Finite Group Databasebeta

← C2×#450×C97 #27138955

Subgroups — C2×#450×C97 #27138955

Order 12416.

996 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 248 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
1C23×D4×C97C211yesmaximal, characteristic
2C22×SD16×C97C212yesmaximal, complemented
3C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
4#450×C97C218yesmaximal, direct product factor
5C2×#102×C97C212yesmaximal, complemented
6C2×C22⋊C8×C97C211yesmaximal, characteristic, complemented
7C25×C97—21no—
8C22×Q8×C97C2211yescharacteristic, complemented
9C22×D4×C97—28no—
10C22×D4×C97—22no—
11C22×D4×C97—21no—
12C22×D4×C97C2214yes—
13C22×D4×C97C2212yes—
14C23×C4×C97C2211yescharacteristic
15C2×SD16×C97—28no—
16C2×SD16×C97C2218yescomplemented
17C22×C8×C97C2212yescomplemented
18C22⋊Q8×C97—22no—
19C22⋊Q8×C97C2214yescomplemented
20C2×C4⋊C4×C97—21no—
21C2×C4⋊C4×C97C2211yescharacteristic, complemented
22C2×C22⋊C4×C97—21no—
23#102×C97C2218yescomplemented
24C22⋊C8×C97C2214yescomplemented
25C24×C97—28no—
26C24×C97—24no—
27C24×C97—22no—
28C24×C97—44no—
29C24×C97—44no—
30C24×C97D411yescharacteristic
31C2×Q8×C97—21no—
32C2×Q8×C97—24no—
33C2×Q8×C97C2312yescomplemented
34C2×D4×C97—28no—
35C2×D4×C97—216no—
36C2×D4×C97—216no—
37C2×D4×C97—24no—
38C2×D4×C97—22no—
39C2×D4×C97—24no—
40C2×D4×C97D418yes—
41C2×D4×C97C2314yes—
42C22×C4×C97—24no—
43C22×C4×C97—21no—
44C22×C4×C97—21no—
45C22×C4×C97D412yes—
46C22×C4×C97D411yescharacteristic
47C22×C4×C97C2312yes—
48C22×C4×C97C2311yescharacteristic
49SD16×C97—216no—
50C2×C8×C97—24no—
51C2×C8×C97C2314yescomplemented
52C4⋊C4×C97—24no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97C2312yescomplemented
55C22⋊C4×C97—24no—
56C23×C97—22no—
57C23×C97—22no—
58C23×C97—216no—
59C23×C97—28no—
60C23×C97—24no—
61C23×C97—48no—
62C23×C97—432no—
63C23×C97—44no—
64C23×C97—48no—
65C23×C97—44no—
66C23×C97SD1614yescomplemented
67C23×C97C2×D412yes—
68C23×C97C2×D411yescharacteristic
69Q8×C97—24no—
70Q8×C97—42no—
71D4×C97—216no—
72D4×C97—28no—
73D4×C97—24no—
74D4×C97C2×D418yes—
75C2×C4×C97—22no—
76C2×C4×C97—24no—
77C2×C4×C97—24no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97—42no—
81C2×C4×C97—42no—
82C2×C4×C97C2×D412yes—
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2×D414yes—
85C2×C4×C97C2411yescharacteristic
86C8×C97—24no—
87C22×C97—22no—
88C22×C97—22no—
89C22×C97—28no—
90C22×C97—28no—
91C22×C97—432no—
92C22×C97—44no—
93C22×C97—44no—
94C22×C97—416no—
95C22×C97#12012yes—
96C22×C97C2×SD1612yes—
97C22×C97C2×SD1614yescomplemented
98C22×C97AGL(1,ℤ/8)12yes—
99C22×C97C22×D411yescharacteristic
100C4×C97—22no—
101C4×C97—42no—
102C4×C97—42no—
103C4×C97#12012yes—
104C4×C97C22×D412yes—
105C2×C97—24no—
106C2×C97—48no—
107C2×C97#45014yesdirect product factor
108C2×C97C2×#12011yescharacteristic
109C2×C97C22×SD1611yescharacteristic
110C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
111C2×#450C9711yesmaximal, characteristic, direct product factor, Sylow
112C97C2×#45011yescharacteristic, direct product factor, Sylow
113C23×D4C2×C9711yescharacteristic
114C22×SD16C2×C9712yescomplemented
115C2×C22⋊Q8C2×C9711yescharacteristic, complemented
116#450C2×C9718yesdirect product factor
117C2×#102C2×C9712yescomplemented
118C2×C22⋊C8C2×C9711yescharacteristic, complemented
119C25—21no—
120C22×Q8C22×C9711yescharacteristic, complemented
121C22×D4—28no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4C22×C9714yes—
125C22×D4C22×C9712yes—
126C23×C4C22×C9711yescharacteristic
127C2×SD16—28no—
128C2×SD16C22×C9718yescomplemented
129C22×C8C22×C9712yescomplemented
130C22⋊Q8—22no—
131C22⋊Q8C22×C9714yescomplemented
132C2×C4⋊C4—21no—
133C2×C4⋊C4C22×C9711yescharacteristic, complemented
134C2×C22⋊C4—21no—
135#102C22×C9718yescomplemented
136C22⋊C8C22×C9714yescomplemented
137C24—28no—
138C24—24no—
139C24—22no—
140C24—44no—
141C24—44no—
142C24D4×C9711yescharacteristic
143C2×Q8—21no—
144C2×Q8—24no—
145C2×Q8C23×C9712yescomplemented
146C2×D4—28no—
147C2×D4—216no—
148C2×D4—216no—
149C2×D4—24no—
150C2×D4—22no—
151C2×D4—24no—
152C2×D4D4×C9718yes—
153C2×D4C23×C9714yes—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4D4×C9711yescharacteristic
158C22×C4D4×C9712yes—
159C22×C4C23×C9712yes—
160C22×C4C23×C9711yescharacteristic
161SD16—216no—
162C2×C8—24no—
163C2×C8C23×C9714yescomplemented
164C4⋊C4—24no—
165C4⋊C4—21no—
166C4⋊C4C23×C9712yescomplemented
167C22⋊C4—24no—
168C23—22no—
169C23—22no—
170C23—216no—
171C23—28no—
172C23—24no—
173C23—48no—
174C23—432no—
175C23—44no—
176C23—48no—
177C23—44no—
178C23SD16×C9714yescomplemented
179C23C2×D4×C9712yes—
180C23C2×D4×C9711yescharacteristic
181Q8—24no—
182Q8—42no—
183D4—216no—
184D4—28no—
185D4—24no—
186D4C2×D4×C9718yes—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4C2×D4×C9712yes—
195C2×C4C2×D4×C9711yescharacteristic
196C2×C4C2×D4×C9714yes—
197C2×C4C24×C9711yescharacteristic
198C8—24no—
199C22—22no—
200C22—22no—
201C22—28no—
202C22—28no—
203C22—432no—
204C22—44no—
205C22—44no—
206C22—416no—
207C22#120×C9712yes—
208C22C2×SD16×C9712yes—
209C22C2×SD16×C9714yescomplemented
210C22AGL(1,ℤ/8)×C9712yes—
211C22C22×D4×C9711yescharacteristic
212C4—22no—
213C4—42no—
214C4—42no—
215C4#120×C9712yes—
216C4C22×D4×C9712yes—
217C2—24no—
218C2—48no—
219C2#450×C9714yesdirect product factor
220C2C2×#120×C9711yescharacteristic
221C2C22×SD16×C9711yescharacteristic
222C2C2×AGL(1,ℤ/8)×C9711yescharacteristic
subgroupG/H
center, socleC23×C97C2×D4
commutator, Frattini, cosocleC2×C4C24×C97
prime radicalC23×D4×C97C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C97 ◁ C23×C4×C97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×SD16×C97C2semidirect
3. C2×C22⋊Q8×C97C2semidirect
4. #450×C97C2direct (N × H)
5. C2×#102×C97C2semidirect
6. C2×C22⋊C8×C97C2semidirect
8. C22×Q8×C97C22semidirect
16. C2×SD16×C97C22semidirect
17. C22×C8×C97C22semidirect
19. C22⋊Q8×C97C22semidirect
21. C2×C4⋊C4×C97C22semidirect
23. #102×C97C22semidirect
24. C22⋊C8×C97C22semidirect
33. C2×Q8×C97C23semidirect
51. C2×C8×C97C23semidirect
54. C4⋊C4×C97C23semidirect
66. C23×C97SD16semidirect
97. C22×C97C2×SD16semidirect
107. C2×C97#450direct (N × H)
111. C2×#450C97direct (N × H)
112. C97C2×#450direct (N × H)
114. C22×SD16C2×C97semidirect
115. C2×C22⋊Q8C2×C97semidirect
116. #450C2×C97direct (N × H)
117. C2×#102C2×C97semidirect
118. C2×C22⋊C8C2×C97semidirect
120. C22×Q8C22×C97semidirect
128. C2×SD16C22×C97semidirect
129. C22×C8C22×C97semidirect
131. C22⋊Q8C22×C97semidirect
133. C2×C4⋊C4C22×C97semidirect
135. #102C22×C97semidirect
136. C22⋊C8C22×C97semidirect
145. C2×Q8C23×C97semidirect
163. C2×C8C23×C97semidirect
166. C4⋊C4C23×C97semidirect
178. C23SD16×C97semidirect
209. C22C2×SD16×C97semidirect
219. C2#450×C97direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4×C972
C22×SD16×C972
C2×C22⋊Q8×C972
#450×C972
C2×#102×C972
C2×C22⋊C8×C972
C2×#45097