Finite Group Databasebeta

← #31903287

Subgroups — #31903287

Order 18176.

1338 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985859C212yesmaximal
2C2×#17985766C214yesmaximal
3C2×C4×C22⋊C4×C71C211yesmaximal, characteristic
4C22×C4⋊Dic71—22no—
5C22×C22⋊Dic71C2214yes—
6#17985859—24no—
7#17985859C418yescomplemented
8C2×C71⋊#95—22no—
9#17985766—28no—
10C24×C4×C71C2211yescharacteristic
11C22×C22⋊C4×C71C2212yes—
12C22×C42×C71C412yescomplemented
13C4×C22⋊C4×C71—22no—
14C4×C22⋊C4×C71—22no—
15C2×#95×C71C412yescomplemented
16C23×Dic71—24no—
17C2×C4⋊Dic71—28no—
18C2×C4⋊Dic71—48no—
19C2×C22⋊Dic71—28no—
20C2×C22⋊Dic71—24no—
21C2×C22⋊Dic71—28no—
22C2×C22⋊Dic71C2×C418yes—
23C71⋊#95—28no—
24C25×C71C2311yescharacteristic
25C23×C4×C71—44no—
26C23×C4×C71C2×C411yescharacteristic
27C23×C4×C71C2×C414yes—
28C23×C4×C71C2×C414yes—
29C23×C4×C71C2×C411yescharacteristic
30C23×C4×C71D412yes—
31C23×C4×C71D412yes—
32C23×C4×C71D412yes—
33C23×C4×C71Q812yes—
34C2×C22⋊C4×C71—24no—
35C2×C22⋊C4×C71—24no—
36C2×C22⋊C4×C71D414yes—
37C2×C22⋊C4×C71Q814yes—
38C2×C42×C71—22no—
39C2×C42×C71—22no—
40C2×C42×C71—44no—
41#95×C71—24no—
42C22×Dic71—28no—
43C22×Dic71—48no—
44C22×Dic71—48no—
45C22×Dic71—48no—
46C4⋊Dic71—416no—
47C22⋊Dic71—216no—
48C22⋊Dic71—44no—
49C22⋊Dic71—44no—
50C24×C71—42no—
51C24×C71—42no—
52C24×C71C22×C412yes—
53C24×C71C22×C411yescharacteristic
54C24×C71C2×D412yes—
55C24×C71C2×D411yescharacteristic
56C24×C71C2×D412yes—
57C24×C71C2×Q811yescharacteristic
58C24×C71C2×Q812yes—
59C24×C71#4114yes—
60C22×C4×C71—24no—
61C22×C4×C71—28no—
62C22×C4×C71—24no—
63C22×C4×C71—24no—
64C22×C4×C71—48no—
65C22×C4×C71—44no—
66C22×C4×C71—44no—
67C22×C4×C71—42no—
68C22×C4×C71—44no—
69C22×C4×C71—44no—
70C22×C4×C71—44no—
71C22×C4×C71C4212yes—
72C22×C4×C71C4212yes—
73C22×C4×C71C22⋊C414yes—
74C22×C4×C71C22⋊C414yes—
75C22×C4×C71C22⋊C412yes—
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C4⋊C414yes—
79C22×C4×C71C4⋊C414yes—
80C22×C4×C71C4⋊C412yes—
81C22×C4×C71C4⋊C412yes—
82C22×C4×C71C4⋊C418yescomplemented
83C22⋊C4×C71—28no—
84C22⋊C4×C71—28no—
85C42×C71—44no—
86C42×C71—44no—
87C23×C71—22no—
88C23×C71—24no—
89C23×C71—22no—
90C23×C71—24no—
91C23×C71—42no—
92C23×C71—41no—
93C23×C71—41no—
94C23×C71—42no—
95C23×C71—44no—
96C23×C71—41no—
97C23×C71—44no—
98C23×C71—41no—
99C23×C71—48no—
100C23×C71C2×C4211yescharacteristic
101C23×C71C2×C22⋊C412yes—
102C23×C71C2×C22⋊C411yescharacteristic
103C23×C71C2×C4⋊C412yes—
104C23×C71C2×C4⋊C411yescharacteristic
105C23×C71C2×C4⋊C412yes—
106C23×C71#11712yes—
107C23×C71C4×D414yes—
108C23×C71C4×Q814yes—
109C23×C71#12114yes—
110C23×C71C22⋊Q814yes—
111C23×C71#12412yes—
112C23×C71#12512yes—
113C23×C71#12711yescharacteristic
114C23×C71C4⋊Q812yes—
115C23×C71C4⋊Q811yescharacteristic
116C2×C4×C71—22no—
117C2×C4×C71—24no—
118C2×C4×C71—24no—
119C2×C4×C71—28no—
120C2×C4×C71—28no—
121C2×C4×C71—24no—
122C2×C4×C71—28no—
123C2×C4×C71—42no—
124C2×C4×C71—44no—
125C2×C4×C71—44no—
126C2×C4×C71—42no—
127C2×C4×C71—48no—
128C2×C4×C71—44no—
129C2×C4×C71—44no—
130C2×C4×C71—48no—
131C2×C4×C71#9512yes—
132C2×C4×C71#9514yes—
133C2×C4×C71#9512yes—
134C2×C4×C71#9914yes—
135C2×Dic71—48no—
136C2×Dic71—48no—
137C2×Dic71—48no—
138C2×Dic71—816no—
139C22×C71—28no—
140C22×C71—24no—
141C22×C71—24no—
142C22×C71—28no—
143C22×C71—22no—
144C22×C71—21no—
145C22×C71—22no—
146C22×C71—21no—
147C22×C71—22no—
148C22×C71—21no—
149C22×C71—22no—
150C22×C71—21no—
151C22×C71—44no—
152C22×C71—44no—
153C22×C71—42no—
154C22×C71—42no—
155C22×C71—41no—
156C22×C71—42no—
157C22×C71—41no—
158C22×C71C2×#9511yescharacteristic
159C22×C71C4×C4⋊C412yes—
160C22×C71#38012yes—
161C22×C71#38211yescharacteristic
162C22×C71#38411yescharacteristic
163C22×C71#38914yes—
164C22×C71#39012yes—
165C22×C71#39112yes—
166C22×C71C2×#9912yes—
167C22×C71#41012yes—
168C4×C71—22no—
169C4×C71—24no—
170C4×C71—48no—
171C4×C71#34214yescomplemented
172Dic71—88no—
173#10846—711nomaximal, Sylow
174C2×C71—22no—
175C2×C71—22no—
176C2×C71—21no—
177C2×C71—21no—
178C2×C71—24no—
179C2×C71—22no—
180C2×C71C4⋊#9511yescharacteristic
181C2×C71C2×#34211yescharacteristic
182C2×C71#174011yescharacteristic
183C2×C71#179914yes—
184C2×#380—712no—
185C2×C4×C22⋊C4D7111yescharacteristic
186C2×#342—714no—
187C71#1084611yescharacteristic, complemented, Sylow
188C24×C4C2×D7111yescharacteristic
189C22×C4⋊C4—1422no—
190C22×C22⋊C4—714no—
191C22×C22⋊C4C2×D7112yes—
192C22×C42Dic7112yescomplemented
193#380—718no—
194#380—1424no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C2×#95—1422no—
198C2×#95Dic7112yescomplemented
199#342—1428no—
200C25C22×D7111yescharacteristic
201C23×C4—44no—
202C23×C4—1424no—
203C23×C4Dic14212yes—
204C23×C4C4×D7114yes—
205C23×C4D28412yes—
206C23×C4C2×Dic7111yescharacteristic
207C23×C4C2×Dic7114yes—
208C23×C4C2×Dic7111yescharacteristic
209C23×C4C22⋊D7112yes—
210C23×C4C22⋊D7112yes—
211C2×C4⋊C4—2848no—
212C2×C4⋊C4—1428no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—718no—
216C2×C22⋊C4—1424no—
217C2×C22⋊C4—1428no—
218C2×C22⋊C4—1428no—
219C2×C22⋊C4Dic14214yes—
220C2×C22⋊C4D28414yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—44no—
224#95—24no—
225#95—1428no—
226C24—42no—
227C24—42no—
228C24C2×C4×D7112yes—
229C24C22×Dic7111yescharacteristic
230C24C2×C22⋊D7112yes—
231C24C2×D28411yescharacteristic
232C24C2×D28412yes—
233C24C2×Dic14211yescharacteristic
234C24C2×Dic14212yes—
235C24#1176586514yes—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—2848no—
248C22×C4—2848no—
249C22×C4—2848no—
250C22×C4—1428no—
251C22×C4C4×Dic7112yes—
252C22×C4C4×Dic7112yes—
253C22×C4(C2×D71)⋊C414yes—
254C22×C4(C2×D71)⋊C414yes—
255C22×C4C22⋊Dic7112yes—
256C22×C4C22⋊Dic7111yescharacteristic
257C22×C4C22⋊Dic7111yescharacteristic
258C22×C4Dic71⋊C414yes—
259C22×C4Dic71⋊C414yes—
260C22×C4C4⋊Dic7112yes—
261C22×C4C4⋊Dic7112yes—
262C22×C4C4⋊Dic7118yescomplemented
263C4⋊C4—28416no—
264C22⋊C4—28no—
265C22⋊C4—28no—
266C22⋊C4—2844no—
267C22⋊C4—2844no—
268C22⋊C4—14216no—
269C42—44no—
270C42—44no—
271C23—22no—
272C23—24no—
273C23—22no—
274C23—24no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—44no—
280C23—41no—
281C23—44no—
282C23—41no—
283C23—48no—
284C23C2×C4×Dic7111yescharacteristic
285C23C2×(C2×D71)⋊C412yes—
286C23C2×C22⋊Dic7111yescharacteristic
287C23C2×Dic71⋊C412yes—
288C23C2×C4⋊Dic7111yescharacteristic
289C23C2×C4⋊Dic7112yes—
290C23#1455608712yes—
291C23C4×D28414yes—
292C23C4×Dic14214yes—
293C23#1455610214yes—
294C23#1455610714yes—
295C23#1455611612yes—
296C23#1455612012yes—
297C23#1455612511yescharacteristic
298C23#1455612812yes—
299C23#1455612811yescharacteristic
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—24no—
306C2×C4—28no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—2848no—
316C2×C4—2848no—
317C2×C4—2848no—
318C2×C4—56816no—
319C2×C4C71⋊#9512yes—
320C2×C4C71⋊#9514yes—
321C2×C4C71⋊#9512yes—
322C2×C4#1455604314yes—
323C22—22no—
324C22—21no—
325C22—22no—
326C22—21no—
327C22—24no—
328C22—24no—
329C22—28no—
330C22—28no—
331C22—22no—
332C22—21no—
333C22—22no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22C2×C71⋊#9511yescharacteristic
343C22C4×C4⋊Dic7112yes—
344C22#1798585912yes—
345C22#1798586511yescharacteristic
346C22#1798586911yescharacteristic
347C22#1798589714yes—
348C22#1798590112yes—
349C22#1798590512yes—
350C22C2×#1455604312yes—
351C22#1798597312yes—
352C4—22no—
353C4—24no—
354C4—48no—
355C4—5688no—
356C4#1798576614yescomplemented
357C2—22no—
358C2—22no—
359C2—21no—
360C2—21no—
361C2—24no—
362C2—22no—
363C2#2258657011yescharacteristic
364C2C2×#1798576611yescharacteristic
365C2#2258807311yescharacteristic
366C2#2258832814yes—
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC23×C71C2×C42
FrattiniC25C22×D71
FittingC2×C4×C22⋊C4×C71C2
socleC23×C71C2×C22⋊C4
prime radical, cosocleC25×C71C23
hypercenterC2×C4×C22⋊C4D71
hypocenterC71#10846
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
7. #17985859C4semidirect
12. C22×C42×C71C4semidirect
15. C2×#95×C71C4semidirect
82. C22×C4×C71C4⋊C4semidirect
171. C4×C71#342semidirect
187. C71#10846semidirect
192. C22×C42Dic71semidirect
198. C2×#95Dic71semidirect
262. C22×C4C4⋊Dic71semidirect
356. C4#17985766semidirect

Maximal subgroups

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

subgroupindex
C2×#179858592
C2×#179857662
C2×C4×C22⋊C4×C712
#1084671