Finite Group Databasebeta

← #31908618

Subgroups — #31908618

Order 18176.

388 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 99 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#22593545C211yesmaximal, characteristic
2#22586664C211yesmaximal, characteristic
3#22586369C214yesmaximal
4C71×#1801C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6#17986052—21no—
7#17985919C412yes—
8#17985919C2212yes—
9#17985906—21no—
10#17985897—21no—
11#17985897—21no—
12#17985868—21no—
13C4×C4⋊Dic71C411yescharacteristic
14#17985725—21no—
15#17985725C2212yes—
16C2×C4×Q8×C71C2211yescharacteristic
17#423×C71—21no—
18C2×#101×C71—21no—
19#343×C71C2212yes—
20C22×Dic142C2×C412yes—
21#14556128—41no—
22#14556128—41no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×Dic71⋊C4—22no—
27C2×Dic71⋊C4—22no—
28C2×C4×Dic71—21no—
29C4⋊(C71⋊C8)—41no—
30C22⋊(C71⋊C8)—22no—
31C4×C71⋊C8—41no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95C2×C412yes—
35C22×Q8×C71D411yescharacteristic
36C2×M16×C71—22no—
37C4×Q8×C71—41no—
38C4×Q8×C71—41no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C4⋊C8×C71—41no—
46C4⋊C8×C71—41no—
47#101×C71—42no—
48C2×Dic142—42no—
49C2×Dic142—82no—
50C22×Dic71—22no—
51C22×Dic71—22no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—24no—
54C4⋊Dic71—42no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57Dic71⋊C4—82no—
58Dic71⋊C4—82no—
59C4×Dic71—42no—
60C2×Q8×C71—41no—
61C2×Q8×C71—41no—
62C2×Q8×C71SD1612yes—
63C2×Q8×C71Q1612yes—
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22×C411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71M16×C71—42no—
72M16×C71—42no—
73C2×C8×C71—42no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—41no—
76C4⋊C4×C71—41no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71—21no—
83C23×C71C2×C22⋊C411yescharacteristic
84Q8×C71—42no—
85Q8×C71—42no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—22no—
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#9911yescharacteristic
100C2×C4×C71#9911yescharacteristic
101C2×C4×C71#10312yes—
102C2×C4×C71#10312yes—
103C2×C4×C71C2×SD1611yescharacteristic
104C2×C4×C71C2×Q1611yescharacteristic
105C2×Dic71—24no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110C2×Dic71—42no—
111Dic142—84no—
112C8×C71—82no—
113C71⋊C8—82no—
114C22×C71—21no—
115C22×C71—21no—
116C22×C71C2×#9911yescharacteristic
117C22×C71C2×#10311yescharacteristic
118C22×C71#42111yescharacteristic
119C4×C71—21no—
120C4×C71—21no—
121C4×C71—22no—
122C4×C71—22no—
123C4×C71—21no—
124C4×C71—41no—
125C4×C71—41no—
126C4×C71—41no—
127C4×C71#35612yes—
128Dic71—44no—
129Dic71—82no—
130#11908—711nomaximal, Sylow
131C2×C71—21no—
132C2×C71—21no—
133C2×C71#152011yescharacteristic
134C2×C71#212211yescharacteristic
135C2×C71C2×#35611yescharacteristic
136#2567—711no—
137#1801D7111yescharacteristic
138#1461—711no—
139#1354—714no—
140C71#1190811yescharacteristic, complemented, Sylow
141C2×C4⋊Q8—711no—
142C2×C4×Q8C2×D7111yescharacteristic
143#431—1421no—
144#423—21no—
145C2×#101—21no—
146#395—712no—
147#395—712no—
148#391—1421no—
149#389—1421no—
150#389—1421no—
151#384—1421no—
152C4×C4⋊C4—711no—
153#343C2×D7112yes—
154#324—712no—
155#324—1421no—
156C22×Q8—712no—
157C22×Q8C22⋊D7111yescharacteristic
158C2×M16—22no—
159C4⋊Q8—2841no—
160C4⋊Q8—2841no—
161C4×Q8—41no—
162C4×Q8—41no—
163C2×C4⋊C4—1421no—
164C2×C4⋊C4—1421no—
165C2×C4⋊C4—1422no—
166C2×C4⋊C4—1422no—
167C2×C4⋊C4—1421no—
168C2×C4⋊C4C4×D7111yescharacteristic
169C2×C4⋊C4D28411yescharacteristic
170C2×C4⋊C4C22×D7111yescharacteristic
171C2×C42—1421no—
172C2×C42C4×D7111yescharacteristic
173C2×C42D28411yescharacteristic
174C2×C42C22⋊D7111yescharacteristic
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—2841no—
178#101—42no—
179C22⋊C8—1422no—
180C4×C8—2841no—
181#95—712no—
182#95—1422no—
183#95—1422no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—2842no—
187C2×Q8—5682no—
188C2×Q8#1176585312yes—
189C2×Q8#1176585512yes—
190C22×C4—1422no—
191C22×C4—1422no—
192C22×C4(C2×D71)⋊C411yescharacteristic
193C22×C4(C2×D71)⋊C411yescharacteristic
194C22×C4(C2×D71)⋊C411yescharacteristic
195C22×C4(C2×D71)⋊C411yescharacteristic
196C22×C4C2×C4×D7111yescharacteristic
197C22×C4C2×C22⋊D7111yescharacteristic
198C22×C4C2×D28411yescharacteristic
199M16—42no—
200M16—42no—
201C2×C8—42no—
202C2×C8—2842no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—2842no—
207C4⋊C4—2841no—
208C4⋊C4—2841no—
209C4⋊C4—5682no—
210C4⋊C4—5682no—
211C4⋊C4—1424no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—2842no—
219C23C2×(C2×D71)⋊C411yescharacteristic
220Q8—42no—
221Q8—42no—
222Q8—5684no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—2842no—
237C2×C4—2842no—
238C2×C4—2842no—
239C2×C4—2842no—
240C2×C4—2842no—
241C2×C4—1424no—
242C2×C4#1455604211yescharacteristic
243C2×C4#1455604211yescharacteristic
244C2×C4#1455605112yes—
245C2×C4#1455605112yes—
246C2×C4C2×#1176585311yescharacteristic
247C2×C4C2×#1176585511yescharacteristic
248C8—82no—
249C8—5682no—
250C22—21no—
251C22—21no—
252C22C2×#1455604211yescharacteristic
253C22C2×#1455605111yescharacteristic
254C22#1798601711yescharacteristic
255C4—21no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—2844no—
264C4—5682no—
265C4#1798579712yes—
266C2—21no—
267C2—21no—
268C2#2258691111yescharacteristic
269C2#2258999111yescharacteristic
270C2C2×#1798579711yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1801C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1801D71
hypocenterC71#11908
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
140. C71#11908semidirect

Maximal subgroups

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

subgroupindex
#225935452
#225866642
#225863692
C71×#18012
#1190871