Finite Group Databasebeta

← #31895156

Subgroups — #31895156

Order 18176.

432 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 107 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#22593552C211yesmaximal, characteristic
2#22587058C212yesmaximal
3#22586684C211yesmaximal, characteristic
4#22586197C212yesmaximal
5C71×#2983C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986056C2212yes—
8#17985938—21no—
9#17985938—21no—
10#17985906C412yes—
11#17985905—21no—
12#17985905—21no—
13#17985864C412yes—
14C4×C4⋊Dic71C2211yescharacteristic
15#17985746—24no—
16#17985725—21no—
17#17985725—21no—
18C2×C4×Q8×C71C2211yescharacteristic
19#446×C71—24no—
20#432×C71—21no—
21#432×C71C2212yes—
22#431×C71—21no—
23C22×Dic142—21no—
24#14556128—22no—
25#14556128—22no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71C2×C412yes—
29C2×C4×Dic71C2×C412yes—
30C4⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32#14556039—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—21no—
38C22×Q8×C71C2×C411yescharacteristic
39C4×Q8×C71—22no—
40C4×Q8×C71—22no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71C2311yescharacteristic
47C4⋊C8×C71—22no—
48C4⋊C8×C71—22no—
49C4⋊C8×C71D414yes—
50C22⋊C8×C71—22no—
51#97×C71—22no—
52#97×C71—22no—
53C4×C8×C71—22no—
54C2×Dic142—44no—
55C2×Dic142—41no—
56C2×Dic142—41no—
57C22×Dic71—22no—
58C22×Dic71—22no—
59C2×C71⋊C8—42no—
60C4⋊Dic71—24no—
61C4⋊Dic71—42no—
62C4⋊Dic71—42no—
63C4×Dic71—42no—
64C2×Q8×C71—22no—
65C2×Q8×C71—22no—
66C2×Q8×C71—41no—
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C2×C8×C71—24no—
75C4⋊C4×C71—22no—
76C4⋊C4×C71—22no—
77C4⋊C4×C71—22no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71C2×D412yes—
83C42×C71#4112yes—
84C23×C71C2×C22⋊C411yescharacteristic
85Q8×C71—44no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71#10112yes—
100C2×C4×C71C4×D412yes—
101C2×C4×C71#12011yescharacteristic
102C2×C4×C71#12112yes—
103C2×C4×C71#12311yescharacteristic
104C2×Dic71—42no—
105C2×Dic71—42no—
106C2×Dic71—44no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109Dic142—82no—
110Dic142—82no—
111C8×C71—44no—
112C71⋊C8—82no—
113C22×C71—21no—
114C22×C71—21no—
115C22×C71#38611yescharacteristic
116C22×C71C2×#10111yescharacteristic
117C22×C71#41311yescharacteristic
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—22no—
122C4×C71—41no—
123C4×C71—41no—
124C4×C71—41no—
125C4×C71#45412yes—
126Dic71—82no—
127Dic71—82no—
128#9456—711nomaximal, Sylow
129C2×C71—21no—
130C2×C71—21no—
131C2×C71#188811yescharacteristic
132C2×C71#190111yescharacteristic
133C2×C71#190911yescharacteristic
134#2983D7111yescharacteristic
135#2568—711no—
136#1556—712no—
137#1467—711no—
138#1285—712no—
139C71#945611yescharacteristic, complemented, Sylow
140C2×C4⋊Q8—711no—
141C2×C4×Q8C2×D7111yescharacteristic
142#446—24no—
143#432—21no—
144#432—712no—
145#432C2×D7112yes—
146#431—21no—
147#400—1421no—
148#400—1421no—
149#391—712no—
150#391—1421no—
151#391—1421no—
152#382—712no—
153C4×C4⋊C4—711no—
154#332—1424no—
155#324—1421no—
156#324—1421no—
157C22×Q8—1421no—
158C22×Q8C4×D7111yescharacteristic
159C4⋊Q8—1422no—
160C4⋊Q8—1422no—
161C4×Q8—22no—
162C4×Q8—22no—
163C2×C4⋊C4—712no—
164C2×C4⋊C4—1421no—
165C2×C4⋊C4—1421no—
166C2×C4⋊C4C4×D7111yescharacteristic
167C2×C4⋊C4D28411yescharacteristic
168C2×C4⋊C4D28411yescharacteristic
169C2×C42—712no—
170C2×C42C22⋊D7111yescharacteristic
171C2×C42C22⋊D7111yescharacteristic
172C2×C42C22×D7111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—1422no—
176C4⋊C8C22⋊D7114yes—
177C22⋊C8—22no—
178C22⋊C8—1422no—
179#97—22no—
180#97—22no—
181#97—1422no—
182C4×C8—22no—
183#95—1422no—
184#95—1422no—
185#95—1422no—
186#95—1422no—
187#95—1421no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—2844no—
192C2×Q8—2841no—
193C2×Q8—2841no—
194C22×C4—1422no—
195C22×C4—1422no—
196C22×C4(C2×D71)⋊C411yescharacteristic
197C22×C4(C2×D71)⋊C411yescharacteristic
198C22×C4(C2×D71)⋊C411yescharacteristic
199C22×C4(C2×D71)⋊C411yescharacteristic
200C22×C4C2×C4×D7111yescharacteristic
201C22×C4C2×C22⋊D7111yescharacteristic
202C22×C4C2×D28411yescharacteristic
203C2×C8—24no—
204C2×C8—2842no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—2842no—
209C4⋊C4—2842no—
210C4⋊C4—1424no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—2842no—
216C42C2×C22⋊D7112yes—
217C42#1176586512yes—
218C23C2×(C2×D71)⋊C411yescharacteristic
219Q8—44no—
220Q8—5682no—
221Q8—5682no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—2842no—
236C2×C4—2844no—
237C2×C4—2842no—
238C2×C4—2842no—
239C2×C4—2842no—
240C2×C4#1455604612yes—
241C2×C4C4×C22⋊D7112yes—
242C2×C4#1455609711yescharacteristic
243C2×C4#1455610212yes—
244C2×C4#1455611111yescharacteristic
245C8—44no—
246C8—5682no—
247C22—21no—
248C22—21no—
249C22#1798587811yescharacteristic
250C22C2×#1455604611yescharacteristic
251C22#1798598111yescharacteristic
252C4—21no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—5682no—
260C4—5682no—
261C4#1798617212yes—
262C2—21no—
263C2—21no—
264C2#2258874311yescharacteristic
265C2#2258880811yescharacteristic
266C2#2258886211yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2983C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2983D71
hypocenterC71#9456
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
139. C71#9456semidirect

Maximal subgroups

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

subgroupindex
#225935522
#225870582
#225866842
#225861972
C71×#29832
#945671