Finite Group Databasebeta

← #31886009

Subgroups — #31886009

Order 18176.

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

normal subgroups only

This group has 394 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#22588483C211yesmaximal, characteristic
2#22588089C211yesmaximal, characteristic
3#22586163C214yesmaximal
4C71×#2859C211yesmaximal, characteristic
5C2×#14556039—21no—
6#17985754—21no—
7#17985754—21no—
8#17985754—21no—
9#17985754C2212yes—
10#17985754C2212yes—
11C2×#125×C71—21no—
12#389×C71—21no—
13#389×C71—21no—
14#389×C71C412yes—
15#389×C71C2212yes—
16#384×C71C411yescharacteristic
17#384×C71C411yescharacteristic
18#382×C71—21no—
19C43×C71C2211yescharacteristic
20C22×C71⋊C8—22no—
21C22×C71⋊C8—22no—
22#14556039—24no—
23#125×C71—44no—
24C2×C4⋊C4×C71—21no—
25C2×C4⋊C4×C71—21no—
26C2×C4⋊C4×C71—22no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71—22no—
30C2×C4⋊C4×C71C2×C412yes—
31C2×C4⋊C4×C71C2×C412yes—
32C2×C42×C71C2×C411yescharacteristic
33C2×C42×C71C2×C411yescharacteristic
34C2×C42×C71D411yescharacteristic
35C2×C42×C71D411yescharacteristic
36C2×C42×C71D411yescharacteristic
37C2×C42×C71D411yescharacteristic
38C2×C42×C71C2311yescharacteristic
39#95×C71—22no—
40#95×C71—22no—
41C2×C71⋊C8—24no—
42C2×C71⋊C8—42no—
43C2×C71⋊C8—42no—
44C2×C71⋊C8—42no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—42no—
47C22×C4×C71—22no—
48C22×C4×C71—22no—
49C22×C4×C71C22⋊C411yescharacteristic
50C22×C4×C71C22⋊C411yescharacteristic
51C22×C4×C71C22⋊C411yescharacteristic
52C22×C4×C71C22⋊C411yescharacteristic
53C22×C4×C71C22×C411yescharacteristic
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71C2×D411yescharacteristic
56C4⋊C4×C71—24no—
57C4⋊C4×C71—42no—
58C4⋊C4×C71—42no—
59C4⋊C4×C71—42no—
60C4⋊C4×C71—42no—
61C4⋊C4×C71—44no—
62C4⋊C4×C71—42no—
63C4⋊C4×C71—44no—
64C42×C71—22no—
65C42×C71—22no—
66C42×C71—22no—
67C42×C71—22no—
68C42×C71—22no—
69C42×C71—41no—
70C42×C71—41no—
71C23×C71C2×C22⋊C411yescharacteristic
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
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—42no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71#10112yes—
95C2×C4×C71#10412yes—
96C2×C4×C71#13112yes—
97C2×C4×C71AGL(1,ℤ/8)12yes—
98C2×C4×C71#13712yes—
99C71⋊C8—44no—
100C71⋊C8—82no—
101C22×C71#40811yescharacteristic
102C22×C71#41011yescharacteristic
103C22×C71C2×#10111yescharacteristic
104C22×C71#41811yescharacteristic
105C22×C71#41911yescharacteristic
106C22×C71C2×#10411yescharacteristic
107C22×C71#42111yescharacteristic
108C4×C71—22no—
109C4×C71—22no—
110C4×C71—22no—
111C4×C71—22no—
112C4×C71—22no—
113C4×C71—41no—
114C4×C71—41no—
115C4×C71—82no—
116C4×C71—82no—
117#7703—711nomaximal, Sylow
118C2×C71#147011yescharacteristic
119C2×C71#148411yescharacteristic
120C2×C71#149711yescharacteristic
121C2×C71#150611yescharacteristic
122C2×C71#151411yescharacteristic
123C2×C71#152811yescharacteristic
124C2×C71#155611yescharacteristic
125#2859D7111yescharacteristic
126#1839—711no—
127#1745—711no—
128#1271—714no—
129C71#770311yescharacteristic, complemented, Sylow
130C2×#125—21no—
131C2×#97—1421no—
132#389—21no—
133#389—21no—
134#389Dic7112yes—
135#389C2×D7112yes—
136#384Dic7111yescharacteristic
137#384Dic7111yescharacteristic
138#382—21no—
139C43C2×D7111yescharacteristic
140#336—712no—
141#336—712no—
142#336—1421no—
143#336—1421no—
144#336—1421no—
145C22×C8—1422no—
146C22×C8—1422no—
147#125—44no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C2×Dic7112yes—
155C2×C4⋊C4C2×Dic7112yes—
156C2×C42C2×Dic7111yescharacteristic
157C2×C42C2×Dic7111yescharacteristic
158C2×C42C22⋊D7111yescharacteristic
159C2×C42C22⋊D7111yescharacteristic
160C2×C42C22⋊D7111yescharacteristic
161C2×C42C22⋊D7111yescharacteristic
162C2×C42C22×D7111yescharacteristic
163#97—1424no—
164#95—22no—
165#95—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4C22⋊Dic7111yescharacteristic
169C22×C4C22⋊Dic7111yescharacteristic
170C22×C4C22⋊Dic7111yescharacteristic
171C22×C4C22⋊Dic7111yescharacteristic
172C22×C4C22×Dic7111yescharacteristic
173C22×C4C2×C22⋊D7111yescharacteristic
174C22×C4C2×C22⋊D7111yescharacteristic
175C2×C8—2842no—
176C2×C8—2842no—
177C2×C8—2842no—
178C2×C8—2842no—
179C2×C8—2842no—
180C2×C8—1424no—
181C4⋊C4—24no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—44no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—41no—
195C42—41no—
196C23C2×C22⋊Dic7111yescharacteristic
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4#1455604712yes—
220C2×C4#1455605512yes—
221C2×C4#1455613512yes—
222C2×C4#1455615412yes—
223C2×C4#1455615912yes—
224C8—2844no—
225C8—5682no—
226C22#1798596611yescharacteristic
227C22#1798597311yescharacteristic
228C22C2×#1455604711yescharacteristic
229C22#1798600411yescharacteristic
230C22#1798600811yescharacteristic
231C22C2×#1455605511yescharacteristic
232C22#1798601611yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—41no—
239C4—41no—
240C4—82no—
241C4—82no—
242C2#2258669611yescharacteristic
243C2#2258675711yescharacteristic
244C2#2258681311yescharacteristic
245C2#2258685211yescharacteristic
246C2#2258688711yescharacteristic
247C2#2258694211yescharacteristic
248C2#2258705711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2859C2
socle, prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2859D71
hypocenterC71#7703
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #2859

Complements

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

NH ≅ G/Ntype
129. C71#7703semidirect

Maximal subgroups

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

subgroupindex
#225884832
#225880892
#225861632
C71×#28592
#770371