Finite Group Databasebeta

← C71×#12859 #31833731

Subgroups — C71×#12859 #31833731

Order 18176.

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

normal subgroups only

This group has 642 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×#2311C211yesmaximal, characteristic
2C71×#1782C212yesmaximal
3C71×#1393C214yesmaximal, complemented
4C22×#41×C71C2211yescharacteristic
5C2×#117×C71C411yescharacteristic, complemented
6C22×C4⋊C4×C71C411yescharacteristic, complemented
7C2×#99×C71—22no—
8C2×#99×C71C2214yes—
9#391×C71C412yescomplemented
10#390×C71C2212yes—
11#386×C71—22no—
12#381×C71—22no—
13#380×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×C71—21no—
25C2×C4⋊C4×C71—21no—
26C2×C4⋊C4×C71—42no—
27C2×C4⋊C4×C71C2×C412yes—
28C2×C22⋊C4×C71—21no—
29C2×C22⋊C4×C71—24no—
30C2×C22⋊C4×C71—24no—
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—24no—
52C22×C4×C71—24no—
53C22×C4×C71—22no—
54C22×C4×C71—22no—
55C22×C4×C71—42no—
56C22×C4×C71—42no—
57C22×C4×C71—44no—
58C22×C4×C71—82no—
59C22×C4×C71C22⋊C412yes—
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22×C411yescharacteristic
62C4⋊C4×C71—42no—
63C4⋊C4×C71—42no—
64C4⋊C4×C71—42no—
65C22⋊C4×C71—42no—
66C22⋊C4×C71—44no—
67C22⋊C4×C71—44no—
68C22⋊C4×C71—84no—
69C42×C71—42no—
70C23×C71—21no—
71C23×C71—21no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—41no—
75C23×C71—41no—
76C23×C71—42no—
77C23×C71—42no—
78C23×C71#9912yes—
79C23×C71C2×C22⋊C411yescharacteristic
80C23×C71#12314yes—
81Q8×C71—44no—
82D4×C71—44no—
83D4×C71—48no—
84C2×C4×C71—28no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—48no—
93C2×C4×C71—42no—
94C2×C4×C71—44no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—84no—
99C2×C4×C71—84no—
100C2×C4×C71—84no—
101C2×C4×C71—84no—
102C2×C4×C71#9912yes—
103C2×C4×C71#11712yes—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71—22no—
107C22×C71—24no—
108C22×C71—42no—
109C22×C71—44no—
110C22×C71—42no—
111C22×C71—41no—
112C22×C71—42no—
113C22×C71—42no—
114C22×C71—41no—
115C22×C71—82no—
116C22×C71#38111yescharacteristic
117C22×C71C2×#9911yescharacteristic
118C22×C71C2×#9911yescharacteristic
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—42no—
122C4×C71—82no—
123C4×C71—82no—
124C4×C71—164no—
125#12859C7111yesmaximal, characteristic, direct product factor, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71—44no—
129C2×C71—42no—
130C2×C71#181911yescharacteristic
131C2×C71#212811yescharacteristic
132C2×C71#212911yescharacteristic
133#2311C2×C7111yescharacteristic
134#1782C2×C7112yes—
135#1393C2×C7114yescomplemented
136C71#1285911yescharacteristic, direct product factor, Sylow
137C22×#41C22×C7111yescharacteristic
138C2×#117C4×C7111yescharacteristic, complemented
139C22×C4⋊C4C4×C7111yescharacteristic, complemented
140C2×#99—22no—
141C2×#99C22×C7114yes—
142#391C4×C7112yescomplemented
143#390C22×C7112yes—
144#386—22no—
145#381—22no—
146#380—22no—
147C2×#41—24no—
148C2×#41—42no—
149C2×#41—42no—
150C22×Q8C2×C4×C7111yescharacteristic
151C22×D4D4×C7112yes—
152C22×D4C23×C7111yescharacteristic
153C23×C4—21no—
154C23×C4C2×C4×C7111yescharacteristic
155C23×C4D4×C7112yes—
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4C2×C4×C7112yes—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C42C2×C4×C7112yes—
165#99—28no—
166#99—44no—
167#95—24no—
168#95—44no—
169C24C22⋊C4×C7111yescharacteristic
170C24C2×D4×C7112yes—
171#41—416no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—42no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4#41×C7114yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—82no—
192C22×C4C22⋊C4×C7112yes—
193C22×C4C22⋊C4×C7111yescharacteristic
194C22×C4C22×C4×C7111yescharacteristic
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C22⋊C4—84no—
202C42—42no—
203C23—21no—
204C23—22no—
205C23—21no—
206C23—22no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23—42no—
211C23#99×C7112yes—
212C23C2×C22⋊C4×C7111yescharacteristic
213C23#123×C7114yes—
214Q8—44no—
215D4—44no—
216D4—48no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—48no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4—84no—
235C2×C4#99×C7112yes—
236C2×C4#117×C7112yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—82no—
249C22#381×C7111yescharacteristic
250C22C2×#99×C7111yescharacteristic
251C22C2×#99×C7111yescharacteristic
252C4—22no—
253C4—22no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—164no—
258C2—21no—
259C2—21no—
260C2—44no—
261C2—42no—
262C2C71×#181911yescharacteristic
263C2C71×#212811yescharacteristic
264C2C71×#212911yescharacteristic
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×C4⋊C4×C71C4semidirect
9. #391×C71C4semidirect
125. #12859C71direct (N × H)
135. #1393C2×C71semidirect
136. C71#12859direct (N × H)
138. C2×#117C4×C71semidirect
139. C22×C4⋊C4C4×C71semidirect
142. #391C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#23112
C71×#17822
C71×#13932
#1285971