Finite Group Databasebeta

← C73×#23897 #32933301

Subgroups — C73×#23897 #32933301

Order 18688.

1952 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 282 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
1C73×#3484C212yesmaximal, complemented
2C73×#3019C214yesmaximal, complemented
3C73×#2010C214yesmaximal, complemented
4C73×#1911C214yesmaximal, complemented
5C73×#1783C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C73C2212yescomplemented
7C23×D4×C73C2211yescharacteristic
8#575×C73C2214yescomplemented
9#546×C73—28no—
10D42×C73—28no—
11#535×C73C2214yes—
12#534×C73—24no—
13#525×C73C2214yescomplemented
14C2×#120×C73—24no—
15#518×C73C2212yescomplemented
16#455×C73—28no—
17#453×C73—28no—
18#421×C73C2218yescomplemented
19C2×#100×C73C2214yescomplemented
20#407×C73C2212yescomplemented
21H(3,ℤ/4)×C73C2214yescomplemented
22C25×C73D412yes—
23H(5,𝔽2)×C73—28no—
24H(5,𝔽2)×C73—44no—
25C2×#41×C73—24no—
26C2×#41×C73C2314yes—
27C22×D4×C73—24no—
28C22×D4×C73—28no—
29C22×D4×C73—28no—
30C22×D4×C73—24no—
31C22×D4×C73—42no—
32C22×D4×C73D412yes—
33C22×D4×C73C2312yes—
34C23×C4×C73C2311yescharacteristic
35#137×C73—28no—
36AGL(1,ℤ/8)×C73—28no—
37#135×C73—44no—
38C2×M16×C73C2314yes—
39#127×C73D418yescomplemented
40#124×C73D418yescomplemented
41#123×C73—28no—
42#123×C73—24no—
43C22⋊Q8×C73—24no—
44#121×C73—216no—
45#121×C73—28no—
46#121×C73—24no—
47#120×C73—216no—
48#120×C73—28no—
49#120×C73—24no—
50#120×C73—28no—
51C4×D4×C73—28no—
52C4×D4×C73—28no—
53#117×C73C2314yes—
54C2×C22⋊C4×C73—28no—
55C2×C22⋊C4×C73—24no—
56#104×C73—216no—
57#100×C73—28no—
58C22⋊C8×C73—42no—
59C24×C73—24no—
60C24×C73—28no—
61C24×C73—42no—
62C24×C73—84no—
63C24×C73C2×D411yescharacteristic
64C24×C73C2×D414yes—
65#41×C73—28no—
66#41×C73—28no—
67#41×C73—416no—
68C2×Q8×C73C2×D414yescomplemented
69C2×D4×C73—28no—
70C2×D4×C73—28no—
71C2×D4×C73—28no—
72C2×D4×C73—24no—
73C2×D4×C73—216no—
74C2×D4×C73—24no—
75C2×D4×C73—24no—
76C2×D4×C73—42no—
77C2×D4×C73—416no—
78C2×D4×C73—416no—
79C2×D4×C73—48no—
80C2×D4×C73—48no—
81C2×D4×C73—816no—
82C2×D4×C73C2×D418yes—
83C2×D4×C73C2×D414yescomplemented
84C22×C4×C73—24no—
85C22×C4×C73—28no—
86C22×C4×C73—24no—
87C22×C4×C73—24no—
88C22×C4×C73—22no—
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2411yescharacteristic
91Q16×C73—44no—
92SD16×C73—48no—
93D8×C73—44no—
94M16×C73—28no—
95C2×C8×C73—24no—
96C4⋊C4×C73—28no—
97C4⋊C4×C73—24no—
98C22⋊C4×C73—28no—
99C22⋊C4×C73—28no—
100C22⋊C4×C73—28no—
101C22⋊C4×C73—216no—
102C22⋊C4×C73—24no—
103C42×C73C2×D418yescomplemented
104C23×C73—22no—
105C23×C73—24no—
106C23×C73—21no—
107C23×C73—24no—
108C23×C73—24no—
109C23×C73—42no—
110C23×C73—48no—
111C23×C73—416no—
112C23×C73—48no—
113C23×C73—84no—
114C23×C73—84no—
115C23×C73—88no—
116C23×C73—88no—
117C23×C73—84no—
118C23×C73#12018yes—
119C23×C73C22×D411yescharacteristic
120Q8×C73—28no—
121D4×C73—216no—
122D4×C73—28no—
123D4×C73—48no—
124D4×C73—416no—
125D4×C73—416no—
126D4×C73—44no—
127D4×C73—44no—
128D4×C73—816no—
129C2×C4×C73—28no—
130C2×C4×C73—24no—
131C2×C4×C73—22no—
132C2×C4×C73—21no—
133C2×C4×C73—48no—
134C2×C4×C73—44no—
135C2×C4×C73—48no—
136C2×C4×C73—88no—
137C2×C4×C73C22×D414yes—
138C2×C4×C73H(5,𝔽2)12yes—
139C8×C73—44no—
140C22×C73—24no—
141C22×C73—22no—
142C22×C73—28no—
143C22×C73—41no—
144C22×C73—48no—
145C22×C73—48no—
146C22×C73—42no—
147C22×C73—41no—
148C22×C73—48no—
149C22×C73—41no—
150C22×C73—44no—
151C22×C73—84no—
152C22×C73—816no—
153C22×C73—168no—
154C22×C73C2×#12012yes—
155C22×C73#53411yescharacteristic
156C4×C73—48no—
157C4×C73—41no—
158C4×C73—44no—
159C4×C73D4214yes—
160#23897C7311yesmaximal, characteristic, direct product factor, Sylow
161C2×C73—22no—
162C2×C73—21no—
163C2×C73—42no—
164C2×C73—48no—
165C2×C73—84no—
166C2×C73#240311yescharacteristic
167#3484C2×C7312yescomplemented
168#3019C2×C7314yescomplemented
169#2010C2×C7314yescomplemented
170#1911C2×C7314yescomplemented
171#1783C2×C7311yescharacteristic, complemented
172C73#2389711yescharacteristic, direct product factor, Sylow
173C2×H(5,𝔽2)C22×C7312yescomplemented
174C23×D4C22×C7311yescharacteristic
175#575C22×C7314yescomplemented
176#546—28no—
177D42—28no—
178#535C22×C7314yes—
179#534—24no—
180#525C22×C7314yescomplemented
181C2×#120—24no—
182#518C22×C7312yescomplemented
183#455—28no—
184#453—28no—
185#421C22×C7318yescomplemented
186C2×#100C22×C7314yescomplemented
187#407C22×C7312yescomplemented
188H(3,ℤ/4)C22×C7314yescomplemented
189C25D4×C7312yes—
190H(5,𝔽2)—28no—
191H(5,𝔽2)—44no—
192C2×#41—24no—
193C2×#41C23×C7314yes—
194C22×D4—24no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—24no—
198C22×D4—42no—
199C22×D4D4×C7312yes—
200C22×D4C23×C7312yes—
201C23×C4C23×C7311yescharacteristic
202#137—28no—
203AGL(1,ℤ/8)—28no—
204#135—44no—
205C2×M16C23×C7314yes—
206#127D4×C7318yescomplemented
207#124D4×C7318yescomplemented
208#123—28no—
209#123—24no—
210C22⋊Q8—24no—
211#121—216no—
212#121—28no—
213#121—24no—
214#120—216no—
215#120—28no—
216#120—24no—
217#120—28no—
218C4×D4—28no—
219C4×D4—28no—
220#117C23×C7314yes—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223#104—216no—
224#100—28no—
225C22⋊C8—42no—
226C24—24no—
227C24—28no—
228C24—42no—
229C24—84no—
230C24C2×D4×C7311yescharacteristic
231C24C2×D4×C7314yes—
232#41—28no—
233#41—28no—
234#41—416no—
235C2×Q8C2×D4×C7314yescomplemented
236C2×D4—28no—
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—24no—
240C2×D4—216no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—42no—
244C2×D4—416no—
245C2×D4—416no—
246C2×D4—48no—
247C2×D4—48no—
248C2×D4—816no—
249C2×D4C2×D4×C7318yes—
250C2×D4C2×D4×C7314yescomplemented
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4C2×D4×C7311yescharacteristic
257C22×C4C24×C7311yescharacteristic
258Q16—44no—
259SD16—48no—
260D8—44no—
261M16—28no—
262C2×C8—24no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C22⋊C4—28no—
266C22⋊C4—28no—
267C22⋊C4—28no—
268C22⋊C4—216no—
269C22⋊C4—24no—
270C42C2×D4×C7318yescomplemented
271C23—22no—
272C23—24no—
273C23—21no—
274C23—24no—
275C23—24no—
276C23—42no—
277C23—48no—
278C23—416no—
279C23—48no—
280C23—84no—
281C23—84no—
282C23—88no—
283C23—88no—
284C23—84no—
285C23#120×C7318yes—
286C23C22×D4×C7311yescharacteristic
287Q8—28no—
288D4—216no—
289D4—28no—
290D4—48no—
291D4—416no—
292D4—416no—
293D4—44no—
294D4—44no—
295D4—816no—
296C2×C4—28no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—88no—
304C2×C4C22×D4×C7314yes—
305C2×C4H(5,𝔽2)×C7312yes—
306C8—44no—
307C22—24no—
308C22—22no—
309C22—28no—
310C22—41no—
311C22—48no—
312C22—48no—
313C22—42no—
314C22—41no—
315C22—48no—
316C22—41no—
317C22—44no—
318C22—84no—
319C22—816no—
320C22—168no—
321C22C2×#120×C7312yes—
322C22#534×C7311yescharacteristic
323C4—48no—
324C4—41no—
325C4—44no—
326C4D42×C7314yes—
327C2—22no—
328C2—21no—
329C2—42no—
330C2—48no—
331C2—84no—
332C2C73×#240311yescharacteristic
subgroupG/H
center, socleC2×C73#2403
commutator, Frattini, cosocleC22×C4C24×C73
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C73 ◁ C22×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3484C2semidirect
2. C73×#3019C2semidirect
3. C73×#2010C2semidirect
4. C73×#1911C2semidirect
5. C73×#1783C2semidirect
6. C2×H(5,𝔽2)×C73C22semidirect
8. #575×C73C22semidirect
13. #525×C73C22semidirect
15. #518×C73C22semidirect
18. #421×C73C22semidirect
19. C2×#100×C73C22semidirect
20. #407×C73C22semidirect
21. H(3,ℤ/4)×C73C22semidirect
39. #127×C73D4semidirect
40. #124×C73D4semidirect
68. C2×Q8×C73C2×D4semidirect
83. C2×D4×C73C2×D4semidirect
103. C42×C73C2×D4semidirect
160. #23897C73direct (N × H)
167. #3484C2×C73semidirect
168. #3019C2×C73semidirect
169. #2010C2×C73semidirect
170. #1911C2×C73semidirect
171. #1783C2×C73semidirect
172. C73#23897direct (N × H)
173. C2×H(5,𝔽2)C22×C73semidirect
175. #575C22×C73semidirect
180. #525C22×C73semidirect
182. #518C22×C73semidirect
185. #421C22×C73semidirect
186. C2×#100C22×C73semidirect
187. #407C22×C73semidirect
188. H(3,ℤ/4)C22×C73semidirect
206. #127D4×C73semidirect
207. #124D4×C73semidirect
235. C2×Q8C2×D4×C73semidirect
250. C2×D4C2×D4×C73semidirect
270. C42C2×D4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#34842
C73×#30192
C73×#20102
C73×#19112
C73×#17832
#2389773