Finite Group Databasebeta

← #31895314

Subgroups — #31895314

Order 18176.

454 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 111 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#22593702C211yesmaximal, characteristic
3#22586768C212yesmaximal
4#22586200C212yesmaximal
5C71×#1801C211yesmaximal, characteristic
6C2×C4×Dic142C2211yescharacteristic
7#17986130—24no—
8#17986056—22no—
9#17986052C2212yes—
10#17985938C412yes—
11#17985919C412yes—
12#17985906—21no—
13#17985895—21no—
14#17985883—22no—
15#17985865C2211yescharacteristic
16#17985729—24no—
17C2×C4×Q8×C71C2211yescharacteristic
18#423×C71C2212yes—
19C2×#101×C71—21no—
20#343×C71—21no—
21C22×Dic142C2×C411yescharacteristic
22C4×Dic142—22no—
23C4×Dic142—22no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×Dic71⋊C4—22no—
27C2×Dic71⋊C4—21no—
28C2×C4×Dic71—21no—
29C4⋊(C71⋊C8)—24no—
30C4⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32#14556039—24no—
33C4×C71⋊C8D414yes—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—21no—
38C71⋊#95C2×C412yes—
39C22×Q8×C71D411yescharacteristic
40C2×M16×C71—22no—
41C4×Q8×C71—22no—
42C4×Q8×C71—22no—
43C2×C4⋊C4×C71C2×C412yes—
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C42×C71D412yes—
46C2×C42×C71C2311yescharacteristic
47C4⋊C8×C71—22no—
48C4⋊C8×C71—22no—
49#101×C71—42no—
50C2×Dic142—22no—
51C2×Dic142—42no—
52C22×Dic71—21no—
53C22×Dic71—22no—
54C22×Dic71—21no—
55C2×C71⋊C8—24no—
56C4⋊Dic71—22no—
57C4⋊Dic71—41no—
58C4⋊Dic71—41no—
59Dic71⋊C4—44no—
60Dic71⋊C4—41no—
61Dic71⋊C4—41no—
62C4×Dic71—42no—
63C2×Q8×C71—22no—
64C2×Q8×C71—21no—
65C2×Q8×C71—22no—
66C2×Q8×C71—21no—
67C22×C4×C71C22⋊C412yes—
68C22×C4×C71C22⋊C412yes—
69C22×C4×C71C22×C411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72M16×C71—42no—
73M16×C71—42no—
74C2×C8×C71—42no—
75C4⋊C4×C71—24no—
76C4⋊C4×C71—22no—
77C42×C71—22no—
78C42×C71—22no—
79C42×C71C2×D412yes—
80C42×C71#4112yes—
81C23×C71C2×C22⋊C411yescharacteristic
82Q8×C71—42no—
83Q8×C71—42no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—42no—
91C2×C4×C71C4×D412yes—
92C2×C4×C71#12011yescharacteristic
93C2×C4×C71#12112yes—
94C2×C4×C71#12311yescharacteristic
95C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C71#13712yes—
97C2×C4×C71#13711yescharacteristic
98C2×Dic71—22no—
99C2×Dic71—41no—
100C2×Dic71—41no—
101C2×Dic71—42no—
102C2×Dic71—42no—
103C2×Dic71—42no—
104C2×Dic71—41no—
105C2×Dic71—42no—
106Dic142—24no—
107Dic142—42no—
108C8×C71—82no—
109C71⋊C8—44no—
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#46812yes—
121C4×C71#47012yes—
122Dic71—42no—
123Dic71—82no—
124Dic71—81no—
125#9488—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71#189111yescharacteristic
129C2×C71#189411yescharacteristic
130C2×C71#191211yescharacteristic
131#2983—711no—
132#2581—711no—
133#1801D7111yescharacteristic
134#1487—712no—
135#1286—712no—
136C71#948811yescharacteristic, complemented, Sylow
137C2×C4×Q8—711no—
138C2×C4×Q8C2×D7111yescharacteristic
139#446—1424no—
140#432—1422no—
141#431—712no—
142#423C2×D7112yes—
143C2×#101—21no—
144#400—712no—
145#395—712no—
146#391—1421no—
147#389—1421no—
148#387—1422no—
149#382—711no—
150#343—21no—
151Q8⋊C8—1424no—
152C22×Q8—711no—
153C22×Q8C22⋊D7111yescharacteristic
154C2×M16—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—1422no—
158C4×Q8—1422no—
159C2×C4⋊C4—711no—
160C2×C4⋊C4—1422no—
161C2×C4⋊C4—1421no—
162C2×C4⋊C4—1421no—
163C2×C4⋊C4C4×D7112yes—
164C2×C4⋊C4C22⋊D7111yescharacteristic
165C2×C42—1421no—
166C2×C42D28412yes—
167C2×C42C22×D7111yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—1424no—
171C4⋊C8—1422no—
172#101—42no—
173C22⋊C8—1422no—
174#97—1424no—
175C4×C8—714no—
176#95—712no—
177#95—1422no—
178#95—1422no—
179#95—1422no—
180#95—1421no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—2842no—
186C2×Q8—1422no—
187C22×C4—1422no—
188C22×C4—1421no—
189C22×C4—1421no—
190C22×C4(C2×D71)⋊C412yes—
191C22×C4(C2×D71)⋊C412yes—
192C22×C4C2×C4×D7111yescharacteristic
193C22×C4C2×C22⋊D7111yescharacteristic
194C22×C4C2×D28411yescharacteristic
195M16—42no—
196M16—42no—
197C2×C8—42no—
198C2×C8—1424no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—2844no—
202C4⋊C4—2841no—
203C4⋊C4—2841no—
204C4⋊C4—2841no—
205C4⋊C4—2841no—
206C4⋊C4—1422no—
207C42—22no—
208C42—22no—
209C42—2842no—
210C42D4×D7112yes—
211C42#1176586412yes—
212C23C2×(C2×D71)⋊C411yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—2842no—
216Q8—1424no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—2841no—
225C2×C4—2841no—
226C2×C4—2842no—
227C2×C4—2842no—
228C2×C4—2842no—
229C2×C4—2841no—
230C2×C4—2842no—
231C2×C4—1422no—
232C2×C4(C22⋊D71)⋊C412yes—
233C2×C4#1455609811yescharacteristic
234C2×C4#1455610012yes—
235C2×C4#1455611211yescharacteristic
236C2×C4#1455615411yescharacteristic
237C2×C4#1455615812yes—
238C2×C4#1455615911yescharacteristic
239C8—82no—
240C8—2844no—
241C22—21no—
242C22—21no—
243C22#1798587611yescharacteristic
244C22#1798599811yescharacteristic
245C22#1798600711yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—2842no—
252C4—5682no—
253C4—5681no—
254C4#1798624312yes—
255C4#1798625512yes—
256C2—21no—
257C2—21no—
258C2#2258875611yescharacteristic
259C2#2258877111yescharacteristic
260C2#2258887511yescharacteristic
subgroupG/H
centerC22#17985876
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1801C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1801D71
hypocenterC71#9488
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

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
#225937022
#225867682
#225862002
C71×#18012
#948871