Finite Group Databasebeta

← #31894127

Subgroups — #31894127

Order 18176.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#22599484C211yesmaximal, characteristic
2#22597277C211yesmaximal, characteristic
3#22589161C211yesmaximal, characteristic
4#22586905C211yesmaximal, characteristic
5#22586200C211yesmaximal, characteristic
6#22586170C211yesmaximal, characteristic
7C71×#1546C211yesmaximal, characteristic
8#17986556C2211yescharacteristic
9#17986395C412yescomplemented
10#17986351C412yescomplemented
11#17986329—22no—
12#17986281—22no—
13#17986128—22no—
14#17986128—22no—
15#17986052C2211yescharacteristic
16#17986027—21no—
17C2×#14556052—21no—
18C2×#14556052—21no—
19#17985957—21no—
20C2×#14556039C2211yescharacteristic
21#17985865C2211yescharacteristic
22#17985737—22no—
23C2×C4⋊Q8×C71C2211yescharacteristic
24#431×C71C2211yescharacteristic
25#423×C71C2211yescharacteristic
26#332×C71—22no—
27C22×C71⋊C8D411yescharacteristic
28C22×C71⋊C8D411yescharacteristic
29#14556128C2×C411yescharacteristic
30#14556128C2×C411yescharacteristic
31#14556120—21no—
32#14556107—21no—
33#14556107D412yes—
34C4×Dic142—21no—
35C4×Dic142D412yes—
36#14556087—21no—
37C2×C4⋊Dic71—21no—
38#14556059—24no—
39C4⋊(C71⋊C8)—21no—
40Dic71⋊C8—24no—
41#14556052—42no—
42#14556052—42no—
43#14556051—24no—
44C22⋊(C71⋊C8)—21no—
45#14556039—21no—
46#14556039—21no—
47#14556038—22no—
48C4×C71⋊C8C2×C412yes—
49C8×Dic71—22no—
50C71⋊#95—21no—
51C71⋊#95—21no—
52C22×Q8×C71—21no—
53C2×M16×C71—21no—
54C4⋊Q8×C71—21no—
55C4⋊Q8×C71—21no—
56C4⋊Q8×C71—21no—
57C4⋊Q8×C71C2×C412yes—
58C2×C4⋊C4×C71—21no—
59C2×C4⋊C4×C71—21no—
60C2×C42×C71C2311yescharacteristic
61C4⋊C8×C71—21no—
62C4⋊C8×C71—21no—
63C4⋊C8×C71D412yes—
64C22⋊C8×C71—21no—
65C4×C8×C71—21no—
66C2×Dic142—21no—
67C2×Dic142C22⋊C412yes—
68C22×Dic71—41no—
69C2×C71⋊C8—21no—
70C2×C71⋊C8—21no—
71C2×C71⋊C8—22no—
72C2×C71⋊C8—21no—
73C2×C71⋊C8—21no—
74C2×C71⋊C8—21no—
75C2×C71⋊C8SD1612yes—
76C4⋊Dic71—21no—
77C4⋊Dic71—21no—
78C4⋊Dic71—21no—
79C4⋊Dic71—41no—
80C4⋊Dic71—41no—
81C4⋊Dic71C22⋊C412yes—
82Dic71⋊C4—22no—
83Dic71⋊C4—22no—
84C22⋊Dic71—22no—
85C4×Dic71—22no—
86C2×Q8×C71—22no—
87C2×Q8×C71—41no—
88C2×Q8×C71—41no—
89C2×Q8×C71—41no—
90C2×Q8×C71—41no—
91C2×Q8×C71—41no—
92C22×C4×C71—41no—
93C22×C4×C71C2×D411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71#4111yescharacteristic
96M16×C71—41no—
97M16×C71—41no—
98C2×C8×C71—22no—
99C2×C8×C71—41no—
100C4⋊C4×C71—22no—
101C4⋊C4×C71—22no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—41no—
104C42×C71C22×C411yescharacteristic
105C42×C71C2×D411yescharacteristic
106C42×C71C2×D411yescharacteristic
107C42×C71#4111yescharacteristic
108C23×C71#12111yescharacteristic
109Q8×C71—42no—
110Q8×C71—42no—
111Q8×C71—81no—
112Q8×C71—81no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—42no—
116C2×C4×C71—81no—
117C2×C4×C71—81no—
118C2×C4×C71C2×C22⋊C411yescharacteristic
119C2×C4×C71C4×D411yescharacteristic
120C2×C4×C71C4×D411yescharacteristic
121C2×C4×C71#12011yescharacteristic
122C2×C4×C71#12111yescharacteristic
123C2×C4×C71#12311yescharacteristic
124C2×C4×C71C2×SD1611yescharacteristic
125C2×C4×C71#13511yescharacteristic
126C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C71#13711yescharacteristic
128C2×Dic71—22no—
129C2×Dic71—22no—
130C2×Dic71—81no—
131C2×Dic71—81no—
132C2×Dic71—81no—
133Dic142—42no—
134Dic142—81no—
135C8×C71—42no—
136C8×C71—81no—
137C71⋊C8—22no—
138C71⋊C8—42no—
139C71⋊C8—41no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71#38611yescharacteristic
143C22×C71#46011yescharacteristic
144C22×C71#46411yescharacteristic
145C4×C71—21no—
146C4×C71—21no—
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—82no—
150C4×C71C4×SD1611yescharacteristic
151C4×C71#44011yescharacteristic
152C4×C71#44811yescharacteristic
153C4×C71#44911yescharacteristic
154Dic71—42no—
155Dic71—42no—
156Dic71—161no—
157#9292—711nomaximal, Sylow
158C2×C71—21no—
159C2×C71—21no—
160C2×C71#188011yescharacteristic
161C2×C71#190211yescharacteristic
162C2×C71#190911yescharacteristic
163#3229—711no—
164#2961—711no—
165#1960—711no—
166#1546D7111yescharacteristic
167#1518—711no—
168#1286—711no—
169#1275—711no—
170C71#929211yescharacteristic, complemented, Sylow
171#533—711no—
172C2×C4⋊Q8C2×D7111yescharacteristic
173#498—712no—
174#487—712no—
175#483—1422no—
176#475—1422no—
177#446—1422no—
178#446—1422no—
179#431—711no—
180#431C2×D7111yescharacteristic
181#424—1421no—
182#423C2×D7111yescharacteristic
183C2×#103—1421no—
184C2×#103—1421no—
185#405—1421no—
186C2×#97—711no—
187#382—711no—
188#332—22no—
189#328—1422no—
190C22×Q8—21no—
191C2×M16—21no—
192C22×C8—711no—
193C22×C8—711no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—711no—
198C4⋊Q8—711no—
199C4⋊Q8C4×D7112yes—
200#125—1421no—
201C22⋊Q8—712no—
202C22⋊Q8—1421no—
203C4×Q8—712no—
204C4×Q8—1421no—
205#117—1421no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—1421no—
209C2×C42C22×D7111yescharacteristic
210#106—1424no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—1424no—
214C4⋊C8—1421no—
215C4⋊C8C22⋊D7112yes—
216#103—2842no—
217#103—2842no—
218#103—1424no—
219C22⋊C8—21no—
220C22⋊C8—1421no—
221#97—1422no—
222#97—1421no—
223#97—1421no—
224C4×C8—21no—
225C4×C8—712no—
226C4×C8—1422no—
227#95—1421no—
228#95—1421no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—712no—
236C2×Q8—1421no—
237C22×C4—41no—
238C22×C4—2841no—
239C22×C4D4×D7111yescharacteristic
240C22×C4D4×D7111yescharacteristic
241C22×C4#1176586511yescharacteristic
242M16—41no—
243M16—41no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—712no—
247C2×C8—1422no—
248C2×C8—1421no—
249C2×C8—1421no—
250C2×C8—1421no—
251C2×C8—1421no—
252C2×C8—1421no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—2841no—
258C4⋊C4—2841no—
259C4⋊C4—712no—
260C4⋊C4—1421no—
261C4⋊C4—1421no—
262C4⋊C4—1421no—
263C4⋊C4—1422no—
264C4⋊C4—1422no—
265C22⋊C4—1422no—
266C42—1422no—
267C42C2×C4×D7111yescharacteristic
268C42D4×D7111yescharacteristic
269C42C2×C22⋊D7111yescharacteristic
270C42#1176586411yescharacteristic
271C23#1455610111yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—81no—
275Q8—81no—
276Q8—2842no—
277Q8—5681no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—5681no—
284C2×C4—5681no—
285C2×C4—5681no—
286C2×C4—1422no—
287C2×C4—1422no—
288C2×C4C22⋊C4×D7111yescharacteristic
289C2×C4(C22⋊D71)⋊C411yescharacteristic
290C2×C4C4×C22⋊D7111yescharacteristic
291C2×C4#1455609611yescharacteristic
292C2×C4#1455609911yescharacteristic
293C2×C4#1455611011yescharacteristic
294C2×C4SD16×D7111yescharacteristic
295C2×C4#1455614711yescharacteristic
296C2×C4#1455615211yescharacteristic
297C2×C4#1455615711yescharacteristic
298C8—42no—
299C8—81no—
300C8—2842no—
301C8—2841no—
302C8—1422no—
303C22—21no—
304C22—21no—
305C22#1798587511yescharacteristic
306C22#1798619811yescharacteristic
307C22#1798622011yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—2842no—
314C4—2842no—
315C4—11361no—
316C4#1798609011yescharacteristic
317C4#1798609611yescharacteristic
318C4#1798613611yescharacteristic
319C4#1798614111yescharacteristic
320C2—21no—
321C2—21no—
322C2#2258869211yescharacteristic
323C2#2258881611yescharacteristic
324C2#2258885911yescharacteristic
subgroupG/H
centerC22#17985875
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1546C2
socleC22×C71#386
prime radicalC23×C71#121
cosocleC2×C42×C71C23
hypercenter#1546D71
hypocenterC71#9292
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
9. #17986395C4semidirect
10. #17986351C4semidirect
170. C71#9292semidirect

Maximal subgroups

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

subgroupindex
#225994842
#225972772
#225891612
#225869052
#225862002
#225861702
C71×#15462
#929271