Finite Group Databasebeta

← #31890864

Subgroups — #31890864

Order 18176.

526 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 107 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#22590746C211yesmaximal, characteristic
2#22588103C212yesmaximal
3#22586222C212yesmaximal, complemented
4#22586221C211yesmaximal, characteristic, complemented
5C71×#1295C211yesmaximal, characteristic, complemented
6C2×#14556086C412yes—
7C2×C42×D71C2211yescharacteristic
8C2×(C2×D71)⋊C8—22no—
9C2×#14556039C2212yes—
10#17985868C412yes—
11#17985865C2211yescharacteristic
12#17985858—22no—
13#17985850—22no—
14#17985753—22no—
15C2×#97×C71C2212yes—
16#382×C71C2211yescharacteristic
17C22×C4×D71C2×C411yescharacteristic
18C22×C4×D71D412yes—
19C22×C71⋊C8—22no—
20#14556086—44no—
21C2×C4⋊Dic71C2×C412yes—
22C2×Dic71⋊C4—22no—
23C2×Dic71⋊C4—22no—
24C2×(C2×D71)⋊C4—22no—
25C2×C4×Dic71—22no—
26C2×C4×Dic71C2×C411yescharacteristic
27C2×C4×Dic71D412yes—
28C42×D71—21no—
29C42×D71—22no—
30C42×D71—21no—
31(C2×D71)⋊C8—28no—
32#14556039—22no—
33#14556039—22no—
34C71⋊#95—22no—
35C22×C8×C71—22no—
36C2×C4⋊C4×C71C2×C412yes—
37C2×C42×C71C2311yescharacteristic
38#97×C71—22no—
39#97×C71—22no—
40#95×C71—22no—
41C23×D71C22⋊C411yescharacteristic
42C22×Dic71—22no—
43C22×Dic71C22⋊C411yescharacteristic
44C22×Dic71C22⋊C412yes—
45C2×C4×D71—22no—
46C2×C4×D71—21no—
47C2×C4×D71—21no—
48C2×C4×D71—22no—
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—24no—
52C2×C4×D71—42no—
53C2×C71⋊C8—24no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—42no—
56C4⋊Dic71—42no—
57Dic71⋊C4—44no—
58Dic71⋊C4—44no—
59(C2×D71)⋊C4—44no—
60C4×Dic71—21no—
61C4×Dic71—21no—
62C4×Dic71—22no—
63C4×Dic71—22no—
64C4×Dic71—44no—
65C22×C4×C71—22no—
66C22×C4×C71C22×C411yescharacteristic
67C22×C4×C71C2×D412yes—
68C2×C8×C71—24no—
69C2×C8×C71—42no—
70C2×C8×C71—42no—
71C4⋊C4×C71—42no—
72C42×C71#4111yescharacteristic
73C42×C71#4112yes—
74C42×C71#4111yescharacteristic
75C23×C71C2×C22⋊C411yescharacteristic
76C22×D71—21no—
77C22×D71—21no—
78C22×D71—22no—
79C22×D71—22no—
80C22×D71#10012yes—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71#11711yescharacteristic
89C2×C4×C71#11711yescharacteristic
90C2×C4×C71#12312yes—
91C2×C4×C71#12312yes—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—21no—
95C2×Dic71—22no—
96C2×Dic71—42no—
97C2×Dic71—42no—
98C2×Dic71—42no—
99C2×Dic71#10112yes—
100C2×Dic71#10414yes—
101C4×D71—24no—
102C4×D71—24no—
103C4×D71—44no—
104C4×D71—44no—
105C8×C71—44no—
106C71⋊C8—44no—
107C22×C71#38111yescharacteristic
108C22×C71C2×#10011yescharacteristic
109C22×C71C2×#10111yescharacteristic
110C22×C71C2×#10412yes—
111C22×C71#42112yes—
112C2×D71—22no—
113C2×D71—24no—
114C2×D71—41no—
115C2×D71—42no—
116C2×D71—41no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—82no—
122Dic71—22no—
123Dic71—24no—
124Dic71—82no—
125#8722—711nomaximal, Sylow
126C2×C71#152611yescharacteristic
127C2×C71#153012yes—
128C2×C71#153711yescharacteristic
129C2×C71#182111yescharacteristic
130C2×C71#182612yes—
131D71—42no—
132#2285—711no—
133#1749—712no—
134#1295—712no—
135#1295—711no—
136#1295D7111yescharacteristic, complemented
137C71#872211yescharacteristic, complemented, Sylow
138C2×#117—712no—
139C22×C42—711no—
140C2×C22⋊C8—1422no—
141C2×#97—712no—
142C2×#97C2×D7112yes—
143#384—712no—
144#382—711no—
145#382C2×D7111yescharacteristic
146#380—1422no—
147C4×C4⋊C4—1422no—
148#336—1422no—
149C23×C4—712no—
150C23×C4—711no—
151C22×C8—22no—
152C22×C8—1422no—
153#117—2844no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4—1422no—
156C2×C4⋊C4—1422no—
157C2×C4⋊C4C4×D7112yes—
158C2×C22⋊C4—1422no—
159C2×C42—711no—
160C2×C42—712no—
161C2×C42—1422no—
162C2×C42—1421no—
163C2×C42—1422no—
164C2×C42—1421no—
165C2×C42C22×D7111yescharacteristic
166C22⋊C8—1428no—
167#97—22no—
168#97—22no—
169#97—1422no—
170#97—1422no—
171#95—22no—
172#95—1422no—
173C24—711no—
174C22×C4—22no—
175C22×C4—2842no—
176C22×C4—711no—
177C22×C4—712no—
178C22×C4—1424no—
179C22×C4—1422no—
180C22×C4—1422no—
181C22×C4—1422no—
182C22×C4—1422no—
183C22×C4—1422no—
184C22×C4—1421no—
185C22×C4—1421no—
186C22×C4C2×C4×D7111yescharacteristic
187C22×C4D4×D7112yes—
188C2×C8—24no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—2842no—
192C2×C8—2842no—
193C2×C8—1424no—
194C4⋊C4—42no—
195C4⋊C4—2842no—
196C4⋊C4—2844no—
197C4⋊C4—2844no—
198C22⋊C4—2844no—
199C42—2844no—
200C42—1421no—
201C42—1421no—
202C42—1422no—
203C42—1422no—
204C42#1176586411yescharacteristic
205C42#1176586512yes—
206C42#1176586611yescharacteristic
207C23—712no—
208C23—1421no—
209C23—1421no—
210C23—1422no—
211C23—1422no—
212C23C22⋊C4×D7111yescharacteristic
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—2844no—
221C2×C4—2844no—
222C2×C4—2842no—
223C2×C4—2842no—
224C2×C4—2842no—
225C2×C4—712no—
226C2×C4—714no—
227C2×C4—1424no—
228C2×C4—1424no—
229C2×C4—1421no—
230C2×C4—1422no—
231C2×C4—1421no—
232C2×C4—1422no—
233C2×C4#1455608511yescharacteristic
234C2×C4#1455608611yescharacteristic
235C2×C4#1455611012yes—
236C2×C4#1455611312yes—
237C8—44no—
238C8—2844no—
239C22—2842no—
240C22—2841no—
241C22—2841no—
242C22—1422no—
243C22—1424no—
244C22#1798586111yescharacteristic
245C22#100×D7111yescharacteristic
246C22#101×D7111yescharacteristic
247C22#104×D7112yes—
248C22#1798601412yes—
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—82no—
254C4—5682no—
255C4—1422no—
256C4—1424no—
257C2—2842no—
258C2#2258693311yescharacteristic
259C2#2258694912yes—
260C2#2258697611yescharacteristic
261C2#2258841911yescharacteristic
262C2#2258843812yes—
subgroupG/H
centerC23C22⋊C4×D71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1295C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1295D71
hypocenterC71#8722
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
3. #22586222C2semidirect
4. #22586221C2semidirect
5. C71×#1295C2semidirect
136. #1295D71semidirect
137. C71#8722semidirect

Maximal subgroups

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

subgroupindex
#225907462
#225881032
#225862222
#225862212
C71×#12952
#872271