Finite Group Databasebeta

← C73×#3337 #22607490

Subgroups — C73×#3337 #22607490

Order 9344.

354 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#553×C73C211yesmaximal, characteristic
2#552×C73C211yesmaximal, characteristic, complemented
3#499×C73C211yesmaximal, characteristic, complemented
4#485×C73C211yesmaximal, characteristic, complemented
5#484×C73C211yesmaximal, characteristic, complemented
6#482×C73C211yesmaximal, characteristic, complemented
7#480×C73C211yesmaximal, characteristic, complemented
8#479×C73C211yesmaximal, characteristic, complemented
9#478×C73C211yesmaximal, characteristic, complemented
10#465×C73C211yesmaximal, characteristic, complemented
11#465×C73C211yesmaximal, characteristic, complemented
12C4×SD16×C73C211yesmaximal, characteristic, complemented
13C8×D4×C73C211yesmaximal, characteristic, complemented
14#427×C73C211yesmaximal, characteristic, complemented
15C2×#106×C73C211yesmaximal, characteristic, complemented
16C2×SD16×C73C2211yescharacteristic
17C22×C8×C73C2211yescharacteristic
18C22×C8×C73C2211yescharacteristic
19#126×C73—21no—
20#126×C73—21no—
21#125×C73—21no—
22#125×C73C2211yescharacteristic
23#125×C73C2211yescharacteristic
24#123×C73—21no—
25#123×C73—21no—
26C22⋊Q8×C73C2211yescharacteristic
27C22⋊Q8×C73C2211yescharacteristic
28#121×C73—21no—
29#121×C73C2211yescharacteristic
30#121×C73C2211yescharacteristic
31C4×Q8×C73C2211yescharacteristic
32C4×D4×C73—21no—
33C4×D4×C73C2211yescharacteristic
34C4×D4×C73C2211yescharacteristic
35#117×C73—21no—
36#117×C73C2211yescharacteristic
37#117×C73C2211yescharacteristic
38C2×C4⋊C4×C73C2211yescharacteristic
39C2×C4⋊C4×C73C2211yescharacteristic
40#107×C73C2211yescharacteristic
41#107×C73C2211yescharacteristic
42#107×C73C2211yescharacteristic
43#107×C73C2211yescharacteristic
44#106×C73C2211yescharacteristic
45#106×C73C2211yescharacteristic
46#106×C73C2211yescharacteristic
47#106×C73C2211yescharacteristic
48#106×C73C2211yescharacteristic
49C4⋊C8×C73C2211yescharacteristic
50#103×C73C2211yescharacteristic
51#103×C73C2211yescharacteristic
52#103×C73C2211yescharacteristic
53#102×C73C2211yescharacteristic
54#102×C73C2211yescharacteristic
55#102×C73C2211yescharacteristic
56C22⋊C8×C73C2211yescharacteristic
57C22⋊C8×C73C2211yescharacteristic
58C4×C8×C73C2211yescharacteristic
59C2×Q8×C73C2311yescharacteristic
60C2×D4×C73—21no—
61C2×D4×C73—21no—
62C2×D4×C73D411yescharacteristic
63C2×D4×C73C2311yescharacteristic
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73C2311yescharacteristic
68C22×C4×C73C2311yescharacteristic
69SD16×C73—22no—
70C2×C8×C73—22no—
71C2×C8×C73—22no—
72C2×C8×C73C2311yescharacteristic
73C2×C8×C73C2311yescharacteristic
74C2×C8×C73C2311yescharacteristic
75C2×C8×C73C2311yescharacteristic
76C4⋊C4×C73—21no—
77C4⋊C4×C73—21no—
78C4⋊C4×C73—21no—
79C4⋊C4×C73—21no—
80C4⋊C4×C73—21no—
81C4⋊C4×C73—21no—
82C4⋊C4×C73—21no—
83C4⋊C4×C73—21no—
84C4⋊C4×C73—21no—
85C4⋊C4×C73D411yescharacteristic
86C4⋊C4×C73C2311yescharacteristic
87C4⋊C4×C73C2311yescharacteristic
88C4⋊C4×C73C2311yescharacteristic
89C4⋊C4×C73C2311yescharacteristic
90C4⋊C4×C73C2311yescharacteristic
91C4⋊C4×C73C2311yescharacteristic
92C22⋊C4×C73—21no—
93C22⋊C4×C73—21no—
94C22⋊C4×C73—21no—
95C22⋊C4×C73—21no—
96C22⋊C4×C73—21no—
97C22⋊C4×C73—21no—
98C22⋊C4×C73—21no—
99C22⋊C4×C73D411yescharacteristic
100C22⋊C4×C73D411yescharacteristic
101C42×C73—21no—
102C42×C73—21no—
103C42×C73—21no—
104C42×C73C2311yescharacteristic
105C23×C73—21no—
106C23×C73C2×D411yescharacteristic
107C23×C73C2×D411yescharacteristic
108Q8×C73—21no—
109Q8×C73—21no—
110D4×C73—22no—
111D4×C73—21no—
112D4×C73—21no—
113D4×C73—42no—
114D4×C73—81no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—42no—
127C2×C4×C73—81no—
128C2×C4×C73—81no—
129C2×C4×C73C2×D411yescharacteristic
130C2×C4×C73C2×D411yescharacteristic
131C2×C4×C73C2×D411yescharacteristic
132C2×C4×C73C2×D411yescharacteristic
133C2×C4×C73C2411yescharacteristic
134C8×C73—21no—
135C8×C73—21no—
136C8×C73—21no—
137C8×C73#4112yes—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—41no—
144C22×C73—41no—
145C22×C73—41no—
146C22×C73#13512yes—
147C22×C73C22×D411yescharacteristic
148C4×C73—21no—
149C4×C73—21no—
150C4×C73—21no—
151C4×C73—21no—
152C4×C73—41no—
153C4×C73—41no—
154C4×C73—41no—
155C4×C73—41no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73—41no—
159C4×C73C2×#4111yescharacteristic
160C4×C732−1+411yescharacteristic
161C2×C73—22no—
162C2×C73—41no—
163C2×C73—81no—
164C2×C73#54711yescharacteristic
165C2×C73C2×#13511yescharacteristic
166C2×C73#57711yescharacteristic
167#3337C7311yesmaximal, characteristic, direct product factor, Sylow
168C73#333711yescharacteristic, direct product factor, Sylow
169#553C2×C7311yescharacteristic
170#552C2×C7311yescharacteristic, complemented
171#499C2×C7311yescharacteristic, complemented
172#485C2×C7311yescharacteristic, complemented
173#484C2×C7311yescharacteristic, complemented
174#482C2×C7311yescharacteristic, complemented
175#480C2×C7311yescharacteristic, complemented
176#479C2×C7311yescharacteristic, complemented
177#478C2×C7311yescharacteristic, complemented
178#465C2×C7311yescharacteristic, complemented
179#465C2×C7311yescharacteristic, complemented
180C4×SD16C2×C7311yescharacteristic, complemented
181C8×D4C2×C7311yescharacteristic, complemented
182#427C2×C7311yescharacteristic, complemented
183C2×#106C2×C7311yescharacteristic, complemented
184C2×SD16C22×C7311yescharacteristic
185C22×C8C22×C7311yescharacteristic
186C22×C8C22×C7311yescharacteristic
187#126—21no—
188#126—21no—
189#125—21no—
190#125C22×C7311yescharacteristic
191#125C22×C7311yescharacteristic
192#123—21no—
193#123—21no—
194C22⋊Q8C22×C7311yescharacteristic
195C22⋊Q8C22×C7311yescharacteristic
196#121—21no—
197#121C22×C7311yescharacteristic
198#121C22×C7311yescharacteristic
199C4×Q8C22×C7311yescharacteristic
200C4×D4—21no—
201C4×D4C22×C7311yescharacteristic
202C4×D4C22×C7311yescharacteristic
203#117—21no—
204#117C22×C7311yescharacteristic
205#117C22×C7311yescharacteristic
206C2×C4⋊C4C22×C7311yescharacteristic
207C2×C4⋊C4C22×C7311yescharacteristic
208#107C22×C7311yescharacteristic
209#107C22×C7311yescharacteristic
210#107C22×C7311yescharacteristic
211#107C22×C7311yescharacteristic
212#106C22×C7311yescharacteristic
213#106C22×C7311yescharacteristic
214#106C22×C7311yescharacteristic
215#106C22×C7311yescharacteristic
216#106C22×C7311yescharacteristic
217C4⋊C8C22×C7311yescharacteristic
218#103C22×C7311yescharacteristic
219#103C22×C7311yescharacteristic
220#103C22×C7311yescharacteristic
221#102C22×C7311yescharacteristic
222#102C22×C7311yescharacteristic
223#102C22×C7311yescharacteristic
224C22⋊C8C22×C7311yescharacteristic
225C22⋊C8C22×C7311yescharacteristic
226C4×C8C22×C7311yescharacteristic
227C2×Q8C23×C7311yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4D4×C7311yescharacteristic
231C2×D4C23×C7311yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C23×C7311yescharacteristic
236C22×C4C23×C7311yescharacteristic
237SD16—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8C23×C7311yescharacteristic
241C2×C8C23×C7311yescharacteristic
242C2×C8C23×C7311yescharacteristic
243C2×C8C23×C7311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4D4×C7311yescharacteristic
254C4⋊C4C23×C7311yescharacteristic
255C4⋊C4C23×C7311yescharacteristic
256C4⋊C4C23×C7311yescharacteristic
257C4⋊C4C23×C7311yescharacteristic
258C4⋊C4C23×C7311yescharacteristic
259C4⋊C4C23×C7311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4D4×C7311yescharacteristic
268C22⋊C4D4×C7311yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42C23×C7311yescharacteristic
273C23—21no—
274C23C2×D4×C7311yescharacteristic
275C23C2×D4×C7311yescharacteristic
276Q8—21no—
277Q8—21no—
278D4—22no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4C2×D4×C7311yescharacteristic
298C2×C4C2×D4×C7311yescharacteristic
299C2×C4C2×D4×C7311yescharacteristic
300C2×C4C2×D4×C7311yescharacteristic
301C2×C4C24×C7311yescharacteristic
302C8—21no—
303C8—21no—
304C8—21no—
305C8#41×C7312yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22#135×C7312yes—
315C22C22×D4×C7311yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4C2×#41×C7311yescharacteristic
328C42−1+4×C7311yescharacteristic
329C2—22no—
330C2—41no—
331C2—81no—
332C2#547×C7311yescharacteristic
333C2C2×#135×C7311yescharacteristic
334C2#577×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C22×D4
commutator, Frattini, cosocleC2×C4C24×C73
prime radical#553×C73C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C4×D4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. #552×C73C2semidirect
3. #499×C73C2semidirect
4. #485×C73C2semidirect
5. #484×C73C2semidirect
6. #482×C73C2semidirect
7. #480×C73C2semidirect
8. #479×C73C2semidirect
9. #478×C73C2semidirect
10. #465×C73C2semidirect
11. #465×C73C2semidirect
12. C4×SD16×C73C2semidirect
13. C8×D4×C73C2semidirect
14. #427×C73C2semidirect
15. C2×#106×C73C2semidirect
167. #3337C73direct (N × H)
168. C73#3337direct (N × H)
170. #552C2×C73semidirect
171. #499C2×C73semidirect
172. #485C2×C73semidirect
173. #484C2×C73semidirect
174. #482C2×C73semidirect
175. #480C2×C73semidirect
176. #479C2×C73semidirect
177. #478C2×C73semidirect
178. #465C2×C73semidirect
179. #465C2×C73semidirect
180. C4×SD16C2×C73semidirect
181. C8×D4C2×C73semidirect
182. #427C2×C73semidirect
183. C2×#106C2×C73semidirect

Maximal subgroups

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

subgroupindex
#553×C732
#552×C732
#499×C732
#485×C732
#484×C732
#482×C732
#480×C732
#479×C732
#478×C732
#465×C732
C4×SD16×C732
C8×D4×C732
#427×C732
C2×#106×C732
#333773