Finite Group Databasebeta

← C71×#19080 #31839952

Subgroups — C71×#19080 #31839952

Order 18176.

936 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3548C211yesmaximal, characteristic
2C71×#3141C212yesmaximal, complemented
3C71×#3139C211yesmaximal, characteristic, complemented
4C71×#1662C218yesmaximal, complemented
5C71×#1526C211yesmaximal, characteristic, complemented
6C71×#1525C212yesmaximal, complemented
7#562×C71C2214yes—
8#554×C71—22no—
9#550×C71—22no—
10#547×C71—22no—
11#540×C71—22no—
12#537×C71—24no—
13#533×C71C2211yescharacteristic
14#532×C71C2212yes—
15C2×C4⋊Q8×C71C2211yescharacteristic
16#491×C71C2214yes—
17#489×C71C2218yes—
18#487×C71—22no—
19#487×C71—22no—
20#486×C71—22no—
21#419×C71—22no—
22#417×C71—24no—
23C4×M16×C71C2211yescharacteristic
24C4×M16×C71C2212yes—
25#330×C71C2214yes—
26#329×C71C2218yes—
27C2×#41×C71—24no—
28C2×#41×C71D414yes—
29C22×Q8×C71—21no—
30C2×M16×C71—22no—
31C2×M16×C71—21no—
32C4⋊Q8×C71—22no—
33C4⋊Q8×C71—21no—
34C4⋊Q8×C71D412yes—
35C4⋊Q8×C71D412yes—
36C4⋊Q8×C71C2312yes—
37#127×C71—22no—
38#127×C71D412yes—
39#125×C71—22no—
40#125×C71C2312yes—
41#124×C71C2314yes—
42#123×C71—28no—
43C22⋊Q8×C71—24no—
44C22⋊Q8×C71—28no—
45C22⋊Q8×C71—22no—
46#121×C71—28no—
47#121×C71—24no—
48#121×C71—28no—
49#121×C71—22no—
50#121×C71—22no—
51C4×Q8×C71—22no—
52C4×D4×C71—24no—
53C4×D4×C71—22no—
54C4×D4×C71—22no—
55#117×C71D412yes—
56C2×C4⋊C4×C71—24no—
57C2×C4⋊C4×C71—22no—
58C2×C42×C71C2311yescharacteristic
59#103×C71—28no—
60#103×C71—28no—
61#102×C71—28no—
62#97×C71C2314yes—
63#97×C71C2312yes—
64C4×C8×C71—22no—
65C4×C8×C71—21no—
66#41×C71—48no—
67#41×C71—88no—
68C2×Q8×C71—22no—
69C2×Q8×C71—22no—
70C2×Q8×C71—44no—
71C2×Q8×C71—41no—
72C2×Q8×C71C2×D414yes—
73C2×D4×C71—28no—
74C2×D4×C71—24no—
75C2×D4×C71—22no—
76C2×D4×C71—22no—
77C2×D4×C71C2×D414yes—
78C22×C4×C71—28no—
79C22×C4×C71—24no—
80C22×C4×C71—41no—
81C22×C4×C71C2×D412yes—
82C22×C4×C71C2×D411yescharacteristic
83M16×C71—42no—
84M16×C71—44no—
85C2×C8×C71—24no—
86C2×C8×C71—22no—
87C4⋊C4×C71—28no—
88C4⋊C4×C71—22no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—22no—
91C4⋊C4×C71—24no—
92C4⋊C4×C71—24no—
93C4⋊C4×C71—42no—
94C4⋊C4×C71C2×D414yes—
95C22⋊C4×C71—28no—
96C22⋊C4×C71—22no—
97C22⋊C4×C71—24no—
98C22⋊C4×C71—24no—
99C42×C71—22no—
100C42×C71C2×D411yescharacteristic
101C42×C71C2×D412yes—
102C42×C71C2411yescharacteristic
103C23×C71—24no—
104C23×C71#12011yescharacteristic
105Q8×C71—44no—
106Q8×C71—44no—
107Q8×C71—44no—
108Q8×C71—82no—
109D4×C71—44no—
110D4×C71—44no—
111D4×C71—44no—
112D4×C71—84no—
113D4×C71—88no—
114C2×C4×C71—22no—
115C2×C4×C71—24no—
116C2×C4×C71—24no—
117C2×C4×C71—24no—
118C2×C4×C71—48no—
119C2×C4×C71—42no—
120C2×C4×C71—88no—
121C2×C4×C71—82no—
122C2×C4×C71#12011yescharacteristic
123C2×C4×C71#12012yes—
124C2×C4×C71C22×D411yescharacteristic
125C2×C4×C71C22×D412yes—
126C8×C71—44no—
127C8×C71—42no—
128C22×C71—22no—
129C22×C71—21no—
130C22×C71—44no—
131C22×C71—44no—
132C22×C71—44no—
133C22×C71C2×#12011yescharacteristic
134C4×C71—22no—
135C4×C71—21no—
136C4×C71—44no—
137C4×C71—44no—
138C4×C71—82no—
139C4×C71#45412yes—
140C4×C71#45514yes—
141#19080C7111yesmaximal, characteristic, direct product factor, Sylow
142C2×C71—41no—
143C2×C71—84no—
144C2×C71C2×#45512yes—
145C2×C71C2×#45411yescharacteristic
146#3548C2×C7111yescharacteristic
147#3141C2×C7112yescomplemented
148#3139C2×C7111yescharacteristic, complemented
149#1662C2×C7118yescomplemented
150#1526C2×C7111yescharacteristic, complemented
151#1525C2×C7112yescomplemented
152C71#1908011yescharacteristic, direct product factor, Sylow
153#562C22×C7114yes—
154#554—22no—
155#550—22no—
156#547—22no—
157#540—22no—
158#537—24no—
159#533C22×C7111yescharacteristic
160#532C22×C7112yes—
161C2×C4⋊Q8C22×C7111yescharacteristic
162#491C22×C7114yes—
163#489C22×C7118yes—
164#487—22no—
165#487—22no—
166#486—22no—
167#419—22no—
168#417—24no—
169C4×M16C22×C7111yescharacteristic
170C4×M16C22×C7112yes—
171#330C22×C7114yes—
172#329C22×C7118yes—
173C2×#41—24no—
174C2×#41D4×C7114yes—
175C22×Q8—21no—
176C2×M16—22no—
177C2×M16—21no—
178C4⋊Q8—22no—
179C4⋊Q8—21no—
180C4⋊Q8D4×C7112yes—
181C4⋊Q8D4×C7112yes—
182C4⋊Q8C23×C7112yes—
183#127—22no—
184#127D4×C7112yes—
185#125—22no—
186#125C23×C7112yes—
187#124C23×C7114yes—
188#123—28no—
189C22⋊Q8—24no—
190C22⋊Q8—28no—
191C22⋊Q8—22no—
192#121—28no—
193#121—24no—
194#121—28no—
195#121—22no—
196#121—22no—
197C4×Q8—22no—
198C4×D4—24no—
199C4×D4—22no—
200C4×D4—22no—
201#117D4×C7112yes—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C42C23×C7111yescharacteristic
205#103—28no—
206#103—28no—
207#102—28no—
208#97C23×C7112yes—
209#97C23×C7114yes—
210C4×C8—22no—
211C4×C8—21no—
212#41—48no—
213#41—88no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—44no—
217C2×Q8—41no—
218C2×Q8C2×D4×C7114yes—
219C2×D4—28no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4C2×D4×C7114yes—
224C22×C4—28no—
225C22×C4—24no—
226C22×C4—41no—
227C22×C4C2×D4×C7112yes—
228C22×C4C2×D4×C7111yescharacteristic
229M16—42no—
230M16—44no—
231C2×C8—24no—
232C2×C8—22no—
233C4⋊C4—28no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4C2×D4×C7114yes—
241C22⋊C4—28no—
242C22⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C42—22no—
246C42C2×D4×C7111yescharacteristic
247C42C2×D4×C7112yes—
248C42C24×C7111yescharacteristic
249C23—24no—
250C23#120×C7111yescharacteristic
251Q8—44no—
252Q8—44no—
253Q8—44no—
254Q8—82no—
255D4—44no—
256D4—44no—
257D4—44no—
258D4—84no—
259D4—88no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—88no—
267C2×C4—82no—
268C2×C4#120×C7111yescharacteristic
269C2×C4#120×C7112yes—
270C2×C4C22×D4×C7111yescharacteristic
271C2×C4C22×D4×C7112yes—
272C8—44no—
273C8—42no—
274C22—22no—
275C22—21no—
276C22—44no—
277C22—44no—
278C22—44no—
279C22C2×#120×C7111yescharacteristic
280C4—22no—
281C4—21no—
282C4—44no—
283C4—44no—
284C4—82no—
285C4#454×C7112yes—
286C4#455×C7114yes—
287C2—41no—
288C2—84no—
289C2C2×#455×C7112yes—
290C2C2×#454×C7111yescharacteristic
subgroupG/H
center, socleC22×C71C2×#120
commutator, Frattini, cosocleC42C24×C71
prime radicalC71×#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#3141C2semidirect
3. C71×#3139C2semidirect
4. C71×#1662C2semidirect
5. C71×#1526C2semidirect
6. C71×#1525C2semidirect
141. #19080C71direct (N × H)
147. #3141C2×C71semidirect
148. #3139C2×C71semidirect
149. #1662C2×C71semidirect
150. #1526C2×C71semidirect
151. #1525C2×C71semidirect
152. C71#19080direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#35482
C71×#31412
C71×#31392
C71×#16622
C71×#15262
C71×#15252
#1908071