Finite Group Databasebeta

← C2×#455×C71 #22585577

Subgroups — C2×#455×C71 #22585577

Order 9088.

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

normal subgroups only

This group has 760 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×H(5,𝔽2)×C71C211yesmaximal, characteristic
2C2×#137×C71C212yesmaximal, complemented
3C2×#124×C71C211yesmaximal, characteristic, complemented
4#455×C71C218yesmaximal, direct product factor
5C2×#104×C71C212yesmaximal, complemented
6C2×#100×C71C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C71—24no—
8H(5,𝔽2)×C71—22no—
9H(5,𝔽2)×C71C2214yes—
10C2×#41×C71—22no—
11C2×#41×C71C2212yes—
12C22×Q8×C71C2211yescharacteristic, complemented
13C22×D4×C71—22no—
14C22×D4×C71—22no—
15C22×D4×C71C2211yescharacteristic
16#137×C71—24no—
17#137×C71C2218yescomplemented
18C2×Q16×C71—22no—
19C2×SD16×C71—22no—
20C2×M16×C71C2212yescomplemented
21#124×C71—22no—
22#124×C71C2214yescomplemented
23C2×C22⋊C4×C71—22no—
24C2×C42×C71C2211yescharacteristic, complemented
25#104×C71C2218yescomplemented
26#100×C71C2214yescomplemented
27C24×C71—22no—
28C24×C71D412yes—
29#41×C71—28no—
30#41×C71—28no—
31#41×C71—22no—
32#41×C71—22no—
33#41×C71D414yes—
34#41×C71C2314yes—
35C2×Q8×C71—21no—
36C2×Q8×C71—22no—
37C2×Q8×C71—22no—
38C2×Q8×C71D412yes—
39C2×Q8×C71C2312yescomplemented
40C2×D4×C71—24no—
41C2×D4×C71—28no—
42C2×D4×C71—28no—
43C2×D4×C71—22no—
44C2×D4×C71—28no—
45C2×D4×C71—28no—
46C2×D4×C71—22no—
47C2×D4×C71—22no—
48C2×D4×C71D412yes—
49C2×D4×C71D412yes—
50C2×D4×C71C2312yes—
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—21no—
54C22×C4×C71—21no—
55C22×C4×C71C2311yescharacteristic
56Q16×C71—28no—
57SD16×C71—28no—
58M16×C71—22no—
59M16×C71C2314yescomplemented
60C2×C8×C71—22no—
61C22⋊C4×C71—28no—
62C42×C71—21no—
63C42×C71C2312yescomplemented
64C23×C71—28no—
65C23×C71—22no—
66C23×C71—22no—
67C23×C71—22no—
68C23×C71—44no—
69C23×C71—84no—
70C23×C71C2×D411yescharacteristic
71C23×C71C2×D414yes—
72Q8×C71—22no—
73Q8×C71—24no—
74Q8×C71—42no—
75Q8×C71C2×D414yes—
76D4×C71—216no—
77D4×C71—24no—
78D4×C71—24no—
79D4×C71—24no—
80D4×C71—22no—
81D4×C71—24no—
82D4×C71C2×D414yes—
83C2×C4×C71—28no—
84C2×C4×C71—22no—
85C2×C4×C71—24no—
86C2×C4×C71—24no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—42no—
90C2×C4×C71—41no—
91C2×C4×C71—41no—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2×D412yes—
94C2×C4×C71C2×D412yes—
95C2×C4×C71C2411yescharacteristic
96C8×C71—24no—
97C22×C71—22no—
98C22×C71—24no—
99C22×C71—24no—
100C22×C71—44no—
101C22×C71—44no—
102C22×C71—44no—
103C22×C71—82no—
104C22×C71—88no—
105C22×C71#12011yescharacteristic
106C22×C71#12011yescharacteristic
107C22×C71C22×D411yescharacteristic
108C4×C71—24no—
109C4×C71—42no—
110C4×C71—42no—
111C4×C71#12012yes—
112C4×C71C22×D412yes—
113C2×C71—21no—
114C2×C71—21no—
115C2×C71—44no—
116C2×C71—44no—
117C2×C71#45512yesdirect product factor
118C2×C71C2×#12011yescharacteristic
119C2×#455C7111yesmaximal, characteristic, direct product factor, Sylow
120C71C2×#45511yescharacteristic, direct product factor, Sylow
121C2×H(5,𝔽2)C2×C7111yescharacteristic
122C2×#137C2×C7112yescomplemented
123C2×#124C2×C7111yescharacteristic, complemented
124#455C2×C7118yesdirect product factor
125C2×#104C2×C7112yescomplemented
126C2×#100C2×C7111yescharacteristic, complemented
127H(5,𝔽2)—24no—
128H(5,𝔽2)—22no—
129H(5,𝔽2)C22×C7114yes—
130C2×#41—22no—
131C2×#41C22×C7112yes—
132C22×Q8C22×C7111yescharacteristic, complemented
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22×C7111yescharacteristic
136#137—24no—
137#137C22×C7118yescomplemented
138C2×Q16—22no—
139C2×SD16—22no—
140C2×M16C22×C7112yescomplemented
141#124—22no—
142#124C22×C7114yescomplemented
143C2×C22⋊C4—22no—
144C2×C42C22×C7111yescharacteristic, complemented
145#104C22×C7118yescomplemented
146#100C22×C7114yescomplemented
147C24—22no—
148C24D4×C7112yes—
149#41—28no—
150#41—28no—
151#41—22no—
152#41—22no—
153#41D4×C7114yes—
154#41C23×C7114yes—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8D4×C7112yes—
159C2×Q8C23×C7112yescomplemented
160C2×D4—24no—
161C2×D4—28no—
162C2×D4—28no—
163C2×D4—22no—
164C2×D4—28no—
165C2×D4—28no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4D4×C7112yes—
169C2×D4D4×C7112yes—
170C2×D4C23×C7112yes—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4C23×C7111yescharacteristic
176Q16—28no—
177SD16—28no—
178M16—22no—
179M16C23×C7114yescomplemented
180C2×C8—22no—
181C22⋊C4—28no—
182C42—21no—
183C42C23×C7112yescomplemented
184C23—28no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—44no—
189C23—84no—
190C23C2×D4×C7111yescharacteristic
191C23C2×D4×C7114yes—
192Q8—22no—
193Q8—24no—
194Q8—42no—
195Q8C2×D4×C7114yes—
196D4—216no—
197D4—24no—
198D4—24no—
199D4—24no—
200D4—22no—
201D4—24no—
202D4C2×D4×C7114yes—
203C2×C4—28no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4C2×D4×C7111yescharacteristic
213C2×C4C2×D4×C7112yes—
214C2×C4C2×D4×C7112yes—
215C2×C4C24×C7111yescharacteristic
216C8—24no—
217C22—22no—
218C22—24no—
219C22—24no—
220C22—44no—
221C22—44no—
222C22—44no—
223C22—82no—
224C22—88no—
225C22#120×C7111yescharacteristic
226C22#120×C7111yescharacteristic
227C22C22×D4×C7111yescharacteristic
228C4—24no—
229C4—42no—
230C4—42no—
231C4#120×C7112yes—
232C4C22×D4×C7112yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—44no—
237C2#455×C7112yesdirect product factor
238C2C2×#120×C7111yescharacteristic
subgroupG/H
center, socleC22×C71#120
commutator, Frattini, cosocleC2×C4C24×C71
prime radicalC2×H(5,𝔽2)×C71C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#137×C71C2semidirect
3. C2×#124×C71C2semidirect
4. #455×C71C2direct (N × H)
5. C2×#104×C71C2semidirect
6. C2×#100×C71C2semidirect
12. C22×Q8×C71C22semidirect
17. #137×C71C22semidirect
20. C2×M16×C71C22semidirect
22. #124×C71C22semidirect
24. C2×C42×C71C22semidirect
25. #104×C71C22semidirect
26. #100×C71C22semidirect
39. C2×Q8×C71C23semidirect
59. M16×C71C23semidirect
63. C42×C71C23semidirect
117. C2×C71#455direct (N × H)
119. C2×#455C71direct (N × H)
120. C71C2×#455direct (N × H)
122. C2×#137C2×C71semidirect
123. C2×#124C2×C71semidirect
124. #455C2×C71direct (N × H)
125. C2×#104C2×C71semidirect
126. C2×#100C2×C71semidirect
132. C22×Q8C22×C71semidirect
137. #137C22×C71semidirect
140. C2×M16C22×C71semidirect
142. #124C22×C71semidirect
144. C2×C42C22×C71semidirect
145. #104C22×C71semidirect
146. #100C22×C71semidirect
159. C2×Q8C23×C71semidirect
179. M16C23×C71semidirect
183. C42C23×C71semidirect
237. C2#455×C71direct (N × H)

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C712
C2×#137×C712
C2×#124×C712
#455×C712
C2×#104×C712
C2×#100×C712
C2×#45571