Finite Group Databasebeta

← C71×#13309 #31834181

Subgroups — C71×#13309 #31834181

Order 18176.

568 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3414C211yesmaximal, characteristic, complemented
2C71×#3050C212yesmaximal, complemented
3C71×#2145C212yesmaximal, complemented
4C71×#2112C211yesmaximal, characteristic, complemented
5C71×#1384C211yesmaximal, characteristic, complemented
6C2×#135×C71C2212yes—
7C23×C8×C71C2211yescharacteristic
8#525×C71—22no—
9#510×C71—22no—
10#509×C71—22no—
11#508×C71—22no—
12C2×M32×C71C2211yescharacteristic
13C2×M32×C71C2211yescharacteristic
14#467×C71—22no—
15#466×C71—22no—
16#465×C71—24no—
17#427×C71C2212yes—
18#416×C71—22no—
19#360×C71—24no—
20C22⋊C16×C71—22no—
21C2×#41×C71—22no—
22C23×C4×C71D411yescharacteristic
23#135×C71—22no—
24#135×C71—22no—
25#135×C71—22no—
26#135×C71—22no—
27C2×Q16×C71D412yescomplemented
28C2×SD16×C71—22no—
29C2×D8×C71D412yescomplemented
30C22×C8×C71—22no—
31C22×C8×C71—22no—
32C22×C8×C71—21no—
33C22×C8×C71—21no—
34C22×C8×C71D411yescharacteristic
35C22×C8×C71C2311yescharacteristic
36#123×C71—42no—
37C22⋊Q8×C71—24no—
38#121×C71—24no—
39#120×C71—42no—
40C4×D4×C71—44no—
41#117×C71—22no—
42Q32×C71—22no—
43SD32×C71—24no—
44D16×C71—22no—
45M32×C71—22no—
46M32×C71—22no—
47C2×C16×C71—21no—
48C2×C16×C71D412yescomplemented
49#107×C71—22no—
50#106×C71D414yescomplemented
51#103×C71—24no—
52#102×C71—24no—
53C24×C71D811yescharacteristic
54#41×C71—42no—
55#41×C71—42no—
56#41×C71—42no—
57#41×C71—42no—
58C2×Q8×C71—22no—
59C2×D4×C71—22no—
60C2×D4×C71—44no—
61C2×D4×C71—42no—
62C22×C4×C71—22no—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—22no—
66C22×C4×C71—42no—
67C22×C4×C71D811yescharacteristic
68C22×C4×C71C2×D411yescharacteristic
69Q16×C71—22no—
70Q16×C71—22no—
71SD16×C71—42no—
72SD16×C71—42no—
73D8×C71—22no—
74D8×C71—22no—
75C2×C8×C71—22no—
76C2×C8×C71—21no—
77C2×C8×C71—41no—
78C2×C8×C71—42no—
79C2×C8×C71—41no—
80C2×C8×C71C2×D412yes—
81C2×C8×C71C2×D411yescharacteristic
82C2×C8×C71C2×D412yes—
83C2×C8×C71#4111yescharacteristic
84C4⋊C4×C71—24no—
85C4⋊C4×C71—44no—
86C22⋊C4×C71—44no—
87C22⋊C4×C71—44no—
88C22⋊C4×C71—42no—
89C42×C71—42no—
90C16×C71—22no—
91C16×C71—41no—
92C23×C71—22no—
93C23×C71—21no—
94C23×C71—42no—
95C23×C71—42no—
96C23×C71C2×D811yescharacteristic
97Q8×C71—42no—
98Q8×C71—42no—
99D4×C71—42no—
100D4×C71—42no—
101D4×C71—88no—
102D4×C71—82no—
103D4×C71—82no—
104C2×C4×C71—22no—
105C2×C4×C71—21no—
106C2×C4×C71—41no—
107C2×C4×C71—42no—
108C2×C4×C71—41no—
109C2×C4×C71—44no—
110C2×C4×C71—42no—
111C2×C4×C71—82no—
112C2×C4×C71—82no—
113C2×C4×C71#12012yes—
114C2×C4×C71#12111yescharacteristic
115C2×C4×C71#13511yescharacteristic
116C2×C4×C71AGL(1,ℤ/8)12yes—
117C8×C71—41no—
118C8×C71#12011yescharacteristic
119C8×C71#12112yes—
120C8×C71#12711yescharacteristic
121C22×C71—24no—
122C22×C71—22no—
123C22×C71—41no—
124C22×C71—42no—
125C22×C71—41no—
126C22×C71—41no—
127C22×C71—84no—
128C22×C71—82no—
129C22×C71#44712yes—
130C22×C71#46611yescharacteristic
131C4×C71—41no—
132C4×C71—84no—
133C4×C71—82no—
134C4×C71#39211yescharacteristic
135C4×C71#44912yes—
136C4×C71#46911yescharacteristic
137#13309C7111yesmaximal, characteristic, direct product factor, Sylow
138C2×C71—22no—
139C2×C71—21no—
140C2×C71—42no—
141C2×C71—162no—
142C2×C71#199911yescharacteristic
143#3414C2×C7111yescharacteristic, complemented
144#3050C2×C7112yescomplemented
145#2145C2×C7112yescomplemented
146#2112C2×C7111yescharacteristic, complemented
147#1384C2×C7111yescharacteristic, complemented
148C71#1330911yescharacteristic, direct product factor, Sylow
149C2×#135C22×C7112yes—
150C23×C8C22×C7111yescharacteristic
151#525—22no—
152#510—22no—
153#509—22no—
154#508—22no—
155C2×M32C22×C7111yescharacteristic
156C2×M32C22×C7111yescharacteristic
157#467—22no—
158#466—22no—
159#465—24no—
160#427C22×C7112yes—
161#416—22no—
162#360—24no—
163C22⋊C16—22no—
164C2×#41—22no—
165C23×C4D4×C7111yescharacteristic
166#135—22no—
167#135—22no—
168#135—22no—
169#135—22no—
170C2×Q16D4×C7112yescomplemented
171C2×SD16—22no—
172C2×D8D4×C7112yescomplemented
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8D4×C7111yescharacteristic
178C22×C8C23×C7111yescharacteristic
179#123—42no—
180C22⋊Q8—24no—
181#121—24no—
182#120—42no—
183C4×D4—44no—
184#117—22no—
185Q32—22no—
186SD32—24no—
187D16—22no—
188M32—22no—
189M32—22no—
190C2×C16—21no—
191C2×C16D4×C7112yescomplemented
192#107—22no—
193#106D4×C7114yescomplemented
194#103—24no—
195#102—24no—
196C24D8×C7111yescharacteristic
197#41—42no—
198#41—42no—
199#41—42no—
200#41—42no—
201C2×Q8—22no—
202C2×D4—22no—
203C2×D4—44no—
204C2×D4—42no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4D8×C7111yescharacteristic
211C22×C4C2×D4×C7111yescharacteristic
212Q16—22no—
213Q16—22no—
214SD16—42no—
215SD16—42no—
216D8—22no—
217D8—22no—
218C2×C8—22no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8C2×D4×C7112yes—
224C2×C8C2×D4×C7111yescharacteristic
225C2×C8C2×D4×C7112yes—
226C2×C8#41×C7111yescharacteristic
227C4⋊C4—24no—
228C4⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—42no—
232C42—42no—
233C16—22no—
234C16—41no—
235C23—22no—
236C23—21no—
237C23—42no—
238C23—42no—
239C23C2×D8×C7111yescharacteristic
240Q8—42no—
241Q8—42no—
242D4—42no—
243D4—42no—
244D4—88no—
245D4—82no—
246D4—82no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#120×C7112yes—
257C2×C4#121×C7111yescharacteristic
258C2×C4#135×C7111yescharacteristic
259C2×C4AGL(1,ℤ/8)×C7112yes—
260C8—41no—
261C8#120×C7111yescharacteristic
262C8#121×C7112yes—
263C8#127×C7111yescharacteristic
264C22—24no—
265C22—22no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—84no—
271C22—82no—
272C22#447×C7112yes—
273C22#466×C7111yescharacteristic
274C4—41no—
275C4—84no—
276C4—82no—
277C4#392×C7111yescharacteristic
278C4#449×C7112yes—
279C4#469×C7111yescharacteristic
280C2—22no—
281C2—21no—
282C2—42no—
283C2—162no—
284C2C71×#199911yescharacteristic
subgroupG/H
centerC4×C71#469
commutator, Frattini, cosocleC22×C8C23×C71
socleC2×C71#1999
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C71 ◁ C22×C4×C71 ◁ C23×C8×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C71×#3414C2semidirect
2. C71×#3050C2semidirect
3. C71×#2145C2semidirect
4. C71×#2112C2semidirect
5. C71×#1384C2semidirect
27. C2×Q16×C71D4semidirect
29. C2×D8×C71D4semidirect
48. C2×C16×C71D4semidirect
50. #106×C71D4semidirect
137. #13309C71direct (N × H)
143. #3414C2×C71semidirect
144. #3050C2×C71semidirect
145. #2145C2×C71semidirect
146. #2112C2×C71semidirect
147. #1384C2×C71semidirect
148. C71#13309direct (N × H)
170. C2×Q16D4×C71semidirect
172. C2×D8D4×C71semidirect
191. C2×C16D4×C71semidirect
193. #106D4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#34142
C71×#30502
C71×#21452
C71×#21122
C71×#13842
#1330971