Finite Group Databasebeta

← #31888950

Subgroups — #31888950

Order 18176.

802 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986556C211yesmaximal, characteristic
2#22588320C211yesmaximal, characteristic
3C2×#17985747C212yesmaximal, complemented
4C2×#17985746C212yesmaximal, complemented
5C71×#1797C211yesmaximal, characteristic
6#17986556—24no—
7#17986556—24no—
8C2×#14556128C2211yescharacteristic
9C2×#14556128C2211yescharacteristic
10C2×#14556120—21no—
11C2×#14556107—21no—
12C2×#14556107C412yes—
13C2×C4×Dic142—21no—
14C2×C4×Dic142C412yes—
15C2×#14556087—21no—
16C2×C4⋊(C71⋊C8)C2212yes—
17C2×C22⋊(C71⋊C8)—21no—
18#17985754—21no—
19#17985747—24no—
20#17985746—24no—
21C22×C42×C71C2211yescharacteristic
22C2×C4⋊C8×C71C2212yes—
23C2×C22⋊C8×C71—21no—
24#336×C71—21no—
25C22×Dic142—21no—
26C22×Dic142C2×C412yes—
27C22×C71⋊C8—22no—
28#14556128—22no—
29#14556128—22no—
30#14556128D414yes—
31#14556128D414yes—
32#14556120—28no—
33#14556107—28no—
34#14556107—24no—
35#14556107—24no—
36C4×Dic142—28no—
37C4×Dic142—24no—
38C4×Dic142—24no—
39#14556087—28no—
40C2×C4⋊Dic71—21no—
41C2×C4⋊Dic71—21no—
42C2×C4⋊Dic71—21no—
43C2×C4⋊Dic71C2×C412yes—
44C2×Dic71⋊C4—22no—
45C2×Dic71⋊C4—22no—
46C2×C22⋊Dic71—22no—
47C2×C4×Dic71—22no—
48C4⋊(C71⋊C8)—22no—
49C4⋊(C71⋊C8)—22no—
50C22⋊(C71⋊C8)—42no—
51C23×C4×C71C2×C411yescharacteristic
52C23×C4×C71D411yescharacteristic
53C23×C4×C71D411yescharacteristic
54C22×C8×C71—22no—
55C2×C42×C71—22no—
56C2×C42×C71—22no—
57C2×C42×C71C2×C411yescharacteristic
58C2×C42×C71D411yescharacteristic
59C2×C42×C71D411yescharacteristic
60C2×C42×C71C2311yescharacteristic
61C4⋊C8×C71—22no—
62C4⋊C8×C71—22no—
63C22⋊C8×C71—42no—
64C2×Dic142—22no—
65C2×Dic142—24no—
66C2×Dic142—42no—
67C2×Dic142—41no—
68C2×Dic142—41no—
69C2×Dic142—42no—
70C2×Dic142—42no—
71C2×Dic142C22⋊C414yes—
72C22×Dic71—22no—
73C22×Dic71—22no—
74C2×C71⋊C8—42no—
75C2×C71⋊C8—42no—
76C2×C71⋊C8—42no—
77C4⋊Dic71—22no—
78C4⋊Dic71—24no—
79C4⋊Dic71—24no—
80C4⋊Dic71—24no—
81C4⋊Dic71C22⋊C414yes—
82Dic71⋊C4—28no—
83Dic71⋊C4—28no—
84C22⋊Dic71—28no—
85C4×Dic71—28no—
86C24×C71C22⋊C411yescharacteristic
87C22×C4×C71—22no—
88C22×C4×C71—22no—
89C22×C4×C71—21no—
90C22×C4×C71—21no—
91C22×C4×C71—22no—
92C22×C4×C71—22no—
93C22×C4×C71—21no—
94C22×C4×C71—21no—
95C22×C4×C71—41no—
96C22×C4×C71—41no—
97C22×C4×C71C22⋊C411yescharacteristic
98C22×C4×C71C22⋊C411yescharacteristic
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71SD1612yes—
101C22×C4×C71Q1612yes—
102C22×C4×C71C22×C411yescharacteristic
103C22×C4×C71C2×D411yescharacteristic
104C22×C4×C71C2×D411yescharacteristic
105C2×C8×C71—42no—
106C2×C8×C71—42no—
107C2×C8×C71—42no—
108C42×C71—21no—
109C42×C71—21no—
110C42×C71—22no—
111C42×C71—22no—
112C42×C71C2×D412yes—
113C42×C71C2×D412yes—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—21no—
117C23×C71—21no—
118C23×C71—21no—
119C23×C71—21no—
120C23×C71#10312yes—
121C23×C71C2×C22⋊C411yescharacteristic
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—24no—
127C2×C4×C71—24no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—42no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71#10312yes—
139C2×C4×C71C2×C22⋊C411yescharacteristic
140C2×C4×C71C2×C22⋊C411yescharacteristic
141C2×C4×C71#12012yes—
142C2×C4×C71#12012yes—
143C2×C4×C71C2×SD1611yescharacteristic
144C2×C4×C71C2×Q1611yescharacteristic
145C2×C4×C71#13512yes—
146C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C71#13712yes—
148C2×C4×C71#13711yescharacteristic
149C2×Dic71—24no—
150C2×Dic71—24no—
151C2×Dic71—42no—
152C2×Dic71—42no—
153C2×Dic71—42no—
154C2×Dic71—42no—
155Dic142—44no—
156Dic142—44no—
157Dic142—44no—
158Dic142—44no—
159C8×C71—82no—
160C71⋊C8—82no—
161C22×C71—22no—
162C22×C71—22no—
163C22×C71—22no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71#37911yescharacteristic
169C22×C71#41311yescharacteristic
170C22×C71#41311yescharacteristic
171C22×C71C2×#10311yescharacteristic
172C22×C71#41611yescharacteristic
173C22×C71#41711yescharacteristic
174C22×C71#41911yescharacteristic
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—22no—
178C4×C71—41no—
179C4×C71—41no—
180C4×C71#44812yes—
181C4×C71#44912yes—
182C4×C71#45112yes—
183C4×C71#45212yes—
184Dic71—44no—
185Dic71—44no—
186#8343—711nomaximal, Sylow
187C2×C71—42no—
188C2×C71#154211yescharacteristic
189C2×C71#155011yescharacteristic
190C2×C71#155411yescharacteristic
191C2×C71#155911yescharacteristic
192C2×C71#178511yescharacteristic
193C2×C71#178711yescharacteristic
194C2×C71#178811yescharacteristic
195C2×#533—711no—
196#1797—711no—
197#1797D7111yescharacteristic
198C2×#333—712no—
199C2×#332—712no—
200C71#834311yescharacteristic, complemented, Sylow
201#533—1424no—
202#533—1424no—
203C2×C4⋊Q8—711no—
204C2×C4⋊Q8—711no—
205C2×#125—1421no—
206C2×C22⋊Q8—712no—
207C2×C22⋊Q8—1421no—
208C2×C4×Q8—712no—
209C2×C4×Q8—1421no—
210C2×#117—1421no—
211C22×C42C2×D7111yescharacteristic
212C2×C4⋊C8—712no—
213C2×C4⋊C8C2×D7112yes—
214C2×C22⋊C8—21no—
215C2×C22⋊C8—1421no—
216#336—21no—
217#336—1421no—
218#333—1424no—
219#332—1424no—
220C22×Q8—712no—
221C22×Q8—1421no—
222C23×C4C4×D7111yescharacteristic
223C23×C4D28411yescharacteristic
224C23×C4C22⋊D7111yescharacteristic
225C22×C8—22no—
226C22×C8—1422no—
227C4⋊Q8—714no—
228C4⋊Q8—714no—
229C4⋊Q8—1422no—
230C4⋊Q8—1422no—
231#125—1428no—
232C22⋊Q8—1424no—
233C22⋊Q8—1424no—
234C22⋊Q8—1428no—
235C4×Q8—1424no—
236C4×Q8—1424no—
237C4×Q8—1428no—
238#117—1428no—
239C2×C4⋊C4—712no—
240C2×C4⋊C4—1422no—
241C2×C4⋊C4—1422no—
242C2×C4⋊C4—1421no—
243C2×C4⋊C4—1421no—
244C2×C4⋊C4—1421no—
245C2×C22⋊C4—1422no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—1422no—
249C2×C42C4×D7111yescharacteristic
250C2×C42D28411yescharacteristic
251C2×C42C22⋊D7111yescharacteristic
252C2×C42C22×D7111yescharacteristic
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—1422no—
256C4⋊C8—1422no—
257C22⋊C8—42no—
258C22⋊C8—2842no—
259C24(C2×D71)⋊C411yescharacteristic
260C2×Q8—2842no—
261C2×Q8—2842no—
262C2×Q8—2842no—
263C2×Q8—2841no—
264C2×Q8—2841no—
265C2×Q8—714no—
266C2×Q8—1422no—
267C2×Q8—1424no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—1422no—
279C22×C4—1422no—
280C22×C4(C2×D71)⋊C411yescharacteristic
281C22×C4(C2×D71)⋊C411yescharacteristic
282C22×C4(C2×D71)⋊C411yescharacteristic
283C22×C4#1176585412yes—
284C22×C4Dic28412yes—
285C22×C4C2×C4×D7111yescharacteristic
286C22×C4C2×C22⋊D7111yescharacteristic
287C22×C4C2×D28411yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—2842no—
292C2×C8—2842no—
293C2×C8—2842no—
294C4⋊C4—714no—
295C4⋊C4—1428no—
296C4⋊C4—1428no—
297C4⋊C4—1422no—
298C4⋊C4—1424no—
299C4⋊C4—1424no—
300C4⋊C4—1424no—
301C22⋊C4—1428no—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—1428no—
307C42D4×D7112yes—
308C42D4×D7112yes—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23#1455605312yes—
316C23C2×(C2×D71)⋊C411yescharacteristic
317Q8—2844no—
318Q8—2844no—
319Q8—2844no—
320Q8—2844no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—2842no—
338C2×C4—2842no—
339C2×C4—2842no—
340C2×C4—2842no—
341C2×C4—1424no—
342C2×C4—1424no—
343C2×C4#1455605312yes—
344C2×C4C22⋊C4×D7111yescharacteristic
345C2×C4C22⋊C4×D7111yescharacteristic
346C2×C4#1455609612yes—
347C2×C4#1455609812yes—
348C2×C4C2×#1176585411yescharacteristic
349C2×C4C2×Dic28411yescharacteristic
350C2×C4#1455614812yes—
351C2×C4#1455615411yescharacteristic
352C2×C4#1455615911yescharacteristic
353C2×C4#1455616112yes—
354C8—82no—
355C8—5682no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22#1798585411yescharacteristic
364C22#1798598111yescharacteristic
365C22#1798598111yescharacteristic
366C22C2×#1455605311yescharacteristic
367C22#1798599311yescharacteristic
368C22#1798599811yescharacteristic
369C22#1798600911yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—2844no—
376C4—2844no—
377C4#1798614012yes—
378C4#1798614212yes—
379C4#1798615712yes—
380C4#1798615912yes—
381C2—42no—
382C2#2258699511yescharacteristic
383C2#2258702811yescharacteristic
384C2#2258705111yescharacteristic
385C2#2258706811yescharacteristic
386C2#2258826611yescharacteristic
387C2#2258827611yescharacteristic
388C2#2258827811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1797C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1797D71
hypocenterC71#8343
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. C2×#17985747C2semidirect
4. C2×#17985746C2semidirect
200. C71#8343semidirect

Maximal subgroups

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

subgroupindex
C2×#179865562
#225883202
C2×#179857472
C2×#179857462
C71×#17972
#834371