Finite Group Databasebeta

← #31885722

Subgroups — #31885722

Order 18176.

468 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 127 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#22588089C211yesmaximal, characteristic
2#22588089C211yesmaximal, characteristic
3#22586163C214yesmaximal
4C4×C4⋊Q8×C71C211yesmaximal, characteristic
5C2×#14556039—21no—
6C2×#14556039—21no—
7#17985754—21no—
8#17985754—21no—
9#17985754C2212yes—
10#17985754C2212yes—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×C4×Q8×C71—21no—
13#391×C71—21no—
14#389×C71—21no—
15#389×C71C2212yes—
16#384×C71C411yescharacteristic
17C4×C4⋊C4×C71—21no—
18C4×C4⋊C4×C71C412yes—
19C43×C71C2211yescharacteristic
20C22×C71⋊C8—22no—
21C22×C71⋊C8—22no—
22#14556039—24no—
23#14556039—24no—
24C22×Q8×C71—21no—
25C4⋊Q8×C71—24no—
26C4×Q8×C71—44no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71C2×C412yes—
32C2×C4⋊C4×C71C2×C412yes—
33C2×C42×C71—22no—
34C2×C42×C71C2×C411yescharacteristic
35C2×C42×C71C2×C411yescharacteristic
36C2×C42×C71D411yescharacteristic
37C2×C42×C71D411yescharacteristic
38C2×C42×C71D411yescharacteristic
39C2×C42×C71D411yescharacteristic
40C2×C42×C71C2311yescharacteristic
41#95×C71—22no—
42#95×C71—22no—
43C2×C71⋊C8—24no—
44C2×C71⋊C8—24no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—42no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—42no—
49C2×Q8×C71—24no—
50C2×Q8×C71—42no—
51C2×Q8×C71—42no—
52C22×C4×C71—22no—
53C22×C4×C71—22no—
54C22×C4×C71C22⋊C411yescharacteristic
55C22×C4×C71C22⋊C411yescharacteristic
56C22×C4×C71C22⋊C411yescharacteristic
57C22×C4×C71C22⋊C411yescharacteristic
58C22×C4×C71C22×C411yescharacteristic
59C22×C4×C71C2×D411yescharacteristic
60C22×C4×C71C2×D411yescharacteristic
61C4⋊C4×C71—24no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—42no—
64C4⋊C4×C71—42no—
65C4⋊C4×C71—42no—
66C4⋊C4×C71—44no—
67C4⋊C4×C71M1614yes—
68C42×C71—22no—
69C42×C71—22no—
70C42×C71—22no—
71C42×C71—22no—
72C42×C71—22no—
73C42×C71—22no—
74C42×C71—41no—
75C42×C71—44no—
76C23×C71C2×C22⋊C411yescharacteristic
77Q8×C71—48no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—24no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71#10112yes—
100C2×C4×C71#10412yes—
101C2×C4×C71#10412yes—
102C2×C4×C71C2×M1612yes—
103C2×C4×C71AGL(1,ℤ/8)12yes—
104C2×C4×C71#13712yes—
105C71⋊C8—44no—
106C71⋊C8—44no—
107C22×C71#40711yescharacteristic
108C22×C71#41011yescharacteristic
109C22×C71C2×#10111yescharacteristic
110C22×C71#41811yescharacteristic
111C22×C71#41911yescharacteristic
112C22×C71C2×#10411yescharacteristic
113C22×C71C2×#10411yescharacteristic
114C4×C71—22no—
115C4×C71—22no—
116C4×C71—22no—
117C4×C71—22no—
118C4×C71—22no—
119C4×C71—22no—
120C4×C71—44no—
121C4×C71—41no—
122C4×C71—82no—
123#7638—711nomaximal, Sylow
124C2×C71#146711yescharacteristic
125C2×C71#149011yescharacteristic
126C2×C71#149111yescharacteristic
127C2×C71#150511yescharacteristic
128C2×C71#150611yescharacteristic
129C2×C71#152611yescharacteristic
130C2×C71#155611yescharacteristic
131C4×C4⋊Q8D7111yescharacteristic
132#1745—711no—
133#1745—711no—
134#1271—714no—
135C71#763811yescharacteristic, complemented, Sylow
136C2×C4⋊Q8Dic7111yescharacteristic
137C2×C4×Q8—21no—
138C2×#97—1421no—
139C2×#97—1421no—
140#391—21no—
141#389—21no—
142#389C2×D7112yes—
143#384Dic7111yescharacteristic
144C4×C4⋊C4—21no—
145C4×C4⋊C4Dic7112yes—
146C43C2×D7111yescharacteristic
147#336—712no—
148#336—712no—
149#336—1421no—
150#336—1421no—
151C22×Q8—21no—
152C22×C8—1422no—
153C22×C8—1422no—
154C4⋊Q8—24no—
155C4×Q8—44no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C2×Dic7112yes—
161C2×C4⋊C4C2×Dic7112yes—
162C2×C42—22no—
163C2×C42C2×Dic7111yescharacteristic
164C2×C42C2×Dic7111yescharacteristic
165C2×C42C22⋊D7111yescharacteristic
166C2×C42C22⋊D7111yescharacteristic
167C2×C42C22⋊D7111yescharacteristic
168C2×C42C22⋊D7111yescharacteristic
169C2×C42C22×D7111yescharacteristic
170#97—1424no—
171#97—1424no—
172#95—22no—
173#95—22no—
174C2×Q8—24no—
175C2×Q8—42no—
176C2×Q8—42no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4C22⋊Dic7111yescharacteristic
180C22×C4C22⋊Dic7111yescharacteristic
181C22×C4C22⋊Dic7111yescharacteristic
182C22×C4C22⋊Dic7111yescharacteristic
183C22×C4C22×Dic7111yescharacteristic
184C22×C4C2×C22⋊D7111yescharacteristic
185C22×C4C2×C22⋊D7111yescharacteristic
186C2×C8—2842no—
187C2×C8—2842no—
188C2×C8—2842no—
189C2×C8—2842no—
190C2×C8—1424no—
191C2×C8—1424no—
192C4⋊C4—24no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4#1176584914yes—
199C42—22no—
200C42—22no—
201C42—22no—
202C42—22no—
203C42—22no—
204C42—22no—
205C42—41no—
206C42—44no—
207C23C2×C22⋊Dic7111yescharacteristic
208Q8—48no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4#1455604712yes—
231C2×C4#1455605512yes—
232C2×C4#1455605512yes—
233C2×C4C2×#1176584912yes—
234C2×C4#1455615412yes—
235C2×C4#1455615912yes—
236C8—2844no—
237C8—2844no—
238C22#1798596311yescharacteristic
239C22#1798597311yescharacteristic
240C22C2×#1455604711yescharacteristic
241C22#1798600411yescharacteristic
242C22#1798600811yescharacteristic
243C22C2×#1455605511yescharacteristic
244C22C2×#1455605511yescharacteristic
245C4—22no—
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—44no—
253C4—82no—
254C2#2258668411yescharacteristic
255C2#2258678311yescharacteristic
256C2#2258678711yescharacteristic
257C2#2258684811yescharacteristic
258C2#2258685211yescharacteristic
259C2#2258693511yescharacteristic
260C2#2258705711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC4×C4⋊Q8×C71C2
socle, prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenterC4×C4⋊Q8D71
hypocenterC71#7638
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
135. C71#7638semidirect

Maximal subgroups

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

subgroupindex
#225880892
#225861632
C4×C4⋊Q8×C712
#763871