Finite Group Databasebeta

← C71×#12857 #31833729

Subgroups — C71×#12857 #31833729

Order 18176.

670 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C71×#2297C211yesmaximal, characteristic
2C71×#1782C212yesmaximal
3C71×#1393C214yesmaximal, complemented
4C22×#41×C71C2211yescharacteristic
5C2×#117×C71C411yescharacteristic, complemented
6C22×C42×C71C411yescharacteristic, complemented
7C2×#99×C71—22no—
8C2×#99×C71C2214yes—
9#391×C71C412yescomplemented
10#390×C71C2212yes—
11#386×C71—22no—
12#381×C71—22no—
13C4×C22⋊C4×C71—22no—
14C2×#41×C71—24no—
15C2×#41×C71—42no—
16C2×#41×C71—42no—
17C22×Q8×C71C2×C411yescharacteristic
18C22×D4×C71D412yes—
19C22×D4×C71C2311yescharacteristic
20C23×C4×C71—21no—
21C23×C4×C71C2×C411yescharacteristic
22C23×C4×C71D412yes—
23#117×C71—42no—
24C2×C4⋊C4×C71C2×C412yes—
25C2×C22⋊C4×C71—21no—
26C2×C22⋊C4×C71—24no—
27C2×C22⋊C4×C71—24no—
28C2×C42×C71—21no—
29C2×C42×C71—21no—
30C2×C42×C71—42no—
31C2×C42×C71C2×C412yes—
32#99×C71—28no—
33#99×C71—44no—
34#95×C71—24no—
35#95×C71—44no—
36C24×C71C22⋊C411yescharacteristic
37C24×C71C2×D412yes—
38#41×C71—416no—
39C2×Q8×C71—22no—
40C2×Q8×C71—41no—
41C2×Q8×C71—41no—
42C2×D4×C71—24no—
43C2×D4×C71—24no—
44C2×D4×C71—42no—
45C2×D4×C71—41no—
46C2×D4×C71—41no—
47C2×D4×C71#4114yes—
48C22×C4×C71—22no—
49C22×C4×C71—22no—
50C22×C4×C71—22no—
51C22×C4×C71—22no—
52C22×C4×C71—24no—
53C22×C4×C71—24no—
54C22×C4×C71—22no—
55C22×C4×C71—42no—
56C22×C4×C71—44no—
57C22×C4×C71—42no—
58C22×C4×C71—44no—
59C22×C4×C71C22⋊C412yes—
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22×C411yescharacteristic
62C4⋊C4×C71—42no—
63C22⋊C4×C71—48no—
64C22⋊C4×C71—42no—
65C22⋊C4×C71—44no—
66C22⋊C4×C71—44no—
67C42×C71—42no—
68C42×C71—42no—
69C42×C71—42no—
70C23×C71—21no—
71C23×C71—21no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—42no—
75C23×C71—44no—
76C23×C71#9912yes—
77C23×C71C2×C22⋊C411yescharacteristic
78C23×C71#12314yes—
79Q8×C71—44no—
80D4×C71—44no—
81D4×C71—48no—
82C2×C4×C71—28no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—24no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—48no—
92C2×C4×C71—42no—
93C2×C4×C71—48no—
94C2×C4×C71—42no—
95C2×C4×C71—44no—
96C2×C4×C71—42no—
97C2×C4×C71—84no—
98C2×C4×C71—88no—
99C2×C4×C71#9912yes—
100C2×C4×C71#11712yes—
101C22×C71—22no—
102C22×C71—22no—
103C22×C71—24no—
104C22×C71—42no—
105C22×C71—44no—
106C22×C71—42no—
107C22×C71—44no—
108C22×C71—42no—
109C22×C71—82no—
110C22×C71#38111yescharacteristic
111C22×C71C2×#9911yescharacteristic
112C22×C71C2×#9911yescharacteristic
113C4×C71—22no—
114C4×C71—22no—
115C4×C71—44no—
116C4×C71—42no—
117C4×C71—82no—
118C4×C71—164no—
119#12857C7111yesmaximal, characteristic, direct product factor, Sylow
120C2×C71—21no—
121C2×C71—21no—
122C2×C71—44no—
123C2×C71—42no—
124C2×C71#181911yescharacteristic
125C2×C71#212612yes—
126#2297C2×C7111yescharacteristic
127#1782C2×C7112yes—
128#1393C2×C7114yescomplemented
129C71#1285711yescharacteristic, direct product factor, Sylow
130C22×#41C22×C7111yescharacteristic
131C2×#117C4×C7111yescharacteristic, complemented
132C22×C42C4×C7111yescharacteristic, complemented
133C2×#99—22no—
134C2×#99C22×C7114yes—
135#391C4×C7112yescomplemented
136#390C22×C7112yes—
137#386—22no—
138#381—22no—
139C4×C22⋊C4—22no—
140C2×#41—24no—
141C2×#41—42no—
142C2×#41—42no—
143C22×Q8C2×C4×C7111yescharacteristic
144C22×D4D4×C7112yes—
145C22×D4C23×C7111yescharacteristic
146C23×C4—21no—
147C23×C4C2×C4×C7111yescharacteristic
148C23×C4D4×C7112yes—
149#117—42no—
150C2×C4⋊C4C2×C4×C7112yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—42no—
157C2×C42C2×C4×C7112yes—
158#99—28no—
159#99—44no—
160#95—24no—
161#95—44no—
162C24C22⋊C4×C7111yescharacteristic
163C24C2×D4×C7112yes—
164#41—416no—
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4#41×C7114yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4C22⋊C4×C7112yes—
186C22×C4C22⋊C4×C7111yescharacteristic
187C22×C4C22×C4×C7111yescharacteristic
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C42—42no—
194C42—42no—
195C42—42no—
196C23—21no—
197C23—22no—
198C23—21no—
199C23—22no—
200C23—42no—
201C23—44no—
202C23#99×C7112yes—
203C23C2×C22⋊C4×C7111yescharacteristic
204C23#123×C7114yes—
205Q8—44no—
206D4—44no—
207D4—48no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—48no—
218C2×C4—42no—
219C2×C4—48no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—84no—
224C2×C4—88no—
225C2×C4#99×C7112yes—
226C2×C4#117×C7112yes—
227C22—22no—
228C22—22no—
229C22—24no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—82no—
236C22#381×C7111yescharacteristic
237C22C2×#99×C7111yescharacteristic
238C22C2×#99×C7111yescharacteristic
239C4—22no—
240C4—22no—
241C4—44no—
242C4—42no—
243C4—82no—
244C4—164no—
245C2—21no—
246C2—21no—
247C2—44no—
248C2—42no—
249C2C71×#181911yescharacteristic
250C2C71×#212612yes—
subgroupG/H
center, socleC22×C71C2×#99
commutatorC22×C4C22×C4×C71
Frattini, cosocleC22×D4C23×C71
prime radicalC22×#41×C71C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C22×#41×C71 ◁ G

Complements

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

NH ≅ G/Ntype
3. C71×#1393C2semidirect
5. C2×#117×C71C4semidirect
6. C22×C42×C71C4semidirect
9. #391×C71C4semidirect
119. #12857C71direct (N × H)
128. #1393C2×C71semidirect
129. C71#12857direct (N × H)
131. C2×#117C4×C71semidirect
132. C22×C42C4×C71semidirect
135. #391C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#22972
C71×#17822
C71×#13932
#1285771