Finite Group Databasebeta

← #31906513

Subgroups — #31906513

Order 18176.

666 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C22×(C2×D71)⋊C8C211yesmaximal, characteristic
2#22589951C211yesmaximal, characteristic, complemented
3#22586263C214yesmaximal, complemented
4C71×#2112C211yesmaximal, characteristic, complemented
5C23×C4×D71C411yescharacteristic
6C23×C71⋊C8C411yescharacteristic
7C2×#14556068—21no—
8C2×(C2×D71)⋊C8—21no—
9C2×(C2×D71)⋊C8—21no—
10C2×(C2×D71)⋊C8—21no—
11C2×(C2×D71)⋊C8—21no—
12C2×(C2×D71)⋊C8C412yes—
13C2×(C2×D71)⋊C8C2212yes—
14C22⋊(C71⋊C16)—21no—
15C22⋊(C71⋊C16)C2212yescomplemented
16C23×C8×C71C2211yescharacteristic
17C2×M32×C71—21no—
18C22⋊C16×C71—21no—
19C22⋊C16×C71C2212yescomplemented
20C24×D71C811yescharacteristic
21C23×Dic71C811yescharacteristic
22C22×C4×D71—21no—
23C22×C4×D71—21no—
24C22×C4×D71—21no—
25C22×C4×D71—21no—
26C22×C4×D71—42no—
27C22×C4×D71—42no—
28C22×C4×D71C812yes—
29C22×C4×D71C2×C412yes—
30C22×C71⋊C8—21no—
31C22×C71⋊C8—21no—
32C22×C71⋊C8—42no—
33C22×C71⋊C8C2×C412yes—
34#14556068—42no—
35C2×C71⋊C16—22no—
36(C2×D71)⋊C8—22no—
37(C2×D71)⋊C8—22no—
38(C2×D71)⋊C8—22no—
39(C2×D71)⋊C8—22no—
40C23×C4×C71C2×C411yescharacteristic
41C22×C8×C71—42no—
42C22×C8×C71C2×C411yescharacteristic
43C22×C8×C71D411yescharacteristic
44C22×C8×C71D411yescharacteristic
45C22×C8×C71D411yescharacteristic
46C22×C8×C71D411yescharacteristic
47C22×C8×C71C2311yescharacteristic
48M32×C71—42no—
49C2×C16×C71—22no—
50C23×D71—21no—
51C23×D71—21no—
52C23×D71—24no—
53C23×D71—42no—
54C23×D71—42no—
55C23×D71C2×C812yes—
56C22×Dic71—21no—
57C22×Dic71—21no—
58C22×Dic71—42no—
59C22×Dic71C2×C812yes—
60C2×C4×D71—22no—
61C2×C4×D71—22no—
62C2×C4×D71—22no—
63C2×C4×D71—22no—
64C2×C4×D71—44no—
65C2×C4×D71—44no—
66C2×C4×D71—42no—
67C2×C4×D71—42no—
68C2×C4×D71—42no—
69C2×C4×D71—42no—
70C2×C4×D71—44no—
71C2×C4×D71—42no—
72C2×C4×D71—42no—
73C2×C71⋊C8—22no—
74C2×C71⋊C8—42no—
75C2×C71⋊C8—42no—
76C2×C71⋊C8—42no—
77C71⋊C16—42no—
78C24×C71C2×C811yescharacteristic
79C22×C4×C71—22no—
80C22×C4×C71—22no—
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C2×C811yescharacteristic
84C22×C4×C71M1611yescharacteristic
85C22×C4×C71M1611yescharacteristic
86C22×C4×C71C22×C411yescharacteristic
87C2×C8×C71—22no—
88C2×C8×C71—22no—
89C2×C8×C71—41no—
90C2×C8×C71—41no—
91C2×C8×C71—41no—
92C2×C8×C71—41no—
93C2×C8×C71C22⋊C411yescharacteristic
94C2×C8×C71C22⋊C411yescharacteristic
95C2×C8×C71C2×D411yescharacteristic
96C2×C8×C71C2×D411yescharacteristic
97C16×C71—42no—
98C23×C71—22no—
99C23×C71—22no—
100C23×C71—22no—
101C23×C71C22⋊C811yescharacteristic
102C23×C71C22⋊C811yescharacteristic
103C23×C71C22×C811yescharacteristic
104C22×D71—24no—
105C22×D71—24no—
106C22×D71—22no—
107C22×D71—22no—
108C22×D71—42no—
109C22×D71—416no—
110C22×D71—42no—
111C22×D71—42no—
112C22×D71—42no—
113C22×D71—42no—
114C22×D71—42no—
115C22×D71M3214yescomplemented
116C2×C4×C71—22no—
117C2×C4×C71—22no—
118C2×C4×C71—22no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71C22⋊C811yescharacteristic
124C2×C4×C71C22⋊C811yescharacteristic
125C2×C4×C71#9912yes—
126C2×C4×C71#10012yes—
127C2×C4×C71C2×C22⋊C411yescharacteristic
128C2×C4×C71C2×M1611yescharacteristic
129C2×Dic71—22no—
130C2×Dic71—42no—
131C2×Dic71—42no—
132C2×Dic71—42no—
133C4×D71—44no—
134C4×D71—44no—
135C4×D71—44no—
136C4×D71—42no—
137C4×D71—42no—
138C8×C71—21no—
139C8×C71—21no—
140C8×C71—21no—
141C8×C71—21no—
142C71⋊C8—42no—
143C22×C71—28no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71#32312yes—
151C22×C71#34912yes—
152C22×C71C2×C22⋊C811yescharacteristic
153C22×C71C2×M3212yes—
154C2×D71—24no—
155C2×D71—48no—
156C2×D71—48no—
157C2×D71—44no—
158C2×D71—42no—
159C2×D71—42no—
160C2×D71—44no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71#32312yes—
164C4×C71C2×#9911yescharacteristic
165C4×C71C2×#10011yescharacteristic
166Dic71—42no—
167#11418—711nomaximal, Sylow
168C2×C71—22no—
169C2×C71—22no—
170C2×C71—22no—
171C2×C71C2×#32311yescharacteristic
172C2×C71#211211yescharacteristic
173C2×C71C2×#34911yescharacteristic
174D71—44no—
175C22×C22⋊C8—711no—
176#2112—711no—
177#2112D7111yescharacteristic, complemented
178#1314—714no—
179C71#1141811yescharacteristic, complemented, Sylow
180C24×C4—711no—
181C23×C8—711no—
182C23×C8C2×D7111yescharacteristic
183C2×M32—21no—
184C2×M32—1421no—
185C2×C22⋊C8—712no—
186C2×C22⋊C8—712no—
187C2×C22⋊C8—1421no—
188C2×C22⋊C8—1421no—
189C2×C22⋊C8—1421no—
190C2×C22⋊C8—1421no—
191C22⋊C16—21no—
192C22⋊C16—712no—
193C22⋊C16—1421no—
194C22⋊C16C2×D7112yescomplemented
195C25—711no—
196C23×C4—2842no—
197C23×C4—2842no—
198C23×C4—711no—
199C23×C4—712no—
200C23×C4—712no—
201C23×C4—1421no—
202C23×C4—1421no—
203C23×C4—1421no—
204C23×C4—1421no—
205C23×C4C4×D7111yescharacteristic
206C22×C8—42no—
207C22×C8—2842no—
208C22×C8—712no—
209C22×C8—1421no—
210C22×C8—1421no—
211C22×C8C4×D7111yescharacteristic
212C22×C8D28411yescharacteristic
213C22×C8D28411yescharacteristic
214C22×C8C22⋊D7111yescharacteristic
215C22×C8C22⋊D7111yescharacteristic
216C22×C8C22×D7111yescharacteristic
217M32—42no—
218M32—2842no—
219C2×C16—22no—
220C2×C16—1422no—
221C22⋊C8—1422no—
222C22⋊C8—1422no—
223C22⋊C8—1422no—
224C22⋊C8—1422no—
225C24—2842no—
226C24—2842no—
227C24—712no—
228C24—1424no—
229C24—1421no—
230C24—1421no—
231C24C8×D7111yescharacteristic
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—2844no—
235C22×C4—2842no—
236C22×C4—2842no—
237C22×C4—2844no—
238C22×C4—2842no—
239C22×C4—2842no—
240C22×C4—2844no—
241C22×C4—2842no—
242C22×C4—2842no—
243C22×C4—2842no—
244C22×C4—712no—
245C22×C4—1421no—
246C22×C4—1421no—
247C22×C4—1422no—
248C22×C4—1422no—
249C22×C4—1422no—
250C22×C4—1422no—
251C22×C4(C2×D71)⋊C411yescharacteristic
252C22×C4(C2×D71)⋊C411yescharacteristic
253C22×C4C8×D7111yescharacteristic
254C22×C4#1176584811yescharacteristic
255C22×C4#1176584811yescharacteristic
256C22×C4C2×C4×D7111yescharacteristic
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—2842no—
264C2×C8—2842no—
265C2×C8—2842no—
266C2×C8—1422no—
267C2×C8(C2×D71)⋊C411yescharacteristic
268C2×C8(C2×D71)⋊C411yescharacteristic
269C2×C8C2×C22⋊D7111yescharacteristic
270C2×C8C2×D28411yescharacteristic
271C16—42no—
272C16—2842no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—2842no—
277C23—2842no—
278C23—2842no—
279C23—28416no—
280C23—2842no—
281C23—2842no—
282C23—2842no—
283C23—714no—
284C23—1424no—
285C23—1424no—
286C23—1422no—
287C23—1422no—
288C23(C2×D71)⋊C811yescharacteristic
289C23(C2×D71)⋊C811yescharacteristic
290C23C2×C8×D7111yescharacteristic
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—2844no—
299C2×C4—2844no—
300C2×C4—2842no—
301C2×C4—2842no—
302C2×C4—2842no—
303C2×C4—2842no—
304C2×C4—2844no—
305C2×C4—2842no—
306C2×C4—1422no—
307C2×C4(C2×D71)⋊C811yescharacteristic
308C2×C4(C2×D71)⋊C811yescharacteristic
309C2×C4#1455604212yes—
310C2×C4#1455604412yes—
311C2×C4C2×(C2×D71)⋊C411yescharacteristic
312C2×C4C2×#1176584811yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—21no—
317C8—2842no—
318C22—28no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—2844no—
326C22—2844no—
327C22—2848no—
328C22—2848no—
329C22—2842no—
330C22—2842no—
331C22—1424no—
332C22#1798572212yes—
333C22#1798578012yes—
334C22C2×(C2×D71)⋊C811yescharacteristic
335C22M32×D7112yes—
336C4—21no—
337C4—21no—
338C4—2842no—
339C4#1798572212yes—
340C4C2×#1455604211yescharacteristic
341C4C2×#1455604411yescharacteristic
342C2—22no—
343C2—22no—
344C2—22no—
345C2—2844no—
346C2C2×#1798572211yescharacteristic
347C2#2258995011yescharacteristic
348C2C2×#1798578011yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D71)⋊C4
commutatorC23×C71C22×C8
FrattiniC22×C8C22×D71
FittingC71×#2112C2
socleC22×C71C2×C22⋊C8
prime radicalC24×D71C8
cosocleC22×C8×C71C23
hypercenter#2112D71
hypocenterC71#11418
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. #22589951C2semidirect
3. #22586263C2semidirect
4. C71×#2112C2semidirect
15. C22⋊(C71⋊C16)C22semidirect
19. C22⋊C16×C71C22semidirect
115. C22×D71M32semidirect
177. #2112D71semidirect
179. C71#11418semidirect
194. C22⋊C16C2×D71semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D71)⋊C82
#225899512
#225862632
C71×#21122
#1141871