Finite Group Databasebeta

← #31892412

Subgroups — #31892412

Order 18176.

502 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 103 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#22597411C211yesmaximal, characteristic, complemented
2#22587002C211yesmaximal, characteristic, complemented
3#22586809C212yesmaximal, complemented
4#22586239C212yesmaximal, complemented
5C71×#3150C211yesmaximal, characteristic
6#17986071—24no—
7#17986056C2212yes—
8#17986052C2212yes—
9#17985966—22no—
10#17985748—21no—
11#17985743—21no—
12D4⋊(C71⋊C8)—24no—
13C2×#135×C71—22no—
14#532×C71C2211yescharacteristic
15#532×C71C2211yescharacteristic
16#497×C71—21no—
17#497×C71C412yes—
18#496×C71—21no—
19#496×C71C412yes—
20C2×#97×C71C2211yescharacteristic
21C2×#11765849—22no—
22C22×C71⋊C8—22no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)D414yes—
25C22⋊(C71⋊C8)—24no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28#14556039—22no—
29C4×C71⋊C8—22no—
30C2×#41×C71—21no—
31C2×#41×C71D412yes—
32#135×C71—44no—
33#135×C71—44no—
34C2×Q16×C71—22no—
35C2×SD16×C71—22no—
36C2×SD16×C71—22no—
37C2×D8×C71—22no—
38C22×C8×C71—21no—
39C4⋊Q8×C71C2×C411yescharacteristic
40C4⋊Q8×C71D411yescharacteristic
41#127×C71C2×C411yescharacteristic
42#127×C71D411yescharacteristic
43#124×C71—21no—
44#124×C71C2×C412yes—
45#121×C71—22no—
46#121×C71—22no—
47C4×D4×C71—22no—
48C4×D4×C71—22no—
49C2×C42×C71C2311yescharacteristic
50#97×C71—21no—
51#97×C71C2×C412yes—
52#11765849—82no—
53C2×C71⋊C8—24no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—82no—
56#41×C71—42no—
57#41×C71—42no—
58#41×C71—42no—
59#41×C71—42no—
60C2×Q8×C71—21no—
61C2×Q8×C71C22⋊C412yes—
62C2×D4×C71—22no—
63C2×D4×C71—21no—
64C2×D4×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71—41no—
67C2×D4×C71C22⋊C412yes—
68C22×C4×C71—22no—
69C22×C4×C71—41no—
70C22×C4×C71C2×D411yescharacteristic
71C22×C4×C71#4112yes—
72Q16×C71—44no—
73SD16×C71—44no—
74SD16×C71—44no—
75D8×C71—44no—
76C2×C8×C71—24no—
77C2×C8×C71—22no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—22no—
80C22⋊C4×C71—24no—
81C22⋊C4×C71—42no—
82C42×C71C22×C411yescharacteristic
83C42×C71C2×D412yes—
84C42×C71C2×D411yescharacteristic
85C23×C71—22no—
86C23×C71—41no—
87C23×C71#12311yescharacteristic
88Q8×C71—42no—
89Q8×C71—42no—
90D4×C71—42no—
91D4×C71—42no—
92D4×C71—44no—
93D4×C71—82no—
94D4×C71—82no—
95D4×C71—81no—
96D4×C71—81no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—41no—
103C2×C4×C71—82no—
104C2×C4×C71C2×C22⋊C411yescharacteristic
105C2×C4×C71C4×D412yes—
106C2×C4×C71#12011yescharacteristic
107C2×C4×C71#12112yes—
108C8×C71—24no—
109C71⋊C8—44no—
110C71⋊C8—82no—
111C22×C71—22no—
112C22×C71—21no—
113C22×C71—44no—
114C22×C71—42no—
115C22×C71—82no—
116C22×C71—81no—
117C22×C71#38611yescharacteristic
118C4×C71—22no—
119C4×C71—22no—
120C4×C71—21no—
121C4×C71—42no—
122C4×C71—81no—
123C4×C71#45312yes—
124#9007—711nomaximal, Sylow
125C2×C71—41no—
126C2×C71—82no—
127C2×C71—161no—
128C2×C71#188112yes—
129C2×C71#188711yescharacteristic
130#3150D7111yescharacteristic
131#2980—711no—
132#1543—711no—
133#1496—712no—
134#1302—712no—
135C71#900711yescharacteristic, complemented, Sylow
136C2×#135—22no—
137#532C2×D7111yescharacteristic
138#532C2×D7111yescharacteristic
139#497—21no—
140#497Dic7112yes—
141#496—21no—
142#496Dic7112yes—
143#435—1424no—
144#432—712no—
145#431—712no—
146#408—1422no—
147C2×#97C2×D7111yescharacteristic
148#333—1421no—
149#331—1421no—
150D4⋊C8—1424no—
151C2×#41—21no—
152C2×#41C22⋊D7112yes—
153#135—44no—
154#135—44no—
155C2×Q16—22no—
156C2×SD16—22no—
157C2×SD16—22no—
158C2×D8—22no—
159C2×M16—1422no—
160C22×C8—21no—
161C22×C8—1422no—
162C4⋊Q8C2×Dic7111yescharacteristic
163C4⋊Q8C22⋊D7111yescharacteristic
164#127C2×Dic7111yescharacteristic
165#127C22⋊D7111yescharacteristic
166#124—21no—
167#124C2×Dic7112yes—
168#121—22no—
169#121—22no—
170C4×D4—22no—
171C4×D4—22no—
172C2×C42C22×D7111yescharacteristic
173C4⋊C8—714no—
174C4⋊C8—1422no—
175C22⋊C8—1422no—
176C22⋊C8—1424no—
177C22⋊C8—1422no—
178#97—21no—
179#97—1422no—
180#97C2×Dic7112yes—
181C4×C8—1422no—
182#41—42no—
183#41—42no—
184#41—42no—
185#41—42no—
186C2×Q8—21no—
187C2×Q8C22⋊Dic7112yes—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—41no—
193C2×D4C22⋊Dic7112yes—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4C2×C22⋊D7111yescharacteristic
197C22×C4#1176586412yes—
198Q16—44no—
199SD16—44no—
200SD16—44no—
201D8—44no—
202M16—5682no—
203C2×C8—24no—
204C2×C8—22no—
205C2×C8—2842no—
206C2×C8—5682no—
207C2×C8—1424no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—42no—
212C42C22×Dic7111yescharacteristic
213C42D4×D7112yes—
214C42C2×C22⋊D7111yescharacteristic
215C23—22no—
216C23—41no—
217C23#1455610911yescharacteristic
218Q8—42no—
219Q8—42no—
220D4—42no—
221D4—42no—
222D4—44no—
223D4—82no—
224D4—82no—
225D4—81no—
226D4—81no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4C2×C22⋊Dic7111yescharacteristic
235C2×C4D4×Dic7112yes—
236C2×C4#1455609611yescharacteristic
237C2×C4#1455610012yes—
238C8—24no—
239C8—2844no—
240C8—5682no—
241C22—22no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22#1798587911yescharacteristic
248C4—22no—
249C4—22no—
250C4—21no—
251C4—42no—
252C4—81no—
253C4#1798616612yes—
254C2—41no—
255C2—82no—
256C2—161no—
257C2#2258870412yes—
258C2#2258873911yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
Fitting, prime radicalC71×#3150C2
socleC22×C71#386
cosocleC2×C42×C71C23
hypercenter#3150D71
hypocenterC71#9007
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3150

Complements

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

NH ≅ G/Ntype
1. #22597411C2semidirect
2. #22587002C2semidirect
3. #22586809C2semidirect
4. #22586239C2semidirect
135. C71#9007semidirect

Maximal subgroups

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

subgroupindex
#225974112
#225870022
#225868092
#225862392
C71×#31502
#900771