Finite Group Databasebeta

← C73×#33798 #32943202

Subgroups — C73×#33798 #32943202

Order 18688.

468 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3168C211yesmaximal, characteristic
2C2×#499×C73C211yesmaximal, characteristic, complemented
3C73×#2253C212yesmaximal, complemented
4C73×#2242C212yesmaximal, complemented
5C73×#2236C211yesmaximal, characteristic, complemented
6C73×#2235C211yesmaximal, characteristic, complemented
7C73×#2233C211yesmaximal, characteristic
8C73×#2232C211yesmaximal, characteristic
9C73×#2229C212yesmaximal, complemented
10C73×#2228C212yesmaximal, complemented
11C73×#2162C211yesmaximal, characteristic, complemented
12#529×C73—21no—
13C2×#125×C73—21no—
14#499×C73—21no—
15#499×C73—21no—
16#499×C73C2212yes—
17#499×C73C2212yes—
18#499×C73C2211yescharacteristic
19#495×C73C2211yescharacteristic
20#488×C73—21no—
21#467×C73C2211yescharacteristic
22#466×C73C2211yescharacteristic
23#465×C73—21no—
24C4×Q16×C73C2211yescharacteristic
25C4×SD16×C73—21no—
26C4×D8×C73C2211yescharacteristic
27C2×#107×C73C2211yescharacteristic
28C2×#107×C73C2211yescharacteristic
29C2×#106×C73—21no—
30C2×C4×C8×C73C2211yescharacteristic
31#367×C73C2212yes—
32#367×C73C2212yes—
33#366×C73C2212yes—
34#366×C73C2212yes—
35C4⋊C16×C73C2212yes—
36#358×C73C2212yes—
37#358×C73C2212yes—
38#357×C73C2212yes—
39#357×C73C2212yes—
40C22⋊C16×C73C2211yescharacteristic
41C22⋊C16×C73C2211yescharacteristic
42C4×C16×C73C2212yes—
43C2×Q16×C73C2311yescharacteristic
44C2×SD16×C73—21no—
45C2×D8×C73C2311yescharacteristic
46C22×C8×C73D411yescharacteristic
47C22×C8×C73C2311yescharacteristic
48#126×C73—41no—
49#125×C73—21no—
50#125×C73—22no—
51#125×C73—22no—
52#125×C73—41no—
53#125×C73—41no—
54#124×C73—21no—
55#123×C73—41no—
56C22⋊Q8×C73—21no—
57#121×C73—21no—
58C4×Q8×C73—21no—
59C4×D4×C73—21no—
60C4×D4×C73—41no—
61#117×C73—41no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73—41no—
65C2×C4⋊C4×C73—41no—
66C2×C42×C73D411yescharacteristic
67C2×C16×C73C2312yes—
68C2×C16×C73C2312yes—
69#107×C73—21no—
70#107×C73—21no—
71#107×C73C2312yes—
72#107×C73C2312yes—
73#107×C73C2311yescharacteristic
74#107×C73C2311yescharacteristic
75#106×C73—22no—
76#106×C73—22no—
77#106×C73—21no—
78#103×C73—21no—
79#103×C73—21no—
80#102×C73—21no—
81#102×C73—21no—
82C4×C8×C73D411yescharacteristic
83C4×C8×C73D411yescharacteristic
84C4×C8×C73C2311yescharacteristic
85C4×C8×C73C2311yescharacteristic
86C2×Q8×C73—21no—
87C2×D4×C73—21no—
88C2×D4×C73—41no—
89C22×C4×C73—41no—
90C22×C4×C73—41no—
91C22×C4×C73—41no—
92C22×C4×C73D811yescharacteristic
93C22×C4×C73D811yescharacteristic
94C22×C4×C73C2×D411yescharacteristic
95Q16×C73—22no—
96SD16×C73—42no—
97D8×C73—22no—
98C2×C8×C73—22no—
99C2×C8×C73—22no—
100C2×C8×C73C2×D411yescharacteristic
101C2×C8×C73C2×D411yescharacteristic
102C2×C8×C73C2×D411yescharacteristic
103C2×C8×C73C2411yescharacteristic
104C4⋊C4×C73—21no—
105C4⋊C4×C73—21no—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—22no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C4⋊C4×C73—42no—
111C4⋊C4×C73—42no—
112C4⋊C4×C73—42no—
113C4⋊C4×C73—42no—
114C4⋊C4×C73—41no—
115C4⋊C4×C73—41no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C22⋊C4×C73—41no—
119C22⋊C4×C73—41no—
120C22⋊C4×C73—41no—
121C22⋊C4×C73—41no—
122C22⋊C4×C73—41no—
123C42×C73—41no—
124C42×C73D811yescharacteristic
125C42×C73D811yescharacteristic
126C42×C73C2×D411yescharacteristic
127C42×C73C2×D411yescharacteristic
128C16×C73—22no—
129C16×C73—22no—
130C23×C73—41no—
131C23×C73C2×D811yescharacteristic
132Q8×C73—41no—
133Q8×C73—41no—
134D4×C73—41no—
135D4×C73—41no—
136D4×C73—82no—
137C2×C4×C73—22no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—42no—
146C2×C4×C73—82no—
147C2×C4×C73—82no—
148C2×C4×C73—82no—
149C2×C4×C73C2×D811yescharacteristic
150C2×C4×C73C2×D811yescharacteristic
151C2×C4×C73C2×D811yescharacteristic
152C2×C4×C73C2×D811yescharacteristic
153C2×C4×C73C2×D811yescharacteristic
154C2×C4×C73C22×D411yescharacteristic
155C8×C73—21no—
156C8×C73—21no—
157C8×C73#13712yes—
158C8×C732−1+412yes—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—81no—
162C22×C73—81no—
163C22×C73—81no—
164C22×C73#50812yes—
165C22×C73C22×D811yescharacteristic
166C4×C73—21no—
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—21no—
171C4×C73—82no—
172C4×C73—81no—
173C4×C73—81no—
174C4×C73—82no—
175C4×C73—81no—
176C4×C73#53611yescharacteristic
177C4×C73C2×#13711yescharacteristic
178#33798C7311yesmaximal, characteristic, direct product factor, Sylow
179C2×C73—22no—
180C2×C73—161no—
181C2×C73#322611yescharacteristic
182C2×C73C2×#50811yescharacteristic
183C2×C73#341411yescharacteristic
184#3168C2×C7311yescharacteristic
185C2×#499C2×C7311yescharacteristic, complemented
186#2253C2×C7312yescomplemented
187#2242C2×C7312yescomplemented
188#2236C2×C7311yescharacteristic, complemented
189#2235C2×C7311yescharacteristic, complemented
190#2233C2×C7311yescharacteristic
191#2232C2×C7311yescharacteristic
192#2229C2×C7312yescomplemented
193#2228C2×C7312yescomplemented
194#2162C2×C7311yescharacteristic, complemented
195C73#3379811yescharacteristic, direct product factor, Sylow
196#529—21no—
197C2×#125—21no—
198#499—21no—
199#499—21no—
200#499C22×C7312yes—
201#499C22×C7312yes—
202#499C22×C7311yescharacteristic
203#495C22×C7311yescharacteristic
204#488—21no—
205#467C22×C7311yescharacteristic
206#466C22×C7311yescharacteristic
207#465—21no—
208C4×Q16C22×C7311yescharacteristic
209C4×SD16—21no—
210C4×D8C22×C7311yescharacteristic
211C2×#107C22×C7311yescharacteristic
212C2×#107C22×C7311yescharacteristic
213C2×#106—21no—
214C2×C4×C8C22×C7311yescharacteristic
215#367C22×C7312yes—
216#367C22×C7312yes—
217#366C22×C7312yes—
218#366C22×C7312yes—
219C4⋊C16C22×C7312yes—
220#358C22×C7312yes—
221#358C22×C7312yes—
222#357C22×C7312yes—
223#357C22×C7312yes—
224C22⋊C16C22×C7311yescharacteristic
225C22⋊C16C22×C7311yescharacteristic
226C4×C16C22×C7312yes—
227C2×Q16C23×C7311yescharacteristic
228C2×SD16—21no—
229C2×D8C23×C7311yescharacteristic
230C22×C8D4×C7311yescharacteristic
231C22×C8C23×C7311yescharacteristic
232#126—41no—
233#125—22no—
234#125—22no—
235#125—21no—
236#125—41no—
237#125—41no—
238#124—21no—
239#123—41no—
240C22⋊Q8—21no—
241#121—21no—
242C4×Q8—21no—
243C4×D4—21no—
244C4×D4—41no—
245#117—41no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C42D4×C7311yescharacteristic
251C2×C16C23×C7312yes—
252C2×C16C23×C7312yes—
253#107—21no—
254#107—21no—
255#107C23×C7312yes—
256#107C23×C7312yes—
257#107C23×C7311yescharacteristic
258#107C23×C7311yescharacteristic
259#106—22no—
260#106—22no—
261#106—21no—
262#103—21no—
263#103—21no—
264#102—21no—
265#102—21no—
266C4×C8D4×C7311yescharacteristic
267C4×C8D4×C7311yescharacteristic
268C4×C8C23×C7311yescharacteristic
269C4×C8C23×C7311yescharacteristic
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4D8×C7311yescharacteristic
277C22×C4D8×C7311yescharacteristic
278C22×C4C2×D4×C7311yescharacteristic
279Q16—22no—
280SD16—42no—
281D8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8C2×D4×C7311yescharacteristic
285C2×C8C2×D4×C7311yescharacteristic
286C2×C8C2×D4×C7311yescharacteristic
287C2×C8C24×C7311yescharacteristic
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—41no—
308C42D8×C7311yescharacteristic
309C42D8×C7311yescharacteristic
310C42C2×D4×C7311yescharacteristic
311C42C2×D4×C7311yescharacteristic
312C16—22no—
313C16—22no—
314C23—41no—
315C23C2×D8×C7311yescharacteristic
316Q8—41no—
317Q8—41no—
318D4—41no—
319D4—41no—
320D4—82no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4C2×D8×C7311yescharacteristic
334C2×C4C2×D8×C7311yescharacteristic
335C2×C4C2×D8×C7311yescharacteristic
336C2×C4C2×D8×C7311yescharacteristic
337C2×C4C2×D8×C7311yescharacteristic
338C2×C4C22×D4×C7311yescharacteristic
339C8—21no—
340C8—21no—
341C8#137×C7312yes—
342C82−1+4×C7312yes—
343C22—21no—
344C22—21no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#508×C7312yes—
349C22C22×D8×C7311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—82no—
357C4—81no—
358C4—81no—
359C4—82no—
360C4#536×C7311yescharacteristic
361C4C2×#137×C7311yescharacteristic
362C2—22no—
363C2—161no—
364C2C73×#322611yescharacteristic
365C2C2×#508×C7311yescharacteristic
366C2C73×#341411yescharacteristic
subgroupG/H
center, socleC22×C73C22×D8
commutator, Frattini, cosocleC2×C8C24×C73
prime radical#466×C73C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#499×C73C2semidirect
3. C73×#2253C2semidirect
4. C73×#2242C2semidirect
5. C73×#2236C2semidirect
6. C73×#2235C2semidirect
9. C73×#2229C2semidirect
10. C73×#2228C2semidirect
11. C73×#2162C2semidirect
178. #33798C73direct (N × H)
185. C2×#499C2×C73semidirect
186. #2253C2×C73semidirect
187. #2242C2×C73semidirect
188. #2236C2×C73semidirect
189. #2235C2×C73semidirect
192. #2229C2×C73semidirect
193. #2228C2×C73semidirect
194. #2162C2×C73semidirect
195. C73#33798direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#31682
C2×#499×C732
C73×#22532
C73×#22422
C73×#22362
C73×#22352
C73×#22332
C73×#22322
C73×#22292
C73×#22282
C73×#21622
#3379873