Finite Group Databasebeta

← C2×#22601999 #32905699

Subgroups — C2×#22601999 #32905699

Order 18176.

11510 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556176C211yesmaximal, characteristic, complemented
2C23×#11765866C211yesmaximal, characteristic, complemented
3#22601999C2132yesmaximal, direct product factor
4C2×#17986671C218yesmaximal, complemented
5C2×#17986661C218yesmaximal, complemented
6C2×#17986568C216yesmaximal, complemented
7C2×#17986511C216yesmaximal, complemented
8C2×#519×C71C211yesmaximal, characteristic, complemented
9C2×#14556176—216no—
10C2×#14556176C22116yescomplemented
11C22×#11765866—216no—
12C22×#11765866C2218yes—
13C22×#11765866C22116yescomplemented
14C22×#11765866C2214yescomplemented
15C22×#11765865C2216yes—
16C22×Q8×D71C2214yescomplemented
17C23×D284C2213yes—
18C23×C4×D71C2213yes—
19#17986671C221128yescomplemented
20#17986661C221128yescomplemented
21#17986568C22196yescomplemented
22C2×#14556125C22112yescomplemented
23C2×#14556116C22112yescomplemented
24#17986511C22196yescomplemented
25C2×#14556112C2214yes—
26C2×#14556108C22112yescomplemented
27C2×#14556103C22124yescomplemented
28C2×#14556103C22112yescomplemented
29C2×#14556098C22112yes—
30C2×C4×D284C22124yescomplemented
31C23×Q8×C71C2211yescharacteristic
32#519×C71C22116yescomplemented
33C2×C4×Q8×C71C2218yescomplemented
34C2×#117×C71C2216yes—
35C24×D71C2313yes—
36#14556176—2128no—
37C2×#11765866—264no—
38C2×#11765866—232no—
39C2×#11765866—2128no—
40C2×#11765866C23164yes—
41C2×#11765866C23132yescomplemented
42C2×#11765865—248no—
43C2×#11765865C23148yes—
44C22×Dic142C2316yes—
45C2×Q8×D71—232no—
46C2×Q8×D71C23132yescomplemented
47C22×D284—224no—
48C22×D284C23124yes—
49C22×D284C23112yes—
50C22×D284C2316yes—
51C22×D284C23124yes—
52C22×C22⋊D71C2314yes—
53C23×Dic71C2311yescharacteristic
54C22×C4×D71—224no—
55C22×C4×D71C23112yes—
56C22×C4×D71C23124yes—
57C22×C4×D71C23112yes—
58#14556125C23196yescomplemented
59#14556116C23196yescomplemented
60#14556112C23132yes—
61#14556108C23196yescomplemented
62#14556103C231192yescomplemented
63#14556103C23196yescomplemented
64#14556098C23196yes—
65C4×D284C231192yescomplemented
66C2×C4⋊Dic71C2318yes—
67C2×(C2×D71)⋊C4C2318yes—
68C2×(C2×D71)⋊C4C23124yes—
69C22×Q8×C71D4116yescomplemented
70C22×Q8×C71C2314yes—
71C22×Q8×C71C2318yes—
72C23×C4×C71C2313yes—
73C4×Q8×C71C23164yescomplemented
74#117×C71C23148yes—
75C2×C4⋊C4×C71C23112yes—
76C2×C22⋊C4×C71C2314yes—
77C2×C42×C71C23112yes—
78C23×D71—212no—
79C23×D71—412no—
80C23×D71C2414yes—
81C23×D71C2416yes—
82C23×D71C24112yes—
83#11765866—2256no—
84#11765866—2128no—
85#11765865—2192no—
86C2×Dic142—224no—
87C2×Dic142C24124yes—
88Q8×D71—2128no—
89C2×D284—248no—
90C2×D284—296no—
91C2×D284—224no—
92C2×D284—448no—
93C2×D284C24148yes—
94C2×D284C24124yes—
95C2×D284C24196yes—
96C2×C22⋊D71—216no—
97C2×C22⋊D71C24116yes—
98C22×Dic71—24no—
99C22×Dic71C2412yes—
100C22×Dic71C2414yes—
101C22×Dic71C2414yes—
102C2×C4×D71—248no—
103C2×C4×D71—296no—
104C2×C4×D71—248no—
105C2×C4×D71C24148yes—
106C2×C4×D71C24148yes—
107C4⋊Dic71C24132yes—
108(C2×D71)⋊C4C24132yes—
109(C2×D71)⋊C4C24196yes—
110C24×C71C2411yescharacteristic
111C2×Q8×C71C2×D4132yes—
112C2×Q8×C71C2×D4164yescomplemented
113C2×Q8×C71C24116yes—
114C22×C4×C71C2×D4124yes—
115C22×C4×C71C2418yes—
116C22×C4×C71C2416yes—
117C22×C4×C71C24112yes—
118C4⋊C4×C71C24148yes—
119C22⋊C4×C71C24116yes—
120C42×C71C24148yes—
121C23×C71—24no—
122C23×C71C22×D414yes—
123C23×C71C2512yes—
124C23×C71C2511yescharacteristic
125C22×D71—28no—
126C22×D71—224no—
127C22×D71—212no—
128C22×D71—216no—
129C22×D71—224no—
130C22×D71—424no—
131C22×D71—448no—
132C22×D71C25112yes—
133C22×D71C2518yes—
134Q8×C71C22×D4164yes—
135C2×C4×C71—216no—
136C2×C4×C71C22×D4124yes—
137C2×C4×C71C22×D4148yes—
138C2×C4×C71C25116yes—
139C2×C4×C71C25112yes—
140C22⋊D71—232no—
141C2×Dic71—28no—
142C2×Dic71—24no—
143C2×Dic71—28no—
144C2×Dic71C2518yes—
145C2×Dic71C2514yes—
146D284—296no—
147D284—248no—
148D284—496no—
149C4×D71—296no—
150C4×D71—296no—
151Dic142—248no—
152C22×C71—24no—
153C22×C71—28no—
154C22×C71C23×D414yes—
155C22×C71C23×D412yes—
156C22×C71#58514yes—
157C22×C71C2611yescharacteristic
158C2×D71—28no—
159C2×D71—212no—
160C2×D71—216no—
161C2×D71—224no—
162C2×D71—448no—
163C2×D71—416no—
164C2×D71—424no—
165C4×C71—216no—
166C4×C71C23×D4124yes—
167Dic71—28no—
168Dic71—24no—
169C2×#3487—711nomaximal, Sylow
170C2×C71—24no—
171C2×C71#348714yescomplemented
172C2×C71C24×D411yescharacteristic
173C2×C71C2×#58512yes—
174D71—48no—
175D71—412no—
176C22×2−1+4—711no—
177C23×#41—711no—
178#3487—7132no—
179C2×#550—718no—
180C2×#548—718no—
181C2×#535—716no—
182C2×#525—716no—
183C2×#519D7111yescharacteristic, complemented
184C71C2×#348711yescharacteristic, complemented, Sylow
185C2×2−1+4—7116no—
186C2×2−1+4—14216no—
187C22×#41—718no—
188C22×#41—7116no—
189C22×#41—714no—
190C22×#41—716no—
191C22×#41—14216no—
192C23×Q8—714no—
193C23×Q8C2×D7111yescharacteristic
194C23×D4—713no—
195C24×C4—713no—
196#550—71128no—
197#548—71128no—
198#535—7196no—
199C2×#127—7112no—
200C2×#124—7112no—
201#525—7196no—
202C2×#123—714no—
203C2×C22⋊Q8—7112no—
204C2×#121—7112no—
205C2×#121—7124no—
206C2×#120—7112no—
207#519C2×D71116yescomplemented
208C2×C4×Q8C2×D7118yescomplemented
209C2×C4×D4—7124no—
210C2×#117C2×D7116yes—
211C25—713no—
2122−1+4—142128no—
213C2×#41—7164no—
214C2×#41—7132no—
215C2×#41—7148no—
216C2×#41—14264no—
217C2×#41—14232no—
218C2×#41—142128no—
219C2×#41—14248no—
220C22×Q8—7132no—
221C22×Q8—716no—
222C22×Q8—14232no—
223C22×Q8D284116yescomplemented
224C22×Q8C22×D7114yes—
225C22×Q8C22×D7118yes—
226C22×D4—714no—
227C22×D4—7124no—
228C22×D4—7112no—
229C22×D4—716no—
230C22×D4—7124no—
231C22×D4—14224no—
232C23×C4—711no—
233C23×C4—7112no—
234C23×C4—7112no—
235C23×C4—7124no—
236C23×C4—14224no—
237C23×C4C22×D7113yes—
238#127—7196no—
239#124—7196no—
240#123—7132no—
241C22⋊Q8—7196no—
242#121—7196no—
243#121—71192no—
244#120—7196no—
245C4×Q8C22×D71164yescomplemented
246C4×D4—71192no—
247#117C22×D71148yes—
248C2×C4⋊C4—718no—
249C2×C4⋊C4C22×D71112yes—
250C2×C22⋊C4—718no—
251C2×C22⋊C4—7124no—
252C2×C22⋊C4C22×D7114yes—
253C2×C42C22×D71112yes—
254C24—28412no—
255C24—7112no—
256C24—716no—
257C24—714no—
258C24—14212no—
259C24C23×D7111yescharacteristic
260#41—142256no—
261#41—142128no—
262#41—142192no—
263C2×Q8—7124no—
264C2×Q8—142128no—
265C2×Q8—14224no—
266C2×Q8C2×D284132yes—
267C2×Q8C2×D284164yescomplemented
268C2×Q8C23×D71116yes—
269C2×D4—28448no—
270C2×D4—7116no—
271C2×D4—7196no—
272C2×D4—7148no—
273C2×D4—7124no—
274C2×D4—14216no—
275C2×D4—14248no—
276C2×D4—14296no—
277C2×D4—14224no—
278C22×C4—714no—
279C22×C4—714no—
280C22×C4—712no—
281C22×C4—7148no—
282C22×C4—7148no—
283C22×C4—1424no—
284C22×C4—14248no—
285C22×C4—14296no—
286C22×C4—14248no—
287C22×C4C2×D284124yes—
288C22×C4C23×D7118yes—
289C22×C4C23×D71112yes—
290C22×C4C23×D7116yes—
291C4⋊C4—7132no—
292C4⋊C4C23×D71148yes—
293C22⋊C4—7132no—
294C22⋊C4—7196no—
295C22⋊C4C23×D71116yes—
296C42C23×D71148yes—
297C23—24no—
298C23—28448no—
299C23—28424no—
300C23—7112no—
301C23—718no—
302C23—1428no—
303C23—14224no—
304C23—14212no—
305C23—14216no—
306C23—14224no—
307C23C22×D28414yes—
308C23C24×D7112yes—
309C23C24×D7111yescharacteristic
310Q8—14248no—
311Q8C22×D284164yes—
312D4—28496no—
313D4—14232no—
314D4—14296no—
315D4—14248no—
316C2×C4—216no—
317C2×C4—718no—
318C2×C4—714no—
319C2×C4—1428no—
320C2×C4—1424no—
321C2×C4—1428no—
322C2×C4—14296no—
323C2×C4—14296no—
324C2×C4C22×D284124yes—
325C2×C4C22×D284148yes—
326C2×C4C24×D71116yes—
327C2×C4C24×D71112yes—
328C22—28no—
329C22—24no—
330C22—28448no—
331C22—28424no—
332C22—28416no—
333C22—1428no—
334C22—14212no—
335C22—14216no—
336C22—14224no—
337C22C23×D28414yes—
338C22C23×D28412yes—
339C22#1798683614yes—
340C22C25×D7111yescharacteristic
341C4—216no—
342C4—1424no—
343C4—1428no—
344C4C23×D284124yes—
345C2—24no—
346C2—28412no—
347C2—2848no—
348C2#2260199914yesdirect product factor
349C2C24×D28411yescharacteristic
350C2C2×#1798683612yes—
subgroupG/H
centerC23C24×D71
commutator, cosocleC22×C71C26
FrattiniC22C25×D71
FittingC2×#519×C71C2
socleC23×C71C25
hypercenterC2×#519D71
hypocenterC71C2×#3487
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#519

Complements

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

NH ≅ G/Ntype
1. C22×#14556176C2semidirect
2. C23×#11765866C2semidirect
3. #22601999C2direct (N × H)
4. C2×#17986671C2semidirect
5. C2×#17986661C2semidirect
6. C2×#17986568C2semidirect
7. C2×#17986511C2semidirect
8. C2×#519×C71C2semidirect
10. C2×#14556176C22semidirect
13. C22×#11765866C22semidirect
14. C22×#11765866C22semidirect
16. C22×Q8×D71C22semidirect
19. #17986671C22semidirect
20. #17986661C22semidirect
21. #17986568C22semidirect
22. C2×#14556125C22semidirect
23. C2×#14556116C22semidirect
24. #17986511C22semidirect
26. C2×#14556108C22semidirect
27. C2×#14556103C22semidirect
28. C2×#14556103C22semidirect
30. C2×C4×D284C22semidirect
32. #519×C71C22semidirect
33. C2×C4×Q8×C71C22semidirect
41. C2×#11765866C23semidirect
46. C2×Q8×D71C23semidirect
58. #14556125C23semidirect
59. #14556116C23semidirect
61. #14556108C23semidirect
62. #14556103C23semidirect
63. #14556103C23semidirect
65. C4×D284C23semidirect
69. C22×Q8×C71D4semidirect
73. C4×Q8×C71C23semidirect
112. C2×Q8×C71C2×D4semidirect
171. C2×C71#3487semidirect
183. C2×#519D71semidirect
184. C71C2×#3487semidirect
207. #519C2×D71semidirect
208. C2×C4×Q8C2×D71semidirect
223. C22×Q8D284semidirect
245. C4×Q8C22×D71semidirect
267. C2×Q8C2×D284semidirect
348. C2#22601999direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561762
C23×#117658662
#226019992
C2×#179866712
C2×#179866612
C2×#179865682
C2×#179865112
C2×#519×C712
C2×#348771