Finite Group Databasebeta

← C22×B(4,𝔽2)×C73 #32969796

Subgroups — C22×B(4,𝔽2)×C73 #32969796

Order 18688.

7014 conjugacy classes of subgroups; only the 1134 normal subgroups are shown.

normal subgroups only

This group has 7014 conjugacy classes of subgroups — too many to draw. Only its 1134 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×H(5,𝔽2)×C73C211yesmaximal, characteristic, complemented
2C22×#120×C73C213yesmaximal, complemented
3C2×B(4,𝔽2)×C73C2124yesmaximal, direct product factor
4C22×#99×C73C213yesmaximal, complemented
5C26×C73C2211yescharacteristic, complemented
6C2×H(5,𝔽2)×C73—218no—
7C2×H(5,𝔽2)×C73C22112yescomplemented
8C22×#41×C73—23no—
9C23×D4×C73—23no—
10C23×D4×C73—23no—
11C23×D4×C73C2213yes—
12C2×#120×C73—218no—
13C2×#120×C73C22136yescomplemented
14C22×C22⋊C4×C73—23no—
15C22×C22⋊C4×C73C2213yescomplemented
16B(4,𝔽2)×C73C22164yesdirect product factor
17C2×#99×C73C22136yescomplemented
18C25×C73—21no—
19C25×C73—23no—
20C25×C73—29no—
21C25×C73—48no—
22C25×C73D413yescomplemented
23C25×C73C2311yescharacteristic
24C25×C73C2316yescomplemented
25H(5,𝔽2)×C73—248no—
26H(5,𝔽2)×C73—272no—
27H(5,𝔽2)×C73C23116yescomplemented
28C2×#41×C73—236no—
29C2×#41×C73—236no—
30C22×Q8×C73—21no—
31C22×D4×C73—23no—
32C22×D4×C73—236no—
33C22×D4×C73—26no—
34C22×D4×C73—218no—
35C22×D4×C73—236no—
36C22×D4×C73—236no—
37C22×D4×C73—424no—
38C22×D4×C73D4118yescomplemented
39C22×D4×C73C23118yes—
40C23×C4×C73—23no—
41C23×C4×C73—23no—
42C23×C4×C73D413yescomplemented
43#120×C73—272no—
44#120×C73C23148yescomplemented
45C2×C22⋊C4×C73—29no—
46C2×C22⋊C4×C73—236no—
47C2×C22⋊C4×C73C23118yescomplemented
48#99×C73C23148yescomplemented
49C24×C73—224no—
50C24×C73—212no—
51C24×C73—23no—
52C24×C73—218no—
53C24×C73—29no—
54C24×C73—218no—
55C24×C73—44no—
56C24×C73—424no—
57C24×C73—412no—
58C24×C73—472no—
59C24×C73—412no—
60C24×C73—49no—
61C24×C73—46no—
62C24×C73—41no—
63C24×C73—84no—
64C24×C73C2×D413yes—
65C24×C73C2×D419yes—
66C24×C73C2×D4118yescomplemented
67C24×C73C2413yes—
68C24×C73C2414yescomplemented
69#41×C73—2144no—
70#41×C73—248no—
71C2×Q8×C73—212no—
72C2×D4×C73—236no—
73C2×D4×C73—2144no—
74C2×D4×C73—236no—
75C2×D4×C73—236no—
76C2×D4×C73—236no—
77C2×D4×C73—236no—
78C2×D4×C73—248no—
79C2×D4×C73—248no—
80C2×D4×C73—4144no—
81C2×D4×C73C2×D4136yescomplemented
82C2×D4×C73C24112yes—
83C22×C4×C73—236no—
84C22×C4×C73—29no—
85C22×C4×C73—23no—
86C22×C4×C73—218no—
87C22×C4×C73—412no—
88C22×C4×C73C2×D4118yescomplemented
89C22⋊C4×C73—218no—
90C22⋊C4×C73—248no—
91C22⋊C4×C73C24112yescomplemented
92C23×C73—212no—
93C23×C73—272no—
94C23×C73—216no—
95C23×C73—218no—
96C23×C73—212no—
97C23×C73—218no—
98C23×C73—418no—
99C23×C73—46no—
100C23×C73—48no—
101C23×C73—436no—
102C23×C73—472no—
103C23×C73—472no—
104C23×C73—436no—
105C23×C73—448no—
106C23×C73—418no—
107C23×C73—46no—
108C23×C73—48no—
109C23×C73—412no—
110C23×C73—818no—
111C23×C73—824no—
112C23×C73#12019yes—
113C23×C73#12016yes—
114C23×C73#12011yescharacteristic
115C23×C73C22×D419yes—
116C23×C73C22×D419yes—
117C23×C73C22×D4112yescomplemented
118C23×C73C2511yescharacteristic
119Q8×C73—216no—
120D4×C73—248no—
121D4×C73—272no—
122D4×C73—224no—
123D4×C73—496no—
124C2×C4×C73—248no—
125C2×C4×C73—218no—
126C2×C4×C73—218no—
127C2×C4×C73—212no—
128C2×C4×C73—436no—
129C2×C4×C73C22×D4112yescomplemented
130C22×C73—218no—
131C22×C73—218no—
132C22×C73—224no—
133C22×C73—212no—
134C22×C73—46no—
135C22×C73—49no—
136C22×C73—41no—
137C22×C73—412no—
138C22×C73—412no—
139C22×C73—472no—
140C22×C73—44no—
141C22×C73—424no—
142C22×C73—436no—
143C22×C73—818no—
144C22×C73—816no—
145C22×C73B(4,𝔽2)14yesdirect product factor
146C22×C73C2×#12019yes—
147C22×C73C2×#12013yes—
148C22×C73C23×D413yes—
149C4×C73—212no—
150C4×C73—412no—
151C22×B(4,𝔽2)C7311yesmaximal, characteristic, direct product factor, Sylow
152C2×C73—23no—
153C2×C73—29no—
154C2×C73—48no—
155C2×C73—412no—
156C2×C73C2×B(4,𝔽2)16yesdirect product factor
157C2×C73C22×#12011yescharacteristic
158C22×H(5,𝔽2)C2×C7311yescharacteristic, complemented
159C22×#120C2×C7313yescomplemented
160C2×B(4,𝔽2)C2×C73124yesdirect product factor
161C22×#99C2×C7313yescomplemented
162C73C22×B(4,𝔽2)11yescharacteristic, direct product factor, Sylow
163C26C22×C7311yescharacteristic, complemented
164C2×H(5,𝔽2)—218no—
165C2×H(5,𝔽2)C22×C73112yescomplemented
166C22×#41—23no—
167C23×D4—23no—
168C23×D4—23no—
169C23×D4C22×C7313yes—
170C2×#120—218no—
171C2×#120C22×C73136yescomplemented
172C22×C22⋊C4—23no—
173C22×C22⋊C4C22×C7313yescomplemented
174B(4,𝔽2)C22×C73164yesdirect product factor
175C2×#99C22×C73136yescomplemented
176C25—21no—
177C25—23no—
178C25—29no—
179C25—48no—
180C25D4×C7313yescomplemented
181C25C23×C7311yescharacteristic
182C25C23×C7316yescomplemented
183H(5,𝔽2)—248no—
184H(5,𝔽2)—272no—
185H(5,𝔽2)C23×C73116yescomplemented
186C2×#41—236no—
187C2×#41—236no—
188C22×Q8—21no—
189C22×D4—23no—
190C22×D4—236no—
191C22×D4—26no—
192C22×D4—218no—
193C22×D4—236no—
194C22×D4—236no—
195C22×D4—424no—
196C22×D4D4×C73118yescomplemented
197C22×D4C23×C73118yes—
198C23×C4—23no—
199C23×C4—23no—
200C23×C4D4×C7313yescomplemented
201#120—272no—
202#120C23×C73148yescomplemented
203C2×C22⋊C4—29no—
204C2×C22⋊C4—236no—
205C2×C22⋊C4C23×C73118yescomplemented
206#99C23×C73148yescomplemented
207C24—224no—
208C24—212no—
209C24—23no—
210C24—218no—
211C24—29no—
212C24—218no—
213C24—44no—
214C24—424no—
215C24—412no—
216C24—472no—
217C24—412no—
218C24—49no—
219C24—46no—
220C24—41no—
221C24—84no—
222C24C2×D4×C7313yes—
223C24C2×D4×C7319yes—
224C24C2×D4×C73118yescomplemented
225C24C24×C7313yes—
226C24C24×C7314yescomplemented
227#41—2144no—
228#41—248no—
229C2×Q8—212no—
230C2×D4—236no—
231C2×D4—2144no—
232C2×D4—236no—
233C2×D4—236no—
234C2×D4—236no—
235C2×D4—236no—
236C2×D4—248no—
237C2×D4—248no—
238C2×D4—4144no—
239C2×D4C2×D4×C73136yescomplemented
240C2×D4C24×C73112yes—
241C22×C4—236no—
242C22×C4—23no—
243C22×C4—29no—
244C22×C4—218no—
245C22×C4—412no—
246C22×C4C2×D4×C73118yescomplemented
247C22⋊C4—218no—
248C22⋊C4—248no—
249C22⋊C4C24×C73112yescomplemented
250C23—212no—
251C23—272no—
252C23—216no—
253C23—218no—
254C23—212no—
255C23—218no—
256C23—418no—
257C23—46no—
258C23—48no—
259C23—436no—
260C23—472no—
261C23—472no—
262C23—436no—
263C23—448no—
264C23—48no—
265C23—412no—
266C23—418no—
267C23—46no—
268C23—818no—
269C23—824no—
270C23#120×C7319yes—
271C23#120×C7316yes—
272C23#120×C7311yescharacteristic
273C23C22×D4×C7319yes—
274C23C22×D4×C7319yes—
275C23C22×D4×C73112yescomplemented
276C23C25×C7311yescharacteristic
277Q8—216no—
278D4—248no—
279D4—272no—
280D4—224no—
281D4—496no—
282C2×C4—248no—
283C2×C4—218no—
284C2×C4—218no—
285C2×C4—212no—
286C2×C4—436no—
287C2×C4C22×D4×C73112yescomplemented
288C22—218no—
289C22—218no—
290C22—224no—
291C22—212no—
292C22—46no—
293C22—49no—
294C22—41no—
295C22—412no—
296C22—412no—
297C22—472no—
298C22—424no—
299C22—436no—
300C22—44no—
301C22—818no—
302C22—816no—
303C22B(4,𝔽2)×C7314yesdirect product factor
304C22C2×#120×C7319yes—
305C22C2×#120×C7313yes—
306C22C23×D4×C7313yes—
307C4—212no—
308C4—412no—
309C2—23no—
310C2—29no—
311C2—48no—
312C2—412no—
313C2C2×B(4,𝔽2)×C7316yesdirect product factor
314C2C22×#120×C7311yescharacteristic
subgroupG/H
center, socleC23×C73#120
commutator, Frattini, cosocleC23C25×C73
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C25×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)×C73C2semidirect
2. C22×#120×C73C2semidirect
3. C2×B(4,𝔽2)×C73C2direct (N × H)
4. C22×#99×C73C2semidirect
5. C26×C73C22semidirect
7. C2×H(5,𝔽2)×C73C22semidirect
13. C2×#120×C73C22semidirect
15. C22×C22⋊C4×C73C22semidirect
16. B(4,𝔽2)×C73C22direct (N × H)
17. C2×#99×C73C22semidirect
22. C25×C73D4semidirect
24. C25×C73C23semidirect
27. H(5,𝔽2)×C73C23semidirect
38. C22×D4×C73D4semidirect
42. C23×C4×C73D4semidirect
44. #120×C73C23semidirect
47. C2×C22⋊C4×C73C23semidirect
48. #99×C73C23semidirect
66. C24×C73C2×D4semidirect
68. C24×C73C24semidirect
81. C2×D4×C73C2×D4semidirect
88. C22×C4×C73C2×D4semidirect
91. C22⋊C4×C73C24semidirect
117. C23×C73C22×D4semidirect
129. C2×C4×C73C22×D4semidirect
145. C22×C73B(4,𝔽2)direct (N × H)
151. C22×B(4,𝔽2)C73direct (N × H)
156. C2×C73C2×B(4,𝔽2)direct (N × H)
158. C22×H(5,𝔽2)C2×C73semidirect
159. C22×#120C2×C73semidirect
160. C2×B(4,𝔽2)C2×C73direct (N × H)
161. C22×#99C2×C73semidirect
162. C73C22×B(4,𝔽2)direct (N × H)
163. C26C22×C73semidirect
165. C2×H(5,𝔽2)C22×C73semidirect
171. C2×#120C22×C73semidirect
173. C22×C22⋊C4C22×C73semidirect
174. B(4,𝔽2)C22×C73direct (N × H)
175. C2×#99C22×C73semidirect
180. C25D4×C73semidirect
182. C25C23×C73semidirect
185. H(5,𝔽2)C23×C73semidirect
196. C22×D4D4×C73semidirect
200. C23×C4D4×C73semidirect
202. #120C23×C73semidirect
205. C2×C22⋊C4C23×C73semidirect
206. #99C23×C73semidirect
224. C24C2×D4×C73semidirect
226. C24C24×C73semidirect
239. C2×D4C2×D4×C73semidirect
246. C22×C4C2×D4×C73semidirect
249. C22⋊C4C24×C73semidirect
275. C23C22×D4×C73semidirect
287. C2×C4C22×D4×C73semidirect
303. C22B(4,𝔽2)×C73direct (N × H)
313. C2C2×B(4,𝔽2)×C73direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)×C732
C22×#120×C732
C2×B(4,𝔽2)×C732
C22×#99×C732
C22×B(4,𝔽2)73