Finite Group Databasebeta

← C2×#21427014 #28747995

Subgroups — C2×#21427014 #28747995

Order 15104.

3158 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic59C211yesmaximal, characteristic
2C22×Q8×Dic59C211yesmaximal, characteristic
3#21427014C2116yesmaximal, direct product factor
4C2×#16837300C213yesmaximal
5C2×#16837298C213yesmaximal
6C2×#16837267C213yesmaximal
7C2×C4×Dic59⋊C4C213yesmaximal
8C22×C4×Q8×C59C211yesmaximal, characteristic
9C2×Q8×Dic59—28no—
10C2×Q8×Dic59C4116yescomplemented
11C2×Q8×Dic59C2218yes—
12C2×Q8×Dic59C2218yes—
13C22×C4⋊Dic59C2213yes—
14C22×C4⋊Dic59C2213yes—
15C22×Dic59⋊C4C2213yes—
16C22×Dic59⋊C4C2213yes—
17C22×C4×Dic59C2213yes—
18C22×C4×Dic59C2213yes—
19#16837300C22124yes—
20#16837298C22124yes—
21#16837267C22124yes—
22C4×Dic59⋊C4C22124yes—
23C2×C59⋊#95C2211yescharacteristic
24C2×C59⋊#95C2213yes—
25C2×C59⋊#95C2216yes—
26C23×Q8×C59C2211yescharacteristic
27C2×C4×Q8×C59—28no—
28C2×C4×Q8×C59C2218yes—
29C22×C4⋊C4×C59C2213yes—
30C22×C42×C59C2213yes—
31C23×Dic59C2313yes—
32C23×Dic59C2311yescharacteristic
33C23×Dic59C2313yes—
34C23×Dic59C2311yescharacteristic
35Q8×Dic59—232no—
36Q8×Dic59C2×C4164yescomplemented
37C2×C4⋊Dic59—212no—
38C2×C4⋊Dic59C2×C4124yes—
39C2×C4⋊Dic59C23112yes—
40C2×C4⋊Dic59C23112yes—
41C2×Dic59⋊C4—212no—
42C2×Dic59⋊C4—212no—
43C2×Dic59⋊C4C23112yes—
44C2×Dic59⋊C4C23112yes—
45C2×C4×Dic59—212no—
46C2×C4×Dic59C2×C4124yes—
47C2×C4×Dic59C23112yes—
48C2×C4×Dic59C23112yes—
49C59⋊#95C2314yes—
50C59⋊#95C23112yes—
51C59⋊#95C23124yes—
52C22×Q8×C59C2×C418yes—
53C22×Q8×C59D418yes—
54C22×Q8×C59Q818yes—
55C22×Q8×C59C2314yes—
56C23×C4×C59C2313yes—
57C23×C4×C59C2311yescharacteristic
58C23×C4×C59C2313yes—
59C4×Q8×C59—232no—
60C2×C4⋊C4×C59—212no—
61C2×C4⋊C4×C59C23112yes—
62C2×C42×C59—212no—
63C2×C42×C59C23112yes—
64C24×C59C2411yescharacteristic
65C2×Q8×C59C4⋊C4164yescomplemented
66C2×Q8×C59C22×C4116yes—
67C2×Q8×C59C2×D4116yes—
68C2×Q8×C59C2×Q8116yes—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—26no—
72C22×C4×C59—26no—
73C22×C4×C59—26no—
74C22×C4×C59C22×C4112yes—
75C22×C4×C59C2×D4112yes—
76C22×C4×C59C2×Q8112yes—
77C22×C4×C59#4114yes—
78C22×C4×C59C2416yes—
79C22×C4×C59C2412yes—
80C22×C4×C59C2416yes—
81C22×Dic59—22no—
82C22×Dic59—26no—
83C22×Dic59—26no—
84C22×Dic59—22no—
85C22×Dic59—22no—
86C22×Dic59—22no—
87C22×Dic59—26no—
88C22×Dic59—26no—
89C22×Dic59—26no—
90C22×Dic59C22×C414yes—
91C22×Dic59C22×C4112yes—
92C22×Dic59#4114yes—
93C22×Dic59C2416yes—
94C22×Dic59C2412yes—
95C22×Dic59C2416yes—
96C22×Dic59C2412yes—
97C4⋊C4×C59—224no—
98C42×C59—224no—
99C4⋊Dic59—224no—
100C4⋊Dic59C22×C4148yes—
101Dic59⋊C4—224no—
102Dic59⋊C4—224no—
103C4×Dic59—224no—
104C4×Dic59C22×C4148yes—
105C23×C59C23×C412yes—
106C23×C59C22×D412yes—
107C23×C59C22×Q812yes—
108C23×C59C2×#4112yes—
109C23×C59C2×#4112yes—
110C23×C592−1+412yes—
111C23×C592−1+412yes—
112C23×C59C2511yescharacteristic
113Q8×C59C2×C4⋊C4164yescomplemented
114C2×C4×C59—28no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—26no—
118C2×C4×C59—26no—
119C2×C4×C59—26no—
120C2×C4×C59—412no—
121C2×C4×C59C2×C4⋊C4148yes—
122C2×C4×C59C23×C4112yes—
123C2×C4×C59C22×D4112yes—
124C2×C4×C59C22×Q8112yes—
125C2×C4×C59C2×#4114yes—
126C2×Dic59—224no—
127C2×Dic59—28no—
128C2×Dic59—28no—
129C2×Dic59—22no—
130C2×Dic59—26no—
131C2×Dic59—26no—
132C2×Dic59—22no—
133C2×Dic59—22no—
134C2×Dic59—22no—
135C2×Dic59—26no—
136C2×Dic59—26no—
137C2×Dic59—26no—
138C2×Dic59—412no—
139C2×Dic59C23×C414yes—
140C2×Dic59C23×C4112yes—
141C2×Dic59C2×#4114yes—
142C2×#2351—591nomaximal, Sylow
143C22×C59C22×C4⋊C414yes—
144C22×C59C4×#4114yes—
145C22×C59#51914yes—
146C22×C59#54814yes—
147C22×C59#54814yes—
148C22×C59#55714yes—
149C22×C59#55714yes—
150C22×C59C24×C411yescharacteristic
151C22×C59C23×D411yescharacteristic
152C22×C59C23×Q811yescharacteristic
153C22×C59C22×#4111yescharacteristic
154C22×C59C22×#4111yescharacteristic
155C22×C59C2×2−1+411yescharacteristic
156C22×C59C2×2−1+411yescharacteristic
157C4×C59—24no—
158C4×C59—46no—
159C4×C59C22×C4⋊C4124yes—
160Dic59—212no—
161Dic59—24no—
162Dic59—24no—
163Dic59—46no—
164C22×C4×Q8—591no—
165C22×C4×Q8—591no—
166C22×C4×Q8D5911yescharacteristic
167#2351—5916no—
168C2×#389—593no—
169C2×#389—593no—
170C2×#382—593no—
171C2×C4×C4⋊C4—593no—
172C2×C59#235118yescomplemented
173C2×C59C23×C4⋊C411yescharacteristic
174C2×C59C2×C4×#4111yescharacteristic
175C2×C59C2×#51911yescharacteristic
176C2×C59C2×#54811yescharacteristic
177C2×C59C2×#54811yescharacteristic
178C2×C59C2×#55711yescharacteristic
179C2×C59C2×#55711yescharacteristic
180C23×Q8C2×D5911yescharacteristic
181C2×C4×Q8—28no—
182C2×C4×Q8—5916no—
183C2×C4×Q8—598no—
184C2×C4×Q8—598no—
185C2×C4×Q8—1188no—
186C2×C4×Q8C2×D5918yes—
187C22×C4⋊C4—593no—
188C22×C4⋊C4—593no—
189C22×C4⋊C4—593no—
190C22×C4⋊C4—593no—
191C22×C4⋊C4C2×D5913yes—
192C22×C42—593no—
193C22×C42—593no—
194C22×C42C2×D5913yes—
195#389—5924no—
196#389—5924no—
197#382—5924no—
198C4×C4⋊C4—5924no—
199C2×#95—591no—
200C2×#95—593no—
201C2×#95—596no—
202C59C2×#235111yescharacteristic, complemented, Sylow
203C22×Q8Dic11818yes—
204C22×Q8C4×D5918yes—
205C22×Q8C22⋊D5918yes—
206C22×Q8C22×D5914yes—
207C23×C4—593no—
208C23×C4—591no—
209C23×C4—591no—
210C23×C4—593no—
211C23×C4C22×D5913yes—
212C23×C4C22×D5911yescharacteristic
213C23×C4C22×D5913yes—
214C4×Q8—232no—
215C4×Q8—5964no—
216C4×Q8—11832no—
217C2×C4⋊C4—212no—
218C2×C4⋊C4—5924no—
219C2×C4⋊C4—5912no—
220C2×C4⋊C4—5912no—
221C2×C4⋊C4—5912no—
222C2×C4⋊C4—5912no—
223C2×C4⋊C4—11812no—
224C2×C4⋊C4—11812no—
225C2×C4⋊C4—11812no—
226C2×C4⋊C4C22×D59112yes—
227C2×C42—212no—
228C2×C42—5924no—
229C2×C42—5912no—
230C2×C42—5912no—
231C2×C42—11812no—
232C2×C42C22×D59112yes—
233#95—594no—
234#95—5912no—
235#95—5924no—
236C24C23×D5911yescharacteristic
237C2×Q8Dic59⋊C4164yescomplemented
238C2×Q8C2×Dic118116yes—
239C2×Q8C2×C4×D59116yes—
240C2×Q8C2×C22⋊D59116yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—594no—
247C22×C4—5912no—
248C22×C4—594no—
249C22×C4—592no—
250C22×C4—596no—
251C22×C4—592no—
252C22×C4—596no—
253C22×C4—1182no—
254C22×C4—1186no—
255C22×C4—1186no—
256C22×C4—1182no—
257C22×C4—1182no—
258C22×C4—1182no—
259C22×C4—1186no—
260C22×C4—1186no—
261C22×C4—1186no—
262C22×C4C2×Dic118112yes—
263C22×C4C2×C4×D59112yes—
264C22×C4#1174568914yes—
265C22×C4C2×C22⋊D59112yes—
266C22×C4C23×D5916yes—
267C22×C4C23×D5912yes—
268C22×C4C23×D5916yes—
269C4⋊C4—224no—
270C4⋊C4—5948no—
271C4⋊C4—11824no—
272C4⋊C4—11824no—
273C4⋊C4—11824no—
274C42—224no—
275C42—5948no—
276C42—11824no—
277C23C22×C4×D5912yes—
278C23C22×C22⋊D5912yes—
279C23C22×Dic11812yes—
280C23#41×D5912yes—
281C23C2×#1174568912yes—
282C23#1427377412yes—
283C23#1427377512yes—
284C23C24×D5911yescharacteristic
285Q8C2×Dic59⋊C4164yescomplemented
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—28no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—412no—
293C2×C4—594no—
294C2×C4—5912no—
295C2×C4—594no—
296C2×C4—1182no—
297C2×C4—1186no—
298C2×C4—1186no—
299C2×C4—1182no—
300C2×C4—1182no—
301C2×C4—1182no—
302C2×C4—1188no—
303C2×C4—11824no—
304C2×C4—1188no—
305C2×C4—1186no—
306C2×C4—1186no—
307C2×C4—1186no—
308C2×C4—23612no—
309C2×C4C2×Dic59⋊C4148yes—
310C2×C4C22×C4×D59112yes—
311C2×C4C22×C22⋊D59112yes—
312C2×C4C22×Dic118112yes—
313C2×C4C2×#1174568914yes—
314C22C22×Dic59⋊C414yes—
315C22C4×#1174568914yes—
316C22#1683787814yes—
317C22#1683805914yes—
318C22#1683806314yes—
319C22#1683812814yes—
320C22#1683813114yes—
321C22C23×C4×D5911yescharacteristic
322C22C23×C22⋊D5911yescharacteristic
323C22C23×Dic11811yescharacteristic
324C22C2×#41×D5911yescharacteristic
325C22C22×#1174568911yescharacteristic
326C22C2×#1427377411yescharacteristic
327C22C2×#1427377511yescharacteristic
328C4—24no—
329C4—46no—
330C4—11812no—
331C4—1184no—
332C4—1184no—
333C4—2366no—
334C4C22×Dic59⋊C4124yes—
335C2#2142701418yesdirect product factor
336C2C23×Dic59⋊C411yescharacteristic
337C2C2×C4×#1174568911yescharacteristic
338C2C2×#1683787811yescharacteristic
339C2C2×#1683805911yescharacteristic
340C2C2×#1683806311yescharacteristic
341C2C2×#1683812811yescharacteristic
342C2C2×#1683813111yescharacteristic
subgroupG/H
centerC24C23×D59
commutatorC22×C59C24×C4
FrattiniC23C24×D59
FittingC22×C4×Q8×C59C2
socle, prime radicalC24×C59C24
cosocleC23×C59C25
hypercenterC22×C4×Q8D59
hypocenterC59C2×#2351
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #21427014C2direct (N × H)
10. C2×Q8×Dic59C4semidirect
36. Q8×Dic59C2×C4semidirect
65. C2×Q8×C59C4⋊C4semidirect
113. Q8×C59C2×C4⋊C4semidirect
172. C2×C59#2351semidirect
202. C59C2×#2351semidirect
237. C2×Q8Dic59⋊C4semidirect
285. Q8C2×Dic59⋊C4semidirect
335. C2#21427014direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×Dic592
#214270142
C2×#168373002
C2×#168372982
C2×#168372672
C2×C4×Dic59⋊C42
C22×C4×Q8×C592
C2×#235159