Finite Group Databasebeta

← #31906649

Subgroups — #31906649

Order 18176.

386 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#17986051C211yesmaximal, characteristic
2#22590136C212yesmaximal, complemented
3#22586280C211yesmaximal, characteristic, complemented
4#22586263C212yesmaximal
5C71×#1322C211yesmaximal, characteristic, complemented
6C2×C42×D71C411yescharacteristic
7#17986051—22no—
8#17986051—22no—
9C2×C4⋊(C71⋊C8)C411yescharacteristic
10C2×(C2×D71)⋊C8C411yescharacteristic
11C2×(C2×D71)⋊C8C2211yescharacteristic
12C2×C8×Dic71C411yescharacteristic
13C2×C8×Dic71C2211yescharacteristic
14Dic71⋊C16—22no—
15C22⋊(C71⋊C16)C2212yes—
16C16×Dic71—22no—
17C2×C4⋊C8×C71C2211yescharacteristic
18C22⋊C16×C71C2212yes—
19C22×C4×D71C811yescharacteristic
20C22×C4×D71C811yescharacteristic
21C22×C4×D71C2×C411yescharacteristic
22C22×C71⋊C8C2×C411yescharacteristic
23C22×C71⋊C8C2×C411yescharacteristic
24C2×C4×Dic71C811yescharacteristic
25C2×C4×Dic71C811yescharacteristic
26C2×C4×Dic71C2×C411yescharacteristic
27C42×D71—21no—
28C42×D71—21no—
29C42×D71—21no—
30C42×D71—21no—
31C2×C71⋊C16—22no—
32C4⋊(C71⋊C8)—22no—
33(C2×D71)⋊C8—22no—
34(C2×D71)⋊C8—22no—
35C8×Dic71—21no—
36C8×Dic71—21no—
37C8×Dic71D412yes—
38C8×Dic71D412yes—
39C22×C8×C71C2×C411yescharacteristic
40C22×C8×C71C2311yescharacteristic
41C2×C42×C71C2×C411yescharacteristic
42C2×C16×C71—22no—
43C4⋊C8×C71—22no—
44C23×D71C2×C811yescharacteristic
45C22×Dic71C2×C811yescharacteristic
46C22×Dic71C2×C811yescharacteristic
47C22×Dic71C2×C811yescharacteristic
48C2×C4×D71—21no—
49C2×C4×D71—21no—
50C2×C4×D71—22no—
51C2×C4×D71—21no—
52C2×C4×D71—21no—
53C2×C4×D71—21no—
54C2×C4×D71—21no—
55C2×C4×D71—42no—
56C2×C4×D71—41no—
57C2×C4×D71—41no—
58C2×C4×D71C1614yescomplemented
59C2×C71⋊C8—21no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8C22⋊C412yes—
63C4×Dic71—21no—
64C4×Dic71—21no—
65C4×Dic71—21no—
66C4×Dic71—21no—
67C4×Dic71C22⋊C411yescharacteristic
68C4×Dic71C22⋊C411yescharacteristic
69C4×Dic71M1611yescharacteristic
70C4×Dic71M1611yescharacteristic
71C71⋊C16—42no—
72C22×C4×C71C2×C811yescharacteristic
73C22×C4×C71C2×C811yescharacteristic
74C22×C4×C71C22×C411yescharacteristic
75C2×C8×C71—21no—
76C2×C8×C71—41no—
77C2×C8×C71—41no—
78C2×C8×C71C2×D412yes—
79C42×C71—21no—
80C42×C71—21no—
81C16×C71—42no—
82C23×C71C22×C811yescharacteristic
83C22×D71—21no—
84C22×D71—21no—
85C22×D71—41no—
86C22×D71—41no—
87C22×D71C2×C1612yes—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—41no—
94C2×C4×C71C2×C1612yes—
95C2×C4×C71C2×C22⋊C411yescharacteristic
96C2×C4×C71C2×M1611yescharacteristic
97C2×Dic71—22no—
98C2×Dic71—21no—
99C2×Dic71—41no—
100C2×Dic71C22⋊C811yescharacteristic
101C2×Dic71C22⋊C811yescharacteristic
102C2×Dic71C22⋊C811yescharacteristic
103C2×Dic71C22⋊C811yescharacteristic
104C2×Dic71C2×C1612yes—
105C2×Dic71M3212yes—
106C4×D71—24no—
107C4×D71—24no—
108C4×D71—42no—
109C4×D71—42no—
110C4×D71—41no—
111C4×D71—41no—
112C4×D71—42no—
113C8×C71—22no—
114C8×C71—41no—
115C71⋊C8—22no—
116C71⋊C8—41no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71C2×C22⋊C811yescharacteristic
120C22×C71C22×C1611yescharacteristic
121C22×C71C2×M3211yescharacteristic
122C2×D71—22no—
123C2×D71—42no—
124C2×D71—41no—
125C2×D71—41no—
126C2×D71—42no—
127C4×C71—22no—
128C4×C71—21no—
129C4×C71—41no—
130C4×C71#41011yescharacteristic
131C4×C71#41311yescharacteristic
132Dic71—21no—
133Dic71—41no—
134Dic71C22⋊C1612yes—
135Dic71C22⋊C1612yes—
136Dic71#34912yes—
137#11459—711nomaximal, Sylow
138C2×C71—21no—
139C2×C71—21no—
140C2×C71#145811yescharacteristic
141C2×C71C2×C22⋊C1611yescharacteristic
142C2×C71C2×#34911yescharacteristic
143D71—42no—
144C2×#431—711no—
145#2162—712no—
146#1322—711no—
147#1322D7111yescharacteristic, complemented
148#1314—712no—
149C71#1145911yescharacteristic, complemented, Sylow
150C22×C42—711no—
151#431—1422no—
152#431—1422no—
153C2×C4⋊C8—711no—
154C2×C4⋊C8C2×D7111yescharacteristic
155C2×C22⋊C8—711no—
156C2×C22⋊C8—711no—
157C2×C4×C8—711no—
158C2×C4×C8—711no—
159C4⋊C16—1422no—
160C22⋊C16—712no—
161C22⋊C16C2×D7112yes—
162C4×C16—1422no—
163C23×C4—711no—
164C23×C4—711no—
165C23×C4—711no—
166C22×C8—711no—
167C22×C8—711no—
168C22×C8C4×D7111yescharacteristic
169C22×C8C22×D7111yescharacteristic
170C2×C42—711no—
171C2×C42—711no—
172C2×C42—711no—
173C2×C42—1421no—
174C2×C42—1421no—
175C2×C42—1421no—
176C2×C42—1421no—
177C2×C42C4×D7111yescharacteristic
178C2×C16—22no—
179C2×C16—1422no—
180C4⋊C8—22no—
181C4⋊C8—1422no—
182C22⋊C8—1422no—
183C22⋊C8—1422no—
184C4×C8—712no—
185C4×C8—712no—
186C4×C8—1421no—
187C4×C8—1421no—
188C24—711no—
189C22×C4—2841no—
190C22×C4—2841no—
191C22×C4—2842no—
192C22×C4—711no—
193C22×C4—711no—
194C22×C4—711no—
195C22×C4—714no—
196C22×C4—1421no—
197C22×C4—1421no—
198C22×C4—1421no—
199C22×C4—1421no—
200C22×C4—1422no—
201C22×C4—1421no—
202C22×C4—1421no—
203C22×C4C8×D7111yescharacteristic
204C22×C4C8×D7111yescharacteristic
205C22×C4C2×C4×D7111yescharacteristic
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—2841no—
210C2×C8—2841no—
211C2×C8—712no—
212C2×C8—1421no—
213C2×C8D4×D7112yes—
214C42—21no—
215C42—21no—
216C42—711no—
217C42—711no—
218C42—711no—
219C42—711no—
220C42—1421no—
221C42—1421no—
222C42—1421no—
223C42—1421no—
224C16—42no—
225C16—2842no—
226C23—2841no—
227C23—2841no—
228C23—712no—
229C23—1421no—
230C23—1421no—
231C23C2×C8×D7111yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—2842no—
239C2×C4—2842no—
240C2×C4—2842no—
241C2×C4—2841no—
242C2×C4—2841no—
243C2×C4—2841no—
244C2×C4—712no—
245C2×C4—712no—
246C2×C4—711no—
247C2×C4—711no—
248C2×C4—711no—
249C2×C4—711no—
250C2×C4—1424no—
251C2×C4—1424no—
252C2×C4—1421no—
253C2×C4—1422no—
254C2×C4C16×D7112yes—
255C2×C4C22⋊C4×D7111yescharacteristic
256C2×C4M16×D7111yescharacteristic
257C8—22no—
258C8—41no—
259C8—2841no—
260C8—1422no—
261C22—21no—
262C22—21no—
263C22—2842no—
264C22—2842no—
265C22—2841no—
266C22—2841no—
267C22—1422no—
268C22C22⋊C8×D7111yescharacteristic
269C22C2×C16×D7111yescharacteristic
270C22M32×D7111yescharacteristic
271C4—21no—
272C4—22no—
273C4—41no—
274C4—2841no—
275C4—712no—
276C4—712no—
277C4—712no—
278C4—1421no—
279C4#1798597111yescharacteristic
280C4#1798598211yescharacteristic
281C2—21no—
282C2—21no—
283C2—2842no—
284C2#2258665311yescharacteristic
285C2C22⋊C16×D7111yescharacteristic
286C2#349×D7111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC22×C71C22×C16
FrattiniC22×C8C22×D71
FittingC71×#1322C2
socleC22×C71C2×C22⋊C8
prime radicalC23×D71C2×C8
cosocleC22×C8×C71C23
hypercenter#1322D71
hypocenterC71#11459
LCSG ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #22590136C2semidirect
3. #22586280C2semidirect
5. C71×#1322C2semidirect
58. C2×C4×D71C16semidirect
147. #1322D71semidirect
149. C71#11459semidirect

Maximal subgroups

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

subgroupindex
C2×#179860512
#225901362
#225862802
#225862632
C71×#13222
#1145971