Finite Group Databasebeta

← C2×#11734333 #13560203

Subgroups — C2×#11734333 #13560203

Order 1792.

3550 conjugacy classes of subgroups; only the 627 normal subgroups are shown.

normal subgroups only

This group has 3550 conjugacy classes of subgroups — too many to draw. Only its 627 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
1C2×C7×#393C211yesmaximal, characteristic, complemented
2C2×#90003C212yesmaximal, complemented
3C2×#89727C213yesmaximal
4C2×#89719C213yesmaximal, complemented
5#11734333C2116yesmaximal, direct product factor
6C2×#89445C216yesmaximal, complemented
7C25×D7C2211yescharacteristic
8C23×C7×Q8C2211yescharacteristic, complemented
9C23×Dic14C2213yescomplemented
10C22×C7×C22⋊C4C2213yes—
11C22×C22⋊Dic7C2213yes—
12C22×(C2×D7)⋊C4C2216yes—
13C22×(C2×D7)⋊C4C2216yes—
14C7×#393C2218yescomplemented
15C2×C7×#95C2213yescomplemented
16#90003C22116yescomplemented
17C2×C7⋊#95C2216yescomplemented
18C2×C7⋊#95C2213yescomplemented
19#89727C22124yes—
20#89719C22124yescomplemented
21#89445C22148yescomplemented
22C2×#2840—71nomaximal, Sylow
23C25×C7C2311yescharacteristic
24C24×D7—46no—
25C24×D7—46no—
26C24×D7Q818yescomplemented
27C24×D7C2312yes—
28C24×D7C2314yes—
29C22×C7×Q8—46no—
30C22×C7×Q8C2314yescomplemented
31C23×C4×C7C2313yes—
32C23×C4×C7C2313yes—
33C23×Dic7C2316yes—
34C22×Dic14—46no—
35C22×Dic14—412no—
36C22×Dic14C23112yescomplemented
37C2×C7×C22⋊C4—46no—
38C2×C7×C22⋊C4C23112yes—
39C2×C22⋊Dic7—46no—
40C2×C22⋊Dic7C23112yes—
41C2×(C2×D7)⋊C4—412no—
42C2×(C2×D7)⋊C4—412no—
43C2×(C2×D7)⋊C4C23124yes—
44C2×(C2×D7)⋊C4C23124yes—
45C7×#95C23112yescomplemented
46C7⋊#95C23112yescomplemented
47C7⋊#95C23124yescomplemented
48#2840—716no—
49C2×#393—76no—
50C2×#393—72no—
51C2×#393—73no—
52C2×#393D711yescharacteristic, complemented
53C2×#379—73no—
54C24×C7—26no—
55C24×C7—26no—
56C24×C7C2×Q814yes—
57C24×C7C2412yes—
58C24×C7C2411yescharacteristic
59C23×D7—212no—
60C23×D7—212no—
61C23×D7—448no—
62C23×D7—412no—
63C23×D7—412no—
64C23×D7—448no—
65C23×D7—416no—
66C23×D7C2×Q818yes—
67C23×D7C2×Q8116yescomplemented
68C23×D7C2414yes—
69C2×C7×Q8—412no—
70C2×C7×Q8—416no—
71C22×C4×C7—212no—
72C22×C4×C7—212no—
73C22×C4×C7—26no—
74C22×C4×C7—26no—
75C22×C4×C7C2416yes—
76C22×C4×C7C2416yes—
77C22×Dic7—212no—
78C22×Dic7—212no—
79C22×Dic7—212no—
80C22×Dic7C24112yes—
81C2×Dic14—412no—
82C2×Dic14—424no—
83C2×Dic14—448no—
84C7×C22⋊C4—412no—
85C22⋊Dic7—412no—
86(C2×D7)⋊C4—424no—
87(C2×D7)⋊C4—424no—
88C26—71no—
89C23×Q8—73no—
90C23×Q8C2×D711yescharacteristic, complemented
91C22×C22⋊C4—73no—
92C22×C22⋊C4—76no—
93C22×C22⋊C4—76no—
94C22×C22⋊C4C2×D713yes—
95#393—716no—
96#393—724no—
97#393—748no—
98#393C2×D718yescomplemented
99#379—724no—
100C2×#95—73no—
101C2×#95—76no—
102C2×#95C2×D713yescomplemented
103C23×C7—224no—
104C23×C7—26no—
105C23×C7—26no—
106C23×C7—412no—
107C23×C7—44no—
108C23×C7C22×Q814yes—
109C23×C7C22×Q812yes—
110C23×C7H(5,𝔽2)16yes—
111C23×C7H(5,𝔽2)16yes—
112C23×C7C2511yescharacteristic
113C22×D7—248no—
114C22×D7—212no—
115C22×D7—212no—
116C22×D7—448no—
117C22×D7—4128no—
118C22×D7—416no—
119C22×D7—448no—
120C22×D7—424no—
121C22×D7—48no—
122C22×D7C22×Q818yes—
123C7×Q8—416no—
124C2×C4×C7—212no—
125C2×C4×C7—212no—
126C2×C4×C7—26no—
127C2×C4×C7—26no—
128C2×C4×C7—412no—
129C2×C4×C7—412no—
130C2×Dic7—212no—
131C2×Dic7—212no—
132C2×Dic7—212no—
133C2×Dic7—424no—
134Dic14—448no—
135C25—72no—
136C25—78no—
137C25—74no—
138C25—286no—
139C25—286no—
140C25C22×D711yescharacteristic
141C22×Q8—46no—
142C22×Q8—712no—
143C22×Q8—286no—
144C22×Q8—2812no—
145C22×Q8C22×D714yescomplemented
146C23×C4—76no—
147C23×C4C22×D713yes—
148C23×C4C22×D713yes—
149C2×C22⋊C4—46no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4—724no—
152C2×C22⋊C4—724no—
153C2×C22⋊C4—2812no—
154C2×C22⋊C4—2812no—
155C2×C22⋊C4—286no—
156C2×C22⋊C4C22×D7112yes—
157#95—724no—
158#95—712no—
159#95C22×D7112yescomplemented
160C22×C7—212no—
161C22×C7—416no—
162C22×C7—42no—
163C22×C7—46no—
164C22×C7#543112yes—
165C22×C7H(3,𝔽4)112yes—
166C22×C7H(3,𝔽4)14yes—
167C22×C7C23×Q811yescharacteristic
168C22×C7C2×H(5,𝔽2)13yes—
169C22×C7C2×H(5,𝔽2)13yes—
170C2×D7—224no—
171C2×D7—464no—
172C2×D7—432no—
173C2×D7—44no—
174C2×D7—412no—
175C4×C7—46no—
176C4×C7—46no—
177Dic7—412no—
178C24—26no—
179C24—26no—
180C24—78no—
181C24—74no—
182C24—716no—
183C24—1412no—
184C24—1412no—
185C24—2848no—
186C24—2812no—
187C24—2812no—
188C24—2848no—
189C24—2816no—
190C24Q8×D714yes—
191C24C23×D712yes—
192C24C23×D711yescharacteristic
193C2×Q8—412no—
194C2×Q8—416no—
195C2×Q8—2812no—
196C2×Q8—2824no—
197C2×Q8—2848no—
198C22×C4—26no—
199C22×C4—26no—
200C22×C4—212no—
201C22×C4—212no—
202C22×C4—712no—
203C22×C4—1412no—
204C22×C4—1412no—
205C22×C4—1412no—
206C22×C4C23×D716yes—
207C22×C4C23×D716yes—
208C22⋊C4—412no—
209C22⋊C4—2824no—
210C22⋊C4—2824no—
211C22⋊C4—2812no—
212C2×C7—44no—
213C2×C7#284018yescomplemented
214C2×C7C2×#54313yes—
215C2×C7C2×H(3,𝔽4)11yescharacteristic
216C2×C7C2×H(3,𝔽4)13yes—
217D7—48no—
218C23—224no—
219C23—26no—
220C23—26no—
221C23—412no—
222C23—44no—
223C23—78no—
224C23—1448no—
225C23—1412no—
226C23—1412no—
227C23—2848no—
228C23—2816no—
229C23—28128no—
230C23—2848no—
231C23—2824no—
232C23—288no—
233C23#654816yes—
234C23C2×Q8×D714yes—
235C23C2×Q8×D712yes—
236C23#655316yes—
237C23C24×D711yescharacteristic
238Q8—416no—
239Q8—2848no—
240C2×C4—26no—
241C2×C4—26no—
242C2×C4—212no—
243C2×C4—212no—
244C2×C4—412no—
245C2×C4—412no—
246C2×C4—1412no—
247C2×C4—1412no—
248C2×C4—1412no—
249C2×C4—2824no—
250C7C2×#284011yescharacteristic, complemented, Sylow
251C22—212no—
252C22—42no—
253C22—416no—
254C22—46no—
255C22—1424no—
256C22—2864no—
257C22—284no—
258C22—2832no—
259C22—2812no—
260C22#9028514yes—
261C22#90325112yes—
262C22#90374112yes—
263C22C2×#654813yes—
264C22C22×Q8×D711yescharacteristic
265C22C2×#655313yes—
266C4—46no—
267C4—46no—
268C4—2812no—
269C2—44no—
270C2—288no—
271C2#1173433318yesdirect product factor
272C2C2×#9028511yescharacteristic
273C2C2×#9032513yes—
274C2C2×#9037413yes—
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#393C2
socleC24×C7C24
prime radicalC25×D7C22
hypercenterC2×#393D7
hypocenterC7C2×#2840
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C2×C7×#393C2semidirect
2. C2×#90003C2semidirect
4. C2×#89719C2semidirect
5. #11734333C2direct (N × H)
6. C2×#89445C2semidirect
8. C23×C7×Q8C22semidirect
9. C23×Dic14C22semidirect
14. C7×#393C22semidirect
15. C2×C7×#95C22semidirect
16. #90003C22semidirect
17. C2×C7⋊#95C22semidirect
18. C2×C7⋊#95C22semidirect
20. #89719C22semidirect
21. #89445C22semidirect
26. C24×D7Q8semidirect
30. C22×C7×Q8C23semidirect
36. C22×Dic14C23semidirect
45. C7×#95C23semidirect
46. C7⋊#95C23semidirect
47. C7⋊#95C23semidirect
52. C2×#393D7semidirect
67. C23×D7C2×Q8semidirect
90. C23×Q8C2×D7semidirect
98. #393C2×D7semidirect
102. C2×#95C2×D7semidirect
145. C22×Q8C22×D7semidirect
159. #95C22×D7semidirect
213. C2×C7#2840semidirect
250. C7C2×#2840semidirect
271. C2#11734333direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3932
C2×#900032
C2×#897272
C2×#897192
#117343332
C2×#894452
C2×#28407