Finite Group Databasebeta

← C2×#22553653 #30675853

Subgroups — C2×#22553653 #30675853

Order 16768.

416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862225C211yesmaximal, characteristic
2#22553653C218yesmaximal, direct product factor
3C2×#16862179C211yesmaximal, characteristic, complemented
4C2×#16862177C211yesmaximal, characteristic, complemented
5C2×#16862169C211yesmaximal, characteristic, complemented
6C2×#16862169C211yesmaximal, characteristic, complemented
7C2×C22⋊(C131⋊C8)C211yesmaximal, characteristic
8C2×#117×C131C211yesmaximal, characteristic
9C22×Dic262C2211yescharacteristic
10C22×C131⋊C8C2211yescharacteristic
11C22×C131⋊C8C2211yescharacteristic
12#16862225—22no—
13#16862225C2214yes—
14C2×C4⋊Dic131C2211yescharacteristic
15C2×Dic131⋊C4—21no—
16C2×C22⋊Dic131—21no—
17#16862179C2214yescomplemented
18#16862177C2214yescomplemented
19#16862169C2214yescomplemented
20#16862169C2214yescomplemented
21C22⋊(C131⋊C8)C2214yes—
22C23×C4×C131C2211yescharacteristic
23#117×C131—22no—
24#117×C131C2214yes—
25C2×C4⋊C4×C131C2211yescharacteristic
26C2×C4⋊C4×C131C2211yescharacteristic
27C2×C22⋊C4×C131—21no—
28C2×C42×C131—21no—
29C2×Dic262—21no—
30C2×Dic262—22no—
31C2×Dic262—22no—
32C2×Dic262C2312yes—
33C22×Dic131—21no—
34C22×Dic131—21no—
35C2×C131⋊C8—22no—
36C2×C131⋊C8—22no—
37C2×C131⋊C8C2312yes—
38C2×C131⋊C8C2312yes—
39C4⋊Dic131—21no—
40C4⋊Dic131C2312yes—
41Dic131⋊C4—24no—
42C22⋊Dic131—24no—
43C24×C131D411yescharacteristic
44C22×C4×C131—22no—
45C22×C4×C131—22no—
46C22×C4×C131—21no—
47C22×C4×C131—21no—
48C22×C4×C131D412yes—
49C22×C4×C131D411yescharacteristic
50C22×C4×C131C2312yes—
51C22×C4×C131C2311yescharacteristic
52C4⋊C4×C131—21no—
53C4⋊C4×C131—21no—
54C4⋊C4×C131C2312yes—
55C4⋊C4×C131C2312yes—
56C22⋊C4×C131—24no—
57C42×C131—24no—
58Dic262—22no—
59Dic262—22no—
60Dic262—42no—
61C2×Dic131—22no—
62C2×Dic131—22no—
63C2×Dic131—42no—
64C2×Dic131—42no—
65C131⋊C8—22no—
66C131⋊C8—22no—
67C23×C131—22no—
68C23×C131—22no—
69C23×C131—22no—
70C23×C131C2×D412yes—
71C23×C131C2×D411yescharacteristic
72C2×C4×C131—22no—
73C2×C4×C131—24no—
74C2×C4×C131—22no—
75C2×C4×C131—22no—
76C2×C4×C131—22no—
77C2×C4×C131—22no—
78C2×C4×C131—42no—
79C2×C4×C131—42no—
80C2×C4×C131C2×D412yes—
81C2×C4×C131C2×D411yescharacteristic
82C2×C4×C131#4112yes—
83C2×C4×C131#4112yes—
84C2×C4×C131C2411yescharacteristic
85C22×C131—22no—
86C22×C131—22no—
87C22×C131—22no—
88C22×C131—44no—
89C22×C131#12312yes—
90C22×C131#13512yes—
91C22×C131#13712yes—
92C22×C131C22×D411yescharacteristic
93C4×C131—22no—
94C4×C131—42no—
95C4×C131—42no—
96C4×C131#12312yes—
97C4×C131C2×#4111yescharacteristic
98C4×C131C2×#4111yescharacteristic
99Dic131—42no—
100Dic131—42no—
101C2×C131—42no—
102C2×C131#48514yescomplemented
103C2×C131C2×#12311yescharacteristic
104C2×C131C2×#13511yescharacteristic
105C2×C131C2×#13711yescharacteristic
106C131C2×#48511yescharacteristic, complemented, Sylow
107C2×#485—1311nomaximal, Sylow
108C2×C22⋊Q8—1311no—
109C2×#117D13111yescharacteristic
110#485—1318no—
111C2×#107—1311no—
112C2×#106—1311no—
113C2×#103—1311no—
114C2×#103—1311no—
115C2×C22⋊C8—1311no—
116C22×Q8—1311no—
117C23×C4C2×D13111yescharacteristic
118C22×C8—1311no—
119C22×C8—1311no—
120C22⋊Q8—2622no—
121C22⋊Q8—1314no—
122#117—22no—
123#117C2×D13114yes—
124C2×C4⋊C4—2621no—
125C2×C4⋊C4—1311no—
126C2×C4⋊C4C2×D13111yescharacteristic
127C2×C4⋊C4C2×D13111yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—2621no—
130C2×C42—21no—
131#107—1314no—
132#106—1314no—
133#103—1314no—
134#103—1314no—
135C22⋊C8—1314no—
136C24C22⋊D13111yescharacteristic
137C2×Q8—2621no—
138C2×Q8—2622no—
139C2×Q8—2622no—
140C2×Q8—1312no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—2621no—
146C22×C4—2621no—
147C22×C4C22⋊D13112yes—
148C22×C4C22⋊D13111yescharacteristic
149C22×C4C22×D13112yes—
150C22×C4C22×D13111yescharacteristic
151C2×C8—2622no—
152C2×C8—2622no—
153C2×C8—1312no—
154C2×C8—1312no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—2624no—
158C4⋊C4—2621no—
159C4⋊C4—1312no—
160C4⋊C4C22×D13112yes—
161C4⋊C4C22×D13112yes—
162C22⋊C4—24no—
163C22⋊C4—2624no—
164C42—24no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23C2×C22⋊D13112yes—
169C23C2×C22⋊D13111yescharacteristic
170Q8—2622no—
171Q8—2622no—
172Q8—5242no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—2622no—
182C2×C4—2622no—
183C2×C4—5242no—
184C2×C4—5242no—
185C2×C4C2×C22⋊D13112yes—
186C2×C4C2×C22⋊D13111yescharacteristic
187C2×C4#1452351312yes—
188C2×C4#1452351312yes—
189C2×C4C23×D13111yescharacteristic
190C8—2622no—
191C8—2622no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—44no—
196C22#1686222912yes—
197C22#1686226612yes—
198C22#1686227612yes—
199C22C22×C22⋊D13111yescharacteristic
200C4—22no—
201C4—42no—
202C4—42no—
203C4—5242no—
204C4—5242no—
205C4#1686222912yes—
206C4C2×#1452351311yescharacteristic
207C4C2×#1452351311yescharacteristic
208C2—42no—
209C2#2255365314yesdirect product factor
210C2C2×#1686222911yescharacteristic
211C2C2×#1686226611yescharacteristic
212C2C2×#1686227611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#117×C131C2
socleC23×C131C2×D4
prime radicalC24×C131D4
hypercenterC2×#117D131
hypocenterC131C2×#485
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22553653C2direct (N × H)
3. C2×#16862179C2semidirect
4. C2×#16862177C2semidirect
5. C2×#16862169C2semidirect
6. C2×#16862169C2semidirect
17. #16862179C22semidirect
18. #16862177C22semidirect
19. #16862169C22semidirect
20. #16862169C22semidirect
102. C2×C131#485semidirect
106. C131C2×#485semidirect
209. C2#22553653direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168622252
#225536532
C2×#168621792
C2×#168621772
C2×#168621692
C2×C22⋊(C131⋊C8)2
C2×#117×C1312
C2×#485131