Finite Group Databasebeta

← #31890789

Subgroups — #31890789

Order 18176.

530 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 119 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×#17985863C211yesmaximal, characteristic
2#22588406C212yesmaximal
3#22586183C211yesmaximal, characteristic
4#22586182C212yesmaximal
5C71×#1280C211yesmaximal, characteristic
6C22×Dic71⋊C4C412yes—
7C22×C4×Dic71C2211yescharacteristic
8C2×C22⋊(C71⋊C8)C2212yes—
9#17985863—22no—
10#17985863—22no—
11C2×C71⋊#95C412yes—
12C2×C71⋊#95C2211yescharacteristic
13#17985753—22no—
14#17985753—22no—
15C2×C22⋊C8×C71C2212yes—
16C2×#95×C71C2211yescharacteristic
17C23×Dic71C2×C411yescharacteristic
18C23×Dic71C2×C411yescharacteristic
19C23×Dic71C2×C412yes—
20C22×C71⋊C8—22no—
21C2×Dic71⋊C4—22no—
22C2×Dic71⋊C4—22no—
23C2×Dic71⋊C4—44no—
24C2×C4×Dic71—21no—
25C2×C4×Dic71—22no—
26C2×C4×Dic71—21no—
27C2×C4×Dic71D412yes—
28C2×C4×Dic71D412yes—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C71⋊#95—24no—
32C71⋊#95—22no—
33C23×C4×C71C2×C412yes—
34C23×C4×C71C2311yescharacteristic
35C22×C8×C71—22no—
36C22⋊C8×C71—22no—
37C22⋊C8×C71—22no—
38#95×C71—22no—
39C22×Dic71—21no—
40C22×Dic71—21no—
41C22×Dic71—22no—
42C22×Dic71—22no—
43C22×Dic71—21no—
44C22×Dic71—21no—
45C22×Dic71—22no—
46C22×Dic71—22no—
47C22×Dic71—22no—
48C22×Dic71—22no—
49C22×Dic71—22no—
50C22×Dic71—42no—
51C22×Dic71—42no—
52C22×Dic71C22⋊C412yes—
53C22×Dic71C22⋊C411yescharacteristic
54C22×Dic71C22⋊C411yescharacteristic
55C2×C71⋊C8—44no—
56C2×C71⋊C8—42no—
57Dic71⋊C4—44no—
58Dic71⋊C4—44no—
59C4×Dic71—22no—
60C4×Dic71—22no—
61C4×Dic71—22no—
62C4×Dic71—22no—
63C24×C71C22×C411yescharacteristic
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—42no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71C2×D412yes—
71C22×C4×C71#4111yescharacteristic
72C22×C4×C71#4112yes—
73C22×C4×C71#4111yescharacteristic
74C2×C8×C71—44no—
75C2×C8×C71—42no—
76C23×C71—22no—
77C23×C71—21no—
78C23×C71—21no—
79C23×C71C2×C22⋊C411yescharacteristic
80C23×C71#11711yescharacteristic
81C23×C71#11711yescharacteristic
82C23×C71#13112yes—
83C23×C71#13112yes—
84C2×C4×C71—24no—
85C2×C4×C71—41no—
86C2×C4×C71—42no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—82no—
92C2×C4×C71—82no—
93C2×C4×C71#12312yes—
94C2×C4×C71#12312yes—
95C2×Dic71—24no—
96C2×Dic71—24no—
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—24no—
103C2×Dic71—41no—
104C2×Dic71—42no—
105C2×Dic71—41no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—82no—
110C2×Dic71—82no—
111C2×Dic71#10012yes—
112C2×Dic71#10112yes—
113C8×C71—82no—
114C71⋊C8—82no—
115C22×C71—21no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71—22no—
120C22×C71—22no—
121C22×C71—22no—
122C22×C71—22no—
123C22×C71#38111yescharacteristic
124C22×C71#40812yes—
125C22×C71#41012yes—
126C22×C71C2×#10011yescharacteristic
127C22×C71C2×#10111yescharacteristic
128C22×C71#43311yescharacteristic
129C22×C71#43312yes—
130C22×C71#43311yescharacteristic
131C4×C71—22no—
132C4×C71—22no—
133C4×C71—82no—
134Dic71—22no—
135Dic71—22no—
136Dic71—42no—
137Dic71—82no—
138#8703—711nomaximal, Sylow
139C2×C71—21no—
140C2×C71—22no—
141C2×C71—21no—
142C2×C71#146912yes—
143C2×C71#147012yes—
144C2×C71#181811yescharacteristic
145C2×C71#182011yescharacteristic
146C2×C71#182111yescharacteristic
147C2×#382—711no—
148#1817—712no—
149#1280—712no—
150#1280—711no—
151#1280D7111yescharacteristic
152C71#870311yescharacteristic, complemented, Sylow
153C22×C4⋊C4—712no—
154C22×C42—711no—
155C2×C22⋊C8—712no—
156C2×C22⋊C8C2×D7112yes—
157#382—1422no—
158#382—1422no—
159C2×#95—712no—
160C2×#95—711no—
161C2×#95C2×D7111yescharacteristic
162#336—1422no—
163#336—1422no—
164C23×C4—712no—
165C23×C4—711no—
166C23×C4—711no—
167C23×C4C4×D7112yes—
168C23×C4C22×D7111yescharacteristic
169C22×C8—22no—
170C22×C8—1422no—
171C2×C4⋊C4—2844no—
172C2×C4⋊C4—1422no—
173C2×C4⋊C4—1422no—
174C2×C42—712no—
175C2×C42—712no—
176C2×C42—1421no—
177C2×C42—1422no—
178C2×C42—1421no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—1422no—
182C22⋊C8—1422no—
183#95—22no—
184#95—1424no—
185#95—1422no—
186C24C2×C4×D7111yescharacteristic
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—2842no—
194C22×C4—2842no—
195C22×C4—711no—
196C22×C4—711no—
197C22×C4—712no—
198C22×C4—1422no—
199C22×C4—1421no—
200C22×C4—1421no—
201C22×C4—1422no—
202C22×C4—1422no—
203C22×C4—1422no—
204C22×C4—1422no—
205C22×C4—1422no—
206C22×C4—1422no—
207C22×C4—1421no—
208C22×C4—1421no—
209C22×C4D4×D7112yes—
210C22×C4#1176586411yescharacteristic
211C22×C4#1176586512yes—
212C22×C4#1176586611yescharacteristic
213C2×C8—44no—
214C2×C8—42no—
215C2×C8—2844no—
216C2×C8—2842no—
217C4⋊C4—2844no—
218C4⋊C4—2844no—
219C42—1422no—
220C42—1422no—
221C42—1422no—
222C42—1422no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23C22⋊C4×D7111yescharacteristic
227C23#1455608511yescharacteristic
228C23#1455608611yescharacteristic
229C23#1455613412yes—
230C23#1455613612yes—
231C2×C4—24no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—2841no—
241C2×C4—2842no—
242C2×C4—2841no—
243C2×C4—2842no—
244C2×C4—2842no—
245C2×C4—2842no—
246C2×C4—5682no—
247C2×C4—5682no—
248C2×C4—712no—
249C2×C4—712no—
250C2×C4—1422no—
251C2×C4—1424no—
252C2×C4—1424no—
253C2×C4—1424no—
254C2×C4—1421no—
255C2×C4—1421no—
256C2×C4—1421no—
257C2×C4—1421no—
258C2×C4#1455611012yes—
259C2×C4#1455611312yes—
260C8—82no—
261C8—5682no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22#1798586111yescharacteristic
271C22#1798596412yes—
272C22#1798597112yes—
273C22#100×D7111yescharacteristic
274C22#101×D7111yescharacteristic
275C22#1798605811yescharacteristic
276C22#1798605912yes—
277C22#1798606111yescharacteristic
278C4—22no—
279C4—22no—
280C4—82no—
281C4—2842no—
282C4—5682no—
283C4—1422no—
284C4—1422no—
285C2—21no—
286C2—22no—
287C2—21no—
288C2#2258669012yes—
289C2#2258669412yes—
290C2#2258841011yescharacteristic
291C2#2258841611yescharacteristic
292C2#2258841911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D71
commutator, prime radicalC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1280C2
socleC23×C71C2×C22⋊C4
cosocleC23×C4×C71C23
hypercenter#1280D71
hypocenterC71#8703
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
152. C71#8703semidirect

Maximal subgroups

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

subgroupindex
C2×#179858632
#225884062
#225861832
#225861822
C71×#12802
#870371