23×D131—26no—
26C23×D131—26no—
27C23×D131D4112yescomplemented
28C23×D131C2313yes—
29C23×D131C2311yescharacteristic
30C2×D524—212no—
31C2×D524—26no—
32C2×C22⋊D131—212no—
33C2×C22⋊D131—212no—
34C2×C22⋊D131—212no—
35D4×D131—296no—
36C22×Dic131C2311yescharacteristic
37C22×Dic131C2313yes—
38C2×C4×D131—212no—
39C2×C4×D131—26no—
40C2×C4×D131D4112yescomplemented
41C22⋊Dic131—26no—
42(C2×D131)⋊C4—26no—
43(C2×D131)⋊C4—26no—
44C24×C131C2313yes—
45C2×D4×C131—212no—
46C2×D4×C131—26no—
47C22×C4×C131C2311yescharacteristic
48C22×C4×C131C2313yes—
49C22⋊C4×C131—26no—
50C22×D131—23no—
51C22×D131—26no—
52C22×D131—26no—
53C22×D131—23no—
54C22×D131—248no—
55C22×D131—224no—
56C22×D131—412no—
57C22×D131—424no—
58C22×D131—412no—
59C22×D131C2×D416yes—
60C22×D131C2×D416yes—
61C22×D131C2×D416yescomplemented
62C22×D131#4112yes—
63D524—224no—
64C22⋊D131—248no—
65C2×Dic131—26no—
66C2×Dic131—23no—
67C2×Dic131C2×D416yescomplemented
68C4×D131—224no—
69C4×D131—44no—
70C23×C131—23no—
71C23×C131—26no—
72C23×C131—26no—
73C23×C131—23no—
74C23×C131C2×D416yescomplemented
75C23×C131C2411yescharacteristic
76D4×C131—224no—
77C2×C4×C131—26no—
78C2×C4×C131—23no—
79C2×C4×C131C2×D416yescomplemented
80C22×C131—212no—
81C22×C131—26no—
82C22×C131—43no—
83C22×C131—46no—
84C22×C131—43no—
85C22×C131C22×D413yes—
86C22×C131C22×D413yes—
87C22×C131C2×#4111yescharacteristic
88C2×D131—212no—
89C2×D131—26no—
90C2×D131—43no—
91C2×D131—46no—
92C2×D131—43no—
93C2×D131—448no—
94C2×D131#120112yes—
95C2×D131#121112yes—
96C2×D131#12714yes—
97C4×C131—26no—
98C4×C131—41no—
99Dic131—26no—
100Dic131—41no—
101C2×C131—46no—
102C2×C131C2×#12013yes—
103C2×C131C2×#12113yes—
104C2×C131C2×#12711yescharacteristic
105D131—46no—
106D131#39218yesdirect product factor
107C131C2×#39211yescharacteristic, complemented, Sylow
108C2×#392—1311nomaximal, Sylow
109C23×D4—1313no—
110C22×C22⋊C4—1313no—
111#392—1313no—
112#392—1313no—
113#392—1311no—
114#392D13111yescharacteristic, direct product factor
115C2×#95—1311no—
116C25—1313no—
117C22×D4—26224no—
118C22×D4—26212no—
119C22×D4—1316no—
120C22×D4—1313no—
121C22×D4C2×D13113yescomplemented
122C23×C4—1311no—
123C23×C4—1313no—
124C2×C22⋊C4—26212no—
125C2×C22⋊C4—1313no—
126C2×C22⋊C4—1313no—
127C2×C22⋊C4—1313no—
128C2×C22⋊C4C2×D13113yescomplemented
129#95—1313no—
130#95C2×D13111yescharacteristic, complemented
131C24—2626no—
132C24—26212no—
133C24—26212no—
134C24—2626no—
135C24—13112no—
136C24—1313no—
137C24—1311no—
138C24C22×D13113yes—
139C2×D4—212no—
140C2×D4—26no—
141C2×D4—26212no—
142C2×D4—26296no—
143C2×D4—26212no—
144C2×D4—26212no—
145C2×D4—26212no—
146C2×D4—2626no—
147C22×C4—26212no—
148C22×C4—2626no—
149C22×C4—1311no—
150C22×C4—1313no—
151C22×C4—13112no—
152C22×C4C22×D13111yescharacteristic
153C22×C4C22×D13113yes—
154C22⋊C4—26no—
155C22⋊C4—2626no—
156C22⋊C4—2626no—
157C22⋊C4—2626no—
158C23—23no—
159C23—26no—
160C23—26no—
161C23—23no—
162C23—26248no—
163C23—26224no—
164C23—2623no—
165C23—2626no—
166C23—2626no—
167C23—2623no—
168C23—52412no—
169C23—52424no—
170C23—52412no—
171C23—1316no—
172C23—1316no—
173C23—1316no—
174C23—1312no—
175C23D4×D13116yescomplemented
176C23C23×D13111yescharacteristic
177D4—224no—
178D4—26224no—
179D4—26248no—
180C2×C4—26no—
181C2×C4—23no—
182C2×C4—26224no—
183C2×C4—2626no—
184C2×C4—2623no—
185C2×C4—5244no—
186C2×C4—1316no—
187C2×C4D4×D13116yescomplemented
188C22—212no—
189C22—26no—
190C22—43no—
191C22—46no—
192C22—43no—
193C22—26212no—
194C22—2626no—
195C22—52448no—
196C22—5243no—
197C22—5246no—
198C22—5243no—
199C22—13112no—
200C22—13112no—
201C22—1314no—
202C22C2×D4×D13113yes—
203C22C2×D4×D13113yes—
204C22#41×D13111yescharacteristic
205C4—26no—
206C4—41no—
207C4—2626no—
208C4—5241no—
209C2—46no—
210C2—5246no—
211C2—1318no—
212C2#120×D13113yes—
213C2#121×D13113yes—
214C2#127×D13111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#392×C131C2
hypercenter#392D131
hypocenterC131C2×#392
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D131C2semidirect
2. C2×C22⋊C4×D131C2semidirect
3. #22553222C2semidirect
4. #22553221C2semidirect
5. #22553220C2semidirect
6. #95×D131C2semidirect
7. #392×C131C2semidirect
9. C22×D524C22semidirect
10. C22×C22⋊D131C22semidirect
15. C2×C22⋊Dic131C22semidirect
16. C2×(C2×D131)⋊C4C22semidirect
17. C2×(C2×D131)⋊C4C22semidirect
19. C131⋊#95C22semidirect
20. C22×D4×C131C22semidirect
21. C2×C22⋊C4×C131C22semidirect
22. #95×C131C22semidirect
27. C23×D131D4semidirect
40. C2×C4×D131D4semidirect
61. C22×D131C2×D4semidirect
67. C2×Dic131C2×D4semidirect
74. C23×C131C2×D4semidirect
79. C2×C4×C131C2×D4semidirect
106. D131#392direct (N × H)
107. C131C2×#392semidirect
114. #392D131direct (N × H)
121. C22×D4C2×D131semidirect
128. C2×C22⋊C4C2×D131semidirect
130. #95C2×D131semidirect
175. C23D4×D131semidirect
187. C2×C4D4×D131semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1312
C2×C22⋊C4×D1312
#225532222
#225532212
#225532202
#95×D1312
#392×C1312
C2×#392131