Finite Group Databasebeta

← #30671127

Subgroups — #30671127

Order 16768.

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

normal subgroups only

This group has 532 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×#16862238C211yesmaximal, characteristic, complemented
2#22553250C212yesmaximal, complemented
3#22553226C212yesmaximal
4#22553206C214yesmaximal, complemented
5#22553172C212yesmaximal, complemented
6#22553172C212yesmaximal, complemented
7C4×C4⋊Dic131C211yesmaximal, characteristic, complemented
8C2×C4×D4×C131C211yesmaximal, characteristic
9#16862238—24no—
10C2×C4⋊Dic131C2212yes—
11C2×C4⋊Dic131C2211yescharacteristic
12C2×C4⋊Dic131C2211yescharacteristic
13C2×Dic131⋊C4C2212yes—
14C2×Dic131⋊C4C2212yes—
15C2×C22⋊Dic131C2214yes—
16C2×C22⋊Dic131C2214yes—
17C2×C4×Dic131C2212yes—
18C131⋊#95C2212yes—
19C131⋊#95C2214yes—
20C131⋊#95C2214yes—
21C22×D4×C131C2211yescharacteristic
22C23×C4×C131C2212yes—
23C4×D4×C131—24no—
24C2×C4⋊C4×C131C2211yescharacteristic
25C2×C22⋊C4×C131C2212yes—
26C2×C42×C131C2211yescharacteristic
27C22×Dic131C2314yes—
28C22×Dic131C2312yes—
29C22×Dic131C2312yes—
30C4⋊Dic131—22no—
31C4⋊Dic131—22no—
32C4⋊Dic131—24no—
33Dic131⋊C4—24no—
34Dic131⋊C4—24no—
35C22⋊Dic131—28no—
36C22⋊Dic131—44no—
37C4×Dic131—42no—
38C24×C131C2312yes—
39C2×D4×C131—24no—
40C2×D4×C131—22no—
41C22×C4×C131—24no—
42C22×C4×C131—24no—
43C22×C4×C131—24no—
44C22×C4×C131C2312yes—
45C22×C4×C131C2311yescharacteristic
46C22×C4×C131C2311yescharacteristic
47C22×C4×C131C2311yescharacteristic
48C4⋊C4×C131—22no—
49C22⋊C4×C131—24no—
50C42×C131D414yes—
51C2×Dic131—22no—
52C2×Dic131—22no—
53C2×Dic131—24no—
54C2×Dic131—24no—
55C2×Dic131—24no—
56C2×Dic131—24no—
57C2×Dic131—22no—
58C2×Dic131—22no—
59C23×C131—22no—
60C23×C131—22no—
61C23×C131—22no—
62C23×C131—22no—
63C23×C131—22no—
64C23×C131—22no—
65C23×C131#4114yes—
66C23×C131C2411yescharacteristic
67D4×C131—28no—
68C2×C4×C131—22no—
69C2×C4×C131—21no—
70C2×C4×C131—21no—
71C2×C4×C131—22no—
72C2×C4×C131—22no—
73C2×C4×C131—22no—
74C2×C4×C131—48no—
75C2×C4×C131C2×D412yes—
76C2×C4×C131C2×D412yes—
77C2×C4×C131C2×D412yes—
78C2×C4×C131#4114yes—
79C22×C131—24no—
80C22×C131—24no—
81C22×C131—24no—
82C22×C131—42no—
83C22×C131—42no—
84C22×C131—42no—
85C22×C131—42no—
86C22×C131C22×D411yescharacteristic
87C22×C131C2×#4112yes—
88C22×C131C2×#4112yes—
89C22×C131H(5,𝔽2)11yescharacteristic
90C22×C1312−1+411yescharacteristic
91C4×C131—22no—
92C4×C131—22no—
93C4×C131—42no—
94C4×C131#12414yes—
95Dic131—42no—
96Dic131—42no—
97Dic131—44no—
98C2×C131—44no—
99C2×C131#52511yescharacteristic
100C2×C131C2×#12411yescharacteristic
101C2×C131#53711yescharacteristic
102C2×C131#55212yes—
103C2×C131#55312yes—
104C131#256211yescharacteristic, complemented, Sylow
105#2562—1311nomaximal, Sylow
106C2×#125—1311no—
107C2×C4×D4D13111yescharacteristic
108#400—1312no—
109#394—1312no—
110#388—1314no—
111#380—1312no—
112#380—1312no—
113C4×C4⋊C4—1311no—
114C22×D4C2×D13111yescharacteristic
115C23×C4C2×D13112yes—
116#125—2624no—
117C4×D4—24no—
118C2×C4⋊C4—1311no—
119C2×C4⋊C4—1311no—
120C2×C4⋊C4—1312no—
121C2×C4⋊C4—1312no—
122C2×C4⋊C4—1312no—
123C2×C4⋊C4C2×D13111yescharacteristic
124C2×C22⋊C4—1314no—
125C2×C22⋊C4—1314no—
126C2×C22⋊C4C2×D13112yes—
127C2×C42—1312no—
128C2×C42C2×D13111yescharacteristic
129#95—1312no—
130#95—1314no—
131#95—1314no—
132C24C22×D13112yes—
133C2×D4—24no—
134C2×D4—22no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—1314no—
139C22×C4—1312no—
140C22×C4—1312no—
141C22×C4C22×D13112yes—
142C22×C4C22×D13111yescharacteristic
143C22×C4C22×D13111yescharacteristic
144C22×C4C22×D13111yescharacteristic
145C4⋊C4—22no—
146C4⋊C4—2624no—
147C4⋊C4—2624no—
148C4⋊C4—2622no—
149C4⋊C4—2622no—
150C4⋊C4—2624no—
151C22⋊C4—24no—
152C22⋊C4—2628no—
153C22⋊C4—5244no—
154C42—5242no—
155C42C22⋊D13114yes—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23#1452351314yes—
163C23C23×D13111yescharacteristic
164D4—28no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—48no—
172C2×C4—2622no—
173C2×C4—2622no—
174C2×C4—2624no—
175C2×C4—2624no—
176C2×C4—2624no—
177C2×C4—2624no—
178C2×C4—2622no—
179C2×C4—2622no—
180C2×C4C2×C22⋊D13112yes—
181C2×C4C2×C22⋊D13112yes—
182C2×C4C2×C22⋊D13112yes—
183C2×C4#1452351214yes—
184C22—24no—
185C22—24no—
186C22—24no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22C22×C22⋊D13111yescharacteristic
192C22C2×#1452351212yes—
193C22C2×#1452351312yes—
194C22#1686229211yescharacteristic
195C22#1686229311yescharacteristic
196C4—22no—
197C4—22no—
198C4—42no—
199C4—5242no—
200C4—5244no—
201C4—5242no—
202C4#1686223314yes—
203C2—44no—
204C2#2255382311yescharacteristic
205C2C2×#1686223311yescharacteristic
206C2#2255389011yescharacteristic
207C2#2255400212yes—
208C2#2255400812yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×C4×D4×C131C2
prime radicalC22×D4×C131C22
hypercenterC2×C4×D4D131
hypocenterC131#2562
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16862238C2semidirect
2. #22553250C2semidirect
4. #22553206C2semidirect
5. #22553172C2semidirect
6. #22553172C2semidirect
7. C4×C4⋊Dic131C2semidirect
104. C131#2562semidirect

Maximal subgroups

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

subgroupindex
C2×#168622382
#225532502
#225532262
#225532062
#225531722
C4×C4⋊Dic1312
C2×C4×D4×C1312
#2562131