Finite Group Databasebeta

← C2×#448×C71 #22585560

Subgroups — C2×#448×C71 #22585560

Order 9088.

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

normal subgroups only

This group has 792 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×Q8×C71C211yesmaximal, characteristic, complemented
2C22×SD16×C71C212yesmaximal, complemented
3C2×#121×C71C211yesmaximal, characteristic
4#448×C71C218yesmaximal, direct product factor
5C2×#103×C71C212yesmaximal, complemented
6C2×C22⋊C8×C71C211yesmaximal, characteristic, complemented
7C22×Q8×C71—28no—
8C22×Q8×C71—22no—
9C22×Q8×C71—21no—
10C22×Q8×C71C2212yes—
11C22×Q8×C71C2214yescomplemented
12C22×D4×C71—21no—
13C22×D4×C71C2211yescharacteristic
14C23×C4×C71—21no—
15C23×C4×C71C2211yescharacteristic
16C2×SD16×C71—28no—
17C2×SD16×C71C2218yescomplemented
18C22×C8×C71C2212yes—
19#121×C71—22no—
20#121×C71C2214yes—
21C2×C4⋊C4×C71C2211yescharacteristic
22C2×C22⋊C4×C71—21no—
23#103×C71C2218yescomplemented
24C22⋊C8×C71C2214yescomplemented
25C24×C71—21no—
26C24×C71D411yescharacteristic
27C2×Q8×C71—216no—
28C2×Q8×C71—28no—
29C2×Q8×C71—216no—
30C2×Q8×C71—24no—
31C2×Q8×C71—22no—
32C2×Q8×C71—24no—
33C2×Q8×C71D418yescomplemented
34C2×Q8×C71C2314yes—
35C2×D4×C71—24no—
36C2×D4×C71—24no—
37C2×D4×C71—21no—
38C2×D4×C71—44no—
39C2×D4×C71C2312yes—
40C22×C4×C71—28no—
41C22×C4×C71—24no—
42C22×C4×C71—22no—
43C22×C4×C71—24no—
44C22×C4×C71—21no—
45C22×C4×C71D412yes—
46C22×C4×C71D411yescharacteristic
47C22×C4×C71C2312yes—
48C22×C4×C71C2311yescharacteristic
49SD16×C71—216no—
50C2×C8×C71—24no—
51C2×C8×C71C2314yes—
52C4⋊C4×C71—21no—
53C4⋊C4×C71C2312yes—
54C22⋊C4×C71—24no—
55C23×C71—22no—
56C23×C71—22no—
57C23×C71—22no—
58C23×C71—42no—
59C23×C71—42no—
60C23×C71—42no—
61C23×C71SD1614yescomplemented
62C23×C71C2×D412yes—
63C23×C71C2×D411yescharacteristic
64Q8×C71—216no—
65Q8×C71—28no—
66Q8×C71—24no—
67Q8×C71C2×D418yescomplemented
68D4×C71—24no—
69D4×C71—48no—
70D4×C71—42no—
71C2×C4×C71—216no—
72C2×C4×C71—28no—
73C2×C4×C71—22no—
74C2×C4×C71—24no—
75C2×C4×C71—24no—
76C2×C4×C71—24no—
77C2×C4×C71—22no—
78C2×C4×C71—42no—
79C2×C4×C71C2×D412yes—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D414yes—
82C2×C4×C71C2411yescharacteristic
83C8×C71—24no—
84C22×C71—22no—
85C22×C71—22no—
86C22×C71—28no—
87C22×C71—42no—
88C22×C71—42no—
89C22×C71—42no—
90C22×C71—84no—
91C22×C71#12012yes—
92C22×C71C2×SD1612yes—
93C22×C71C2×SD1614yescomplemented
94C22×C71#13712yes—
95C22×C71C22×D411yescharacteristic
96C4×C71—28no—
97C4×C71—22no—
98C4×C71—42no—
99C4×C71#12012yes—
100C4×C71C22×D412yes—
101C2×C71—24no—
102C2×C71—82no—
103C2×C71#44814yesdirect product factor
104C2×C71C2×#12011yescharacteristic
105C2×C71C22×SD1611yescharacteristic
106C2×C71C2×#13711yescharacteristic
107C2×#448C7111yesmaximal, characteristic, direct product factor, Sylow
108C71C2×#44811yescharacteristic, direct product factor, Sylow
109C23×Q8C2×C7111yescharacteristic, complemented
110C22×SD16C2×C7112yescomplemented
111C2×#121C2×C7111yescharacteristic
112#448C2×C7118yesdirect product factor
113C2×#103C2×C7112yescomplemented
114C2×C22⋊C8C2×C7111yescharacteristic, complemented
115C22×Q8—28no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8C22×C7112yes—
119C22×Q8C22×C7114yescomplemented
120C22×D4—21no—
121C22×D4C22×C7111yescharacteristic
122C23×C4—21no—
123C23×C4C22×C7111yescharacteristic
124C2×SD16—28no—
125C2×SD16C22×C7118yescomplemented
126C22×C8C22×C7112yes—
127#121—22no—
128#121C22×C7114yes—
129C2×C4⋊C4C22×C7111yescharacteristic
130C2×C22⋊C4—21no—
131#103C22×C7118yescomplemented
132C22⋊C8C22×C7114yescomplemented
133C24—21no—
134C24D4×C7111yescharacteristic
135C2×Q8—216no—
136C2×Q8—28no—
137C2×Q8—216no—
138C2×Q8—24no—
139C2×Q8—22no—
140C2×Q8—24no—
141C2×Q8D4×C7118yescomplemented
142C2×Q8C23×C7114yes—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—21no—
146C2×D4—44no—
147C2×D4C23×C7112yes—
148C22×C4—28no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—21no—
153C22×C4D4×C7111yescharacteristic
154C22×C4D4×C7112yes—
155C22×C4C23×C7112yes—
156C22×C4C23×C7111yescharacteristic
157SD16—216no—
158C2×C8—24no—
159C2×C8C23×C7114yes—
160C4⋊C4—21no—
161C4⋊C4C23×C7112yes—
162C22⋊C4—24no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23SD16×C7114yescomplemented
170C23C2×D4×C7112yes—
171C23C2×D4×C7111yescharacteristic
172Q8—216no—
173Q8—28no—
174Q8—24no—
175Q8C2×D4×C7118yescomplemented
176D4—24no—
177D4—48no—
178D4—42no—
179C2×C4—216no—
180C2×C4—28no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4C2×D4×C7112yes—
188C2×C4C2×D4×C7111yescharacteristic
189C2×C4C2×D4×C7114yes—
190C2×C4C24×C7111yescharacteristic
191C8—24no—
192C22—28no—
193C22—22no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—84no—
199C22#120×C7112yes—
200C22C2×SD16×C7112yes—
201C22C2×SD16×C7114yescomplemented
202C22#137×C7112yes—
203C22C22×D4×C7111yescharacteristic
204C4—28no—
205C4—22no—
206C4—42no—
207C4#120×C7112yes—
208C4C22×D4×C7112yes—
209C2—24no—
210C2—82no—
211C2#448×C7114yesdirect product factor
212C2C2×#120×C7111yescharacteristic
213C2C22×SD16×C7111yescharacteristic
214C2C2×#137×C7111yescharacteristic
subgroupG/H
center, socleC23×C71C2×D4
commutator, Frattini, cosocleC2×C4C24×C71
prime radicalC2×#121×C71C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×Q8×C71C2semidirect
2. C22×SD16×C71C2semidirect
4. #448×C71C2direct (N × H)
5. C2×#103×C71C2semidirect
6. C2×C22⋊C8×C71C2semidirect
11. C22×Q8×C71C22semidirect
17. C2×SD16×C71C22semidirect
23. #103×C71C22semidirect
24. C22⋊C8×C71C22semidirect
33. C2×Q8×C71D4semidirect
61. C23×C71SD16semidirect
67. Q8×C71C2×D4semidirect
93. C22×C71C2×SD16semidirect
103. C2×C71#448direct (N × H)
107. C2×#448C71direct (N × H)
108. C71C2×#448direct (N × H)
109. C23×Q8C2×C71semidirect
110. C22×SD16C2×C71semidirect
112. #448C2×C71direct (N × H)
113. C2×#103C2×C71semidirect
114. C2×C22⋊C8C2×C71semidirect
119. C22×Q8C22×C71semidirect
125. C2×SD16C22×C71semidirect
131. #103C22×C71semidirect
132. C22⋊C8C22×C71semidirect
141. C2×Q8D4×C71semidirect
169. C23SD16×C71semidirect
175. Q8C2×D4×C71semidirect
201. C22C2×SD16×C71semidirect
211. C2#448×C71direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Q8×C712
C22×SD16×C712
C2×#121×C712
#448×C712
C2×#103×C712
C2×C22⋊C8×C712
C2×#44871