Finite Group Databasebeta

← C71×#7810 #31828682

Subgroups — C71×#7810 #31828682

Order 18176.

478 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2733C211yesmaximal, characteristic
2C71×#2096C211yesmaximal, characteristic, complemented
3C71×#2071C212yesmaximal, complemented
4C71×#1272C211yesmaximal, characteristic, complemented
5C71×#1270C212yesmaximal, complemented
6C2×#124×C71—21no—
7C2×#106×C71—21no—
8C2×#103×C71—21no—
9C2×#102×C71—22no—
10#394×C71—21no—
11#392×C71—22no—
12#391×C71C2211yescharacteristic
13#390×C71C2212yes—
14#388×C71—22no—
15#384×C71C2211yescharacteristic
16#340×C71—22no—
17#336×C71C2211yescharacteristic
18#336×C71C2212yes—
19C22×Q8×C71D411yescharacteristic
20C22×D4×C71—21no—
21C22×D4×C71—21no—
22C22×D4×C71D412yes—
23C22×C8×C71—21no—
24C22×C8×C71—22no—
25#124×C71—42no—
26#124×C71—42no—
27C2×C4⋊C4×C71—22no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71D411yescharacteristic
30C2×C4⋊C4×C71D412yes—
31C2×C22⋊C4×C71—22no—
32C2×C22⋊C4×C71—22no—
33C2×C22⋊C4×C71—22no—
34C2×C22⋊C4×C71—22no—
35C2×C42×C71—21no—
36C2×C42×C71C2311yescharacteristic
37#106×C71—42no—
38#103×C71—42no—
39#102×C71—44no—
40#95×C71—21no—
41#95×C71—22no—
42C24×C71—22no—
43C2×Q8×C71—21no—
44C2×Q8×C71—41no—
45C2×Q8×C71—41no—
46C2×Q8×C71SD1612yes—
47C2×D4×C71—22no—
48C2×D4×C71—22no—
49C2×D4×C71—42no—
50C2×D4×C71—42no—
51C2×D4×C71—42no—
52C2×D4×C71—81no—
53C2×D4×C71—81no—
54C2×D4×C71—81no—
55C2×D4×C71—82no—
56C22×C4×C71—21no—
57C22×C4×C71—21no—
58C22×C4×C71—22no—
59C22×C4×C71—41no—
60C22×C4×C71C2×D411yescharacteristic
61C22×C4×C71C2×D412yes—
62C2×C8×C71—22no—
63C2×C8×C71—41no—
64C2×C8×C71—41no—
65C2×C8×C71—42no—
66C2×C8×C71—42no—
67C2×C8×C71—42no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—42no—
73C4⋊C4×C71—42no—
74C22⋊C4×C71—44no—
75C22⋊C4×C71—42no—
76C22⋊C4×C71—42no—
77C22⋊C4×C71—42no—
78C22⋊C4×C71—42no—
79C22⋊C4×C71—82no—
80C42×C71—41no—
81C42×C71—42no—
82C23×C71—42no—
83C23×C71—42no—
84C23×C71—42no—
85C23×C71—42no—
86C23×C71—42no—
87C23×C71—42no—
88C23×C71—42no—
89C23×C71#12011yescharacteristic
90Q8×C71—42no—
91Q8×C71—81no—
92D4×C71—44no—
93D4×C71—82no—
94D4×C71—84no—
95D4×C71—84no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—21no—
100C2×C4×C71—22no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—41no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—82no—
109C2×C4×C71—81no—
110C2×C4×C71C2×SD1611yescharacteristic
111C2×C4×C71#13512yes—
112C2×C4×C71AGL(1,ℤ/8)12yes—
113C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
114C8×C71—42no—
115C8×C71—82no—
116C22×C71—82no—
117C22×C71—82no—
118C22×C71—82no—
119C22×C71—82no—
120C22×C71—82no—
121C22×C71—82no—
122C22×C71—82no—
123C22×C71#44912yes—
124C22×C71#45011yescharacteristic
125C22×C71#45311yescharacteristic
126C22×C71#45512yes—
127C22×C71#45811yescharacteristic
128C4×C71—22no—
129C4×C71—22no—
130C4×C71—21no—
131C4×C71—21no—
132C4×C71—42no—
133C4×C71—82no—
134C4×C71—81no—
135C4×C71—161no—
136#7810C7111yesmaximal, characteristic, direct product factor, Sylow
137C2×C71—162no—
138C2×C71#161212yes—
139C2×C71#161311yescharacteristic
140C2×C71#162411yescharacteristic
141C2×C71#164212yes—
142C2×C71#165911yescharacteristic
143#2733C2×C7111yescharacteristic
144#2096C2×C7111yescharacteristic, complemented
145#2071C2×C7112yescomplemented
146#1272C2×C7111yescharacteristic, complemented
147#1270C2×C7112yescomplemented
148C71#781011yescharacteristic, direct product factor, Sylow
149C2×#124—21no—
150C2×#106—21no—
151C2×#103—21no—
152C2×#102—22no—
153#394—21no—
154#392—22no—
155#391C22×C7111yescharacteristic
156#390C22×C7112yes—
157#388—22no—
158#384C22×C7111yescharacteristic
159#340—22no—
160#336C22×C7112yes—
161#336C22×C7111yescharacteristic
162C22×Q8D4×C7111yescharacteristic
163C22×D4—21no—
164C22×D4—21no—
165C22×D4D4×C7112yes—
166C22×C8—21no—
167C22×C8—22no—
168#124—42no—
169#124—42no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4D4×C7112yes—
173C2×C4⋊C4D4×C7111yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42—21no—
179C2×C42C23×C7111yescharacteristic
180#106—42no—
181#103—42no—
182#102—44no—
183#95—21no—
184#95—22no—
185C24—22no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8SD16×C7112yes—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—81no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4—82no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4C2×D4×C7111yescharacteristic
204C22×C4C2×D4×C7112yes—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C22⋊C4—44no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—82no—
223C42—41no—
224C42—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23#120×C7111yescharacteristic
233Q8—42no—
234Q8—81no—
235D4—44no—
236D4—82no—
237D4—84no—
238D4—84no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4C2×SD16×C7111yescharacteristic
254C2×C4#135×C7112yes—
255C2×C4AGL(1,ℤ/8)×C7112yes—
256C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
257C8—42no—
258C8—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22#449×C7112yes—
267C22#450×C7111yescharacteristic
268C22#453×C7111yescharacteristic
269C22#455×C7112yes—
270C22#458×C7111yescharacteristic
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—82no—
277C4—81no—
278C4—161no—
279C2—162no—
280C2C71×#161212yes—
281C2C71×#161311yescharacteristic
282C2C71×#162411yescharacteristic
283C2C71×#164212yes—
284C2C71×#165911yescharacteristic
subgroupG/H
center, socleC23×C71#120
commutator, Frattini, cosocleC2×C42C23×C71
prime radicalC71×#2733C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2096C2semidirect
3. C71×#2071C2semidirect
4. C71×#1272C2semidirect
5. C71×#1270C2semidirect
136. #7810C71direct (N × H)
144. #2096C2×C71semidirect
145. #2071C2×C71semidirect
146. #1272C2×C71semidirect
147. #1270C2×C71semidirect
148. C71#7810direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#27332
C71×#20962
C71×#20712
C71×#12722
C71×#12702
#781071