Finite Group Databasebeta

← #31892950

Subgroups — #31892950

Order 18176.

498 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#22597421C211yesmaximal, characteristic
2#22591056C211yesmaximal, characteristic
3#22586734C212yesmaximal
4#22586168C212yesmaximal
5C71×#1461C211yesmaximal, characteristic
6C2×#14556126C411yescharacteristic
7C2×C4×Dic142C2211yescharacteristic
8Q8×C71⋊C8—24no—
9#17986052—22no—
10#17986052C2212yes—
11#17985906C411yescharacteristic
12#17985904C411yescharacteristic
13#17985883—22no—
14#17985880—22no—
15#17985864C411yescharacteristic
16#17985845C2211yescharacteristic
17#17985729—24no—
18C2×C4×Q8×C71C2211yescharacteristic
19#431×C71C2212yes—
20#324×C71—22no—
21C22×Dic142C2×C411yescharacteristic
22#14556126—22no—
23#14556126C814yescomplemented
24C4×Dic142—22no—
25C4×Dic142—22no—
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—21no—
29C2×C4×Dic71—21no—
30C2×C4×Dic71C2×C411yescharacteristic
31C2×C4×Dic71C2×C411yescharacteristic
32C4⋊(C71⋊C8)—24no—
33C4⋊(C71⋊C8)—22no—
34C22⋊(C71⋊C8)—22no—
35C4×C71⋊C8—24no—
36C4×C71⋊C8D414yes—
37C71⋊#95—22no—
38C71⋊#95—22no—
39C71⋊#95—22no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C22×Q8×C71C2×C411yescharacteristic
43C4×Q8×C71—22no—
44C4×Q8×C71—22no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D412yes—
47C2×C42×C71D412yes—
48C2×C42×C71C2311yescharacteristic
49C4⋊C8×C71—22no—
50C22⋊C8×C71—22no—
51C4×C8×C71—22no—
52C2×Dic142—22no—
53C2×Dic142—22no—
54C2×Dic142—21no—
55C2×Dic142C2×C812yes—
56C22×Dic71—21no—
57C22×Dic71—21no—
58C22×Dic71—21no—
59C22×Dic71—21no—
60C2×C71⋊C8—24no—
61C4⋊Dic71—22no—
62Dic71⋊C4—28no—
63Dic71⋊C4—41no—
64Dic71⋊C4—41no—
65C4×Dic71—21no—
66C4×Dic71—22no—
67C4×Dic71—42no—
68C4×Dic71C2×C812yes—
69C2×Q8×C71—22no—
70C2×Q8×C71—22no—
71C2×Q8×C71C2×C812yes—
72C2×Q8×C71M1612yes—
73C22×C4×C71C22⋊C412yes—
74C22×C4×C71C22⋊C412yes—
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C2×C8×C71—42no—
79C4⋊C4×C71—22no—
80C4⋊C4×C71—24no—
81C42×C71—22no—
82C42×C71—22no—
83C42×C71C2×D412yes—
84C42×C71#4112yes—
85C23×C71C2×C22⋊C411yescharacteristic
86Q8×C71—24no—
87Q8×C71—24no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—42no—
94C2×C4×C71C22⋊C814yes—
95C2×C4×C71C4×D412yes—
96C2×C4×C71#12011yescharacteristic
97C2×C4×C71#12112yes—
98C2×C4×C71#12311yescharacteristic
99C2×C4×C71C22×C811yescharacteristic
100C2×C4×C71C2×M1611yescharacteristic
101C2×C4×C71#13112yes—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—41no—
105C2×Dic71—41no—
106C2×Dic71—41no—
107C2×Dic71—41no—
108C2×Dic71—41no—
109C2×Dic71—41no—
110C2×Dic71—42no—
111C2×Dic71—41no—
112C2×Dic71—41no—
113Dic142—24no—
114Dic142—42no—
115C8×C71—82no—
116C71⋊C8—44no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71#38611yescharacteristic
120C22×C71C2×C22⋊C811yescharacteristic
121C22×C71#40811yescharacteristic
122C4×C71—24no—
123C4×C71—22no—
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—42no—
127C4×C71C8×D412yes—
128C4×C71#43512yes—
129Dic71—42no—
130Dic71—42no—
131Dic71—81no—
132Dic71—81no—
133#9100—711nomaximal, Sylow
134C2×C71—21no—
135C2×C71—21no—
136C2×C71#186911yescharacteristic
137C2×C71#190911yescharacteristic
138C2×C71#191211yescharacteristic
139#2982—711no—
140#2327—711no—
141#1479—712no—
142#1461D7111yescharacteristic
143#1274—712no—
144C71#910011yescharacteristic, complemented, Sylow
145C2×C4⋊Q8—711no—
146C2×C4×Q8—711no—
147C2×C4×Q8C2×D7111yescharacteristic
148C8×Q8—1424no—
149#431—712no—
150#431—1422no—
151#431C2×D7112yes—
152#391—711no—
153#391—711no—
154#387—1422no—
155#387—1422no—
156#382—711no—
157#376—711no—
158Q8⋊C8—1424no—
159#324—22no—
160C22×Q8—711no—
161C22×Q8C4×D7111yescharacteristic
162C4⋊Q8—714no—
163C4⋊Q8—1422no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—1422no—
167C4×Q8—1422no—
168C2×C4⋊C4—711no—
169C2×C4⋊C4—1421no—
170C2×C4⋊C4—1422no—
171C2×C4⋊C4C4×D7111yescharacteristic
172C2×C4⋊C4C22⋊D7112yes—
173C2×C42—711no—
174C2×C42—711no—
175C2×C42—1421no—
176C2×C42D28412yes—
177C2×C42C22×D7111yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—1424no—
180C4⋊C8—1422no—
181C22⋊C8—22no—
182C22⋊C8—1422no—
183C4×C8—22no—
184C4×C8—714no—
185C4×C8—1424no—
186#95—1422no—
187#95—1422no—
188#95—1422no—
189#95—1421no—
190#95—1421no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—712no—
194C2×Q8—1422no—
195C2×Q8—1422no—
196C2×Q8—1421no—
197C2×Q8C8×D7112yes—
198C2×Q8#1176584812yes—
199C22×C4—1421no—
200C22×C4—1421no—
201C22×C4—1421no—
202C22×C4—1421no—
203C22×C4(C2×D71)⋊C412yes—
204C22×C4(C2×D71)⋊C412yes—
205C22×C4C2×C4×D7111yescharacteristic
206C22×C4C2×C22⋊D7111yescharacteristic
207C22×C4C2×D28411yescharacteristic
208C2×C8—42no—
209C2×C8—1424no—
210C4⋊C4—22no—
211C4⋊C4—24no—
212C4⋊C4—2841no—
213C4⋊C4—2841no—
214C4⋊C4—1422no—
215C4⋊C4—1428no—
216C42—22no—
217C42—22no—
218C42—2842no—
219C42—712no—
220C42—1422no—
221C42—1421no—
222C42D4×D7112yes—
223C42#1176586412yes—
224C23C2×(C2×D71)⋊C411yescharacteristic
225Q8—24no—
226Q8—24no—
227Q8—2842no—
228Q8—1424no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—2841no—
236C2×C4—2841no—
237C2×C4—2841no—
238C2×C4—2841no—
239C2×C4—2841no—
240C2×C4—2841no—
241C2×C4—2842no—
242C2×C4—2841no—
243C2×C4—2841no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4(C2×D71)⋊C814yes—
247C2×C4(C22⋊D71)⋊C412yes—
248C2×C4#1455609811yescharacteristic
249C2×C4#1455610012yes—
250C2×C4#1455611211yescharacteristic
251C2×C4C2×C8×D7111yescharacteristic
252C2×C4C2×#1176584811yescharacteristic
253C2×C4#1455613612yes—
254C8—82no—
255C8—2844no—
256C22—21no—
257C22—21no—
258C22#1798587611yescharacteristic
259C22C2×(C2×D71)⋊C811yescharacteristic
260C22#1798596711yescharacteristic
261C4—22no—
262C4—21no—
263C4—21no—
264C4—24no—
265C4—42no—
266C4—2842no—
267C4—2842no—
268C4—5681no—
269C4—5681no—
270C4(C22⋊D71)⋊C812yes—
271C4#1798606712yes—
272C2—21no—
273C2—21no—
274C2#2258863011yescharacteristic
275C2#2258886011yescharacteristic
276C2#2258887511yescharacteristic
subgroupG/H
centerC22#17985876
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC71×#1461C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1461D71
hypocenterC71#9100
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
23. #14556126C8semidirect
144. C71#9100semidirect

Maximal subgroups

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

subgroupindex
#225974212
#225910562
#225867342
#225861682
C71×#14612
#910071