Finite Group Databasebeta

← C71×#7649 #31828521

Subgroups — C71×#7649 #31828521

Order 18176.

468 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×#124×C71C211yesmaximal, characteristic
2C71×#1839C212yesmaximal, complemented
3C71×#1272C212yesmaximal, complemented
4C71×#1270C212yesmaximal
5C2×#124×C71C411yescharacteristic
6C2×C4×Q8×C71C411yescharacteristic
7C2×C4×D4×C71C411yescharacteristic
8#391×C71C2211yescharacteristic
9#390×C71C2211yescharacteristic
10#388×C71—22no—
11#382×C71C411yescharacteristic
12C4×C22⋊C4×C71—22no—
13C43×C71C2211yescharacteristic
14#336×C71—22no—
15#336×C71—22no—
16#336×C71C2214yes—
17C22×Q8×C71C2×C411yescharacteristic
18C22×D4×C71C2×C411yescharacteristic
19C23×C4×C71—21no—
20C22×C8×C71—24no—
21#124×C71—22no—
22#124×C71—22no—
23C4×Q8×C71—42no—
24C4×D4×C71—42no—
25C2×C4⋊C4×C71—21no—
26C2×C4⋊C4×C71C2×C411yescharacteristic
27C2×C4⋊C4×C71C2×C411yescharacteristic
28C2×C22⋊C4×C71—22no—
29C2×C22⋊C4×C71—21no—
30C2×C22⋊C4×C71—22no—
31C2×C42×C71—21no—
32C2×C42×C71C2×C411yescharacteristic
33C2×C42×C71C2×C411yescharacteristic
34C2×C42×C71D412yes—
35C2×C42×C71D412yes—
36C2×C42×C71C2311yescharacteristic
37#95×C71—22no—
38#95×C71—22no—
39C24×C71—21no—
40C2×Q8×C71—21no—
41C2×Q8×C71—41no—
42C2×Q8×C71—41no—
43C2×Q8×C71M1612yes—
44C2×D4×C71—21no—
45C2×D4×C71—41no—
46C2×D4×C71—41no—
47C2×D4×C71M1612yes—
48C22×C4×C71—21no—
49C22×C4×C71—21no—
50C22×C4×C71—21no—
51C22×C4×C71—44no—
52C22×C4×C71—42no—
53C22×C4×C71C22⋊C412yes—
54C22×C4×C71C22⋊C411yescharacteristic
55C22×C4×C71C22⋊C411yescharacteristic
56C22×C4×C71C22×C411yescharacteristic
57C22×C4×C71C2×D412yes—
58C2×C8×C71—44no—
59C2×C8×C71—44no—
60C2×C8×C71—44no—
61C4⋊C4×C71—41no—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—42no—
64C22⋊C4×C71—28no—
65C22⋊C4×C71—42no—
66C22⋊C4×C71—44no—
67C42×C71—22no—
68C42×C71—21no—
69C42×C71—21no—
70C42×C71—24no—
71C42×C71—42no—
72C42×C71—41no—
73C42×C71—42no—
74C23×C71—22no—
75C23×C71—41no—
76C23×C71—41no—
77C23×C71—42no—
78C23×C71—42no—
79C23×C71C2×C22⋊C411yescharacteristic
80Q8×C71—42no—
81Q8×C71—42no—
82D4×C71—42no—
83D4×C71—42no—
84C2×C4×C71—22no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—84no—
102C2×C4×C71#10012yes—
103C2×C4×C71C2×M1611yescharacteristic
104C2×C4×C71C2×M1611yescharacteristic
105C2×C4×C71#13514yes—
106C8×C71—84no—
107C22×C71—42no—
108C22×C71—44no—
109C22×C71—81no—
110C22×C71—82no—
111C22×C71—81no—
112C22×C71#40711yescharacteristic
113C22×C71#41011yescharacteristic
114C22×C71C2×#10011yescharacteristic
115C22×C71#41612yes—
116C22×C71#42112yes—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—24no—
120C4×C71—22no—
121C4×C71—41no—
122C4×C71—42no—
123C4×C71—42no—
124C4×C71—81no—
125C4×C71—81no—
126#7649C7111yesmaximal, characteristic, direct product factor, Sylow
127C2×C71—82no—
128C2×C71#146611yescharacteristic
129C2×C71#148812yes—
130C2×C71#152112yes—
131C2×C71#153311yescharacteristic
132C2×C71#154311yescharacteristic
133C4×#124C2×C7111yescharacteristic
134#1839C2×C7112yescomplemented
135#1272C2×C7112yescomplemented
136#1270C2×C7112yes—
137C71#764911yescharacteristic, direct product factor, Sylow
138C2×#124C4×C7111yescharacteristic
139C2×C4×Q8C4×C7111yescharacteristic
140C2×C4×D4C4×C7111yescharacteristic
141#391C22×C7111yescharacteristic
142#390C22×C7111yescharacteristic
143#388—22no—
144#382C4×C7111yescharacteristic
145C4×C22⋊C4—22no—
146C43C22×C7111yescharacteristic
147#336—22no—
148#336—22no—
149#336C22×C7114yes—
150C22×Q8C2×C4×C7111yescharacteristic
151C22×D4C2×C4×C7111yescharacteristic
152C23×C4—21no—
153C22×C8—24no—
154#124—22no—
155#124—22no—
156C4×Q8—42no—
157C4×D4—42no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×C4×C7111yescharacteristic
160C2×C4⋊C4C2×C4×C7111yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42—21no—
165C2×C42C2×C4×C7111yescharacteristic
166C2×C42C2×C4×C7111yescharacteristic
167C2×C42D4×C7112yes—
168C2×C42D4×C7112yes—
169C2×C42C23×C7111yescharacteristic
170#95—22no—
171#95—22no—
172C24—21no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8M16×C7112yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4M16×C7112yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4C22⋊C4×C7112yes—
187C22×C4C22⋊C4×C7111yescharacteristic
188C22×C4C22⋊C4×C7111yescharacteristic
189C22×C4C22×C4×C7111yescharacteristic
190C22×C4C2×D4×C7112yes—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—44no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C22⋊C4—28no—
198C22⋊C4—42no—
199C22⋊C4—44no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42—24no—
204C42—42no—
205C42—41no—
206C42—42no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23C2×C22⋊C4×C7111yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4#100×C7112yes—
236C2×C4C2×M16×C7111yescharacteristic
237C2×C4C2×M16×C7111yescharacteristic
238C2×C4#135×C7114yes—
239C8—84no—
240C22—42no—
241C22—44no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22#407×C7111yescharacteristic
246C22#410×C7111yescharacteristic
247C22C2×#100×C7111yescharacteristic
248C22#416×C7112yes—
249C22#421×C7112yes—
250C4—24no—
251C4—22no—
252C4—21no—
253C4—21no—
254C4—41no—
255C4—42no—
256C4—42no—
257C4—81no—
258C4—81no—
259C2—82no—
260C2C71×#146611yescharacteristic
261C2C71×#148812yes—
262C2C71×#152112yes—
263C2C71×#153311yescharacteristic
264C2C71×#154311yescharacteristic
subgroupG/H
center, socleC23×C71C2×C22⋊C4
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC22×D4×C71C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C71 ◁ C43×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#1839C2semidirect
3. C71×#1272C2semidirect
126. #7649C71direct (N × H)
134. #1839C2×C71semidirect
135. #1272C2×C71semidirect
137. C71#7649direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#124×C712
C71×#18392
C71×#12722
C71×#12702
#764971