Finite Group Databasebeta

← #31893298

Subgroups — #31893298

Order 18176.

436 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597429C211yesmaximal, characteristic
2#22588170C211yesmaximal, characteristic
3#22586955C211yesmaximal, characteristic
4#22586839C211yesmaximal, characteristic
5#22586238C211yesmaximal, characteristic
6#22586197C211yesmaximal, characteristic
7C71×#3113C211yesmaximal, characteristic
8#17986130—22no—
9#17986130—21no—
10Q8×C71⋊C8—21no—
11#17986056—21no—
12#17986056C2211yescharacteristic
13#17986052—21no—
14#17986052C2211yescharacteristic
15C2×#14556052—21no—
16C2×#14556039C2211yescharacteristic
17C4×C4⋊Dic71C2211yescharacteristic
18#17985761—21no—
19#17985741—22no—
20#17985736—21no—
21#17985736—21no—
22#533×C71C2211yescharacteristic
23C2×C4×Q8×C71C2211yescharacteristic
24#483×C71—21no—
25#475×C71C411yescharacteristic, complemented
26#475×C71C411yescharacteristic, complemented
27#462×C71C411yescharacteristic, complemented
28#462×C71C411yescharacteristic, complemented
29#451×C71—21no—
30#441×C71—21no—
31#441×C71—21no—
32#432×C71C2211yescharacteristic
33C22×C71⋊C8—21no—
34C2×C4⋊Dic71—21no—
35C2×C4×Dic71—21no—
36C4⋊(C71⋊C8)—22no—
37C4⋊(C71⋊C8)—22no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40#14556052—24no—
41C22⋊(C71⋊C8)—21no—
42C22⋊(C71⋊C8)D411yescharacteristic
43C22⋊(C71⋊C8)D411yescharacteristic
44#14556039—22no—
45#14556039—21no—
46#14556039—21no—
47#14556039D412yes—
48C4×C71⋊C8—22no—
49C4×C71⋊C8—21no—
50C71⋊#95—21no—
51C22×Q8×C71D411yescharacteristic
52C2×Q16×C71—21no—
53C2×Q16×C71—21no—
54C4⋊Q8×C71C2×C411yescharacteristic
55C4⋊Q8×C71C2×C411yescharacteristic
56#125×C71—21no—
57C22⋊Q8×C71—21no—
58C22⋊Q8×C71—21no—
59C4×Q8×C71—21no—
60C4×Q8×C71—21no—
61C4×Q8×C71—21no—
62C4×Q8×C71—21no—
63C4×Q8×C71C2×C411yescharacteristic
64C4×Q8×C71C2×C411yescharacteristic
65C4×Q8×C71D411yescharacteristic
66C4×Q8×C71D411yescharacteristic
67#117×C71—21no—
68C2×C4⋊C4×C71—21no—
69C2×C4⋊C4×C71D411yescharacteristic
70C2×C42×C71—21no—
71C2×C42×C71C2311yescharacteristic
72#106×C71—21no—
73#106×C71—21no—
74C4⋊C8×C71C2×C411yescharacteristic
75C4⋊C8×C71C2×C411yescharacteristic
76#103×C71—21no—
77#103×C71—21no—
78#103×C71—21no—
79#103×C71—21no—
80C22⋊C8×C71—21no—
81#97×C71—21no—
82C22×Dic71—41no—
83C2×C71⋊C8—22no—
84C2×C71⋊C8—21no—
85C2×C71⋊C8—21no—
86C2×C71⋊C8—22no—
87C2×C71⋊C8—41no—
88C2×C71⋊C8—41no—
89C4⋊Dic71—22no—
90C4⋊Dic71—41no—
91C4×Dic71—81no—
92C2×Q8×C71—21no—
93C2×Q8×C71—22no—
94C2×Q8×C71—21no—
95C2×Q8×C71—21no—
96C2×Q8×C71—41no—
97C2×Q8×C71C22⋊C411yescharacteristic
98C2×Q8×C71C22⋊C411yescharacteristic
99C22×C4×C71—21no—
100C22×C4×C71—21no—
101C22×C4×C71C2×D411yescharacteristic
102C22×C4×C71C2×D411yescharacteristic
103C22×C4×C71#4111yescharacteristic
104Q16×C71—42no—
105Q16×C71—42no—
106C2×C8×C71—21no—
107C2×C8×C71—21no—
108C4⋊C4×C71—21no—
109C4⋊C4×C71—21no—
110C4⋊C4×C71—21no—
111C4⋊C4×C71—21no—
112C4⋊C4×C71—21no—
113C4⋊C4×C71—21no—
114C4⋊C4×C71—21no—
115C4⋊C4×C71—21no—
116C4⋊C4×C71—21no—
117C4⋊C4×C71—41no—
118C4⋊C4×C71—41no—
119C4⋊C4×C71C22⋊C411yescharacteristic
120C4⋊C4×C71C22⋊C411yescharacteristic
121C22⋊C4×C71—41no—
122C42×C71—21no—
123C42×C71—21no—
124C42×C71—21no—
125C42×C71—21no—
126C42×C71—41no—
127C42×C71C22×C411yescharacteristic
128C42×C71C2×D411yescharacteristic
129C42×C71C2×D411yescharacteristic
130C42×C71#4111yescharacteristic
131C23×C71#12111yescharacteristic
132Q8×C71—21no—
133Q8×C71—21no—
134Q8×C71—21no—
135Q8×C71—42no—
136Q8×C71—41no—
137Q8×C71—41no—
138Q8×C71—81no—
139Q8×C71#10412yes—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—41no—
149C2×C4×C71—42no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—41no—
154C2×C4×C71C2×C22⋊C411yescharacteristic
155C2×C4×C71C4×D411yescharacteristic
156C2×C4×C71C4×D411yescharacteristic
157C2×C4×C71#12011yescharacteristic
158C2×C4×C71#12111yescharacteristic
159C2×C4×C71#12311yescharacteristic
160C2×Dic71—42no—
161C2×Dic71—81no—
162C2×Dic71—81no—
163C8×C71—41no—
164C8×C71—41no—
165C71⋊C8—41no—
166C71⋊C8—41no—
167C71⋊C8—42no—
168C71⋊C8—42no—
169C22×C71—21no—
170C22×C71—21no—
171C22×C71#38611yescharacteristic
172C22×C71#47111yescharacteristic
173C22×C71#47311yescharacteristic
174C4×C71—21no—
175C4×C71—22no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—81no—
184C4×C71—81no—
185C4×C71C2×#10411yescharacteristic
186C4×C71#42111yescharacteristic
187Dic71—82no—
188#9166—711nomaximal, Sylow
189C2×C71—21no—
190C2×C71—21no—
191C2×C71#187111yescharacteristic
192C2×C71#190511yescharacteristic
193C2×C71#190811yescharacteristic
194#3113D7111yescharacteristic
195#2984—711no—
196#1763—711no—
197#1531—711no—
198#1503—711no—
199#1302—711no—
200#1285—711no—
201C71#916611yescharacteristic, complemented, Sylow
202#533C2×D7111yescharacteristic
203C2×C4×Q8C2×D7111yescharacteristic
204#483—21no—
205#475Dic7111yescharacteristic, complemented
206#475Dic7111yescharacteristic, complemented
207#462Dic7111yescharacteristic, complemented
208#462Dic7111yescharacteristic, complemented
209#451—21no—
210#446—1422no—
211#446—1421no—
212C8×Q8—1421no—
213#441—21no—
214#441—21no—
215#432—711no—
216#432—1421no—
217#432C2×D7111yescharacteristic
218#431—711no—
219#431—1421no—
220C2×#103—1421no—
221C2×#97—711no—
222C4×C4⋊C4—711no—
223#340—1421no—
224#330—1422no—
225#328—1421no—
226#328—1421no—
227C22×Q8C22⋊D7111yescharacteristic
228C2×Q16—21no—
229C2×Q16—21no—
230C22×C8—1421no—
231C4⋊Q8C2×Dic7111yescharacteristic
232C4⋊Q8C2×Dic7111yescharacteristic
233#125—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C2×Dic7111yescharacteristic
241C4×Q8C2×Dic7111yescharacteristic
242C4×Q8C22⋊D7111yescharacteristic
243C4×Q8C22⋊D7111yescharacteristic
244#117—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—1421no—
247C2×C4⋊C4C22⋊D7111yescharacteristic
248C2×C42—21no—
249C2×C42—1421no—
250C2×C42C22×D7111yescharacteristic
251#106—21no—
252#106—21no—
253C4⋊C8—1422no—
254C4⋊C8—1422no—
255C4⋊C8—1421no—
256C4⋊C8—1421no—
257C4⋊C8C2×Dic7111yescharacteristic
258C4⋊C8C2×Dic7111yescharacteristic
259#103—21no—
260#103—21no—
261#103—21no—
262#103—21no—
263#103—1424no—
264C22⋊C8—21no—
265C22⋊C8—711no—
266C22⋊C8—711no—
267C22⋊C8—1421no—
268#97—21no—
269#97—712no—
270#97—1422no—
271#97—1421no—
272#97—1421no—
273C4×C8—1422no—
274C4×C8—1421no—
275#95—1421no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8C22⋊Dic7111yescharacteristic
282C2×Q8C22⋊Dic7111yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—2841no—
286C22×C4D4×D7111yescharacteristic
287C22×C4C2×C22⋊D7111yescharacteristic
288C22×C4#1176586411yescharacteristic
289Q16—42no—
290Q16—42no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—2841no—
294C2×C8—2841no—
295C2×C8—1422no—
296C2×C8—1421no—
297C2×C8—1422no—
298C2×C8—1421no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—2841no—
311C4⋊C4—1422no—
312C4⋊C4C22⋊Dic7111yescharacteristic
313C4⋊C4C22⋊Dic7111yescharacteristic
314C22⋊C4—41no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—5681no—
321C42C22×Dic7111yescharacteristic
322C42D4×D7111yescharacteristic
323C42C2×C22⋊D7111yescharacteristic
324C42#1176586411yescharacteristic
325C23#1455610011yescharacteristic
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—41no—
331Q8—41no—
332Q8—81no—
333Q8#1455605512yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—2842no—
349C2×C4—5681no—
350C2×C4—5681no—
351C2×C4C2×C22⋊Dic7111yescharacteristic
352C2×C4D4×Dic7111yescharacteristic
353C2×C4D4×Dic7111yescharacteristic
354C2×C4#1455609611yescharacteristic
355C2×C4#1455610011yescharacteristic
356C2×C4#1455610911yescharacteristic
357C8—41no—
358C8—41no—
359C8—2842no—
360C8—2841no—
361C8—2842no—
362C8—2841no—
363C22—21no—
364C22—21no—
365C22#1798587911yescharacteristic
366C22#1798626411yescharacteristic
367C22#1798627211yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—5682no—
380C4C2×#1455605511yescharacteristic
381C4#1798601611yescharacteristic
382C2—21no—
383C2—21no—
384C2#2258864311yescharacteristic
385C2#2258883611yescharacteristic
386C2#2258885711yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3113C2
socleC22×C71#386
prime radicalC23×C71#121
cosocleC2×C42×C71C23
hypercenter#3113D71
hypocenterC71#9166
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
25. #475×C71C4semidirect
26. #475×C71C4semidirect
27. #462×C71C4semidirect
28. #462×C71C4semidirect
201. C71#9166semidirect
205. #475Dic71semidirect
206. #475Dic71semidirect
207. #462Dic71semidirect
208. #462Dic71semidirect

Maximal subgroups

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

subgroupindex
#225974292
#225881702
#225869552
#225868392
#225862382
#225861972
C71×#31132
#916671