Finite Group Databasebeta

← #30664896

Subgroups — #30664896

Order 16768.

310 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 57 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#22553445C212yesmaximal, complemented
2#22553229C211yesmaximal, characteristic, complemented
3#22553046C212yesmaximal, complemented
4#22553038C211yesmaximal, characteristic, complemented
5#537×C131C211yesmaximal, characteristic, complemented
6C22×D524C2211yescharacteristic
7C2×#14523498—22no—
8C2×(C2×D131)⋊C4—21no—
9C2×(C2×D131)⋊C4—21no—
10#16862166—22no—
11C22⋊(C131⋊C8)C2212yes—
12C131⋊#95C2211yescharacteristic
13C2×#41×C131—21no—
14C22⋊Q8×C131—22no—
15#121×C131—22no—
16#121×C131—21no—
17#121×C131C2212yes—
18C2×C4⋊C4×C131C2211yescharacteristic
19C23×D131—21no—
20C2×D524—22no—
21C2×D524—42no—
22C22×Dic131—21no—
23#14523498—44no—
24C2×C131⋊C8—22no—
25(C2×D131)⋊C4—42no—
26(C2×D131)⋊C4—42no—
27#41×C131—42no—
28#41×C131—42no—
29C2×Q8×C131—21no—
30C2×D4×C131—21no—
31C2×D4×C131—22no—
32C2×D4×C131—21no—
33C2×D4×C131D412yes—
34C22×C4×C131—22no—
35C22×C4×C131D411yescharacteristic
36C22×C4×C131D411yescharacteristic
37C22×C4×C131C2311yescharacteristic
38C4⋊C4×C131—21no—
39C4⋊C4×C131—22no—
40C4⋊C4×C131D412yes—
41C22⋊C4×C131—22no—
42C22⋊C4×C131—22no—
43C22×D131—22no—
44C22×D131—41no—
45C22×D131—41no—
46C22×D131—41no—
47C22×D131—41no—
48C22×D131—41no—
49C22×D131—41no—
50D524—24no—
51D524—42no—
52C2×Dic131—41no—
53C2×Dic131—41no—
54C2×Dic131—41no—
55C131⋊C8—42no—
56C23×C131—22no—
57C23×C131C2×D411yescharacteristic
58Q8×C131—42no—
59D4×C131—42no—
60D4×C131—42no—
61D4×C131—42no—
62D4×C131—42no—
63D4×C131—81no—
64C2×C4×C131—21no—
65C2×C4×C131—21no—
66C2×C4×C131—21no—
67C2×C4×C131—22no—
68C2×C4×C131—41no—
69C2×C4×C131—44no—
70C2×C4×C131—41no—
71C2×C4×C131D812yescomplemented
72C2×C4×C131C2×D412yes—
73C22×C131—21no—
74C22×C131—21no—
75C22×C131—42no—
76C22×C131—42no—
77C22×C131—42no—
78C22×C131#12011yescharacteristic
79C22×C131C2×D811yescharacteristic
80C22×C131AGL(1,ℤ/8)11yescharacteristic
81C2×D131—42no—
82C2×D131—42no—
83C2×D131—42no—
84C2×D131—81no—
85C2×D131—81no—
86C2×D131—81no—
87C2×D131—81no—
88C4×C131—22no—
89C4×C131—22no—
90C4×C131—42no—
91C4×C131—41no—
92Dic131—81no—
93C2×C131—21no—
94C2×C131—21no—
95C2×C131—82no—
96C2×C131#44711yescharacteristic
97C2×C131#45411yescharacteristic
98C2×C131B(4,𝔽2)11yescharacteristic
99D131—82no—
100C131#159811yescharacteristic, complemented, Sylow
101#1598—1311nomaximal, Sylow
102#537D13111yescharacteristic, complemented
103#447—1312no—
104#394—1311no—
105#327—1312no—
106#324—1311no—
107C2×#41—21no—
108C22×D4—1311no—
109C2×D8—2622no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D13112yes—
114C2×C4⋊C4C2×D13111yescharacteristic
115C2×C22⋊C4—2621no—
116C2×C22⋊C4—2621no—
117#102—2622no—
118C22⋊C8—1312no—
119#95—1311no—
120C24—2621no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—2622no—
128C2×D4—5242no—
129C2×D4C22⋊D13112yes—
130C22×C4—22no—
131C22×C4—2621no—
132C22×C4C22⋊D13111yescharacteristic
133C22×C4C22⋊D13111yescharacteristic
134C22×C4C22×D13111yescharacteristic
135D8—5244no—
136C2×C8—2622no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D13112yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—5242no—
143C22⋊C4—5242no—
144C23—22no—
145C23—2622no—
146C23—5241no—
147C23—5241no—
148C23—5241no—
149C23—5241no—
150C23—5241no—
151C23—5241no—
152C23C2×C22⋊D13111yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—2624no—
159D4—81no—
160D4—5242no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—5241no—
169C2×C4—5241no—
170C2×C4—5241no—
171C2×C4#1452349812yescomplemented
172C2×C4C2×C22⋊D13112yes—
173C8—5242no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—5242no—
180C22—5242no—
181C22—5242no—
182C22—10481no—
183C22—10481no—
184C22—10481no—
185C22—10481no—
186C22#1686221511yescharacteristic
187C22C2×#1452349811yescharacteristic
188C22#1686227211yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—41no—
193C4—10481no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—10482no—
198C2#2255344611yescharacteristic
199C2#2255348511yescharacteristic
200C2#2255349911yescharacteristic
subgroupG/H
centerC22#16862215
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#537×C131C2
socleC22×C131#120
hypercenter#537D131
hypocenterC131#1598
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #22553445C2semidirect
2. #22553229C2semidirect
3. #22553046C2semidirect
4. #22553038C2semidirect
5. #537×C131C2semidirect
71. C2×C4×C131D8semidirect
100. C131#1598semidirect
102. #537D131semidirect
171. C2×C4#14523498semidirect

Maximal subgroups

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

subgroupindex
#225534452
#225532292
#225530462
#225530382
#537×C1312
#1598131