Finite Group Databasebeta

← C2×#11731305 #13559914

Subgroups — C2×#11731305 #13559914

Order 1792.

4438 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 4438 conjugacy classes of subgroups — too many to draw. Only its 667 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6516C212yesmaximal, complemented
2C2×C7×#379C211yesmaximal, characteristic, complemented
3C2×#89732C213yesmaximal, complemented
4C2×#89725C213yesmaximal, complemented
5C2×#89733C216yesmaximal, complemented
6#11731305C2116yesmaximal, direct product factor
7C26×C7C2211yescharacteristic
8C23×C22⋊D7C2216yes—
9C23×D28C2211yescharacteristic, complemented
10C22×C7×C22⋊C4C2213yes—
11C22×C7×C22⋊C4C2213yes—
12C2×#6516—216no—
13C2×#6516C22116yescomplemented
14C22×C22⋊Dic7C2216yes—
15C22×(C2×D7)⋊C4C2216yescomplemented
16C22×C4⋊Dic7C2213yescomplemented
17C7×#379C2218yescomplemented
18C2×C7⋊#95C2213yescomplemented
19C2×C7⋊#95C2213yescomplemented
20#89733C22148yescomplemented
21#89732C22124yescomplemented
22#89725C22124yescomplemented
23C2×#2812—71nomaximal, Sylow
24C25×C7—46no—
25C25×C7—46no—
26C25×C7D418yescomplemented
27C25×C7C2313yes—
28C25×C7C2314yes—
29C24×D7C2312yes—
30C23×C4×C7C2313yes—
31C23×C4×C7C2311yescharacteristic
32C22×C22⋊D7—224no—
33C22×C22⋊D7—412no—
34C22×C22⋊D7—412no—
35C22×C22⋊D7C23124yes—
36C23×Dic7C2316yes—
37C22×D28—46no—
38C22×D28C2314yescomplemented
39C2×C7×C22⋊C4—46no—
40C2×C7×C22⋊C4—46no—
41C2×C7×C22⋊C4C23112yes—
42C2×C7×C22⋊C4C23112yes—
43#6516—264no—
44C2×C22⋊Dic7—224no—
45C2×C22⋊Dic7C23124yes—
46C2×(C2×D7)⋊C4—412no—
47C2×(C2×D7)⋊C4C23124yescomplemented
48C2×C4⋊Dic7—46no—
49C2×C4⋊Dic7C23112yescomplemented
50C7⋊#95C23112yescomplemented
51C7⋊#95C23112yescomplemented
52C22×#120—72no—
53#2812—716no—
54C2×#397—73no—
55C2×#394—76no—
56C2×#392—73no—
57C2×#379D711yescharacteristic, complemented
58C24×C7—26no—
59C24×C7—26no—
60C24×C7—212no—
61C24×C7—212no—
62C24×C7—448no—
63C24×C7—412no—
64C24×C7—412no—
65C24×C7—448no—
66C24×C7—416no—
67C24×C7C2×D4112yes—
68C24×C7C2×D4116yescomplemented
69C24×C7C2416yes—
70C24×C7C2411yescharacteristic
71C23×D7—24no—
72C23×D7—212no—
73C23×D7—212no—
74C23×D7C2414yes—
75C22×C4×C7—26no—
76C22×C4×C7—212no—
77C22×C4×C7—26no—
78C22×C4×C7C2412yes—
79C22×C4×C7C2416yes—
80C2×C22⋊D7—248no—
81C2×C22⋊D7—424no—
82C2×C22⋊D7—424no—
83C2×C22⋊D7—496no—
84C22×Dic7—212no—
85C22×Dic7—212no—
86C22×Dic7—212no—
87C22×Dic7C24112yes—
88C2×D28—412no—
89C2×D28—416no—
90C7×C22⋊C4—412no—
91C7×C22⋊C4—412no—
92C22⋊Dic7—248no—
93(C2×D7)⋊C4—424no—
94C4⋊Dic7—412no—
95C26C2×D711yescharacteristic
96C23×D4—76no—
97C23×D4—71no—
98C2×#120—716no—
99C2×#120—1416no—
100C22×C4⋊C4—73no—
101C22×C22⋊C4—76no—
102C22×C22⋊C4—76no—
103C22×C22⋊C4C2×D713yes—
104C22×C22⋊C4C2×D713yes—
105#397—724no—
106#394—748no—
107#392—724no—
108#379C2×D718yescomplemented
109C2×#95—73no—
110C2×#95—73no—
111C23×C7—248no—
112C23×C7—224no—
113C23×C7—26no—
114C23×C7—26no—
115C23×C7—212no—
116C23×C7—212no—
117C23×C7—448no—
118C23×C7—448no—
119C23×C7—416no—
120C23×C7—4128no—
121C23×C7—424no—
122C23×C7—412no—
123C23×C7—412no—
124C23×C7C22×D4112yes—
125C23×C7C22×D412yes—
126C23×C7H(5,𝔽2)16yes—
127C23×C7H(5,𝔽2)16yes—
128C23×C7C2511yescharacteristic
129C22×D7—24no—
130C22×D7—212no—
131C22×D7—212no—
132C22×D7—424no—
133C22×D7—424no—
134C22×D7—48no—
135C2×C4×C7—26no—
136C2×C4×C7—212no—
137C2×C4×C7—26no—
138C2×C4×C7—412no—
139C2×C4×C7—44no—
140C22⋊D7—496no—
141C2×Dic7—212no—
142C2×Dic7—212no—
143C2×Dic7—212no—
144C2×Dic7—424no—
145D28—416no—
146C25—46no—
147C25—46no—
148C25—72no—
149C25D2818yescomplemented
150C25C22×D713yes—
151C25C22×D714yes—
152C22×D4—724no—
153C22×D4—74no—
154C22×D4—1424no—
155C22×D4—286no—
156C22×D4—2812no—
157C22×D4—2812no—
158C23×C4—76no—
159C23×C4C22×D713yes—
160C23×C4C22×D711yescharacteristic
161#120—1464no—
162C2×C4⋊C4—712no—
163C2×C4⋊C4—286no—
164C2×C22⋊C4—46no—
165C2×C22⋊C4—46no—
166C2×C22⋊C4—724no—
167C2×C22⋊C4—724no—
168C2×C22⋊C4—1424no—
169C2×C22⋊C4—2812no—
170C2×C22⋊C4C22×D7112yes—
171C2×C22⋊C4C22×D7112yes—
172#95—712no—
173#95—712no—
174C22×C7—224no—
175C22×C7—212no—
176C22×C7—464no—
177C22×C7—412no—
178C22×C7—46no—
179C22×C7—46no—
180C22×C7—448no—
181C22×C7#534112yes—
182C22×C7#560112yes—
183C22×C7H(3,𝔽4)14yes—
184C22×C7C23×D411yescharacteristic
185C22×C7C2×H(5,𝔽2)13yes—
186C22×C7C2×H(5,𝔽2)13yes—
187C2×D7—412no—
188C2×D7—412no—
189C2×D7—44no—
190C2×D7—816no—
191C4×C7—46no—
192C4×C7—42no—
193Dic7—412no—
194C24—26no—
195C24—26no—
196C24—212no—
197C24—212no—
198C24—448no—
199C24—412no—
200C24—412no—
201C24—416no—
202C24—448no—
203C24—74no—
204C24—144no—
205C24—1412no—
206C24—1412no—
207C24C2×D28112yes—
208C24C2×D28116yescomplemented
209C24C23×D716yes—
210C24C23×D711yescharacteristic
211C2×D4—1448no—
212C2×D4—2812no—
213C2×D4—2816no—
214C2×D4—2824no—
215C2×D4—2824no—
216C2×D4—2896no—
217C22×C4—212no—
218C22×C4—26no—
219C22×C4—26no—
220C22×C4—712no—
221C22×C4—1412no—
222C22×C4—1412no—
223C22×C4—1412no—
224C22×C4C23×D712yes—
225C22×C4C23×D716yes—
226C4⋊C4—2812no—
227C22⋊C4—412no—
228C22⋊C4—412no—
229C22⋊C4—1448no—
230C22⋊C4—2824no—
231C2×C7—412no—
232C2×C7#281218yescomplemented
233C2×C7C2×#53413yes—
234C2×C7C2×#56013yes—
235C2×C7C2×H(3,𝔽4)11yescharacteristic
236D7—84no—
237C23—248no—
238C23—224no—
239C23—26no—
240C23—26no—
241C23—212no—
242C23—212no—
243C23—448no—
244C23—416no—
245C23—448no—
246C23—4128no—
247C23—412no—
248C23—412no—
249C23—424no—
250C23—144no—
251C23—1412no—
252C23—1412no—
253C23—2824no—
254C23—288no—
255C23—2824no—
256C23C22×D28112yes—
257C23C22×D2812yes—
258C23#654816yes—
259C23#654816yes—
260C23C24×D711yescharacteristic
261D4—2816no—
262D4—2896no—
263C2×C4—212no—
264C2×C4—26no—
265C2×C4—26no—
266C2×C4—44no—
267C2×C4—412no—
268C2×C4—1412no—
269C2×C4—1412no—
270C2×C4—1412no—
271C2×C4—2824no—
272C7C2×#281211yescharacteristic, complemented, Sylow
273C22—224no—
274C22—212no—
275C22—464no—
276C22—46no—
277C22—46no—
278C22—412no—
279C22—448no—
280C22—284no—
281C22—2812no—
282C22—2812no—
283C22—5616no—
284C22#90184112yes—
285C22#9028514yes—
286C22#90286112yes—
287C22C23×D2811yescharacteristic
288C22C2×#654813yes—
289C22C2×#654813yes—
290C4—42no—
291C4—46no—
292C4—2812no—
293C2—412no—
294C2—564no—
295C2#1173130518yesdirect product factor
296C2C2×#9018413yes—
297C2C2×#9028511yescharacteristic
298C2C2×#9028613yes—
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#379C2
socleC24×C7C24
hypercenterC2×#379D7
hypocenterC7C2×#2812
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#6516C2semidirect
2. C2×C7×#379C2semidirect
3. C2×#89732C2semidirect
4. C2×#89725C2semidirect
5. C2×#89733C2semidirect
6. #11731305C2direct (N × H)
9. C23×D28C22semidirect
13. C2×#6516C22semidirect
15. C22×(C2×D7)⋊C4C22semidirect
16. C22×C4⋊Dic7C22semidirect
17. C7×#379C22semidirect
18. C2×C7⋊#95C22semidirect
19. C2×C7⋊#95C22semidirect
20. #89733C22semidirect
21. #89732C22semidirect
22. #89725C22semidirect
26. C25×C7D4semidirect
38. C22×D28C23semidirect
47. C2×(C2×D7)⋊C4C23semidirect
49. C2×C4⋊Dic7C23semidirect
50. C7⋊#95C23semidirect
51. C7⋊#95C23semidirect
57. C2×#379D7semidirect
68. C24×C7C2×D4semidirect
108. #379C2×D7semidirect
149. C25D28semidirect
208. C24C2×D28semidirect
232. C2×C7#2812semidirect
272. C7C2×#2812semidirect
295. C2#11731305direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#65162
C2×C7×#3792
C2×#897322
C2×#897252
C2×#897332
#117313052
C2×#28127