Finite Group Databasebeta

← #31893460

Subgroups — #31893460

Order 18176.

458 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 115 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#22594161C211yesmaximal, characteristic
2#22587022C212yesmaximal, complemented
3#22586654C211yesmaximal, characteristic
4#22586200C212yesmaximal, complemented
5C71×#2921C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986052C2212yes—
8#17985938—21no—
9#17985935—21no—
10#17985919C412yes—
11#17985911C412yes—
12#17985897—21no—
13#17985865C2211yescharacteristic
14#17985859—21no—
15#17985747—24no—
16#17985725—21no—
17#17985723—21no—
18C2×#117×C71C2211yescharacteristic
19#433×C71—24no—
20#423×C71C2212yes—
21#407×C71—21no—
22C4×M16×C71—21no—
23C22×Dic142C2×C412yes—
24#14556128—22no—
25#14556128—22no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71—21no—
29C2×Dic71⋊C4—22no—
30C2×C22⋊Dic71—22no—
31C4⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C4×C71⋊C8—22no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—21no—
37C71⋊#95C2×C412yes—
38C23×C4×C71D411yescharacteristic
39C2×M16×C71—22no—
40#117×C71—22no—
41#117×C71—22no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71C2311yescharacteristic
48C4⋊C8×C71—22no—
49C4⋊C8×C71D414yes—
50C22⋊C8×C71—24no—
51#97×C71—22no—
52C4×C8×C71—22no—
53C2×Dic142—24no—
54C2×Dic142—82no—
55C22×Dic71—22no—
56C22×Dic71—22no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—41no—
59C4⋊Dic71—42no—
60C4⋊Dic71—41no—
61C4⋊Dic71—42no—
62Dic71⋊C4—82no—
63C22⋊Dic71—82no—
64C24×C71C22⋊C411yescharacteristic
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22×C411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75M16×C71—42no—
76M16×C71—42no—
77C2×C8×C71—24no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—22no—
80C22⋊C4×C71—22no—
81C22⋊C4×C71—22no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71C2×D412yes—
85C42×C71#4112yes—
86C23×C71—21no—
87C23×C71—21no—
88C23×C71—21no—
89C23×C71—41no—
90C23×C71—41no—
91C23×C71C2×C22⋊C411yescharacteristic
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71C4×D412yes—
104C2×C4×C71#12011yescharacteristic
105C2×C4×C71#12112yes—
106C2×C4×C71#12311yescharacteristic
107C2×C4×C71#13512yes—
108C2×C4×C71#13712yes—
109C2×Dic71—42no—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115Dic142—84no—
116C8×C71—44no—
117C71⋊C8—82no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—42no—
121C22×C71—41no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71—81no—
125C22×C71#38611yescharacteristic
126C22×C71#41611yescharacteristic
127C22×C71#41911yescharacteristic
128C4×C71—22no—
129C4×C71—22no—
130C4×C71—41no—
131C4×C71—41no—
132C4×C71—41no—
133C4×C71#45212yes—
134C4×C71#48512yes—
135Dic71—82no—
136Dic71—82no—
137#9192—711nomaximal, Sylow
138C2×C71—21no—
139C2×C71—21no—
140C2×C71—81no—
141C2×C71#187311yescharacteristic
142C2×C71#190911yescharacteristic
143C2×C71#191411yescharacteristic
144#2921D7111yescharacteristic
145#2628—711no—
146#1548—712no—
147#1458—711no—
148#1286—712no—
149C71#919211yescharacteristic, complemented, Sylow
150C2×C4⋊Q8—711no—
151C2×#117C2×D7111yescharacteristic
152#433—24no—
153#431—712no—
154#423C2×D7112yes—
155#407—21no—
156C4×M16—21no—
157#400—1421no—
158#399—1421no—
159#395—712no—
160#393—712no—
161#389—1421no—
162#382—711no—
163#380—1421no—
164#333—1424no—
165#324—1421no—
166#323—1421no—
167C22×Q8—712no—
168C23×C4C22⋊D7111yescharacteristic
169C2×M16—22no—
170C4⋊Q8—1422no—
171C4⋊Q8—1422no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—1421no—
175C2×C4⋊C4—1421no—
176C2×C4⋊C4—1421no—
177C2×C4⋊C4—1422no—
178C2×C4⋊C4C4×D7111yescharacteristic
179C2×C4⋊C4D28411yescharacteristic
180C2×C22⋊C4—1422no—
181C2×C22⋊C4C4×D7111yescharacteristic
182C2×C22⋊C4D28411yescharacteristic
183C2×C42C22⋊D7111yescharacteristic
184C2×C42C22×D7111yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—1422no—
187C4⋊C8C22⋊D7114yes—
188C22⋊C8—24no—
189C22⋊C8—1422no—
190#97—22no—
191C4×C8—22no—
192C4×C8—1422no—
193#95—712no—
194#95—1422no—
195#95—1422no—
196#95—1421no—
197C24(C2×D71)⋊C411yescharacteristic
198C2×Q8—5682no—
199C2×Q8—1424no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—1422no—
205C22×C4—1422no—
206C22×C4(C2×D71)⋊C411yescharacteristic
207C22×C4(C2×D71)⋊C411yescharacteristic
208C22×C4(C2×D71)⋊C411yescharacteristic
209C22×C4C2×C4×D7111yescharacteristic
210C22×C4C2×C22⋊D7111yescharacteristic
211C22×C4C2×D28411yescharacteristic
212M16—42no—
213M16—42no—
214C2×C8—24no—
215C2×C8—2842no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—2841no—
219C4⋊C4—2842no—
220C4⋊C4—2841no—
221C4⋊C4—2842no—
222C4⋊C4—5682no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—5682no—
226C42—21no—
227C42—21no—
228C42C2×C22⋊D7112yes—
229C42#1176586512yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×(C2×D71)⋊C411yescharacteristic
236Q8—5684no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—2842no—
249C2×C4—2842no—
250C2×C4—2842no—
251C2×C4—2842no—
252C2×C4—2842no—
253C2×C4—2842no—
254C2×C4C4×C22⋊D7112yes—
255C2×C4#1455609711yescharacteristic
256C2×C4#1455610212yes—
257C2×C4#1455611111yescharacteristic
258C2×C4#1455614812yes—
259C2×C4#1455615812yes—
260C8—44no—
261C8—5682no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1798587811yescharacteristic
270C22#1798599311yescharacteristic
271C22#1798600711yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—5682no—
278C4—5682no—
279C4#1798616112yes—
280C4#1798634012yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2258865211yescharacteristic
285C2#2258886211yescharacteristic
286C2#2258888711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2921C2
socleC22×C71#386
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2921D71
hypocenterC71#9192
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #22587022C2semidirect
4. #22586200C2semidirect
149. C71#9192semidirect

Maximal subgroups

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

subgroupindex
#225941612
#225870222
#225866542
#225862002
C71×#29212
#919271