Finite Group Databasebeta

← C71×#13611 #31834483

Subgroups — C71×#13611 #31834483

Order 18176.

316 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3221C211yesmaximal, characteristic
2C71×#2135C211yesmaximal, characteristic, complemented
3C71×#2133C211yesmaximal, characteristic, complemented
4C71×#1413C212yesmaximal, complemented
5C71×#1410C212yesmaximal, complemented
6#540×C71C411yescharacteristic
7#536×C71C2211yescharacteristic
8#496×C71C411yescharacteristic
9#492×C71C2211yescharacteristic
10#491×C71C2211yescharacteristic
11#488×C71C411yescharacteristic
12#463×C71—21no—
13#461×C71—21no—
14#449×C71—21no—
15#448×C71—21no—
16#433×C71C411yescharacteristic
17#413×C71—21no—
18C2×#101×C71—21no—
19#356×C71—21no—
20#356×C71C2212yes—
21#356×C71C2212yes—
22#355×C71—21no—
23C2×#41×C71D411yescharacteristic
24C22×Q8×C71D411yescharacteristic
25C2×SD16×C71—21no—
26C2×SD16×C71—21no—
27C2×SD16×C71—21no—
28C2×D8×C71—21no—
29C2×M16×C71—41no—
30C2×M16×C71—41no—
31C4⋊Q8×C71D411yescharacteristic
32C4⋊Q8×C71C2311yescharacteristic
33#127×C71C2×C411yescharacteristic
34#125×C71C2×C411yescharacteristic
35#124×C71C2×C411yescharacteristic
36#124×C71D411yescharacteristic
37#123×C71—41no—
38#123×C71—41no—
39C22⋊Q8×C71—41no—
40#121×C71—21no—
41#121×C71—41no—
42C4×D4×C71—21no—
43#117×C71C2×C411yescharacteristic
44C4⋊C8×C71—21no—
45#103×C71—21no—
46#103×C71—21no—
47#103×C71—21no—
48#102×C71—21no—
49#101×C71—22no—
50#101×C71—22no—
51#101×C71—41no—
52#100×C71—41no—
53C22⋊C8×C71—21no—
54#97×C71C2×C411yescharacteristic
55C4×C8×C71C2×C411yescharacteristic
56#41×C71—81no—
57C2×Q8×C71—21no—
58C2×Q8×C71—81no—
59C2×Q8×C71C22⋊C411yescharacteristic
60C2×Q8×C71C2×D411yescharacteristic
61C2×Q8×C71C2×D411yescharacteristic
62C2×D4×C71—21no—
63C2×D4×C71—21no—
64C2×D4×C71—41no—
65C2×D4×C71—41no—
66C2×D4×C71C22⋊C411yescharacteristic
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—41no—
70C22×C4×C71C22⋊C411yescharacteristic
71SD16×C71—81no—
72SD16×C71—81no—
73SD16×C71—81no—
74D8×C71—81no—
75M16×C71—42no—
76M16×C71—42no—
77M16×C71—81no—
78M16×C71—81no—
79C2×C8×C71—21no—
80C2×C8×C71—21no—
81C2×C8×C71—81no—
82C2×C8×C71—81no—
83C4⋊C4×C71—21no—
84C4⋊C4×C71—21no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C22⋊C4×C71—21no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—41no—
92C22⋊C4×C71—41no—
93C42×C71C22⋊C411yescharacteristic
94C42×C71C22×C411yescharacteristic
95C23×C71—21no—
96C23×C71—41no—
97C23×C71#9911yescharacteristic
98Q8×C71—41no—
99Q8×C71—41no—
100Q8×C71—41no—
101Q8×C71—81no—
102D4×C71—41no—
103D4×C71—41no—
104D4×C71—41no—
105D4×C71—81no—
106D4×C71—161no—
107D4×C71—161no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—161no—
120C2×C4×C71#9911yescharacteristic
121C2×C4×C71C2×C22⋊C411yescharacteristic
122C8×C71—41no—
123C8×C71—41no—
124C8×C71—82no—
125C8×C71—82no—
126C22×C71—21no—
127C22×C71—41no—
128C22×C71—41no—
129C22×C71—81no—
130C22×C71—81no—
131C22×C71—81no—
132C22×C71—81no—
133C22×C71C2×#9911yescharacteristic
134C4×C71—21no—
135C4×C71—21no—
136C4×C71—41no—
137C4×C71—41no—
138C4×C71—41no—
139C4×C71—41no—
140C4×C71—41no—
141C4×C71—81no—
142#13611C7111yesmaximal, characteristic, direct product factor, Sylow
143C2×C71—21no—
144C2×C71—41no—
145C2×C71—81no—
146C2×C71—161no—
147C2×C71#212711yescharacteristic
148#3221C2×C7111yescharacteristic
149#2135C2×C7111yescharacteristic, complemented
150#2133C2×C7111yescharacteristic, complemented
151#1413C2×C7112yescomplemented
152#1410C2×C7112yescomplemented
153C71#1361111yescharacteristic, direct product factor, Sylow
154#540C4×C7111yescharacteristic
155#536C22×C7111yescharacteristic
156#496C4×C7111yescharacteristic
157#492C22×C7111yescharacteristic
158#491C22×C7111yescharacteristic
159#488C4×C7111yescharacteristic
160#463—21no—
161#461—21no—
162#449—21no—
163#448—21no—
164#433C4×C7111yescharacteristic
165#413—21no—
166C2×#101—21no—
167#356—21no—
168#356C22×C7112yes—
169#356C22×C7112yes—
170#355—21no—
171C2×#41D4×C7111yescharacteristic
172C22×Q8D4×C7111yescharacteristic
173C2×SD16—21no—
174C2×SD16—21no—
175C2×SD16—21no—
176C2×D8—21no—
177C2×M16—41no—
178C2×M16—41no—
179C4⋊Q8D4×C7111yescharacteristic
180C4⋊Q8C23×C7111yescharacteristic
181#127C2×C4×C7111yescharacteristic
182#125C2×C4×C7111yescharacteristic
183#124C2×C4×C7111yescharacteristic
184#124D4×C7111yescharacteristic
185#123—41no—
186#123—41no—
187C22⋊Q8—41no—
188#121—21no—
189#121—41no—
190C4×D4—21no—
191#117C2×C4×C7111yescharacteristic
192C4⋊C8—21no—
193#103—21no—
194#103—21no—
195#103—21no—
196#102—21no—
197#101—22no—
198#101—22no—
199#101—41no—
200#100—41no—
201C22⋊C8—21no—
202#97C2×C4×C7111yescharacteristic
203C4×C8C2×C4×C7111yescharacteristic
204#41—81no—
205C2×Q8—21no—
206C2×Q8—81no—
207C2×Q8C22⋊C4×C7111yescharacteristic
208C2×Q8C2×D4×C7111yescharacteristic
209C2×Q8C2×D4×C7111yescharacteristic
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4C22⋊C4×C7111yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4C22⋊C4×C7111yescharacteristic
219SD16—81no—
220SD16—81no—
221SD16—81no—
222D8—81no—
223M16—42no—
224M16—42no—
225M16—81no—
226M16—81no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—81no—
230C2×C8—81no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—21no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C42C22⋊C4×C7111yescharacteristic
242C42C22×C4×C7111yescharacteristic
243C23—21no—
244C23—41no—
245C23#99×C7111yescharacteristic
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—81no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—161no—
255D4—161no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—161no—
268C2×C4#99×C7111yescharacteristic
269C2×C4C2×C22⋊C4×C7111yescharacteristic
270C8—41no—
271C8—41no—
272C8—82no—
273C8—82no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22C2×#99×C7111yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—161no—
294C2C71×#212711yescharacteristic
subgroupG/H
center, socleC2×C71#2127
commutatorC42C22×C4×C71
Frattini, cosocleC4⋊Q8C23×C71
prime radicalC71×#3221C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C22×C71 ◁ C22×C4×C71 ◁ #536×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2135C2semidirect
3. C71×#2133C2semidirect
4. C71×#1413C2semidirect
5. C71×#1410C2semidirect
142. #13611C71direct (N × H)
149. #2135C2×C71semidirect
150. #2133C2×C71semidirect
151. #1413C2×C71semidirect
152. #1410C2×C71semidirect
153. C71#13611direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32212
C71×#21352
C71×#21332
C71×#14132
C71×#14102
#1361171