Finite Group Databasebeta

← C2×#22596521 #32203824

Subgroups — C2×#22596521 #32203824

Order 18176.

1718 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#22596521C2116yesmaximal, direct product factor
2C2×C71⋊#401C212yesmaximal
3C2×#17985941C212yesmaximal
4C2×#17985938C212yesmaximal
5C2×#17985931C212yesmaximal
6C2×#17985929C211yesmaximal, characteristic
7C2×#17985920C212yesmaximal
8C2×#17985904C211yesmaximal, characteristic
9C2×#17985881C212yesmaximal
10C2×#400×C71C211yesmaximal, characteristic
11C23×Dic142C2211yescharacteristic
12C22×C4⋊Dic71C2212yes—
13C22×C4⋊Dic71C2212yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C22×Dic71⋊C4C2212yes—
16C22×Dic71⋊C4C2212yes—
17C22×C4×Dic71C2211yescharacteristic
18C71⋊#401C22116yes—
19#17985941C22116yes—
20#17985938C22116yes—
21#17985931C22116yes—
22#17985929C2218yes—
23#17985920C22116yes—
24#17985904C2218yes—
25#17985881C22116yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2211yescharacteristic
33C2×C71⋊#95C2212yes—
34C2×C71⋊#95C2212yes—
35C22×C4⋊C4×C71C2211yescharacteristic
36C22×C4⋊C4×C71C2211yescharacteristic
37#400×C71C2218yes—
38C2×#95×C71C2212yes—
39C2×#95×C71C2212yes—
40C2×#95×C71C2211yescharacteristic
41C22×Dic142—24no—
42C22×Dic142—42no—
43C22×Dic142—42no—
44C22×Dic142C2314yes—
45C23×Dic71C2312yes—
46C23×Dic71C2312yes—
47C23×Dic71C2312yes—
48C23×Dic71C2312yes—
49C2×C4⋊Dic71—44no—
50C2×C4⋊Dic71—42no—
51C2×C4⋊Dic71—44no—
52C2×C4⋊Dic71C2318yes—
53C2×C4⋊Dic71C2318yes—
54C2×C4⋊Dic71C2314yes—
55C2×Dic71⋊C4—28no—
56C2×Dic71⋊C4—44no—
57C2×Dic71⋊C4C2318yes—
58C2×Dic71⋊C4C2318yes—
59C2×C4×Dic71—42no—
60C2×C4×Dic71C2314yes—
61C71⋊#95C2318yes—
62C71⋊#95C2318yes—
63C71⋊#95C2318yes—
64C71⋊#95C2318yes—
65C71⋊#95C2318yes—
66C71⋊#95C2318yes—
67C71⋊#95C2318yes—
68C71⋊#95C2318yes—
69C71⋊#95C2314yes—
70C23×C4×C71C2311yescharacteristic
71C23×C4×C71C2312yes—
72C23×C4×C71C2311yescharacteristic
73C23×C4×C71C2312yes—
74C23×C4×C71C2311yescharacteristic
75C2×C4⋊C4×C71—42no—
76C2×C4⋊C4×C71—42no—
77C2×C4⋊C4×C71C2314yes—
78C2×C4⋊C4×C71C2314yes—
79#95×C71C2318yes—
80#95×C71C2318yes—
81#95×C71C2314yes—
82C2×Dic142—28no—
83C2×Dic142—44no—
84C2×Dic142—44no—
85C2×Dic142—416no—
86C22×Dic71—24no—
87C22×Dic71—24no—
88C22×Dic71—24no—
89C22×Dic71—24no—
90C22×Dic71—24no—
91C22×Dic71—24no—
92C22×Dic71—24no—
93C22×Dic71—24no—
94C22×Dic71—24no—
95C22×Dic71—24no—
96C22×Dic71—24no—
97C22×Dic71—24no—
98C22×Dic71C2414yes—
99C22×Dic71C2414yes—
100C22×Dic71C2414yes—
101C22×Dic71C2414yes—
102C4⋊Dic71—48no—
103C4⋊Dic71—44no—
104C4⋊Dic71—48no—
105Dic71⋊C4—216no—
106Dic71⋊C4—48no—
107C4×Dic71—44no—
108C24×C71C2411yescharacteristic
109C22×C4×C71—22no—
110C22×C4×C71—22no—
111C22×C4×C71—22no—
112C22×C4×C71—24no—
113C22×C4×C71—22no—
114C22×C4×C71—24no—
115C22×C4×C71—24no—
116C22×C4×C71—28no—
117C22×C4×C71—24no—
118C22×C4×C71—24no—
119C22×C4×C71—24no—
120C22×C4×C71#4114yes—
121C22×C4×C71C2414yes—
122C22×C4×C71C2414yes—
123C22×C4×C71C2412yes—
124C22×C4×C71C2412yes—
125C22×C4×C71C2412yes—
126C4⋊C4×C71—44no—
127C4⋊C4×C71—44no—
128C23×C71C2×#4112yes—
129C23×C71H(5,𝔽2)12yes—
130C23×C71H(5,𝔽2)12yes—
131C23×C712−1+414yes—
132C23×C712−1+414yes—
133C23×C71C2511yescharacteristic
134C2×C4×C71—28no—
135C2×C4×C71—22no—
136C2×C4×C71—22no—
137C2×C4×C71—22no—
138C2×C4×C71—24no—
139C2×C4×C71—22no—
140C2×C4×C71—24no—
141C2×C4×C71—24no—
142C2×C4×C71—24no—
143C2×C4×C71—28no—
144C2×C4×C71—24no—
145C2×C4×C71—24no—
146C2×C4×C71—48no—
147C2×C4×C71—48no—
148C2×C4×C71—44no—
149C2×C4×C71—44no—
150C2×C4×C71C2×#4114yes—
151C2×Dic71—24no—
152C2×Dic71—24no—
153C2×Dic71—24no—
154C2×Dic71—24no—
155C2×Dic71—24no—
156C2×Dic71—24no—
157C2×Dic71—24no—
158C2×Dic71—24no—
159C2×Dic71—24no—
160C2×Dic71—24no—
161C2×Dic71—24no—
162C2×Dic71—24no—
163C2×Dic71—48no—
164C2×Dic71—48no—
165C2×Dic71—48no—
166C2×Dic71—48no—
167Dic142—416no—
168C22×C71#53814yes—
169C22×C71#54118yes—
170C22×C71#56314yes—
171C22×C71#56318yes—
172C22×C71#56314yes—
173C22×C71C22×#4111yescharacteristic
174C22×C71C2×H(5,𝔽2)11yescharacteristic
175C22×C71C2×H(5,𝔽2)11yescharacteristic
176C22×C71C2×2−1+412yes—
177C22×C71C2×2−1+412yes—
178C4×C71—24no—
179C4×C71—44no—
180C4×C71—44no—
181C4×C71—42no—
182C4×C71—42no—
183Dic71—44no—
184Dic71—44no—
185Dic71—44no—
186Dic71—44no—
187C2×#2833—711nomaximal, Sylow
188C2×C71#283318yescomplemented
189C2×C71C2×#53811yescharacteristic
190C2×C71C2×#54112yes—
191C2×C71C2×#56311yescharacteristic
192C2×C71C2×#56312yes—
193C2×C71C2×#56311yescharacteristic
194#2833—7116no—
195C2×#401—712no—
196C2×#400—712no—
197C2×#400—712no—
198C2×#400D7111yescharacteristic
199C2×#398—712no—
200C2×#398—711no—
201C2×#395—712no—
202C2×#391—711no—
203C2×#387—712no—
204C71C2×#283311yescharacteristic, complemented, Sylow
205C23×Q8—711no—
206C22×C4⋊C4—712no—
207C22×C4⋊C4—712no—
208C22×C4⋊C4—711no—
209C22×C4⋊C4—712no—
210C22×C4⋊C4—712no—
211C22×C4⋊C4C2×D7111yescharacteristic
212C22×C4⋊C4C2×D7111yescharacteristic
213C22×C42—711no—
214#401—7116no—
215#400—7116no—
216#400—7116no—
217#400C2×D7118yes—
218#398—718no—
219#398—7116no—
220#395—7116no—
221#391—718no—
222#387—7116no—
223C2×#95—712no—
224C2×#95—712no—
225C2×#95—712no—
226C2×#95—712no—
227C2×#95—711no—
228C2×#95—712no—
229C2×#95—712no—
230C2×#95—712no—
231C2×#95—712no—
232C2×#95C2×D7112yes—
233C2×#95C2×D7112yes—
234C2×#95C2×D7111yescharacteristic
235C22×Q8—2842no—
236C22×Q8—2842no—
237C22×Q8—714no—
238C22×Q8—1424no—
239C23×C4—712no—
240C23×C4—712no—
241C23×C4—712no—
242C23×C4—712no—
243C23×C4C22×D7112yes—
244C23×C4C22×D7112yes—
245C23×C4C22×D7111yescharacteristic
246C23×C4C22×D7111yescharacteristic
247C23×C4C22×D7111yescharacteristic
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—2844no—
251C2×C4⋊C4—2842no—
252C2×C4⋊C4—2844no—
253C2×C4⋊C4—2844no—
254C2×C4⋊C4—718no—
255C2×C4⋊C4—718no—
256C2×C4⋊C4—718no—
257C2×C4⋊C4—714no—
258C2×C4⋊C4—718no—
259C2×C4⋊C4—1428no—
260C2×C4⋊C4C22×D7114yes—
261C2×C4⋊C4C22×D7114yes—
262C2×C42—2842no—
263C2×C42—714no—
264#95—718no—
265#95—718no—
266#95—718no—
267#95—718no—
268#95—718no—
269#95—718no—
270#95—718no—
271#95—718no—
272#95—714no—
273#95C22×D7118yes—
274#95C22×D7118yes—
275#95C22×D7114yes—
276C24C23×D7111yescharacteristic
277C2×Q8—28416no—
278C2×Q8—2844no—
279C2×Q8—2844no—
280C2×Q8—1428no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—28no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—714no—
293C22×C4—714no—
294C22×C4—714no—
295C22×C4—714no—
296C22×C4—1424no—
297C22×C4—1424no—
298C22×C4—1424no—
299C22×C4—1424no—
300C22×C4—1424no—
301C22×C4—1424no—
302C22×C4—1424no—
303C22×C4—1424no—
304C22×C4—1424no—
305C22×C4—1424no—
306C22×C4—1424no—
307C22×C4—1424no—
308C22×C4#1176586614yes—
309C22×C4C23×D7112yes—
310C22×C4C23×D7114yes—
311C22×C4C23×D7112yes—
312C22×C4C23×D7114yes—
313C22×C4C23×D7112yes—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—2848no—
317C4⋊C4—2844no—
318C4⋊C4—2848no—
319C4⋊C4—2848no—
320C4⋊C4—14216no—
321C42—2844no—
322C23C2×#1176586612yes—
323C23#1455617312yes—
324C23#1455617412yes—
325C23#1455617514yes—
326C23#1455617514yes—
327C23C24×D7111yescharacteristic
328Q8—28416no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—28no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—2848no—
346C2×C4—2848no—
347C2×C4—2848no—
348C2×C4—2848no—
349C2×C4—1424no—
350C2×C4—1424no—
351C2×C4—1424no—
352C2×C4—1424no—
353C2×C4—1424no—
354C2×C4—1424no—
355C2×C4—1424no—
356C2×C4—1424no—
357C2×C4—1424no—
358C2×C4—1424no—
359C2×C4—1424no—
360C2×C4—1424no—
361C2×C4C2×#1176586614yes—
362C22#1798658914yes—
363C22#1798661218yes—
364C22#1798674914yes—
365C22#1798675118yes—
366C22#1798675214yes—
367C22C22×#1176586611yescharacteristic
368C22C2×#1455617311yescharacteristic
369C22C2×#1455617411yescharacteristic
370C22C2×#1455617512yes—
371C22C2×#1455617512yes—
372C4—24no—
373C4—44no—
374C4—44no—
375C4—42no—
376C4—42no—
377C4—2844no—
378C4—2844no—
379C4—2844no—
380C4—2844no—
381C2#2259652118yesdirect product factor
382C2C2×#1798658911yescharacteristic
383C2C2×#1798661212yes—
384C2C2×#1798674911yescharacteristic
385C2C2×#1798675112yes—
386C2C2×#1798675211yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#400×C71C2
socle, prime radicalC24×C71C24
hypercenterC2×#400D71
hypocenterC71C2×#2833
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
1. #22596521C2direct (N × H)
188. C2×C71#2833semidirect
204. C71C2×#2833semidirect
381. C2#22596521direct (N × H)

Maximal subgroups

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

subgroupindex
#225965212
C2×C71⋊#4012
C2×#179859412
C2×#179859382
C2×#179859312
C2×#179859292
C2×#179859202
C2×#179859042
C2×#179858812
C2×#400×C712
C2×#283371