Finite Group Databasebeta

← #31886035

Subgroups — #31886035

Order 18176.

372 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588457C211yesmaximal, characteristic
2#22588089C211yesmaximal, characteristic
3#22586166C212yesmaximal
4#22586163C212yesmaximal
5C71×#2863C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7C2×#14556039—21no—
8#17985754—21no—
9#17985754—21no—
10#17985754C2212yes—
11#17985754C2212yes—
12#391×C71C411yescharacteristic
13#391×C71C411yescharacteristic
14#391×C71C2211yescharacteristic
15#389×C71C411yescharacteristic
16#389×C71C411yescharacteristic
17#389×C71C2211yescharacteristic
18#387×C71—21no—
19#387×C71—21no—
20#387×C71—21no—
21#383×C71—21no—
22C43×C71C2211yescharacteristic
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—42no—
26#14556039—24no—
27C22×Q8×C71C2×C411yescharacteristic
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71C2×C411yescharacteristic
32C2×C4⋊C4×C71C2×C411yescharacteristic
33C2×C4⋊C4×C71C2×C411yescharacteristic
34C2×C42×C71C2×C411yescharacteristic
35C2×C42×C71C2×C411yescharacteristic
36C2×C42×C71D411yescharacteristic
37C2×C42×C71D411yescharacteristic
38C2×C42×C71D411yescharacteristic
39C2×C42×C71D411yescharacteristic
40C2×C42×C71C2311yescharacteristic
41#95×C71—21no—
42#95×C71—21no—
43#95×C71—21no—
44#95×C71—21no—
45#95×C71—21no—
46#95×C71—21no—
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50C2×C71⋊C8—24no—
51C2×C71⋊C8—42no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8—42no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—42no—
56C2×Q8×C71—22no—
57C2×Q8×C71—41no—
58C2×Q8×C71—41no—
59C22×C4×C71—21no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22×C411yescharacteristic
68C22×C4×C71C2×D411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C4⋊C4×C71—22no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—42no—
74C4⋊C4×C71—42no—
75C4⋊C4×C71—41no—
76C4⋊C4×C71—41no—
77C42×C71—22no—
78C42×C71—22no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—22no—
82C42×C71—41no—
83C42×C71—41no—
84C42×C71—41no—
85C23×C71C2×C22⋊C411yescharacteristic
86Q8×C71—82no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71#10412yes—
117C2×C4×C71#13112yes—
118C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C71#13711yescharacteristic
120C2×C4×C71#13712yes—
121C71⋊C8—44no—
122C71⋊C8—82no—
123C22×C71#40811yescharacteristic
124C22×C71#41011yescharacteristic
125C22×C71#41311yescharacteristic
126C22×C71#41711yescharacteristic
127C22×C71#41911yescharacteristic
128C22×C71C2×#10411yescharacteristic
129C22×C71#42111yescharacteristic
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—22no—
133C4×C71—21no—
134C4×C71—21no—
135C4×C71—41no—
136C4×C71—41no—
137C4×C71—41no—
138C4×C71—81no—
139C4×C71—81no—
140C4×C71—81no—
141C4×C71—81no—
142#7709—711nomaximal, Sylow
143C2×C71#147311yescharacteristic
144C2×C71#148311yescharacteristic
145C2×C71#149711yescharacteristic
146C2×C71#150611yescharacteristic
147C2×C71#151211yescharacteristic
148C2×C71#152911yescharacteristic
149C2×C71#155911yescharacteristic
150#2863D7111yescharacteristic
151#1831—711no—
152#1745—711no—
153#1272—712no—
154#1271—712no—
155C71#770911yescharacteristic, complemented, Sylow
156C2×C4⋊C8—1421no—
157C2×#97—1421no—
158#391Dic7111yescharacteristic
159#391Dic7111yescharacteristic
160#391C2×D7111yescharacteristic
161#389Dic7111yescharacteristic
162#389Dic7111yescharacteristic
163#389C2×D7111yescharacteristic
164#387—21no—
165#387—21no—
166#387—21no—
167#383—21no—
168C43C2×D7111yescharacteristic
169#336—712no—
170#336—712no—
171#336—1421no—
172#336—1421no—
173C22×Q8C2×Dic7111yescharacteristic
174C22×C8—1422no—
175C22×C8—1422no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C2×Dic7111yescharacteristic
180C2×C4⋊C4C2×Dic7111yescharacteristic
181C2×C4⋊C4C2×Dic7111yescharacteristic
182C2×C42C2×Dic7111yescharacteristic
183C2×C42C2×Dic7111yescharacteristic
184C2×C42C22⋊D7111yescharacteristic
185C2×C42C22⋊D7111yescharacteristic
186C2×C42C22⋊D7111yescharacteristic
187C2×C42C22⋊D7111yescharacteristic
188C2×C42C22×D7111yescharacteristic
189C4⋊C8—2842no—
190#97—1424no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×Q8—41no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4C22⋊Dic7111yescharacteristic
208C22×C4C22⋊Dic7111yescharacteristic
209C22×C4C22⋊Dic7111yescharacteristic
210C22×C4C22⋊Dic7111yescharacteristic
211C22×C4C22×Dic7111yescharacteristic
212C22×C4C2×C22⋊D7111yescharacteristic
213C22×C4C2×C22⋊D7111yescharacteristic
214C2×C8—2842no—
215C2×C8—2842no—
216C2×C8—2842no—
217C2×C8—2842no—
218C2×C8—2842no—
219C2×C8—1424no—
220C4⋊C4—22no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C42—22no—
228C42—22no—
229C42—21no—
230C42—21no—
231C42—22no—
232C42—41no—
233C42—41no—
234C42—41no—
235C23C2×C22⋊Dic7111yescharacteristic
236Q8—82no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4#1455605512yes—
267C2×C4#1455613512yes—
268C2×C4#1455615311yescharacteristic
269C2×C4#1455615811yescharacteristic
270C2×C4#1455615912yes—
271C8—2844no—
272C8—5682no—
273C22#1798596611yescharacteristic
274C22#1798597311yescharacteristic
275C22#1798598311yescharacteristic
276C22#1798600011yescharacteristic
277C22#1798600811yescharacteristic
278C22C2×#1455605511yescharacteristic
279C22#1798601611yescharacteristic
280C4—22no—
281C4—22no—
282C4—22no—
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C2#2258670911yescharacteristic
293C2#2258675311yescharacteristic
294C2#2258681311yescharacteristic
295C2#2258685211yescharacteristic
296C2#2258687811yescharacteristic
297C2#2258694711yescharacteristic
298C2#2258707011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2863C2
socle, prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2863D71
hypocenterC71#7709
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #2863

Complements

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

NH ≅ G/Ntype
155. C71#7709semidirect

Maximal subgroups

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

subgroupindex
#225884572
#225880892
#225861662
#225861632
C71×#28632
#770971