Finite Group Databasebeta

← C2×#454×C131 #30663199

Subgroups — C2×#454×C131 #30663199

Order 16768.

728 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 216 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
1C2×2−1+4×C131C211yesmaximal, characteristic, complemented
2C2×AGL(1,ℤ/8)×C131C212yesmaximal, complemented
3C2×#124×C131C211yesmaximal, characteristic, complemented
4#454×C131C218yesmaximal, direct product factor
5C2×#104×C131C212yesmaximal, complemented
6C2×#101×C131C211yesmaximal, characteristic, complemented
72−1+4×C131—24no—
82−1+4×C131—22no—
92−1+4×C131C2214yescomplemented
10C2×#41×C131—22no—
11C2×#41×C131—22no—
12C2×#41×C131C2212yes—
13C22×Q8×C131—22no—
14C22×Q8×C131C2211yescharacteristic
15C22×D4×C131C2211yescharacteristic
16AGL(1,ℤ/8)×C131—24no—
17AGL(1,ℤ/8)×C131C2218yescomplemented
18C2×SD16×C131—22no—
19C2×D8×C131—22no—
20C2×M16×C131C2212yes—
21#124×C131—22no—
22#124×C131C2214yescomplemented
23C2×C22⋊C4×C131—22no—
24C2×C42×C131C2211yescharacteristic
25#104×C131C2218yescomplemented
26#101×C131C2214yescomplemented
27C24×C131—21no—
28#41×C131—28no—
29#41×C131—28no—
30#41×C131—28no—
31#41×C131—28no—
32#41×C131—22no—
33#41×C131—22no—
34#41×C131D414yes—
35#41×C131C2314yes—
36C2×Q8×C131—22no—
37C2×Q8×C131—28no—
38C2×Q8×C131—28no—
39C2×Q8×C131—22no—
40C2×Q8×C131—22no—
41C2×Q8×C131D412yes—
42C2×Q8×C131D412yes—
43C2×Q8×C131C2312yes—
44C2×D4×C131—22no—
45C2×D4×C131—22no—
46C2×D4×C131—22no—
47C2×D4×C131—22no—
48C2×D4×C131—21no—
49C2×D4×C131D412yes—
50C2×D4×C131C2312yes—
51C22×C4×C131—24no—
52C22×C4×C131—22no—
53C22×C4×C131—21no—
54C22×C4×C131D412yes—
55C22×C4×C131C2311yescharacteristic
56SD16×C131—28no—
57D8×C131—28no—
58M16×C131—22no—
59M16×C131C2314yes—
60C2×C8×C131—22no—
61C22⋊C4×C131—28no—
62C42×C131—21no—
63C42×C131C2312yes—
64C23×C131—22no—
65C23×C131—22no—
66C23×C131—44no—
67C23×C131—41no—
68C23×C131—41no—
69C23×C131C2×D411yescharacteristic
70Q8×C131—28no—
71Q8×C131—22no—
72Q8×C131—24no—
73Q8×C131—24no—
74Q8×C131C2×D414yes—
75D4×C131—28no—
76D4×C131—24no—
77D4×C131—24no—
78D4×C131—22no—
79D4×C131—24no—
80D4×C131—42no—
81D4×C131C2×D414yes—
82C2×C4×C131—216no—
83C2×C4×C131—22no—
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—24no—
87C2×C4×C131—24no—
88C2×C4×C131—22no—
89C2×C4×C131—42no—
90C2×C4×C131C2×D414yes—
91C2×C4×C131C2×D411yescharacteristic
92C2×C4×C131C2×D412yes—
93C2×C4×C131C2×D412yes—
94C2×C4×C131C2411yescharacteristic
95C8×C131—24no—
96C22×C131—22no—
97C22×C131—24no—
98C22×C131—44no—
99C22×C131—44no—
100C22×C131—42no—
101C22×C131—82no—
102C22×C131—82no—
103C22×C131#12011yescharacteristic
104C22×C131#12011yescharacteristic
105C22×C131C22×D411yescharacteristic
106C4×C131—24no—
107C4×C131—24no—
108C4×C131—42no—
109C4×C131#12012yes—
110C4×C131C22×D412yes—
111C2×C131—21no—
112C2×C131—21no—
113C2×C131—44no—
114C2×C131—82no—
115C2×C131#45412yesdirect product factor
116C2×C131C2×#12011yescharacteristic
117C131C2×#45411yescharacteristic, direct product factor, Sylow
118C2×#454C13111yesmaximal, characteristic, direct product factor, Sylow
119C2×2−1+4C2×C13111yescharacteristic, complemented
120C2×AGL(1,ℤ/8)C2×C13112yescomplemented
121C2×#124C2×C13111yescharacteristic, complemented
122#454C2×C13118yesdirect product factor
123C2×#104C2×C13112yescomplemented
124C2×#101C2×C13111yescharacteristic, complemented
1252−1+4—24no—
1262−1+4—22no—
1272−1+4C22×C13114yescomplemented
128C2×#41—22no—
129C2×#41—22no—
130C2×#41C22×C13112yes—
131C22×Q8—22no—
132C22×Q8C22×C13111yescharacteristic
133C22×D4C22×C13111yescharacteristic
134AGL(1,ℤ/8)—24no—
135AGL(1,ℤ/8)C22×C13118yescomplemented
136C2×SD16—22no—
137C2×D8—22no—
138C2×M16C22×C13112yes—
139#124—22no—
140#124C22×C13114yescomplemented
141C2×C22⋊C4—22no—
142C2×C42C22×C13111yescharacteristic
143#104C22×C13118yescomplemented
144#101C22×C13114yescomplemented
145C24—21no—
146#41—28no—
147#41—28no—
148#41—28no—
149#41—28no—
150#41—22no—
151#41—22no—
152#41D4×C13114yes—
153#41C23×C13114yes—
154C2×Q8—22no—
155C2×Q8—28no—
156C2×Q8—28no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8D4×C13112yes—
160C2×Q8D4×C13112yes—
161C2×Q8C23×C13112yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—21no—
167C2×D4D4×C13112yes—
168C2×D4C23×C13112yes—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4D4×C13112yes—
173C22×C4C23×C13111yescharacteristic
174SD16—28no—
175D8—28no—
176M16—22no—
177M16C23×C13114yes—
178C2×C8—22no—
179C22⋊C4—28no—
180C42—21no—
181C42C23×C13112yes—
182C23—22no—
183C23—22no—
184C23—44no—
185C23—41no—
186C23—41no—
187C23C2×D4×C13111yescharacteristic
188Q8—28no—
189Q8—22no—
190Q8—24no—
191Q8—24no—
192Q8C2×D4×C13114yes—
193D4—28no—
194D4—24no—
195D4—24no—
196D4—22no—
197D4—24no—
198D4—42no—
199D4C2×D4×C13114yes—
200C2×C4—216no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4C2×D4×C13114yes—
209C2×C4C2×D4×C13111yescharacteristic
210C2×C4C2×D4×C13112yes—
211C2×C4C2×D4×C13112yes—
212C2×C4C24×C13111yescharacteristic
213C8—24no—
214C22—22no—
215C22—24no—
216C22—44no—
217C22—44no—
218C22—42no—
219C22—82no—
220C22—82no—
221C22#120×C13111yescharacteristic
222C22#120×C13111yescharacteristic
223C22C22×D4×C13111yescharacteristic
224C4—24no—
225C4—24no—
226C4—42no—
227C4#120×C13112yes—
228C4C22×D4×C13112yes—
229C2—21no—
230C2—21no—
231C2—44no—
232C2—82no—
233C2#454×C13112yesdirect product factor
234C2C2×#120×C13111yescharacteristic
subgroupG/H
center, socleC22×C131#120
commutator, Frattini, cosocleC2×C4C24×C131
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C22×C4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4×C131C2semidirect
2. C2×AGL(1,ℤ/8)×C131C2semidirect
3. C2×#124×C131C2semidirect
4. #454×C131C2direct (N × H)
5. C2×#104×C131C2semidirect
6. C2×#101×C131C2semidirect
9. 2−1+4×C131C22semidirect
17. AGL(1,ℤ/8)×C131C22semidirect
22. #124×C131C22semidirect
25. #104×C131C22semidirect
26. #101×C131C22semidirect
115. C2×C131#454direct (N × H)
117. C131C2×#454direct (N × H)
118. C2×#454C131direct (N × H)
119. C2×2−1+4C2×C131semidirect
120. C2×AGL(1,ℤ/8)C2×C131semidirect
121. C2×#124C2×C131semidirect
122. #454C2×C131direct (N × H)
123. C2×#104C2×C131semidirect
124. C2×#101C2×C131semidirect
127. 2−1+4C22×C131semidirect
135. AGL(1,ℤ/8)C22×C131semidirect
140. #124C22×C131semidirect
143. #104C22×C131semidirect
144. #101C22×C131semidirect
233. C2#454×C131direct (N × H)

Maximal subgroups

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

subgroupindex
C2×2−1+4×C1312
C2×AGL(1,ℤ/8)×C1312
C2×#124×C1312
#454×C1312
C2×#104×C1312
C2×#101×C1312
C2×#454131