Finite Group Databasebeta

← #31895176

Subgroups — #31895176

Order 18176.

512 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 103 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#22592821C211yesmaximal, characteristic
2#22588420C211yesmaximal, characteristic
3#22586872C212yesmaximal
4#22586192C212yesmaximal
5C71×#2313C211yesmaximal, characteristic
6C2×#14556107C2211yescharacteristic
7#17985963C2212yes—
8#17985937—22no—
9#17985904C412yescomplemented
10#17985864C412yescomplemented
11#17985862—21no—
12#17985862—21no—
13C4×C22⋊Dic71C2211yescharacteristic
14#17985768—22no—
15#17985735—24no—
16C22×C4⋊C4×C71C2211yescharacteristic
17#387×C71—24no—
18#381×C71C2212yes—
19#380×C71—21no—
20C4×C22⋊C4×C71—21no—
21C22×Dic142C2×C411yescharacteristic
22C2×#11765849—22no—
23#14556107—22no—
24#14556107—22no—
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×Dic71⋊C4—21no—
27C2×C22⋊Dic71—21no—
28C2×C22⋊Dic71—21no—
29C2×C4×Dic71C2×C412yes—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—21no—
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71C2×C412yes—
44C2×C4⋊C4×C71D412yes—
45C2×C22⋊C4×C71—22no—
46C2×C42×C71—22no—
47#95×C71—22no—
48#95×C71—22no—
49#95×C71—22no—
50#95×C71D414yes—
51C2×Dic142—22no—
52C2×Dic142—42no—
53C22×Dic71—21no—
54C22×Dic71—22no—
55C22×Dic71—21no—
56#11765849—44no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—22no—
59Dic71⋊C4—42no—
60C22⋊Dic71—42no—
61C22⋊Dic71—41no—
62C22⋊Dic71—41no—
63C4×Dic71—42no—
64C24×C71C2×D411yescharacteristic
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—24no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71C22⋊C412yes—
74C22×C4×C71C22⋊C412yes—
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D412yes—
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71#4112yes—
79C4⋊C4×C71—24no—
80C4⋊C4×C71—24no—
81C4⋊C4×C71—82no—
82C22⋊C4×C71—44no—
83C42×C71—82no—
84C23×C71—22no—
85C23×C71—22no—
86C23×C71—41no—
87C23×C71C2×C22⋊C411yescharacteristic
88C23×C71#12011yescharacteristic
89C23×C71#12311yescharacteristic
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—44no—
101C2×C4×C71—44no—
102C2×C4×C71—44no—
103C2×C4×C71—81no—
104C2×C4×C71—81no—
105C2×C4×C71C4×D412yes—
106C2×C4×C71#12112yes—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110C2×Dic71—41no—
111C2×Dic71—41no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114Dic142—82no—
115C71⋊C8—82no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71—44no—
124C22×C71#38611yescharacteristic
125C22×C71#41012yes—
126C22×C71#45512yes—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—42no—
130C4×C71—42no—
131C4×C71—84no—
132Dic71—82no—
133Dic71—81no—
134Dic71—81no—
135#9460—711nomaximal, Sylow
136C2×C71—22no—
137C2×C71—22no—
138C2×C71—41no—
139C2×C71#188511yescharacteristic
140C2×C71#189911yescharacteristic
141C2×C71#191211yescharacteristic
142#2503—711no—
143#2313D7111yescharacteristic
144#1821—711no—
145#1511—712no—
146#1284—712no—
147C71#946011yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—711no—
149C22×C4⋊C4C2×D7111yescharacteristic
150#407—712no—
151#400—1422no—
152#391—712no—
153#387—24no—
154#382—712no—
155#381—1421no—
156#381—1421no—
157#381C2×D7112yes—
158#380—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—711no—
161#343—1422no—
162#328—1424no—
163C22×Q8—711no—
164C23×C4C22⋊D7111yescharacteristic
165C23×C4C22⋊D7111yescharacteristic
166C23×C4C22×D7111yescharacteristic
167C2×M16—1422no—
168C22⋊Q8—1422no—
169C22⋊Q8—1422no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—711no—
174C2×C4⋊C4—1421no—
175C2×C4⋊C4C4×D7112yes—
176C2×C4⋊C4D28412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1421no—
179C2×C22⋊C4—1421no—
180C2×C42—22no—
181C2×C42—712no—
182C22⋊C8—1422no—
183C22⋊C8—1422no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—1422no—
188#95—1422no—
189#95—1422no—
190#95—1422no—
191#95—1421no—
192#95C22⋊D7114yes—
193C24C2×C22⋊D7111yescharacteristic
194C2×Q8—2842no—
195C2×Q8—1422no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—1422no—
205C22×C4—1421no—
206C22×C4—1421no—
207C22×C4(C2×D71)⋊C412yes—
208C22×C4(C2×D71)⋊C412yes—
209C22×C4C2×C4×D7111yescharacteristic
210C22×C4C2×C22⋊D7112yes—
211C22×C4C2×D28411yescharacteristic
212C22×C4#1176586512yes—
213M16—2844no—
214C2×C8—2842no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—82no—
218C4⋊C4—2842no—
219C4⋊C4—1422no—
220C22⋊C4—44no—
221C22⋊C4—2842no—
222C22⋊C4—2841no—
223C22⋊C4—2841no—
224C42—82no—
225C42—2842no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23C2×(C2×D71)⋊C411yescharacteristic
230C23#1455609711yescharacteristic
231C23#1455611111yescharacteristic
232Q8—5682no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—2842no—
249C2×C4—2842no—
250C2×C4—2842no—
251C2×C4—2841no—
252C2×C4—2841no—
253C2×C4—2842no—
254C2×C4—2842no—
255C2×C4C4×C22⋊D7112yes—
256C2×C4#1455610212yes—
257C8—5682no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#1798587811yescharacteristic
267C22#1798597212yes—
268C22#1798617712yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—84no—
274C4—5682no—
275C4—5681no—
276C4—5681no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#2258872811yescharacteristic
281C2#2258879811yescharacteristic
282C2#2258887711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2313C2
socleC22×C71#386
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2313D71
hypocenterC71#9460
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
9. #17985904C4semidirect
10. #17985864C4semidirect
147. C71#9460semidirect

Maximal subgroups

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

subgroupindex
#225928212
#225884202
#225868722
#225861922
C71×#23132
#946071