Finite Group Databasebeta

← #31888501

Subgroups — #31888501

Order 18176.

392 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597316C211yesmaximal, characteristic
2#22588093C211yesmaximal, characteristic, complemented
3#22586826C211yesmaximal, characteristic
4#22586627C211yesmaximal, characteristic, complemented
5#22586239C211yesmaximal, characteristic, complemented
6#22586234C211yesmaximal, characteristic, complemented
7C71×#1524C211yesmaximal, characteristic
8#17986130C411yescharacteristic, complemented
9Q8×C71⋊C8C411yescharacteristic, complemented
10#17986077C411yescharacteristic, complemented
11#17986071—21no—
12D4×C71⋊C8—21no—
13D4×C71⋊C8C411yescharacteristic, complemented
14#17986060—21no—
15#17986056C2211yescharacteristic
16#17986052C2211yescharacteristic
17#17985955—21no—
18C2×C4×C71⋊C8C2211yescharacteristic
19#17985845C2211yescharacteristic
20#17985753—21no—
21#17985753—21no—
22#17985736C412yes—
23#17985733C412yes—
24#17985721—21no—
25#17985721—21no—
26#529×C71C2211yescharacteristic
27C4×M16×C71C2211yescharacteristic
28C2×#97×C71C2211yescharacteristic
29#330×C71C412yescomplemented
30#329×C71C412yescomplemented
31C2×#11765849—21no—
32C22×C71⋊C8—21no—
33C22×C71⋊C8D411yescharacteristic
34C22×C71⋊C8D411yescharacteristic
35C2×C4×Dic71—21no—
36C4⋊(C71⋊C8)—21no—
37C4⋊(C71⋊C8)—21no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)C2×C411yescharacteristic
40C4⋊(C71⋊C8)C2×C411yescharacteristic
41C22⋊(C71⋊C8)—21no—
42C22⋊(C71⋊C8)—21no—
43C22⋊(C71⋊C8)—21no—
44C22⋊(C71⋊C8)C2×C412yes—
45#14556039—21no—
46#14556039—21no—
47C4×C71⋊C8—21no—
48C4×C71⋊C8—21no—
49C4×C71⋊C8C2×C411yescharacteristic
50C4×C71⋊C8C2×C411yescharacteristic
51C4×C71⋊C8D411yescharacteristic
52C4×C71⋊C8D411yescharacteristic
53C71⋊#95—21no—
54C71⋊#95C2×C412yes—
55C2×M16×C71—21no—
56C22×C8×C71—21no—
57#126×C71—21no—
58#125×C71C2×C411yescharacteristic
59#124×C71C2×C411yescharacteristic
60#123×C71—21no—
61C22⋊Q8×C71C2×C411yescharacteristic
62#121×C71C2×C411yescharacteristic
63C4×Q8×C71C2×C411yescharacteristic
64C4×D4×C71—21no—
65C4×D4×C71C2×C411yescharacteristic
66#117×C71—21no—
67C2×C42×C71C2311yescharacteristic
68#97×C71—21no—
69#97×C71C2×C412yes—
70#97×C71C2×C412yes—
71C4×C8×C71—21no—
72C22×Dic71—22no—
73#11765849—42no—
74C2×C71⋊C8—21no—
75C2×C71⋊C8—21no—
76C2×C71⋊C8—22no—
77C2×C71⋊C8—41no—
78C2×C71⋊C8—41no—
79C2×C71⋊C8—42no—
80C2×C71⋊C8C22⋊C411yescharacteristic
81C2×C71⋊C8C22⋊C411yescharacteristic
82C2×C71⋊C8C22⋊C411yescharacteristic
83C2×C71⋊C8C22⋊C411yescharacteristic
84C4×Dic71—41no—
85C4×Dic71—41no—
86C2×Q8×C71C4211yescharacteristic
87C2×D4×C71—21no—
88C2×D4×C71C4211yescharacteristic
89C22×C4×C71—21no—
90C22×C4×C71C22×C411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71#4111yescharacteristic
93M16×C71—42no—
94C2×C8×C71—22no—
95C2×C8×C71—22no—
96C2×C8×C71—41no—
97C2×C8×C71—41no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71—21no—
102C4⋊C4×C71C4211yescharacteristic
103C4⋊C4×C71C4211yescharacteristic
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71—21no—
109C42×C71—21no—
110C42×C71C22×C411yescharacteristic
111C42×C71C22×C411yescharacteristic
112C42×C71C2×D411yescharacteristic
113C42×C71#4111yescharacteristic
114C23×C71—21no—
115C23×C71C4×D411yescharacteristic
116Q8×C71—41no—
117D4×C71—41no—
118D4×C71—42no—
119C2×C4×C71—22no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71C2×C4211yescharacteristic
129C2×C4×C71C2×C22⋊C411yescharacteristic
130C2×C4×C71#11711yescharacteristic
131C2×C4×C71C4×D411yescharacteristic
132C2×C4×C71C4×D411yescharacteristic
133C2×C4×C71C4×D411yescharacteristic
134C2×Dic71—42no—
135C2×Dic71—42no—
136C2×Dic71—42no—
137C8×C71—42no—
138C8×C71—42no—
139C71⋊C8—21no—
140C71⋊C8—21no—
141C71⋊C8—21no—
142C71⋊C8—21no—
143C71⋊C8—42no—
144C71⋊C8—41no—
145C71⋊C8—41no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71C4×C22⋊C411yescharacteristic
152C22×C71#44412yes—
153C4×C71—21no—
154C4×C71—21no—
155C4×C71—21no—
156C4×C71—21no—
157C4×C71—21no—
158C4×C71—41no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71#42111yescharacteristic
162C4×C71#42111yescharacteristic
163Dic71—82no—
164#8248—711nomaximal, Sylow
165C2×C71—22no—
166C2×C71—81no—
167C2×C71#175711yescharacteristic
168C2×C71#175911yescharacteristic
169C2×C71#176511yescharacteristic
170#2966—711no—
171#1746—711no—
172#1524D7111yescharacteristic
173#1500—711no—
174#1451—711no—
175#1302—711no—
176#1300—711no—
177C71#824811yescharacteristic, complemented, Sylow
178#529C2×D7111yescharacteristic
179#446—711no—
180C8×Q8—711no—
181#436—711no—
182#435—1421no—
183C8×D4—711no—
184C8×D4—1421no—
185#433—1421no—
186#432—711no—
187#431—711no—
188#405—1421no—
189C4×M16C2×D7111yescharacteristic
190C2×#97C2×D7111yescharacteristic
191C2×C4×C8—711no—
192#376—711no—
193#336—1421no—
194#336—1421no—
195#330Dic7112yescomplemented
196#329Dic7112yescomplemented
197#328—712no—
198#327—712no—
199#322—1421no—
200#322—1421no—
201C2×M16—21no—
202C2×M16—1421no—
203C22×C8—21no—
204C22×C8—711no—
205C22×C8—711no—
206C22×C8—1421no—
207#126—21no—
208#125C2×Dic7111yescharacteristic
209#124C2×Dic7111yescharacteristic
210#123—21no—
211C22⋊Q8C4×D7111yescharacteristic
212#121C4×D7111yescharacteristic
213C4×Q8C4×D7111yescharacteristic
214C4×D4—21no—
215C4×D4C4×D7111yescharacteristic
216#117—21no—
217C2×C42—1421no—
218C2×C42C22×D7111yescharacteristic
219C4⋊C8—711no—
220C4⋊C8—711no—
221C4⋊C8—1421no—
222C4⋊C8—1421no—
223C4⋊C8—1421no—
224C22⋊C8—712no—
225C22⋊C8—1421no—
226C22⋊C8—1421no—
227C22⋊C8—1421no—
228#97—21no—
229#97—1421no—
230#97—1421no—
231#97C2×Dic7112yes—
232#97C2×Dic7112yes—
233C4×C8—21no—
234C4×C8—711no—
235C4×C8—711no—
236C4×C8—711no—
237C4×C8—711no—
238C4×C8—1421no—
239C4×C8—1421no—
240#95—712no—
241#95—1421no—
242C2×Q8C4×Dic7111yescharacteristic
243C2×D4—21no—
244C2×D4C4×Dic7111yescharacteristic
245C22×C4—21no—
246C22×C4—1422no—
247C22×C4C2×C4×D7111yescharacteristic
248C22×C4D4×D7111yescharacteristic
249C22×C4#1176586411yescharacteristic
250M16—42no—
251M16—2842no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—2842no—
257C2×C8—2841no—
258C2×C8—2841no—
259C2×C8—711no—
260C2×C8—711no—
261C2×C8—711no—
262C2×C8—711no—
263C2×C8—1421no—
264C2×C8—1421no—
265C2×C8—1422no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4C4×Dic7111yescharacteristic
271C4⋊C4C4×Dic7111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C42—21no—
278C42—2841no—
279C42—2841no—
280C42C2×C4×D7111yescharacteristic
281C42C22×Dic7111yescharacteristic
282C42D4×D7111yescharacteristic
283C42#1176586411yescharacteristic
284C23—21no—
285C23(C22⋊D71)⋊C411yescharacteristic
286Q8—41no—
287D4—41no—
288D4—42no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2842no—
299C2×C4—2842no—
300C2×C4—2842no—
301C2×C4C2×C4×Dic7111yescharacteristic
302C2×C4C22⋊C4×D7111yescharacteristic
303C2×C4#1455608411yescharacteristic
304C2×C4(C22⋊D71)⋊C411yescharacteristic
305C2×C4D4×Dic7111yescharacteristic
306C2×C4D4×Dic7111yescharacteristic
307C8—42no—
308C8—42no—
309C8—2842no—
310C8—2841no—
311C8—2841no—
312C8—1421no—
313C8—1421no—
314C8—1421no—
315C8—1421no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C22⋊C4×Dic7111yescharacteristic
322C22#1798611812yes—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—5682no—
332C4#1798601411yescharacteristic
333C4#1798601411yescharacteristic
334C2—22no—
335C2—81no—
336C2#2258814311yescharacteristic
337C2#2258815211yescharacteristic
338C2#2258818511yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC2×C4×C71C2×C42
FrattiniC2×C42C22×D71
FittingC71×#1524C2
socleC22×C71C4×C22⋊C4
prime radical#121×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#1524D71
hypocenterC71#8248
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
2. #22588093C2semidirect
4. #22586627C2semidirect
5. #22586239C2semidirect
6. #22586234C2semidirect
8. #17986130C4semidirect
9. Q8×C71⋊C8C4semidirect
10. #17986077C4semidirect
13. D4×C71⋊C8C4semidirect
29. #330×C71C4semidirect
30. #329×C71C4semidirect
177. C71#8248semidirect
195. #330Dic71semidirect
196. #329Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973162
#225880932
#225868262
#225866272
#225862392
#225862342
C71×#15242
#824871