Finite Group Databasebeta

← #30664949

Subgroups — #30664949

Order 16768.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 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#22553874C211yesmaximal, characteristic
2#22553626C212yesmaximal, complemented
3#22553237C211yesmaximal, characteristic, complemented
4#22553047C212yesmaximal, complemented
5#323×C131C211yesmaximal, characteristic, complemented
6C22×C22⋊D131—21no—
7#16862230—22no—
8#16862220—21no—
9#16862220C2212yes—
10#16862216—22no—
11C2×C4⋊Dic131C2211yescharacteristic, complemented
12C2×C22⋊Dic131—21no—
13#16862180—22no—
14#16862168—22no—
15C131⋊#95—21no—
16C131⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C131C2211yescharacteristic
18C22⋊C8×C131C2212yescomplemented
19C23×D131—21no—
20C2×D524—21no—
21C2×D524D412yes—
22C2×C22⋊D131—22no—
23C2×C22⋊D131—22no—
24C2×C22⋊D131—42no—
25C2×C22⋊D131—42no—
26C22×Dic131—21no—
27C22×Dic131—21no—
28C22×Dic131—21no—
29C4⋊Dic131—21no—
30C4⋊Dic131—22no—
31C4⋊Dic131D412yes—
32C22⋊Dic131—42no—
33(C2×D131)⋊C4—22no—
34(C2×D131)⋊C4—22no—
35C24×C131D411yescharacteristic
36C22×C4×C131D411yescharacteristic
37C22×C4×C131C2311yescharacteristic
38C2×C8×C131—22no—
39C22⋊C4×C131—22no—
40C22×D131—22no—
41C22×D131—42no—
42C22×D131—81no—
43C22×D131—81no—
44D524—42no—
45D524—81no—
46C22⋊D131—44no—
47C22⋊D131—42no—
48C22⋊D131—42no—
49C2×Dic131—22no—
50C2×Dic131—42no—
51C2×Dic131—41no—
52C2×Dic131—41no—
53C2×Dic131—41no—
54C2×Dic131—41no—
55C2×Dic131—41no—
56C2×Dic131—41no—
57C23×C131—21no—
58C23×C131—21no—
59C23×C131—41no—
60C23×C131—41no—
61C23×C131D812yes—
62C23×C131C2×D411yescharacteristic
63C2×C4×C131—21no—
64C2×C4×C131—41no—
65C2×C4×C131—41no—
66C2×C4×C131C2×D412yes—
67C8×C131—42no—
68C22×C131—21no—
69C22×C131—21no—
70C22×C131—22no—
71C22×C131—42no—
72C22×C131—41no—
73C22×C131—41no—
74C22×C131—81no—
75C22×C131#12011yescharacteristic
76C22×C131C2×D811yescharacteristic
77C22×C131AGL(1,ℤ/8)11yescharacteristic
78C2×D131—42no—
79C2×D131—42no—
80C2×D131—42no—
81C2×D131—84no—
82C4×C131—22no—
83C4×C131—41no—
84Dic131—42no—
85Dic131—81no—
86Dic131—81no—
87C2×C131—21no—
88C2×C131—21no—
89C2×C131—42no—
90C2×C131#44711yescharacteristic
91C2×C131#45511yescharacteristic
92C2×C131#45811yescharacteristic
93D131—82no—
94C131#160711yescharacteristic, complemented, Sylow
95#1607—1311nomaximal, Sylow
96#534—1311no—
97#480—1312no—
98#397—1311no—
99#327—1312no—
100#323D13111yescharacteristic, complemented
101C22×D4—2621no—
102#123—2622no—
103#121—2621no—
104#121—1312no—
105#120—2622no—
106C2×C4⋊C4—1311no—
107C2×C22⋊C4—2621no—
108C2×C22⋊C4C2×D13111yescharacteristic
109#107—2622no—
110#102—2622no—
111C22⋊C8C2×D13112yescomplemented
112#95—2621no—
113#95—1311no—
114C24—2621no—
115C24D52411yescharacteristic
116C2×D4—2621no—
117C2×D4—2622no—
118C2×D4—2622no—
119C2×D4—5242no—
120C2×D4—5242no—
121C2×D4—1312no—
122C22×C4—2621no—
123C22×C4—2621no—
124C22×C4—2621no—
125C22×C4D52411yescharacteristic
126C22×C4C22×D13111yescharacteristic
127C2×C8—22no—
128C4⋊C4—2621no—
129C4⋊C4—2622no—
130C4⋊C4—1312no—
131C22⋊C4—22no—
132C22⋊C4—2622no—
133C22⋊C4—2622no—
134C22⋊C4—5242no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—2622no—
140C23—5242no—
141C23—10481no—
142C23—10481no—
143C23D104812yes—
144C23C2×D52411yescharacteristic
145D4—5242no—
146D4—5244no—
147D4—5242no—
148D4—5242no—
149D4—10481no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—2622no—
154C2×C4—5242no—
155C2×C4—5241no—
156C2×C4—5241no—
157C2×C4—5241no—
158C2×C4—5241no—
159C2×C4—5241no—
160C2×C4—5241no—
161C2×C4D4×D13112yes—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—5242no—
171C22—5242no—
172C22—5242no—
173C22—10484no—
174C22#1686221611yescharacteristic
175C22C2×D104811yescharacteristic
176C22#1686227411yescharacteristic
177C4—22no—
178C4—41no—
179C4—5242no—
180C4—10481no—
181C4—10481no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—10482no—
186C2#2255344811yescharacteristic
187C2#2255349211yescharacteristic
188C2#2255350311yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#323×C131C2
socleC22×C131#120
prime radical#22553874C2
hypercenter#323D131
hypocenterC131#1607
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22553626C2semidirect
3. #22553237C2semidirect
4. #22553047C2semidirect
5. #323×C131C2semidirect
11. C2×C4⋊Dic131C22semidirect
16. C131⋊#95C22semidirect
18. C22⋊C8×C131C22semidirect
94. C131#1607semidirect
100. #323D131semidirect
111. C22⋊C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225538742
#225536262
#225532372
#225530472
#323×C1312
#1607131