Finite Group Databasebeta

← C71×#7736 #31828608

Subgroups — C71×#7736 #31828608

Order 18176.

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

normal subgroups only

This group has 560 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×#2724C211yesmaximal, characteristic, complemented
2C71×#2041C212yesmaximal, complemented
3C71×#2011C211yesmaximal, characteristic
4C71×#1271C212yesmaximal, complemented
5C71×#1270C211yesmaximal, characteristic, complemented
6C22×D8×C71—21no—
7C2×#127×C71C2211yescharacteristic
8C2×#123×C71—21no—
9C2×#102×C71—22no—
10C2×#102×C71—22no—
11C2×#102×C71—21no—
12#398×C71—21no—
13#397×C71—22no—
14#390×C71C2211yescharacteristic
15#389×C71C2212yes—
16#385×C71—22no—
17#336×C71C2212yes—
18#336×C71C2211yescharacteristic
19C22×D4×C71—22no—
20C22×D4×C71—21no—
21C22×D4×C71D411yescharacteristic
22C23×C4×C71—21no—
23C2×D8×C71—42no—
24C2×D8×C71—42no—
25C2×D8×C71—42no—
26C22×C8×C71—21no—
27C22×C8×C71—22no—
28#127×C71—42no—
29#123×C71—42no—
30#123×C71—42no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71D412yes—
34C2×C4⋊C4×C71D412yes—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C22⋊C4×C71—22no—
37C2×C22⋊C4×C71—21no—
38C2×C22⋊C4×C71—22no—
39C2×C42×C71C2311yescharacteristic
40#102×C71—44no—
41#102×C71—42no—
42#102×C71—44no—
43#95×C71—22no—
44#95×C71—21no—
45#95×C71—22no—
46C24×C71—21no—
47C24×C71—41no—
48C2×D4×C71—24no—
49C2×D4×C71—22no—
50C2×D4×C71—21no—
51C2×D4×C71—41no—
52C2×D4×C71—41no—
53C2×D4×C71—41no—
54C2×D4×C71—41no—
55C2×D4×C71—42no—
56C2×D4×C71—42no—
57C2×D4×C71—42no—
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60C2×D4×C71—42no—
61C2×D4×C71—42no—
62C2×D4×C71—41no—
63C2×D4×C71D812yes—
64C22×C4×C71—22no—
65C22×C4×C71—21no—
66C22×C4×C71—22no—
67C22×C4×C71—42no—
68C22×C4×C71—81no—
69C22×C4×C71—81no—
70C22×C4×C71C2×D411yescharacteristic
71C22×C4×C71C2×D412yes—
72D8×C71—48no—
73C2×C8×C71—22no—
74C2×C8×C71—41no—
75C2×C8×C71—41no—
76C2×C8×C71—42no—
77C2×C8×C71—42no—
78C2×C8×C71—42no—
79C4⋊C4×C71—22no—
80C4⋊C4×C71—22no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—42no—
83C4⋊C4×C71—42no—
84C4⋊C4×C71—42no—
85C4⋊C4×C71—81no—
86C22⋊C4×C71—42no—
87C22⋊C4×C71—44no—
88C22⋊C4×C71—42no—
89C22⋊C4×C71—42no—
90C42×C71—41no—
91C23×C71—22no—
92C23×C71—42no—
93C23×C71—41no—
94C23×C71—42no—
95C23×C71—41no—
96C23×C71—42no—
97C23×C71—81no—
98C23×C71—82no—
99C23×C71—81no—
100C23×C71—82no—
101C23×C71#12011yescharacteristic
102D4×C71—42no—
103D4×C71—42no—
104D4×C71—42no—
105D4×C71—44no—
106D4×C71—44no—
107D4×C71—81no—
108D4×C71—81no—
109D4×C71—81no—
110D4×C71—82no—
111D4×C71—82no—
112C2×C4×C71—21no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—21no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—41no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—84no—
125C2×C4×C71C2×D811yescharacteristic
126C2×C4×C71#13512yes—
127C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C71AGL(1,ℤ/8)12yes—
129C8×C71—42no—
130C8×C71—82no—
131C22×C71—42no—
132C22×C71—44no—
133C22×C71—82no—
134C22×C71—81no—
135C22×C71—81no—
136C22×C71—82no—
137C22×C71—84no—
138C22×C71—161no—
139C22×C71—162no—
140C22×C71—161no—
141C22×C71#44711yescharacteristic
142C22×C71#44912yes—
143C22×C71#45311yescharacteristic
144C22×C71#45412yes—
145C22×C71B(4,𝔽2)11yescharacteristic
146C4×C71—21no—
147C4×C71—21no—
148C4×C71—22no—
149C4×C71—22no—
150C4×C71—82no—
151C4×C71—81no—
152C4×C71—82no—
153#7736C7111yesmaximal, characteristic, direct product factor, Sylow
154C2×C71—82no—
155C2×C71—162no—
156C2×C71#159511yescharacteristic
157C2×C71#159712yes—
158C2×C71#162011yescharacteristic
159C2×C71#162512yes—
160C2×C71#165711yescharacteristic
161#2724C2×C7111yescharacteristic, complemented
162#2041C2×C7112yescomplemented
163#2011C2×C7111yescharacteristic
164#1271C2×C7112yescomplemented
165#1270C2×C7111yescharacteristic, complemented
166C71#773611yescharacteristic, direct product factor, Sylow
167C22×D8—21no—
168C2×#127C22×C7111yescharacteristic
169C2×#123—21no—
170C2×#102—21no—
171C2×#102—22no—
172C2×#102—22no—
173#398—21no—
174#397—22no—
175#390C22×C7111yescharacteristic
176#389C22×C7112yes—
177#385—22no—
178#336C22×C7111yescharacteristic
179#336C22×C7112yes—
180C22×D4—21no—
181C22×D4—22no—
182C22×D4D4×C7111yescharacteristic
183C23×C4—21no—
184C2×D8—42no—
185C2×D8—42no—
186C2×D8—42no—
187C22×C8—21no—
188C22×C8—22no—
189#127—42no—
190#123—42no—
191#123—42no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4D4×C7112yes—
195C2×C4⋊C4D4×C7112yes—
196C2×C4⋊C4D4×C7111yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C42C23×C7111yescharacteristic
201#102—44no—
202#102—42no—
203#102—44no—
204#95—22no—
205#95—21no—
206#95—22no—
207C24—21no—
208C24—41no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4D8×C7112yes—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4C2×D4×C7111yescharacteristic
232C22×C4C2×D4×C7112yes—
233D8—48no—
234C2×C8—22no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C42—41no—
252C23—22no—
253C23—42no—
254C23—41no—
255C23—42no—
256C23—41no—
257C23—42no—
258C23—81no—
259C23—82no—
260C23—81no—
261C23—82no—
262C23#120×C7111yescharacteristic
263D4—42no—
264D4—42no—
265D4—42no—
266D4—44no—
267D4—44no—
268D4—81no—
269D4—81no—
270D4—81no—
271D4—82no—
272D4—82no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—84no—
286C2×C4C2×D8×C7111yescharacteristic
287C2×C4#135×C7112yes—
288C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
289C2×C4AGL(1,ℤ/8)×C7112yes—
290C8—42no—
291C8—82no—
292C22—42no—
293C22—44no—
294C22—82no—
295C22—81no—
296C22—81no—
297C22—82no—
298C22—84no—
299C22—161no—
300C22—162no—
301C22—161no—
302C22#447×C7111yescharacteristic
303C22#449×C7112yes—
304C22#453×C7111yescharacteristic
305C22#454×C7112yes—
306C22B(4,𝔽2)×C7111yescharacteristic
307C4—21no—
308C4—21no—
309C4—22no—
310C4—22no—
311C4—82no—
312C4—81no—
313C4—82no—
314C2—82no—
315C2—162no—
316C2C71×#159511yescharacteristic
317C2C71×#159712yes—
318C2C71×#162011yescharacteristic
319C2C71×#162512yes—
320C2C71×#165711yescharacteristic
subgroupG/H
center, socleC23×C71#120
commutator, Frattini, cosocleC2×C42C23×C71
prime radicalC71×#2011C2
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
1. C71×#2724C2semidirect
2. C71×#2041C2semidirect
4. C71×#1271C2semidirect
5. C71×#1270C2semidirect
153. #7736C71direct (N × H)
161. #2724C2×C71semidirect
162. #2041C2×C71semidirect
164. #1271C2×C71semidirect
165. #1270C2×C71semidirect
166. C71#7736direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#27242
C71×#20412
C71×#20112
C71×#12712
C71×#12702
#773671