Finite Group Databasebeta

← #30666372

Subgroups — #30666372

Order 16768.

386 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16862227C211yesmaximal, characteristic
2#22553787C211yesmaximal, characteristic
3#22553173C212yesmaximal, complemented
4#22553078C212yesmaximal, complemented
5C2×#99×C131C211yesmaximal, characteristic
6C23×Dic131C2211yescharacteristic
7#16862227—22no—
8#16862227C414yescomplemented
9D4×Dic131—24no—
10#16862205—21no—
11C2×Dic131⋊C4—21no—
12C2×C22⋊Dic131—21no—
13C2×C22⋊Dic131—21no—
14C2×C22⋊Dic131C2212yes—
15C2×C22⋊Dic131C2211yescharacteristic
16C131⋊#95—24no—
17C22×D4×C131C2211yescharacteristic
18C2×C22⋊C4×C131C2212yes—
19#99×C131—22no—
20C22×Dic131—22no—
21C22×Dic131—21no—
22C22×Dic131—22no—
23C22×Dic131—21no—
24C22×Dic131—21no—
25C22×Dic131—22no—
26C22×Dic131C2×C412yescomplemented
27C22×Dic131D414yes—
28C4⋊Dic131—22no—
29Dic131⋊C4—24no—
30C22⋊Dic131—22no—
31C22⋊Dic131—24no—
32C22⋊Dic131—22no—
33C22⋊Dic131—21no—
34C22⋊Dic131—22no—
35C22⋊Dic131C2×C412yescomplemented
36C4×Dic131—22no—
37C24×C131D411yescharacteristic
38C24×C131C2311yescharacteristic
39C2×D4×C131—22no—
40C2×D4×C131—22no—
41C2×D4×C131C2×C412yes—
42C2×D4×C131D412yes—
43C22×C4×C131—22no—
44C22×C4×C131D411yescharacteristic
45C22⋊C4×C131—22no—
46C22⋊C4×C131—22no—
47C2×Dic131—22no—
48C2×Dic131—22no—
49C2×Dic131—24no—
50C2×Dic131—24no—
51C2×Dic131—42no—
52C2×Dic131—41no—
53C2×Dic131—42no—
54C2×Dic131—41no—
55C2×Dic131—44no—
56C2×Dic131—82no—
57C23×C131—21no—
58C23×C131—21no—
59C23×C131—42no—
60C23×C131—42no—
61C23×C131C22⋊C412yes—
62C23×C131C22×C411yescharacteristic
63C23×C131C2×D412yes—
64C23×C131C2×D411yescharacteristic
65C23×C131C2×D411yescharacteristic
66C23×C131#4112yes—
67D4×C131—24no—
68D4×C131—24no—
69C2×C4×C131—21no—
70C2×C4×C131—21no—
71C2×C4×C131—44no—
72C2×C4×C131—42no—
73C2×C4×C131C22⋊C412yes—
74C22×C131—24no—
75C22×C131—22no—
76C22×C131—22no—
77C22×C131—41no—
78C22×C131—41no—
79C22×C131—42no—
80C22×C131—42no—
81C22×C131—42no—
82C22×C131—81no—
83C22×C131C2×C22⋊C411yescharacteristic
84C22×C131C4×D412yes—
85C22×C131#12011yescharacteristic
86C22×C131#12112yes—
87C22×C131#12311yescharacteristic
88C4×C131—22no—
89C4×C131—82no—
90Dic131—42no—
91Dic131—42no—
92Dic131—44no—
93C2×C131—21no—
94C2×C131—22no—
95C2×C131—21no—
96C2×C131—22no—
97C2×C131—42no—
98C2×C131#38611yescharacteristic
99C2×C131#45812yes—
100C131#189011yescharacteristic, complemented, Sylow
101#1890—1311nomaximal, Sylow
102C2×#123—1311no—
103#518—1311no—
104C2×#99D13111yescharacteristic
105#381—1312no—
106#342—1312no—
107C22×D4C2×D13111yescharacteristic
108C23×C4—1311no—
109#123—2622no—
110#123—1314no—
111C4×D4—2624no—
112#117—2621no—
113C2×C4⋊C4—2621no—
114C2×C22⋊C4—2621no—
115C2×C22⋊C4—2621no—
116C2×C22⋊C4—1312no—
117C2×C22⋊C4—1311no—
118C2×C22⋊C4C2×D13112yes—
119#99—22no—
120#95—2624no—
121C24D52411yescharacteristic
122C24C22×D13111yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C4×D13112yes—
126C2×D4C22⋊D13112yes—
127C22×C4—22no—
128C22×C4—2622no—
129C22×C4—2621no—
130C22×C4—2621no—
131C22×C4—2621no—
132C22×C4—2622no—
133C22×C4—2622no—
134C22×C4—1312no—
135C22×C4—1314no—
136C22×C4D52411yescharacteristic
137C4⋊C4—2622no—
138C4⋊C4—2624no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—2622no—
142C22⋊C4—2624no—
143C22⋊C4—2622no—
144C22⋊C4—2621no—
145C22⋊C4—2622no—
146C22⋊C4—1312no—
147C42—2622no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23(C2×D131)⋊C412yes—
153C23C2×C4×D13111yescharacteristic
154C23D4×D13112yes—
155C23C2×C22⋊D13111yescharacteristic
156C23C2×D52411yescharacteristic
157C23#1452351212yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4—2624no—
165C2×C4—2622no—
166C2×C4—2622no—
167C2×C4—2624no—
168C2×C4—5242no—
169C2×C4—5241no—
170C2×C4—5242no—
171C2×C4—5241no—
172C2×C4—5244no—
173C2×C4—10482no—
174C2×C4(C2×D131)⋊C412yes—
175C22—24no—
176C22—22no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×(C2×D131)⋊C411yescharacteristic
185C22(C22⋊D131)⋊C412yes—
186C22#1686221611yescharacteristic
187C22#1686221812yes—
188C22#1686223011yescharacteristic
189C4—22no—
190C4—82no—
191C4—5242no—
192C4—5242no—
193C4—5244no—
194C2—22no—
195C2—21no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#2255318911yescharacteristic
200C2#2255350312yes—
subgroupG/H
centerC22#16862216
commutatorC23×C131C22×C4
FrattiniC24C22×D131
FittingC2×#99×C131C2
socleC22×C131#120
prime radicalC22×D4×C131C22
cosocleC24×C131C23
hypercenterC2×#99D131
hypocenterC131#1890
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #22553173C2semidirect
4. #22553078C2semidirect
8. #16862227C4semidirect
26. C22×Dic131C2×C4semidirect
35. C22⋊Dic131C2×C4semidirect
100. C131#1890semidirect

Maximal subgroups

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

subgroupindex
C2×#168622272
#225537872
#225531732
#225530782
C2×#99×C1312
#1890131