Finite Group Databasebeta

← #30666409

Subgroups — #30666409

Order 16768.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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#22553823C211yesmaximal, characteristic
2C2×#16862174C212yesmaximal, complemented
3#22553276C211yesmaximal, characteristic, complemented
4#22553072C212yesmaximal, complemented
5C2×#117×C131C211yesmaximal, characteristic, complemented
6C2×#14523513C2211yescharacteristic
7C2×#14523497C2212yes—
8#16862229—21no—
9#16862225C412yescomplemented
10#16862220C412yescomplemented
11#16862215—21no—
12C4×C22⋊D131—22no—
13#16862205C2211yescharacteristic
14#16862174—24no—
15C22⋊(C131⋊C8)—22no—
16C23×C4×C131C2211yescharacteristic
17#117×C131—24no—
18C2×C4⋊C4×C131—21no—
19C2×C22⋊C4×C131—21no—
20C2×C42×C131C2212yes—
21#14523513—22no—
22#14523513—22no—
23C2×Dic262C2×C411yescharacteristic
24C2×D524C2×C411yescharacteristic
25C2×C22⋊D131—22no—
26C2×C22⋊D131—21no—
27C2×C4×D131—21no—
28#14523497—22no—
29#14523497—22no—
30C2×C131⋊C8—22no—
31C4⋊Dic131C2×C412yes—
32Dic131⋊C4—22no—
33C22⋊Dic131—22no—
34C22⋊Dic131—21no—
35(C2×D131)⋊C4—22no—
36C4×Dic131—21no—
37C24×C131D411yescharacteristic
38C22×C4×C131—22no—
39C22×C4×C131—22no—
40C22×C4×C131—22no—
41C22×C4×C131C2×C412yes—
42C22×C4×C131D411yescharacteristic
43C22×C4×C131D412yes—
44C22×C4×C131C2311yescharacteristic
45C4⋊C4×C131—24no—
46C22⋊C4×C131—24no—
47C42×C131—22no—
48C42×C131D414yescomplemented
49C22×D131—21no—
50Dic262—22no—
51D524—22no—
52C22⋊D131—44no—
53C22⋊D131—41no—
54C22⋊D131—41no—
55C2×Dic131—22no—
56C2×Dic131—21no—
57C4×D131—42no—
58C131⋊C8—42no—
59C23×C131—21no—
60C23×C131—21no—
61C23×C131—42no—
62C23×C131C22⋊C412yes—
63C23×C131C2×D411yescharacteristic
64C2×C4×C131—21no—
65C2×C4×C131—21no—
66C2×C4×C131—24no—
67C2×C4×C131—24no—
68C2×C4×C131—24no—
69C2×C4×C131—44no—
70C2×C4×C131C22⋊C412yes—
71C2×C4×C131C22×C411yescharacteristic
72C2×C4×C131C2×D412yes—
73C2×C4×C131C2×D411yescharacteristic
74C2×C4×C131#4112yes—
75C22×C131—22no—
76C22×C131—22no—
77C22×C131—22no—
78C22×C131—22no—
79C22×C131—42no—
80C22×C131—42no—
81C22×C131C2×C22⋊C411yescharacteristic
82C22×C131#12011yescharacteristic
83C22×C131#12311yescharacteristic
84C2×D131—41no—
85C2×D131—42no—
86C4×C131—22no—
87C4×C131—44no—
88C4×C131C4×D412yes—
89C4×C131#12112yes—
90Dic131—41no—
91Dic131—42no—
92C2×C131—21no—
93C2×C131—21no—
94C2×C131—42no—
95C2×C131#38611yescharacteristic
96C2×C131#42112yes—
97D131—81no—
98C131#189711yescharacteristic, complemented, Sylow
99#1897—1311nomaximal, Sylow
100#525—1311no—
101C2×#117D13111yescharacteristic, complemented
102C2×#104—1312no—
103#407—1311no—
104#339—1312no—
105C2×#41—1311no—
106C23×C4C2×D13111yescharacteristic
107C2×M16—1312no—
108#123—2621no—
109C22⋊Q8—1312no—
110#121—1312no—
111#120—2621no—
112C4×D4—2622no—
113#117—24no—
114#117—1311no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D13112yes—
118#104—2624no—
119C22⋊C8—2622no—
120C24C22⋊D13111yescharacteristic
121#41—2622no—
122#41—2622no—
123C2×Q8—1311no—
124C2×D4—2622no—
125C2×D4—2621no—
126C2×D4—1311no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—2621no—
131C22×C4C4×D13112yes—
132C22×C4C22⋊D13111yescharacteristic
133C22×C4D52412yes—
134C22×C4C22×D13111yescharacteristic
135M16—2622no—
136M16—2622no—
137C2×C8—2622no—
138C4⋊C4—24no—
139C4⋊C4—2622no—
140C4⋊C4—1312no—
141C22⋊C4—24no—
142C22⋊C4—2622no—
143C22⋊C4—2621no—
144C22⋊C4—2622no—
145C42—22no—
146C42—2621no—
147C42C22⋊D13114yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—2621no—
152C23(C2×D131)⋊C412yes—
153C23C2×C22⋊D13111yescharacteristic
154Q8—2622no—
155D4—2622no—
156D4—5244no—
157D4—5241no—
158D4—5241no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—2622no—
166C2×C4—2621no—
167C2×C4—5242no—
168C2×C4(C2×D131)⋊C412yes—
169C2×C4C2×C4×D13111yescharacteristic
170C2×C4C2×C22⋊D13112yes—
171C2×C4C2×D52411yescharacteristic
172C2×C4#1452351312yes—
173C8—5242no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—5241no—
181C22—5242no—
182C22C2×(C2×D131)⋊C411yescharacteristic
183C22#1686221511yescharacteristic
184C22#1686222911yescharacteristic
185C4—22no—
186C4—44no—
187C4—5241no—
188C4—5242no—
189C4C4×C22⋊D13112yes—
190C4#1686222012yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—10481no—
195C2#2255319111yescharacteristic
196C2#2255333012yes—
subgroupG/H
centerC2×C4C2×D524
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
FittingC2×#117×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553823C2
cosocleC22×C4×C131C23
hypercenterC2×#117D131
hypocenterC131#1897
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#16862174C2semidirect
3. #22553276C2semidirect
4. #22553072C2semidirect
5. C2×#117×C131C2semidirect
9. #16862225C4semidirect
10. #16862220C4semidirect
48. C42×C131D4semidirect
98. C131#1897semidirect
101. C2×#117D131semidirect
147. C42C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
#225538232
C2×#168621742
#225532762
#225530722
C2×#117×C1312
#1897131