Finite Group Databasebeta

← #30664947

Subgroups — #30664947

Order 16768.

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

normal subgroups only

This group has 280 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#22553623C212yesmaximal, complemented
2#22553237C211yesmaximal, characteristic, complemented
3#22553046C212yesmaximal, complemented
4#22553036C211yesmaximal, characteristic, complemented
5#534×C131C211yesmaximal, characteristic
6C2×C4⋊Dic131C2211yescharacteristic, complemented
7C2×C22⋊Dic131—21no—
8#16862179—22no—
9#16862167—22no—
10C22⋊(C131⋊C8)C2212yescomplemented
11C131⋊#95—21no—
12C131⋊#95C2211yescharacteristic, complemented
13C22×D4×C131—21no—
14#123×C131—22no—
15#121×C131—21no—
16#121×C131C2212yes—
17#120×C131—22no—
18C2×C22⋊C4×C131C2211yescharacteristic
19C22×Dic131—21no—
20C22×Dic131—21no—
21C2×C131⋊C8—22no—
22C4⋊Dic131—22no—
23C22⋊Dic131—42no—
24C24×C131—21no—
25C24×C131D411yescharacteristic
26C2×D4×C131—22no—
27C2×D4×C131—22no—
28C2×D4×C131—21no—
29C2×D4×C131—42no—
30C2×D4×C131—42no—
31C2×D4×C131D412yes—
32C22×C4×C131—21no—
33C22×C4×C131D411yescharacteristic
34C22×C4×C131C2311yescharacteristic
35C4⋊C4×C131—21no—
36C4⋊C4×C131D412yes—
37C22⋊C4×C131—22no—
38C22⋊C4×C131—22no—
39C22⋊C4×C131—22no—
40C2×Dic131—41no—
41C2×Dic131—41no—
42C2×Dic131—41no—
43C2×Dic131—41no—
44C2×Dic131—41no—
45C2×Dic131—41no—
46C131⋊C8—42no—
47C23×C131—21no—
48C23×C131—21no—
49C23×C131—22no—
50C23×C131—41no—
51C23×C131—41no—
52C23×C131—42no—
53C23×C131—81no—
54C23×C131—81no—
55C23×C131D812yes—
56C23×C131C2×D411yescharacteristic
57D4×C131—42no—
58D4×C131—44no—
59D4×C131—42no—
60D4×C131—42no—
61D4×C131—81no—
62C2×C4×C131—21no—
63C2×C4×C131—22no—
64C2×C4×C131—41no—
65C2×C4×C131—42no—
66C2×C4×C131—41no—
67C2×C4×C131C2×D412yes—
68C22×C131—21no—
69C22×C131—21no—
70C22×C131—22no—
71C22×C131—42no—
72C22×C131—41no—
73C22×C131—41no—
74C22×C131—42no—
75C22×C131—42no—
76C22×C131—42no—
77C22×C131—81no—
78C22×C131—84no—
79C22×C131#12011yescharacteristic
80C22×C131C2×D811yescharacteristic
81C22×C131AGL(1,ℤ/8)11yescharacteristic
82C4×C131—22no—
83C4×C131—42no—
84C4×C131—41no—
85Dic131—81no—
86Dic131—81no—
87C2×C131—21no—
88C2×C131—21no—
89C2×C131—42no—
90C2×C131—82no—
91C2×C131#44711yescharacteristic
92C2×C131#45511yescharacteristic
93C2×C131#45811yescharacteristic
94C131#160711yescharacteristic, complemented, Sylow
95#1607—1311nomaximal, Sylow
96#534D13111yescharacteristic
97#480—1312no—
98#397—1311no—
99#327—1312no—
100#323—1311no—
101C22×D4—21no—
102#123—22no—
103#121—21no—
104#121C2×D13112yes—
105#120—22no—
106C2×C4⋊C4—1311no—
107C2×C22⋊C4—2621no—
108C2×C22⋊C4C2×D13111yescharacteristic
109#107—2622no—
110#102—2622no—
111C22⋊C8—1312no—
112#95—2621no—
113#95—1311no—
114C24—21no—
115C24C22⋊D13111yescharacteristic
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4C22⋊D13112yes—
122C22×C4—21no—
123C22×C4—2621no—
124C22×C4—2621no—
125C22×C4C22⋊D13111yescharacteristic
126C22×C4C22×D13111yescharacteristic
127C2×C8—2622no—
128C4⋊C4—21no—
129C4⋊C4—2622no—
130C4⋊C4C22⋊D13112yes—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—5242no—
135C23—21no—
136C23—21no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—42no—
141C23—81no—
142C23—81no—
143C23#1452349812yes—
144C23C2×C22⋊D13111yescharacteristic
145D4—42no—
146D4—44no—
147D4—42no—
148D4—42no—
149D4—81no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—5241no—
156C2×C4—5241no—
157C2×C4—5241no—
158C2×C4—5241no—
159C2×C4—5241no—
160C2×C4—5241no—
161C2×C4C2×C22⋊D13112yes—
162C8—5242no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—81no—
173C22—84no—
174C22#1686221511yescharacteristic
175C22C2×#1452349811yescharacteristic
176C22#1686227211yescharacteristic
177C4—22no—
178C4—42no—
179C4—41no—
180C4—10481no—
181C4—10481no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—82no—
186C2#2255344611yescharacteristic
187C2#2255349011yescharacteristic
188C2#2255350211yescharacteristic
subgroupG/H
centerC22#16862215
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting, prime radical#534×C131C2
socleC22×C131#120
hypercenter#534D131
hypocenterC131#1607
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #22553623C2semidirect
2. #22553237C2semidirect
3. #22553046C2semidirect
4. #22553036C2semidirect
6. C2×C4⋊Dic131C22semidirect
10. C22⋊(C131⋊C8)C22semidirect
12. C131⋊#95C22semidirect
94. C131#1607semidirect

Maximal subgroups

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

subgroupindex
#225536232
#225532372
#225530462
#225530362
#534×C1312
#1607131