Finite Group Databasebeta

← #31885572

Subgroups — #31885572

Order 18176.

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

normal subgroups only

This group has 608 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
1C4×#14556125C211yesmaximal, characteristic
2#22588483C211yesmaximal, characteristic, complemented
3#22586160C214yesmaximal, complemented
4C71×#1839C211yesmaximal, characteristic, complemented
5C2×#14556125C411yescharacteristic
6C2×C4×D284—21no—
7C2×C4×D284—21no—
8C2×C4×D284C412yes—
9#17985901—21no—
10#17985901—21no—
11#17985901C2212yes—
12#17985869C411yescharacteristic
13#17985754—21no—
14#17985754—21no—
15#17985754C2212yes—
16C43×C71C2211yescharacteristic
17#336×C71—21no—
18#336×C71—21no—
19#336×C71C2212yes—
20C22×D284—21no—
21C22×D284—21no—
22C22×D284C2×C412yes—
23C22×C4×D71—22no—
24C22×C71⋊C8—22no—
25#14556125—22no—
26#14556125C814yescomplemented
27C4×D284—44no—
28C4×D284—44no—
29C4×D284—44no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71C2×C412yes—
33C2×(C2×D71)⋊C4—22no—
34C2×(C2×D71)⋊C4—22no—
35C2×(C2×D71)⋊C4—22no—
36C22×C8×C71—22no—
37C2×C42×C71C2×C411yescharacteristic
38C2×C42×C71C2×C411yescharacteristic
39C2×C42×C71D411yescharacteristic
40C2×C42×C71D411yescharacteristic
41C2×C42×C71D411yescharacteristic
42C2×C42×C71D411yescharacteristic
43C2×C42×C71C2311yescharacteristic
44C23×D71—22no—
45C2×D284—24no—
46C2×D284—24no—
47C2×D284—22no—
48C2×D284—42no—
49C2×D284—42no—
50C2×D284—42no—
51C2×D284—42no—
52C2×D284—42no—
53C2×D284—42no—
54C2×D284C2×C814yes—
55C22×Dic71—22no—
56C2×C4×D71—44no—
57C2×C4×D71—44no—
58C2×C4×D71—44no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—42no—
62C4⋊Dic71—42no—
63C4⋊Dic71—42no—
64C4⋊Dic71—42no—
65(C2×D71)⋊C4—44no—
66(C2×D71)⋊C4—44no—
67(C2×D71)⋊C4—44no—
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22×C411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C2×C8×C71—42no—
76C2×C8×C71—42no—
77C2×C8×C71—42no—
78C42×C71—22no—
79C42×C71—22no—
80C42×C71—22no—
81C42×C71—41no—
82C42×C71—41no—
83C42×C71—41no—
84C42×C71C2×C812yes—
85C42×C71M1612yes—
86C23×C71C2×C22⋊C411yescharacteristic
87C22×D71—24no—
88C22×D71—42no—
89C22×D71—42no—
90C22×D71—42no—
91C22×D71—42no—
92C22×D71—42no—
93C22×D71—42no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
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×C71C22⋊C812yes—
110C2×C4×C71C22⋊C812yes—
111C2×C4×C71#10012yes—
112C2×C4×C71C22×C811yescharacteristic
113C2×C4×C71C2×M1611yescharacteristic
114C2×C4×C71AGL(1,ℤ/8)12yes—
115C2×C4×C71AGL(1,ℤ/8)12yes—
116C2×Dic71—42no—
117C2×Dic71—42no—
118C2×Dic71—42no—
119D284—44no—
120D284—44no—
121D284—48no—
122D284—48no—
123C4×D71—88no—
124C8×C71—82no—
125C71⋊C8—82no—
126C22×C71C2×C22⋊C811yescharacteristic
127C22×C71#41011yescharacteristic
128C22×C71C2×#10011yescharacteristic
129C22×C71#41811yescharacteristic
130C22×C71#41811yescharacteristic
131C22×C71#42111yescharacteristic
132C22×C71#42111yescharacteristic
133C2×D71—44no—
134C2×D71—44no—
135C2×D71—44no—
136C2×D71—82no—
137C2×D71—82no—
138C2×D71—82no—
139C2×D71—82no—
140C4×C71—22no—
141C4×C71—21no—
142C4×C71—21no—
143C4×C71—22no—
144C4×C71—22no—
145C4×C71—22no—
146C4×C71—22no—
147C4×C71—41no—
148Dic71—82no—
149#7603—711nomaximal, Sylow
150C2×C71#146011yescharacteristic
151C2×C71#147811yescharacteristic
152C2×C71#147811yescharacteristic
153C2×C71#151411yescharacteristic
154C2×C71#151411yescharacteristic
155C2×C71#153611yescharacteristic
156C2×C71#155511yescharacteristic
157D71—84no—
158C4×#127—711no—
159#1839—711no—
160#1839D7111yescharacteristic, complemented
161#1270—714no—
162C71#760311yescharacteristic, complemented, Sylow
163C2×#127—711no—
164C2×C4×D4—712no—
165C2×C4×D4—1421no—
166C2×C4×D4—1421no—
167#390—712no—
168#390—1421no—
169#390—1421no—
170#384—711no—
171C43C2×D7111yescharacteristic
172#336—21no—
173#336—21no—
174#336—712no—
175#336—1421no—
176#336—1421no—
177#336C2×D7112yes—
178C22×D4—712no—
179C22×D4—1421no—
180C22×D4—1421no—
181C23×C4—1422no—
182C22×C8—22no—
183C22×C8—1422no—
184#127—714no—
185#127—1422no—
186C4×D4—2844no—
187C4×D4—2844no—
188C4×D4—2844no—
189C2×C4⋊C4—712no—
190C2×C4⋊C4—1421no—
191C2×C4⋊C4—1421no—
192C2×C22⋊C4—1422no—
193C2×C22⋊C4—1422no—
194C2×C22⋊C4—1422no—
195C2×C42C4×D7111yescharacteristic
196C2×C42C4×D7111yescharacteristic
197C2×C42D28411yescharacteristic
198C2×C42D28411yescharacteristic
199C2×C42C22⋊D7111yescharacteristic
200C2×C42C22⋊D7111yescharacteristic
201C2×C42C22×D7111yescharacteristic
202C24—1422no—
203C2×D4—2842no—
204C2×D4—2842no—
205C2×D4—2842no—
206C2×D4—2842no—
207C2×D4—2842no—
208C2×D4—2842no—
209C2×D4—714no—
210C2×D4—1424no—
211C2×D4—1424no—
212C2×D4—1422no—
213C22×C4—2844no—
214C22×C4—2844no—
215C22×C4—2844no—
216C22×C4—1422no—
217C22×C4(C2×D71)⋊C411yescharacteristic
218C22×C4(C2×D71)⋊C411yescharacteristic
219C22×C4(C2×D71)⋊C411yescharacteristic
220C22×C4(C2×D71)⋊C411yescharacteristic
221C22×C4C2×C4×D7111yescharacteristic
222C22×C4C2×C22⋊D7111yescharacteristic
223C22×C4C2×D28411yescharacteristic
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—2842no—
228C2×C8—2842no—
229C2×C8—2842no—
230C4⋊C4—2842no—
231C4⋊C4—2842no—
232C4⋊C4—2842no—
233C22⋊C4—2844no—
234C22⋊C4—2844no—
235C22⋊C4—2844no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42C8×D7112yes—
243C42#1176584812yes—
244C23—2842no—
245C23—2842no—
246C23—2842no—
247C23—2842no—
248C23—2842no—
249C23—2842no—
250C23—1424no—
251C23C2×(C2×D71)⋊C411yescharacteristic
252D4—2844no—
253D4—2844no—
254D4—2848no—
255D4—2848no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
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—21no—
271C2×C4—2842no—
272C2×C4—2842no—
273C2×C4—2842no—
274C2×C4—5688no—
275C2×C4(C2×D71)⋊C812yes—
276C2×C4(C2×D71)⋊C812yes—
277C2×C4#1455604412yes—
278C2×C4C2×C8×D7111yescharacteristic
279C2×C4C2×#1176584811yescharacteristic
280C2×C4#1455615312yes—
281C2×C4#1455615612yes—
282C8—82no—
283C8—5682no—
284C22—2844no—
285C22—2844no—
286C22—2844no—
287C22—5682no—
288C22—5682no—
289C22—5682no—
290C22—5682no—
291C22C2×(C2×D71)⋊C811yescharacteristic
292C22#1798597211yescharacteristic
293C22C2×#1455604411yescharacteristic
294C22#1798600311yescharacteristic
295C22#1798600511yescharacteristic
296C22#1798601511yescharacteristic
297C22#1798601711yescharacteristic
298C4—22no—
299C4—22no—
300C4—21no—
301C4—21no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—41no—
306C4—5682no—
307C2—5684no—
308C2#2258666011yescharacteristic
309C2#2258673011yescharacteristic
310C2#2258673211yescharacteristic
311C2#2258688611yescharacteristic
312C2#2258688811yescharacteristic
313C2#2258697411yescharacteristic
314C2#2258705311yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC71×#1839C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#14556125C4
cosocleC2×C42×C71C23
hypercenter#1839D71
hypocenterC71#7603
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #1839

Complements

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

NH ≅ G/Ntype
2. #22588483C2semidirect
3. #22586160C2semidirect
4. C71×#1839C2semidirect
26. #14556125C8semidirect
160. #1839D71semidirect
162. C71#7603semidirect

Maximal subgroups

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

subgroupindex
C4×#145561252
#225884832
#225861602
C71×#18392
#760371