Finite Group Databasebeta

← C2×#21437078 #29469591

Subgroups — C2×#21437078 #29469591

Order 15104.

2238 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838131C211yesmaximal, characteristic, complemented
2C2×#16838074C211yesmaximal, characteristic
3#21437078C2116yesmaximal, direct product factor
4C2×#16837775C213yesmaximal, complemented
5C2×#16837603C213yesmaximal, complemented
6C2×#16837608C213yesmaximal, complemented
7C2×C4×#11745660C213yesmaximal, complemented
8C2×C8×Q8×C59C211yesmaximal, characteristic, complemented
9C22×#11745689—21no—
10C22×#11745660C2213yes—
11C22×#11745660C2216yes—
12#16838131—24no—
13#16838131C2218yescomplemented
14#16838074—24no—
15#16838074C2218yes—
16C2×#14273727C2213yes—
17C2×#14273724C2213yes—
18C2×#14273718—23no—
19C2×#14273702—23no—
20C2×Q8×Dic59—21no—
21C2×C4×Dic118C2213yes—
22C2×C4×D236C2213yes—
23#16837775C22124yescomplemented
24#16837608C22124yescomplemented
25#16837603C22124yescomplemented
26C4×#11745660C22124yescomplemented
27C2×#14273659C2211yescharacteristic
28C2×#14273652C2213yes—
29C2×#14273649C2213yes—
30C2×C4×Q8×C59C2211yescharacteristic
31C8×Q8×C59C2218yescomplemented
32C2×C4⋊C8×C59C2213yes—
33C2×C4×C8×C59C2213yes—
34C2×#11745689—28no—
35C2×#11745689—48no—
36C22×Dic118C2313yes—
37C22×D236—23no—
38C22×D236—23no—
39C22×D236C2313yes—
40C22×C4×D59—23no—
41C2×#11745660—212no—
42C2×#11745660—412no—
43C2×#11745660C23112yes—
44C2×#11745660C23124yes—
45#14273727—26no—
46#14273727C23112yes—
47#14273724—26no—
48#14273724C23112yes—
49#14273718—224no—
50#14273702—224no—
51Q8×Dic59—28no—
52C4×Dic118—26no—
53C4×Dic118C23112yes—
54C4×D236—26no—
55C4×D236C23112yes—
56C2×C4⋊Dic59—23no—
57C2×C4⋊Dic59—23no—
58C2×C4⋊Dic59C2311yescharacteristic
59C2×C4⋊Dic59C2311yescharacteristic
60C2×Dic59⋊C4—23no—
61C2×(C2×D59)⋊C4—23no—
62C2×C4×Dic59—23no—
63#14273659C2314yes—
64#14273652C23112yes—
65#14273649C23112yes—
66C22×Q8×C59D411yescharacteristic
67C22×C8×C59C2313yes—
68C22×C8×C59C2311yescharacteristic
69C4×Q8×C59D414yes—
70C4×Q8×C59C2314yes—
71C2×C4⋊C4×C59D413yes—
72C2×C42×C59C2313yes—
73C4⋊C8×C59C23112yes—
74C4×C8×C59C23112yes—
75C23×D59—23no—
76C2×Q8×C59SD1618yescomplemented
77C2×Q8×C59C2×D414yes—
78C22×C4×C59C2×D413yes—
79C22×C4×C59C2×D413yes—
80C22×C4×C59C2411yescharacteristic
81C22×Dic59—21no—
82C22×Dic59—23no—
83C22×Dic59—21no—
84#11745689—432no—
85C2×D236—26no—
86C2×D236—26no—
87C2×D236—23no—
88C2×D236—212no—
89C2×D236—212no—
90C2×D236—412no—
91C2×D236—86no—
92C2×D236C2416yes—
93C2×C4×D59—212no—
94C2×C4×D59—412no—
95C2×Dic118—26no—
96C2×Dic118—26no—
97C2×Dic118—23no—
98C2×Dic118C2416yes—
99C2×C8×C59—26no—
100C2×C8×C59#4114yescomplemented
101C2×C8×C59C2416yes—
102C2×C8×C59C2412yes—
103C4⋊C4×C59C2×D4112yes—
104C42×C59C2×D416yes—
105C42×C59C2416yes—
106(C2×D59)⋊C4—212no—
107C4⋊Dic59—212no—
108C4⋊Dic59—212no—
109C4⋊Dic59—21no—
110C4⋊Dic59—21no—
111C4⋊Dic59C2412yes—
112C4⋊Dic59C2412yes—
113Dic59⋊C4—212no—
114C4×Dic59—212no—
115#11745660—224no—
116#11745660—424no—
117C23×C59C22×D411yescharacteristic
118C22×D59—26no—
119C22×D59—46no—
120C22×D59—46no—
121C22×D59—46no—
122Q8×C59C2×SD16116yescomplemented
123C2×C4×C59—26no—
124C2×C4×C59C2×SD16112yes—
125C2×C4×C59C22×D416yes—
126C2×C4×C59C22×D411yescharacteristic
127C2×C4×C59C22×D416yes—
128C2×C4×C59C2×#4112yes—
129C2×C4×C59H(5,𝔽2)12yes—
130C2×C4×C59C2511yescharacteristic
131C2×Dic59—22no—
132C2×Dic59—26no—
133C2×Dic59—22no—
134C2×Dic59—46no—
135C2×Dic59—42no—
136C2×Dic59—42no—
137D236—26no—
138D236—26no—
139D236—46no—
140D236—424no—
141D236—812no—
142C4×D59—424no—
143Dic118—26no—
144Dic118—26no—
145Dic118—46no—
146C8×C59—26no—
147C8×C59C2×#4114yescomplemented
148C2×#3392—591nomaximal, Sylow
149C22×C59#55012yes—
150C22×C59C22×SD1612yes—
151C22×C59#57712yes—
152C22×C59C23×D411yescharacteristic
153C2×D59—46no—
154C2×D59—46no—
155C2×D59—46no—
156C2×D59—812no—
157C4×C59—26no—
158C4×C59#55012yes—
159C4×C59C22×SD16112yes—
160C4×C59C22×#4111yescharacteristic
161C4×C59C2×H(5,𝔽2)11yescharacteristic
162Dic59—46no—
163Dic59—42no—
164Dic59—42no—
165C2×#557—591no—
166C2×#550—591no—
167#3392—5916no—
168C2×#492—593no—
169C2×#460—593no—
170C2×#461—593no—
171C2×C8×Q8D5911yescharacteristic, complemented
172C2×C4×SD16—593no—
173C2×C59#339214yescomplemented
174C2×C59C2×#55011yescharacteristic
175C2×C59C23×SD1611yescharacteristic
176C2×C59C2×#57711yescharacteristic
177D59—86no—
178C22×#41—1181no—
179C22×SD16—593no—
180C22×SD16—596no—
181#557—598no—
182#557—1184no—
183#550—598no—
184#550—1184no—
185C2×C4⋊Q8—593no—
186C2×#127—593no—
187C2×#125—1183no—
188C2×#121—1183no—
189C2×C4×Q8—593no—
190C2×C4×Q8—1181no—
191C2×C4×Q8C2×D5911yescharacteristic
192C2×C4×D4—593no—
193#492—5924no—
194#461—5924no—
195#460—5924no—
196C8×Q8C2×D5918yescomplemented
197C4×SD16—5924no—
198C2×#106—591no—
199C2×C4⋊C8C2×D5913yes—
200C2×#103—593no—
201C2×#102—593no—
202C2×C4×C8C2×D5913yes—
203C59C2×#339211yescharacteristic, complemented, Sylow
204C2×#41—1188no—
205C2×#41—2368no—
206C22×Q8—593no—
207C22×Q8D23611yescharacteristic
208C22×D4—593no—
209C22×D4—1183no—
210C22×D4—1183no—
211C23×C4—1183no—
212C2×SD16—5912no—
213C2×SD16—5924no—
214C2×SD16—11812no—
215C2×SD16—23612no—
216C22×C8C22×D5913yes—
217C22×C8C22×D5911yescharacteristic
218C4⋊Q8—5912no—
219C4⋊Q8—1186no—
220#127—5912no—
221#127—1186no—
222#125—11824no—
223#121—11824no—
224C4×Q8—5912no—
225C4×Q8—1188no—
226C4×Q8—1186no—
227C4×Q8D23614yes—
228C4×Q8C22×D5914yes—
229C4×D4—5912no—
230C4×D4—1186no—
231C2×C4⋊C4—591no—
232C2×C4⋊C4—591no—
233C2×C4⋊C4—1183no—
234C2×C4⋊C4—1183no—
235C2×C4⋊C4—1183no—
236C2×C4⋊C4D23613yes—
237C2×C22⋊C4—1183no—
238C2×C42—1183no—
239C2×C42C22×D5913yes—
240#106—594no—
241C4⋊C8C22×D59112yes—
242#103—5912no—
243#102—5912no—
244C4×C8C22×D59112yes—
245C24—1183no—
246#41—23632no—
247C2×Q8—596no—
248C2×Q8—1183no—
249C2×Q8—1186no—
250C2×Q8—1186no—
251C2×Q8#1174566018yescomplemented
252C2×Q8C2×D23614yes—
253C2×D4—596no—
254C2×D4—11812no—
255C2×D4—1183no—
256C2×D4—11812no—
257C2×D4—1186no—
258C2×D4—1186no—
259C2×D4—4726no—
260C2×D4—23612no—
261C22×C4—1181no—
262C22×C4—1183no—
263C22×C4—1181no—
264C22×C4—11812no—
265C22×C4—23612no—
266C22×C4C2×D23613yes—
267C22×C4C2×D23613yes—
268C22×C4C23×D5911yescharacteristic
269SD16—11824no—
270SD16—23624no—
271C2×C8—26no—
272C2×C8#1174568914yescomplemented
273C2×C8C23×D5916yes—
274C2×C8C23×D5912yes—
275C4⋊C4—592no—
276C4⋊C4—592no—
277C4⋊C4—1181no—
278C4⋊C4—1181no—
279C4⋊C4—11812no—
280C4⋊C4—11812no—
281C4⋊C4—11812no—
282C4⋊C4C2×D236112yes—
283C22⋊C4—11812no—
284C42—11812no—
285C42C2×D23616yes—
286C42C23×D5916yes—
287C23—1186no—
288C23—2366no—
289C23—2366no—
290C23—2366no—
291C23C22×D23611yescharacteristic
292Q8—1186no—
293Q8—1186no—
294Q8—2366no—
295Q8C2×#11745660116yescomplemented
296D4—1186no—
297D4—1186no—
298D4—47212no—
299D4—23624no—
300D4—2366no—
301C2×C4—26no—
302C2×C4—1182no—
303C2×C4—1186no—
304C2×C4—1182no—
305C2×C4—2362no—
306C2×C4—2362no—
307C2×C4—2366no—
308C2×C4—23624no—
309C2×C4C2×#11745660112yes—
310C2×C4C22×D23616yes—
311C2×C4C22×D23616yes—
312C2×C4C22×D23611yescharacteristic
313C2×C4C2×#1174568912yes—
314C2×C4#1427377312yes—
315C2×C4C24×D5911yescharacteristic
316C8—26no—
317C8C2×#1174568914yescomplemented
318C22—47212no—
319C22—2366no—
320C22—2366no—
321C22—2366no—
322C22#1683807412yes—
323C22C22×#1174566012yes—
324C22#1683821412yes—
325C22C23×D23611yescharacteristic
326C4—26no—
327C4—2362no—
328C4—2366no—
329C4—2362no—
330C4#1683807412yes—
331C4C22×#11745660112yes—
332C4C22×#1174568911yescharacteristic
333C4C2×#1427377311yescharacteristic
334C2—4726no—
335C2#2143707814yesdirect product factor
336C2C2×#1683807411yescharacteristic
337C2C23×#1174566011yescharacteristic
338C2C2×#1683821411yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC2×C8×Q8×C59C2
socleC23×C59C22×D4
prime radicalC2×#16838074C2
hypercenterC2×C8×Q8D59
hypocenterC59C2×#3392
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16838131C2semidirect
3. #21437078C2direct (N × H)
4. C2×#16837775C2semidirect
5. C2×#16837603C2semidirect
6. C2×#16837608C2semidirect
7. C2×C4×#11745660C2semidirect
8. C2×C8×Q8×C59C2semidirect
13. #16838131C22semidirect
23. #16837775C22semidirect
24. #16837608C22semidirect
25. #16837603C22semidirect
26. C4×#11745660C22semidirect
31. C8×Q8×C59C22semidirect
76. C2×Q8×C59SD16semidirect
100. C2×C8×C59#41semidirect
122. Q8×C59C2×SD16semidirect
147. C8×C59C2×#41semidirect
171. C2×C8×Q8D59semidirect
173. C2×C59#3392semidirect
196. C8×Q8C2×D59semidirect
203. C59C2×#3392semidirect
251. C2×Q8#11745660semidirect
272. C2×C8#11745689semidirect
295. Q8C2×#11745660semidirect
317. C8C2×#11745689semidirect
335. C2#21437078direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168381312
C2×#168380742
#214370782
C2×#168377752
C2×#168376032
C2×#168376082
C2×C4×#117456602
C2×C8×Q8×C592
C2×#339259