Finite Group Databasebeta

← #31906851

Subgroups — #31906851

Order 18176.

578 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 95 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×#17985868C211yesmaximal, characteristic
2#22586670C211yesmaximal, characteristic
3#22586285C214yesmaximal, complemented
4C71×#1463C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic
6C22×Dic71⋊C4—21no—
7C22×Dic71⋊C4—21no—
8C22×C4×Dic71C411yescharacteristic
9#17985963—21no—
10#17985868—21no—
11#17985868—21no—
12#17985868C412yes—
13#17985868C2212yes—
14#17985725—21no—
15#17985725C2212yes—
16C22×C4⋊C4×C71C2211yescharacteristic
17#407×C71—21no—
18#324×C71—21no—
19#324×C71C2212yes—
20C23×Dic71—21no—
21C23×Dic71—21no—
22C2×#11765849—41no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—41no—
25C2×C4⋊Dic71—41no—
26C2×C4⋊Dic71C2×C412yes—
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—22no—
29C2×Dic71⋊C4—22no—
30C2×Dic71⋊C4—22no—
31C2×Dic71⋊C4—44no—
32C2×Dic71⋊C4—44no—
33C2×C4×Dic71—24no—
34C2×C4×Dic71—21no—
35C2×C4×Dic71C2×C412yes—
36C22⋊(C71⋊C8)—22no—
37C22⋊(C71⋊C8)—41no—
38C23×C4×C71C2×C411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C2×M16×C71—41no—
42C2×C4⋊C4×C71—41no—
43C2×C4⋊C4×C71—41no—
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71C2311yescharacteristic
48C22⋊C8×C71—22no—
49C22⋊C8×C71—41no—
50C22×Dic71—24no—
51C22×Dic71—24no—
52C22×Dic71—24no—
53C22×Dic71—22no—
54C22×Dic71—22no—
55C22×Dic71—42no—
56C22×Dic71—42no—
57C22×Dic71—41no—
58C22×Dic71—41no—
59#11765849—82no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—42no—
62C4⋊Dic71—42no—
63Dic71⋊C4—44no—
64Dic71⋊C4—44no—
65Dic71⋊C4—44no—
66Dic71⋊C4—44no—
67C4×Dic71—24no—
68C4×Dic71—22no—
69C4×Dic71M1614yescomplemented
70C24×C71C22⋊C411yescharacteristic
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—22no—
74C22×C4×C71—22no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—42no—
80C22×C4×C71—42no—
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87M16×C71—82no—
88C2×C8×C71—42no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C23×C71—22no—
94C23×C71—22no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71#9911yescharacteristic
98C23×C71#9911yescharacteristic
99C23×C71C2×C22⋊C411yescharacteristic
100C2×C4×C71—24no—
101C2×C4×C71—24no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—81no—
117C2×C4×C71—81no—
118C2×C4×C71#10112yes—
119C2×C4×C71C2×M1612yes—
120C2×Dic71—24no—
121C2×Dic71—216no—
122C2×Dic71—24no—
123C2×Dic71—24no—
124C2×Dic71—42no—
125C2×Dic71—42no—
126C2×Dic71—42no—
127C2×Dic71—82no—
128C2×Dic71—82no—
129C8×C71—82no—
130C71⋊C8—82no—
131C22×C71—24no—
132C22×C71—24no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71#35312yes—
140C22×C71#35612yes—
141C22×C71#40711yescharacteristic
142C22×C71C2×#9911yescharacteristic
143C22×C71C2×#10111yescharacteristic
144C4×C71—22no—
145C4×C71—22no—
146C4×C71—41no—
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—41no—
150Dic71—28no—
151Dic71—82no—
152#11512—711nomaximal, Sylow
153C2×C71—21no—
154C2×C71—22no—
155C2×C71—22no—
156C2×C71—21no—
157C2×C71#146311yescharacteristic
158C2×C71C2×#35311yescharacteristic
159C2×C71C2×#35611yescharacteristic
160C2×#384—711no—
161#1463—711no—
162#1463D7111yescharacteristic
163#1325—714no—
164C71#1151211yescharacteristic, complemented, Sylow
165C22×C4⋊C4—711no—
166C22×C4⋊C4—1421no—
167C22×C4⋊C4—1421no—
168C22×C4⋊C4C2×D7111yescharacteristic
169C22×C42—711no—
170#407—21no—
171#407—1421no—
172#384—712no—
173#384—712no—
174#384—1421no—
175#384—1421no—
176#324—21no—
177#324—712no—
178#324—1421no—
179#324C2×D7112yes—
180C23×C4—1421no—
181C23×C4—1421no—
182C23×C4C4×D7111yescharacteristic
183C23×C4D28411yescharacteristic
184C23×C4C22⋊D7111yescharacteristic
185C2×M16—41no—
186C2×M16—2841no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—2841no—
190C2×C4⋊C4—2841no—
191C2×C4⋊C4—2844no—
192C2×C4⋊C4—2844no—
193C2×C4⋊C4—712no—
194C2×C4⋊C4—1421no—
195C2×C4⋊C4—1422no—
196C2×C4⋊C4—1422no—
197C2×C4⋊C4—1422no—
198C2×C4⋊C4—1422no—
199C2×C4⋊C4C4×D7111yescharacteristic
200C2×C4⋊C4D28411yescharacteristic
201C2×C4⋊C4C22⋊D7111yescharacteristic
202C2×C4⋊C4C22×D7111yescharacteristic
203C2×C42—712no—
204C2×C42—1424no—
205C2×C42—1421no—
206C22⋊C8—22no—
207C22⋊C8—41no—
208C22⋊C8—2841no—
209C22⋊C8—1422no—
210C24(C2×D71)⋊C411yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—2842no—
222C22×C4—2842no—
223C22×C4—2841no—
224C22×C4—2841no—
225C22×C4—1422no—
226C22×C4—1424no—
227C22×C4—1424no—
228C22×C4—1424no—
229C22×C4—1422no—
230C22×C4(C2×D71)⋊C411yescharacteristic
231C22×C4(C2×D71)⋊C411yescharacteristic
232C22×C4(C2×D71)⋊C411yescharacteristic
233C22×C4C2×C4×D7111yescharacteristic
234C22×C4C2×C22⋊D7111yescharacteristic
235C22×C4C2×D28411yescharacteristic
236M16—82no—
237M16—5682no—
238C2×C8—42no—
239C2×C8—2842no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—2842no—
245C4⋊C4—2842no—
246C4⋊C4—2844no—
247C4⋊C4—2844no—
248C4⋊C4—2844no—
249C4⋊C4—2844no—
250C42—714no—
251C42—1424no—
252C42—1422no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23#1455604211yescharacteristic
258C23#1455604211yescharacteristic
259C23C2×(C2×D71)⋊C411yescharacteristic
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—2842no—
279C2×C4—2842no—
280C2×C4—2842no—
281C2×C4—5682no—
282C2×C4—5682no—
283C2×C4—1424no—
284C2×C4—14216no—
285C2×C4—1424no—
286C2×C4—1424no—
287C2×C4#1455604612yes—
288C2×C4M16×D7112yes—
289C8—82no—
290C8—5682no—
291C22—24no—
292C22—24no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22#1798579112yes—
300C22#1798579712yes—
301C22#1798596211yescharacteristic
302C22C2×#1455604211yescharacteristic
303C22C2×#1455604611yescharacteristic
304C4—22no—
305C4—22no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—5682no—
311C4—1428no—
312C2—21no—
313C2—22no—
314C2—21no—
315C2—22no—
316C2#2258666911yescharacteristic
317C2C2×#1798579111yescharacteristic
318C2C2×#1798579711yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1463C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1463D71
hypocenterC71#11512
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #22586285C2semidirect
69. C4×Dic71M16semidirect
164. C71#11512semidirect

Maximal subgroups

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

subgroupindex
C2×#179858682
#225866702
#225862852
C71×#14632
#1151271