Finite Group Databasebeta

← #31885553

Subgroups — #31885553

Order 18176.

514 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 155 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#22588320C212yesmaximal
2#22586163C214yesmaximal, complemented
3C71×#2296C211yesmaximal, characteristic
4C2×C4⋊(C71⋊C8)—22no—
5C2×C22⋊(C71⋊C8)—22no—
6#17985754C2214yes—
7C2×#117×C71C411yescharacteristic
8C2×#117×C71C411yescharacteristic
9C22×C42×C71C2211yescharacteristic
10#389×C71C412yes—
11#389×C71C2212yes—
12#385×C71—22no—
13C4×C4⋊C4×C71—22no—
14C22×C71⋊C8—24no—
15C4⋊(C71⋊C8)—44no—
16C22⋊(C71⋊C8)—44no—
17C23×C4×C71C2×C411yescharacteristic
18C23×C4×C71D412yes—
19#117×C71—22no—
20#117×C71—42no—
21#117×C71C814yescomplemented
22C2×C4⋊C4×C71—22no—
23C2×C4⋊C4×C71—22no—
24C2×C4⋊C4×C71C2×C412yes—
25C2×C4⋊C4×C71C2×C412yes—
26C2×C22⋊C4×C71—21no—
27C2×C22⋊C4×C71—21no—
28C2×C42×C71—21no—
29C2×C42×C71—21no—
30C2×C42×C71—42no—
31C2×C42×C71C2×C411yescharacteristic
32C2×C42×C71D412yes—
33C2×C42×C71C2311yescharacteristic
34#95×C71—22no—
35#95×C71—22no—
36C2×C71⋊C8—44no—
37C2×C71⋊C8—44no—
38C2×C71⋊C8—44no—
39C24×C71C22⋊C411yescharacteristic
40C22×C4×C71—22no—
41C22×C4×C71—22no—
42C22×C4×C71—22no—
43C22×C4×C71—22no—
44C22×C4×C71—22no—
45C22×C4×C71—22no—
46C22×C4×C71—42no—
47C22×C4×C71—42no—
48C22×C4×C71C22⋊C412yes—
49C22×C4×C71C22⋊C411yescharacteristic
50C22×C4×C71C2×C812yes—
51C22×C4×C71M1612yes—
52C22×C4×C71C22×C411yescharacteristic
53C22×C4×C71C2×D412yes—
54C4⋊C4×C71—22no—
55C4⋊C4×C71—42no—
56C4⋊C4×C71—44no—
57C4⋊C4×C71—42no—
58C4⋊C4×C71—42no—
59C4⋊C4×C71C2×C814yes—
60C22⋊C4×C71—24no—
61C22⋊C4×C71—42no—
62C42×C71—24no—
63C42×C71—24no—
64C42×C71—41no—
65C42×C71—41no—
66C42×C71—42no—
67C23×C71—21no—
68C23×C71—21no—
69C23×C71—22no—
70C23×C71—22no—
71C23×C71C22⋊C812yes—
72C23×C71C2×C22⋊C411yescharacteristic
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71—24no—
78C2×C4×C71—24no—
79C2×C4×C71—24no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—44no—
84C2×C4×C71—44no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71C22⋊C812yes—
91C2×C4×C71#9912yes—
92C2×C4×C71#10112yes—
93C2×C4×C71C22×C811yescharacteristic
94C2×C4×C71C2×M1611yescharacteristic
95C2×C4×C71AGL(1,ℤ/8)12yes—
96C2×C4×C71#13712yes—
97C71⋊C8—84no—
98C22×C71—22no—
99C22×C71—24no—
100C22×C71—41no—
101C22×C71—41no—
102C22×C71—42no—
103C22×C71C2×C22⋊C811yescharacteristic
104C22×C71C2×#9911yescharacteristic
105C22×C71C2×#10111yescharacteristic
106C22×C71#41712yes—
107C22×C71#42112yes—
108C4×C71—21no—
109C4×C71—21no—
110C4×C71—22no—
111C4×C71—22no—
112C4×C71—44no—
113C4×C71—42no—
114C4×C71—82no—
115C4×C71#32414yes—
116#7598—711nomaximal, Sylow
117C2×C71—42no—
118C2×C71C2×#32411yescharacteristic
119C2×C71#147712yes—
120C2×C71#151312yes—
121C2×C71#153711yescharacteristic
122C2×C71#156111yescharacteristic
123#2296D7111yescharacteristic
124#1797—712no—
125#1271—714no—
126C71#759811yescharacteristic, complemented, Sylow
127C2×#117Dic7111yescharacteristic
128C2×#117Dic7111yescharacteristic
129C22×C42C2×D7111yescharacteristic
130C2×C4⋊C8—1422no—
131C2×C22⋊C8—1422no—
132#389Dic7112yes—
133#389C2×D7112yes—
134#385—22no—
135C4×C4⋊C4—22no—
136#336—714no—
137C23×C4C2×Dic7111yescharacteristic
138C23×C4C22⋊D7112yes—
139C22×C8—1424no—
140#117—22no—
141#117—42no—
142#117C71⋊C814yescomplemented
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×Dic7112yes—
146C2×C4⋊C4C2×Dic7112yes—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—42no—
152C2×C42C2×Dic7111yescharacteristic
153C2×C42C22⋊D7112yes—
154C2×C42C22×D7111yescharacteristic
155C4⋊C8—2844no—
156C22⋊C8—2844no—
157#95—22no—
158#95—22no—
159C24C22⋊Dic7111yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C22⋊Dic7112yes—
169C22×C4C22⋊Dic7111yescharacteristic
170C22×C4C2×C71⋊C812yes—
171C22×C4#1176584912yes—
172C22×C4C22×Dic7111yescharacteristic
173C22×C4C2×C22⋊D7112yes—
174C2×C8—2844no—
175C2×C8—2844no—
176C2×C8—2844no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4C2×C71⋊C814yes—
183C22⋊C4—24no—
184C22⋊C4—42no—
185C42—24no—
186C42—24no—
187C42—41no—
188C42—41no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23C22⋊(C71⋊C8)12yes—
195C23C2×C22⋊Dic7111yescharacteristic
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4C22⋊(C71⋊C8)12yes—
214C2×C4#1455604312yes—
215C2×C4#1455604712yes—
216C2×C4C22×C71⋊C811yescharacteristic
217C2×C4C2×#1176584911yescharacteristic
218C2×C4#1455615312yes—
219C2×C4#1455615812yes—
220C8—5684no—
221C22—22no—
222C22—24no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22C2×C22⋊(C71⋊C8)11yescharacteristic
227C22C2×#1455604311yescharacteristic
228C22C2×#1455604711yescharacteristic
229C22#1798600012yes—
230C22#1798601612yes—
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4#1798572514yes—
239C2—42no—
240C2C2×#1798572511yescharacteristic
241C2#2258672712yes—
242C2#2258688312yes—
243C2#2258697811yescharacteristic
244C2#2258707811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC71×#2296C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2296D71
hypocenterC71#7598
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #22586163C2semidirect
21. #117×C71C8semidirect
126. C71#7598semidirect
142. #117C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225883202
#225861632
C71×#22962
#759871