Finite Group Databasebeta

← #31895317

Subgroups — #31895317

Order 18176.

454 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 119 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#22597425C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic
3#22586769C212yesmaximal
4#22586198C212yesmaximal
5C71×#2581C211yesmaximal, characteristic
6#17986130—24no—
7#17986056—22no—
8#17986052C2212yes—
9#17986023C2212yes—
10C2×#14556047—21no—
11#17985768—21no—
12Q8⋊(C71⋊C8)—24no—
13C2×C4×Q8×C71C2211yescharacteristic
14C2×C4×Q8×C71C2211yescharacteristic
15#400×C71C412yes—
16#395×C71C412yes—
17#391×C71—21no—
18#389×C71—21no—
19#387×C71—22no—
20#382×C71C2211yescharacteristic
21C2×#11765849—22no—
22C4⋊(C71⋊C8)—24no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26#14556047—42no—
27C22⋊(C71⋊C8)—22no—
28#14556039—24no—
29C4×C71⋊C8D414yes—
30C22×Q8×C71C2×C411yescharacteristic
31C22×Q8×C71D411yescharacteristic
32C4×Q8×C71—22no—
33C4×Q8×C71—22no—
34C4×Q8×C71—22no—
35C4×Q8×C71—22no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C412yes—
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C42×C71—21no—
43C2×C42×C71D412yes—
44C2×C42×C71C2311yescharacteristic
45#95×C71—22no—
46#95×C71—22no—
47#95×C71—22no—
48#95×C71—21no—
49#95×C71C2×C412yes—
50#11765849—42no—
51#11765849—42no—
52C2×C71⋊C8—24no—
53C2×C71⋊C8—42no—
54C2×Q8×C71—22no—
55C2×Q8×C71—21no—
56C2×Q8×C71—21no—
57C2×Q8×C71—22no—
58C2×Q8×C71—22no—
59C2×Q8×C71—42no—
60C22×C4×C71—21no—
61C22×C4×C71—22no—
62C22×C4×C71—21no—
63C22×C4×C71C22⋊C412yes—
64C22×C4×C71C22⋊C412yes—
65C22×C4×C71C22×C411yescharacteristic
66C22×C4×C71C2×D411yescharacteristic
67C22×C4×C71C2×D411yescharacteristic
68C4⋊C4×C71—24no—
69C4⋊C4×C71—22no—
70C4⋊C4×C71—22no—
71C4⋊C4×C71—44no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—41no—
76C42×C71—22no—
77C42×C71—22no—
78C42×C71—42no—
79C42×C71C2×D412yes—
80C42×C71#4112yes—
81C23×C71C2×C22⋊C411yescharacteristic
82Q8×C71—24no—
83Q8×C71—42no—
84Q8×C71—42no—
85Q8×C71—42no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—41no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71—41no—
100C2×C4×C71—42no—
101C2×C4×C71C4×D412yes—
102C2×C4×C71#12011yescharacteristic
103C2×C4×C71#12112yes—
104C2×C4×C71#12311yescharacteristic
105C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
106C2×C4×C71#13711yescharacteristic
107C2×C4×C71#13712yes—
108C71⋊C8—44no—
109C71⋊C8—82no—
110C22×C71—21no—
111C22×C71—21no—
112C22×C71#38611yescharacteristic
113C22×C71#41711yescharacteristic
114C22×C71#41911yescharacteristic
115C4×C71—22no—
116C4×C71—21no—
117C4×C71—21no—
118C4×C71—42no—
119C4×C71—42no—
120C4×C71—42no—
121C4×C71—82no—
122C4×C71—81no—
123C4×C71#46812yes—
124C4×C71#47012yes—
125#9488—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71#189111yescharacteristic
129C2×C71#189411yescharacteristic
130C2×C71#191211yescharacteristic
131#2983—711no—
132#2581D7111yescharacteristic
133#1801—711no—
134#1487—712no—
135#1286—712no—
136C71#948811yescharacteristic, complemented, Sylow
137C2×C4×Q8C2×D7111yescharacteristic
138C2×C4×Q8C2×D7111yescharacteristic
139#446—1424no—
140#432—1422no—
141#431—712no—
142#423—712no—
143C2×#101—1421no—
144#400Dic7112yes—
145#395Dic7112yes—
146#391—21no—
147#389—21no—
148#387—22no—
149#382C2×D7111yescharacteristic
150#343—1421no—
151Q8⋊C8—1424no—
152C22×Q8C2×Dic7111yescharacteristic
153C22×Q8C22⋊D7111yescharacteristic
154C2×M16—1422no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×Dic7111yescharacteristic
163C2×C4⋊C4C2×Dic7112yes—
164C2×C4⋊C4C22⋊D7111yescharacteristic
165C2×C42—21no—
166C2×C42C22⋊D7112yes—
167C2×C42C22×D7111yescharacteristic
168C4⋊C8—1424no—
169C4⋊C8—1422no—
170C4⋊C8—1422no—
171C4⋊C8—1422no—
172#101—2842no—
173C22⋊C8—1422no—
174#97—1424no—
175C4×C8—714no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—21no—
180#95C2×Dic7112yes—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—42no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4C22⋊Dic7112yes—
191C22×C4C22⋊Dic7112yes—
192C22×C4C22×Dic7111yescharacteristic
193C22×C4C2×C22⋊D7111yescharacteristic
194C22×C4C2×C22⋊D7111yescharacteristic
195M16—2842no—
196M16—2842no—
197C2×C8—2842no—
198C2×C8—1424no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—44no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C42—22no—
208C42—22no—
209C42—42no—
210C42D4×D7112yes—
211C42#1176586412yes—
212C23C2×C22⋊Dic7111yescharacteristic
213Q8—24no—
214Q8—42no—
215Q8—42no—
216Q8—42no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4D4×Dic7112yes—
233C2×C4#1455609611yescharacteristic
234C2×C4#1455610012yes—
235C2×C4#1455610911yescharacteristic
236C2×C4#1455615311yescharacteristic
237C2×C4#1455615811yescharacteristic
238C2×C4#1455615912yes—
239C8—2844no—
240C8—5682no—
241C22—21no—
242C22—21no—
243C22#1798587911yescharacteristic
244C22#1798600011yescharacteristic
245C22#1798600811yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—42no—
252C4—82no—
253C4—81no—
254C4#1798624712yes—
255C4#1798625512yes—
256C2—21no—
257C2—21no—
258C2#2258875911yescharacteristic
259C2#2258877411yescharacteristic
260C2#2258887811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2581C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2581D71
hypocenterC71#9488
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
136. C71#9488semidirect

Maximal subgroups

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

subgroupindex
#225974252
#225883362
#225867692
#225861982
C71×#25812
#948871