Finite Group Databasebeta

← #31891963

Subgroups — #31891963

Order 18176.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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#22597441C211yesmaximal, characteristic
2#22589316C211yesmaximal, characteristic
3#22587131C211yesmaximal, characteristic
4#22587004C211yesmaximal, characteristic
5#22586172C211yesmaximal, characteristic
6#22586172C211yesmaximal, characteristic
7C71×#1574C211yesmaximal, characteristic
8#17986556C2211yescharacteristic
9#17986129—21no—
10#17986126—21no—
11C8×Dic142—21no—
12#17986123—21no—
13#17986057—22no—
14#17986052C2211yescharacteristic
15#17986052C2211yescharacteristic
16C42×Dic71C2211yescharacteristic
17#17985760—22no—
18#17985751—21no—
19#17985749—21no—
20#17985746C411yescharacteristic, complemented
21#17985746C411yescharacteristic, complemented
22#17985744C411yescharacteristic, complemented
23#17985744C411yescharacteristic, complemented
24#431×C71C2211yescharacteristic
25#431×C71C2211yescharacteristic
26#423×C71C2211yescharacteristic
27#335×C71—21no—
28#334×C71—21no—
29#14556128C2×C411yescharacteristic
30#14556128C2×C411yescharacteristic
31#14556120Q812yes—
32#14556107—21no—
33#14556107—21no—
34C4×Dic142—21no—
35C4×Dic142—21no—
36#14556087D412yes—
37C2×C4×Dic71—21no—
38C2×C4×Dic71—21no—
39C2×C4×Dic71—21no—
40C4⋊(C71⋊C8)C2×C411yescharacteristic
41C4⋊(C71⋊C8)C2×C411yescharacteristic
42C4⋊(C71⋊C8)D411yescharacteristic
43C4⋊(C71⋊C8)D411yescharacteristic
44Dic71⋊C8—22no—
45Dic71⋊C8—22no—
46C22⋊(C71⋊C8)—21no—
47C22⋊(C71⋊C8)—21no—
48#14556038—22no—
49C4×C71⋊C8—21no—
50C4×C71⋊C8—21no—
51C8×Dic71—22no—
52C2×M16×C71—21no—
53C2×C42×C71C2311yescharacteristic
54C4⋊C8×C71—21no—
55C4⋊C8×C71—21no—
56C4⋊C8×C71C2×C411yescharacteristic
57C4⋊C8×C71C2×C411yescharacteristic
58C4⋊C8×C71D411yescharacteristic
59C4⋊C8×C71D411yescharacteristic
60C22⋊C8×C71—21no—
61C22⋊C8×C71—21no—
62C4×C8×C71—21no—
63C4×C8×C71—21no—
64C2×Dic142—21no—
65C2×Dic142—21no—
66C22×Dic71—41no—
67C2×C71⋊C8—21no—
68C2×C71⋊C8—21no—
69C2×C71⋊C8—21no—
70C2×C71⋊C8—21no—
71C4⋊Dic71—21no—
72C4⋊Dic71—21no—
73C4⋊Dic71C4⋊C412yes—
74C4⋊Dic71C4⋊C412yes—
75Dic71⋊C4—22no—
76Dic71⋊C4—22no—
77C22⋊Dic71—22no—
78C4×Dic71—22no—
79C4×Dic71—22no—
80C4×Dic71—22no—
81C4×Dic71—22no—
82C4×Dic71—41no—
83C4×Dic71—41no—
84C4×Dic71—41no—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4111yescharacteristic
87C22×C4×C71#4111yescharacteristic
88M16×C71—42no—
89C2×C8×C71—21no—
90C2×C8×C71—21no—
91C2×C8×C71—21no—
92C2×C8×C71—21no—
93C2×C8×C71—41no—
94C42×C71C22×C411yescharacteristic
95C42×C71C2×D411yescharacteristic
96C42×C71C2×D411yescharacteristic
97C42×C71C2×Q811yescharacteristic
98C23×C71#12311yescharacteristic
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×C71C2×C4⋊C411yescharacteristic
106C2×C4×C71C4×D411yescharacteristic
107C2×C4×C71C4×D411yescharacteristic
108C2×C4×C71#12011yescharacteristic
109C2×C4×C71C22⋊Q811yescharacteristic
110C2×C4×C71C22⋊Q811yescharacteristic
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71—42no—
116Dic142—41no—
117Dic142—41no—
118Dic142—41no—
119Dic142—41no—
120C8×C71—41no—
121C8×C71—41no—
122C8×C71—41no—
123C8×C71—41no—
124C8×C71—81no—
125C71⋊C8—41no—
126C71⋊C8—41no—
127C71⋊C8—41no—
128C71⋊C8—41no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71#38511yescharacteristic
133C4×C71—21no—
134C4×C71—21no—
135C4×C71—21no—
136C4×C71#44311yescharacteristic
137C4×C71#44311yescharacteristic
138C4×C71#44411yescharacteristic
139C4×C71#44411yescharacteristic
140C4×C71#45411yescharacteristic
141C4×C71#45611yescharacteristic
142Dic71—44no—
143Dic71—42no—
144Dic71—42no—
145#8936—711nomaximal, Sylow
146C2×C71—41no—
147C2×C71#185711yescharacteristic
148C2×C71#186611yescharacteristic
149C2×C71#186611yescharacteristic
150#2986—711no—
151#1994—711no—
152#1574—711no—
153#1574D7111yescharacteristic
154#1544—711no—
155#1276—711no—
156#1276—711no—
157C71#893611yescharacteristic, complemented, Sylow
158#533—711no—
159#446—1421no—
160#446—1421no—
161C8×Q8—1421no—
162C8×Q8—1421no—
163#433—1422no—
164#431—711no—
165#431—711no—
166#431C2×D7111yescharacteristic
167#431C2×D7111yescharacteristic
168#423C2×D7111yescharacteristic
169C43—711no—
170#339—1422no—
171#335—21no—
172#335—1421no—
173#334—21no—
174#334—1421no—
175#332—711no—
176#332—711no—
177#332—711no—
178#332—711no—
179C2×M16—21no—
180C4⋊Q8—711no—
181C4⋊Q8—711no—
182#125—712no—
183C22⋊Q8—1421no—
184C22⋊Q8—1421no—
185C4×Q8—1421no—
186C4×Q8—1421no—
187#117—712no—
188C2×C42—1421no—
189C2×C42—1421no—
190C2×C42—1421no—
191C2×C42C22×D7111yescharacteristic
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—711no—
195C4⋊C8—711no—
196C4⋊C8—711no—
197C4⋊C8—711no—
198C4⋊C8—1422no—
199C4⋊C8—1422no—
200C4⋊C8C4×D7111yescharacteristic
201C4⋊C8C4×D7111yescharacteristic
202C4⋊C8C22⋊D7111yescharacteristic
203C4⋊C8C22⋊D7111yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—1421no—
207C22⋊C8—1421no—
208#97—1422no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—1422no—
212C4×C8—1421no—
213C4×C8—1421no—
214C2×Q8—1421no—
215C2×Q8—1421no—
216C22×C4—2841no—
217C22×C4D4×D7111yescharacteristic
218C22×C4#1176586511yescharacteristic
219C22×C4#1176586611yescharacteristic
220M16—42no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—41no—
226C2×C8—1421no—
227C2×C8—1421no—
228C2×C8—1421no—
229C2×C8—1421no—
230C4⋊C4—712no—
231C4⋊C4—712no—
232C4⋊C4—1421no—
233C4⋊C4—1421no—
234C4⋊C4—1422no—
235C4⋊C4—1422no—
236C22⋊C4—1422no—
237C42—2841no—
238C42—2841no—
239C42—2841no—
240C42—1422no—
241C42—1422no—
242C42—1422no—
243C42—1422no—
244C42C2×C4×D7111yescharacteristic
245C42D4×D7111yescharacteristic
246C42C2×C22⋊D7111yescharacteristic
247C42Q8×D7111yescharacteristic
248C23#1455611311yescharacteristic
249Q8—2841no—
250Q8—2841no—
251Q8—2841no—
252Q8—2841no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—2842no—
260C2×C4—2842no—
261C2×C4—2842no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4C4⋊C4×D7111yescharacteristic
265C2×C4C4×C22⋊D7111yescharacteristic
266C2×C4#1455609011yescharacteristic
267C2×C4#1455609611yescharacteristic
268C2×C4#1455610411yescharacteristic
269C2×C4#1455610611yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—81no—
275C8—2841no—
276C8—2841no—
277C8—2841no—
278C8—2841no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1798587011yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—2844no—
287C4—2842no—
288C4—2842no—
289C4#1798611411yescharacteristic
290C4#1798611511yescharacteristic
291C4#1798611911yescharacteristic
292C4#1798612011yescharacteristic
293C4#1798617011yescharacteristic
294C4#1798618011yescharacteristic
295C2—41no—
296C2#2258855311yescharacteristic
297C2#2258861211yescharacteristic
298C2#2258861411yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1574C2
socleC22×C71#385
prime radicalC23×C71#123
cosocleC2×C42×C71C23
hypercenter#1574D71
hypocenterC71#8936
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
20. #17985746C4semidirect
21. #17985746C4semidirect
22. #17985744C4semidirect
23. #17985744C4semidirect
157. C71#8936semidirect

Maximal subgroups

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

subgroupindex
#225974412
#225893162
#225871312
#225870042
#225861722
C71×#15742
#893671