Finite Group Databasebeta

← C73×#3178 #22607331

Subgroups — C73×#3178 #22607331

Order 9344.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#542×C73C211yesmaximal, characteristic
2#541×C73C211yesmaximal, characteristic, complemented
3#501×C73C211yesmaximal, characteristic, complemented
4#497×C73C211yesmaximal, characteristic, complemented
5#488×C73C211yesmaximal, characteristic, complemented
6#487×C73C211yesmaximal, characteristic, complemented
7#470×C73C211yesmaximal, characteristic, complemented
8#468×C73C211yesmaximal, characteristic
9#467×C73C211yesmaximal, characteristic, complemented
10#465×C73C211yesmaximal, characteristic
11#441×C73C211yesmaximal, characteristic, complemented
12#440×C73C211yesmaximal, characteristic, complemented
13C4×Q16×C73C211yesmaximal, characteristic, complemented
14C4×SD16×C73C211yesmaximal, characteristic, complemented
15#405×C73C211yesmaximal, characteristic, complemented
16C2×Q16×C73C2211yescharacteristic
17C2×Q16×C73C2211yescharacteristic
18C2×SD16×C73C2211yescharacteristic
19C2×SD16×C73C2211yescharacteristic
20C2×M16×C73C2211yescharacteristic
21C22×C8×C73C2211yescharacteristic
22C4⋊Q8×C73C2211yescharacteristic
23C4⋊Q8×C73C2211yescharacteristic
24#126×C73—21no—
25#126×C73—21no—
26#126×C73—21no—
27#125×C73—21no—
28#125×C73—21no—
29#125×C73C2211yescharacteristic
30#124×C73—21no—
31#124×C73C2211yescharacteristic
32#123×C73—21no—
33C22⋊Q8×C73—21no—
34C22⋊Q8×C73C2211yescharacteristic
35C22⋊Q8×C73C2211yescharacteristic
36C22⋊Q8×C73C2211yescharacteristic
37#121×C73C2211yescharacteristic
38C4×Q8×C73C2211yescharacteristic
39C4×Q8×C73C2211yescharacteristic
40C4×Q8×C73C2211yescharacteristic
41C4×D4×C73C2211yescharacteristic
42#117×C73C2211yescharacteristic
43#107×C73C2211yescharacteristic
44#107×C73C2211yescharacteristic
45#106×C73C2211yescharacteristic
46#106×C73C2211yescharacteristic
47#103×C73C2211yescharacteristic
48#103×C73C2211yescharacteristic
49#103×C73C2211yescharacteristic
50#103×C73C2211yescharacteristic
51#103×C73C2211yescharacteristic
52#103×C73C2211yescharacteristic
53#102×C73C2211yescharacteristic
54#102×C73C2211yescharacteristic
55#97×C73C2211yescharacteristic
56#97×C73C2211yescharacteristic
57C4×C8×C73C2211yescharacteristic
58C4×C8×C73C2211yescharacteristic
59C2×Q8×C73—21no—
60C2×Q8×C73C2311yescharacteristic
61C2×Q8×C73C2311yescharacteristic
62C2×Q8×C73C2311yescharacteristic
63C2×D4×C73—21no—
64C2×D4×C73C2311yescharacteristic
65C22×C4×C73—21no—
66C22×C4×C73C2311yescharacteristic
67Q16×C73—22no—
68Q16×C73—22no—
69SD16×C73—22no—
70SD16×C73—22no—
71M16×C73—22no—
72C2×C8×C73—22no—
73C2×C8×C73C2311yescharacteristic
74C2×C8×C73C2311yescharacteristic
75C2×C8×C73C2311yescharacteristic
76C2×C8×C73C2311yescharacteristic
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×C73—21no—
86C4⋊C4×C73—21no—
87C4⋊C4×C73—21no—
88C4⋊C4×C73D411yescharacteristic
89C4⋊C4×C73D411yescharacteristic
90C4⋊C4×C73C2311yescharacteristic
91C4⋊C4×C73C2311yescharacteristic
92C4⋊C4×C73C2311yescharacteristic
93C4⋊C4×C73C2311yescharacteristic
94C22⋊C4×C73—21no—
95C22⋊C4×C73—21no—
96C22⋊C4×C73—21no—
97C22⋊C4×C73—21no—
98C22⋊C4×C73—21no—
99C22⋊C4×C73—21no—
100C22⋊C4×C73—21no—
101C22⋊C4×C73D411yescharacteristic
102C22⋊C4×C73D411yescharacteristic
103C42×C73—21no—
104C42×C73—21no—
105C42×C73—21no—
106C42×C73C2311yescharacteristic
107C42×C73C2311yescharacteristic
108C23×C73—21no—
109C23×C73C2×D411yescharacteristic
110Q8×C73—21no—
111Q8×C73—21no—
112Q8×C73—21no—
113Q8×C73—21no—
114Q8×C73—21no—
115Q8×C73—21no—
116Q8×C73—81no—
117D4×C73—21no—
118D4×C73—21no—
119D4×C73—81no—
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—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—81no—
130C2×C4×C73C2×D411yescharacteristic
131C2×C4×C73C2×D411yescharacteristic
132C2×C4×C73C2×D411yescharacteristic
133C2×C4×C73C2×D411yescharacteristic
134C2×C4×C73C2×D411yescharacteristic
135C2×C4×C73C2411yescharacteristic
136C8×C73—21no—
137C8×C73—21no—
138C8×C73#4112yes—
139C8×C73#4112yes—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—41no—
144C22×C73—41no—
145C22×C73—41no—
146C22×C73C22×D411yescharacteristic
147C4×C73—21no—
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×C73C2×#4111yescharacteristic
161C2×C73—41no—
162C2×C73—81no—
163C2×C73#53211yescharacteristic
164C2×C73#57711yescharacteristic
165C2×C73#57811yescharacteristic
166#3178C7311yesmaximal, characteristic, direct product factor, Sylow
167C73#317811yescharacteristic, direct product factor, Sylow
168#542C2×C7311yescharacteristic
169#541C2×C7311yescharacteristic, complemented
170#501C2×C7311yescharacteristic, complemented
171#497C2×C7311yescharacteristic, complemented
172#488C2×C7311yescharacteristic, complemented
173#487C2×C7311yescharacteristic, complemented
174#470C2×C7311yescharacteristic, complemented
175#468C2×C7311yescharacteristic
176#467C2×C7311yescharacteristic, complemented
177#465C2×C7311yescharacteristic
178#441C2×C7311yescharacteristic, complemented
179#440C2×C7311yescharacteristic, complemented
180C4×Q16C2×C7311yescharacteristic, complemented
181C4×SD16C2×C7311yescharacteristic, complemented
182#405C2×C7311yescharacteristic, complemented
183C2×Q16C22×C7311yescharacteristic
184C2×Q16C22×C7311yescharacteristic
185C2×SD16C22×C7311yescharacteristic
186C2×SD16C22×C7311yescharacteristic
187C2×M16C22×C7311yescharacteristic
188C22×C8C22×C7311yescharacteristic
189C4⋊Q8C22×C7311yescharacteristic
190C4⋊Q8C22×C7311yescharacteristic
191#126—21no—
192#126—21no—
193#126—21no—
194#125—21no—
195#125—21no—
196#125C22×C7311yescharacteristic
197#124—21no—
198#124C22×C7311yescharacteristic
199#123—21no—
200C22⋊Q8—21no—
201C22⋊Q8C22×C7311yescharacteristic
202C22⋊Q8C22×C7311yescharacteristic
203C22⋊Q8C22×C7311yescharacteristic
204#121C22×C7311yescharacteristic
205C4×Q8C22×C7311yescharacteristic
206C4×Q8C22×C7311yescharacteristic
207C4×Q8C22×C7311yescharacteristic
208C4×D4C22×C7311yescharacteristic
209#117C22×C7311yescharacteristic
210#107C22×C7311yescharacteristic
211#107C22×C7311yescharacteristic
212#106C22×C7311yescharacteristic
213#106C22×C7311yescharacteristic
214#103C22×C7311yescharacteristic
215#103C22×C7311yescharacteristic
216#103C22×C7311yescharacteristic
217#103C22×C7311yescharacteristic
218#103C22×C7311yescharacteristic
219#103C22×C7311yescharacteristic
220#102C22×C7311yescharacteristic
221#102C22×C7311yescharacteristic
222#97C22×C7311yescharacteristic
223#97C22×C7311yescharacteristic
224C4×C8C22×C7311yescharacteristic
225C4×C8C22×C7311yescharacteristic
226C2×Q8—21no—
227C2×Q8C23×C7311yescharacteristic
228C2×Q8C23×C7311yescharacteristic
229C2×Q8C23×C7311yescharacteristic
230C2×D4—21no—
231C2×D4C23×C7311yescharacteristic
232C22×C4—21no—
233C22×C4C23×C7311yescharacteristic
234Q16—22no—
235Q16—22no—
236SD16—22no—
237SD16—22no—
238M16—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⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4D4×C7311yescharacteristic
256C4⋊C4D4×C7311yescharacteristic
257C4⋊C4C23×C7311yescharacteristic
258C4⋊C4C23×C7311yescharacteristic
259C4⋊C4C23×C7311yescharacteristic
260C4⋊C4C23×C7311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4D4×C7311yescharacteristic
269C22⋊C4D4×C7311yescharacteristic
270C42—21no—
271C42—21no—
272C42—21no—
273C42C23×C7311yescharacteristic
274C42C23×C7311yescharacteristic
275C23—21no—
276C23C2×D4×C7311yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—81no—
284D4—21no—
285D4—21no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—81no—
297C2×C4C2×D4×C7311yescharacteristic
298C2×C4C2×D4×C7311yescharacteristic
299C2×C4C2×D4×C7311yescharacteristic
300C2×C4C2×D4×C7311yescharacteristic
301C2×C4C2×D4×C7311yescharacteristic
302C2×C4C24×C7311yescharacteristic
303C8—21no—
304C8—21no—
305C8#41×C7312yes—
306C8#41×C7312yes—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C22×D4×C7311yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4C2×#41×C7311yescharacteristic
327C4C2×#41×C7311yescharacteristic
328C2—41no—
329C2—81no—
330C2#532×C7311yescharacteristic
331C2#577×C7311yescharacteristic
332C2#578×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C22×D4
commutator, Frattini, cosocleC2×C4C24×C73
prime radical#121×C73C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ #117×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. #541×C73C2semidirect
3. #501×C73C2semidirect
4. #497×C73C2semidirect
5. #488×C73C2semidirect
6. #487×C73C2semidirect
7. #470×C73C2semidirect
9. #467×C73C2semidirect
11. #441×C73C2semidirect
12. #440×C73C2semidirect
13. C4×Q16×C73C2semidirect
14. C4×SD16×C73C2semidirect
15. #405×C73C2semidirect
166. #3178C73direct (N × H)
167. C73#3178direct (N × H)
169. #541C2×C73semidirect
170. #501C2×C73semidirect
171. #497C2×C73semidirect
172. #488C2×C73semidirect
173. #487C2×C73semidirect
174. #470C2×C73semidirect
176. #467C2×C73semidirect
178. #441C2×C73semidirect
179. #440C2×C73semidirect
180. C4×Q16C2×C73semidirect
181. C4×SD16C2×C73semidirect
182. #405C2×C73semidirect

Maximal subgroups

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

subgroupindex
#542×C732
#541×C732
#501×C732
#497×C732
#488×C732
#487×C732
#470×C732
#468×C732
#467×C732
#465×C732
#441×C732
#440×C732
C4×Q16×C732
C4×SD16×C732
#405×C732
#317873