Finite Group Databasebeta

← C73×#33814 #32943218

Subgroups — C73×#33814 #32943218

Order 18688.

496 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3170C211yesmaximal, characteristic, complemented
2C73×#3168C211yesmaximal, characteristic
3C73×#2229C212yesmaximal, complemented
4C73×#2225C211yesmaximal, characteristic, complemented
5C73×#2224C211yesmaximal, characteristic, complemented
6C73×#2217C212yesmaximal
7C73×#2216C212yesmaximal, complemented
8C73×#2211C212yesmaximal, complemented
9C73×#2209C211yesmaximal, characteristic, complemented
10C73×#2208C211yesmaximal, characteristic, complemented
11C2×C4⋊C16×C73C211yesmaximal, characteristic, complemented
12#533×C73—21no—
13#529×C73—21no—
14C2×Q32×C73C2212yes—
15C2×SD32×C73C2212yes—
16C22×C16×C73C2212yes—
17#500×C73C2211yescharacteristic
18#499×C73C2211yescharacteristic
19#495×C73C2211yescharacteristic
20#494×C73C2211yescharacteristic
21#488×C73—21no—
22#487×C73—21no—
23#467×C73—21no—
24#467×C73C2211yescharacteristic
25#467×C73C2212yes—
26#466×C73C2211yescharacteristic
27#465×C73—21no—
28C4×Q16×C73—21no—
29C4×Q16×C73C2211yescharacteristic
30C4×Q16×C73C2212yes—
31C4×SD16×C73—21no—
32C4×D8×C73C2211yescharacteristic
33C2×C4×C8×C73C2211yescharacteristic
34#367×C73C2212yes—
35#366×C73C2212yes—
36C4⋊C16×C73C2211yescharacteristic
37C4⋊C16×C73C2211yescharacteristic
38C4⋊C16×C73C2212yes—
39#358×C73C2212yes—
40#358×C73C2212yes—
41#358×C73C2212yes—
42#357×C73C2212yes—
43C2×Q16×C73—21no—
44C2×Q16×C73C2312yes—
45C2×Q16×C73C2311yescharacteristic
46C2×SD16×C73—21no—
47C2×D8×C73C2311yescharacteristic
48C22×C8×C73D411yescharacteristic
49C22×C8×C73C2311yescharacteristic
50C4⋊Q8×C73—21no—
51C4⋊Q8×C73—21no—
52#126×C73—41no—
53#125×C73—21no—
54#125×C73—41no—
55#124×C73—21no—
56#123×C73—41no—
57C22⋊Q8×C73—21no—
58C22⋊Q8×C73—22no—
59C22⋊Q8×C73—41no—
60#121×C73—21no—
61C4×Q8×C73—21no—
62C4×Q8×C73—22no—
63C4×Q8×C73—41no—
64C4×D4×C73—21no—
65C4×D4×C73—41no—
66#117×C73—41no—
67#117×C73—41no—
68C2×C42×C73D411yescharacteristic
69Q32×C73—42no—
70SD32×C73—42no—
71C2×C16×C73—42no—
72C2×C16×C73C2312yes—
73C2×C16×C73C2312yes—
74#107×C73—21no—
75#107×C73C2311yescharacteristic
76#107×C73C2311yescharacteristic
77#107×C73C2312yes—
78#106×C73—21no—
79#103×C73—21no—
80#103×C73—22no—
81#103×C73—21no—
82#103×C73—22no—
83#102×C73—21no—
84#102×C73—21no—
85C4×C8×C73D411yescharacteristic
86C4×C8×C73D411yescharacteristic
87C4×C8×C73C2311yescharacteristic
88C4×C8×C73C2311yescharacteristic
89C2×Q8×C73—21no—
90C2×Q8×C73—22no—
91C2×Q8×C73—41no—
92C2×D4×C73—21no—
93C2×D4×C73—41no—
94C22×C4×C73—41no—
95C22×C4×C73D811yescharacteristic
96C22×C4×C73D811yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98Q16×C73—22no—
99Q16×C73—21no—
100Q16×C73—22no—
101Q16×C73—21no—
102Q16×C73—42no—
103SD16×C73—42no—
104D8×C73—21no—
105D8×C73—21no—
106C2×C8×C73—21no—
107C2×C8×C73—21no—
108C2×C8×C73—41no—
109C2×C8×C73C2×D411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73C2411yescharacteristic
113C4⋊C4×C73—21no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—42no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73—42no—
124C4⋊C4×C73—41no—
125C22⋊C4×C73—42no—
126C22⋊C4×C73—41no—
127C22⋊C4×C73—41no—
128C22⋊C4×C73—41no—
129C22⋊C4×C73—41no—
130C22⋊C4×C73—41no—
131C42×C73—41no—
132C42×C73—42no—
133C42×C73D811yescharacteristic
134C42×C73D811yescharacteristic
135C42×C73C2×D411yescharacteristic
136C42×C73C2×D411yescharacteristic
137C16×C73—22no—
138C16×C73—22no—
139C23×C73—41no—
140C23×C73C2×D811yescharacteristic
141Q8×C73—42no—
142Q8×C73—41no—
143Q8×C73—41no—
144Q8×C73—42no—
145Q8×C73—82no—
146D4×C73—41no—
147D4×C73—41no—
148D4×C73—82no—
149C2×C4×C73—22no—
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—22no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—42no—
156C2×C4×C73—42no—
157C2×C4×C73—41no—
158C2×C4×C73—82no—
159C2×C4×C73C2×D811yescharacteristic
160C2×C4×C73C2×D811yescharacteristic
161C2×C4×C73C2×D811yescharacteristic
162C2×C4×C73C2×D811yescharacteristic
163C2×C4×C73C2×D811yescharacteristic
164C2×C4×C73C22×D411yescharacteristic
165C8×C73—21no—
166C8×C73—21no—
167C8×C73—21no—
168C8×C73H(5,𝔽2)11yescharacteristic
169C8×C732−1+411yescharacteristic
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73—81no—
175C22×C73#51012yes—
176C22×C73C22×D811yescharacteristic
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—81no—
182C4×C73—82no—
183C4×C73—81no—
184C4×C73—81no—
185C4×C73—82no—
186C4×C73#50812yes—
187C4×C73#53711yescharacteristic
188C4×C73#57511yescharacteristic
189#33814C7311yesmaximal, characteristic, direct product factor, Sylow
190C2×C73—22no—
191C2×C73—161no—
192C2×C73#324511yescharacteristic
193C2×C73C2×#50811yescharacteristic
194C2×C73C2×#51011yescharacteristic
195#3170C2×C7311yescharacteristic, complemented
196#3168C2×C7311yescharacteristic
197#2229C2×C7312yescomplemented
198#2225C2×C7311yescharacteristic, complemented
199#2224C2×C7311yescharacteristic, complemented
200#2217C2×C7312yes—
201#2216C2×C7312yescomplemented
202#2211C2×C7312yescomplemented
203#2209C2×C7311yescharacteristic, complemented
204#2208C2×C7311yescharacteristic, complemented
205C2×C4⋊C16C2×C7311yescharacteristic, complemented
206C73#3381411yescharacteristic, direct product factor, Sylow
207#533—21no—
208#529—21no—
209C2×Q32C22×C7312yes—
210C2×SD32C22×C7312yes—
211C22×C16C22×C7312yes—
212#500C22×C7311yescharacteristic
213#499C22×C7311yescharacteristic
214#495C22×C7311yescharacteristic
215#494C22×C7311yescharacteristic
216#488—21no—
217#487—21no—
218#467—21no—
219#467C22×C7311yescharacteristic
220#467C22×C7312yes—
221#466C22×C7311yescharacteristic
222#465—21no—
223C4×Q16—21no—
224C4×Q16C22×C7311yescharacteristic
225C4×Q16C22×C7312yes—
226C4×SD16—21no—
227C4×D8C22×C7311yescharacteristic
228C2×C4×C8C22×C7311yescharacteristic
229#367C22×C7312yes—
230#366C22×C7312yes—
231C4⋊C16C22×C7311yescharacteristic
232C4⋊C16C22×C7311yescharacteristic
233C4⋊C16C22×C7312yes—
234#358C22×C7312yes—
235#358C22×C7312yes—
236#358C22×C7312yes—
237#357C22×C7312yes—
238C2×Q16—21no—
239C2×Q16C23×C7312yes—
240C2×Q16C23×C7311yescharacteristic
241C2×SD16—21no—
242C2×D8C23×C7311yescharacteristic
243C22×C8D4×C7311yescharacteristic
244C22×C8C23×C7311yescharacteristic
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247#126—41no—
248#125—21no—
249#125—41no—
250#124—21no—
251#123—41no—
252C22⋊Q8—21no—
253C22⋊Q8—22no—
254C22⋊Q8—41no—
255#121—21no—
256C4×Q8—21no—
257C4×Q8—22no—
258C4×Q8—41no—
259C4×D4—21no—
260C4×D4—41no—
261#117—41no—
262#117—41no—
263C2×C42D4×C7311yescharacteristic
264Q32—42no—
265SD32—42no—
266C2×C16—42no—
267C2×C16C23×C7312yes—
268C2×C16C23×C7312yes—
269#107—21no—
270#107C23×C7312yes—
271#107C23×C7311yescharacteristic
272#107C23×C7311yescharacteristic
273#106—21no—
274#103—21no—
275#103—22no—
276#103—21no—
277#103—22no—
278#102—21no—
279#102—21no—
280C4×C8D4×C7311yescharacteristic
281C4×C8D4×C7311yescharacteristic
282C4×C8C23×C7311yescharacteristic
283C4×C8C23×C7311yescharacteristic
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—41no—
290C22×C4D8×C7311yescharacteristic
291C22×C4D8×C7311yescharacteristic
292C22×C4C2×D4×C7311yescharacteristic
293Q16—22no—
294Q16—21no—
295Q16—22no—
296Q16—21no—
297Q16—42no—
298SD16—42no—
299D8—21no—
300D8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8C2×D4×C7311yescharacteristic
305C2×C8C2×D4×C7311yescharacteristic
306C2×C8C2×D4×C7311yescharacteristic
307C2×C8C24×C7311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C42—41no—
327C42—42no—
328C42D8×C7311yescharacteristic
329C42D8×C7311yescharacteristic
330C42C2×D4×C7311yescharacteristic
331C42C2×D4×C7311yescharacteristic
332C16—22no—
333C16—22no—
334C23—41no—
335C23C2×D8×C7311yescharacteristic
336Q8—42no—
337Q8—41no—
338Q8—41no—
339Q8—42no—
340Q8—82no—
341D4—41no—
342D4—41no—
343D4—82no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—82no—
354C2×C4C2×D8×C7311yescharacteristic
355C2×C4C2×D8×C7311yescharacteristic
356C2×C4C2×D8×C7311yescharacteristic
357C2×C4C2×D8×C7311yescharacteristic
358C2×C4C2×D8×C7311yescharacteristic
359C2×C4C22×D4×C7311yescharacteristic
360C8—21no—
361C8—21no—
362C8—21no—
363C8H(5,𝔽2)×C7311yescharacteristic
364C82−1+4×C7311yescharacteristic
365C22—21no—
366C22—21no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#510×C7312yes—
371C22C22×D8×C7311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—81no—
377C4—82no—
378C4—81no—
379C4—82no—
380C4—81no—
381C4#508×C7312yes—
382C4#537×C7311yescharacteristic
383C4#575×C7311yescharacteristic
384C2—22no—
385C2—161no—
386C2C73×#324511yescharacteristic
387C2C2×#508×C7311yescharacteristic
388C2C2×#510×C7311yescharacteristic
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
1. C73×#3170C2semidirect
3. C73×#2229C2semidirect
4. C73×#2225C2semidirect
5. C73×#2224C2semidirect
7. C73×#2216C2semidirect
8. C73×#2211C2semidirect
9. C73×#2209C2semidirect
10. C73×#2208C2semidirect
11. C2×C4⋊C16×C73C2semidirect
189. #33814C73direct (N × H)
195. #3170C2×C73semidirect
197. #2229C2×C73semidirect
198. #2225C2×C73semidirect
199. #2224C2×C73semidirect
201. #2216C2×C73semidirect
202. #2211C2×C73semidirect
203. #2209C2×C73semidirect
204. #2208C2×C73semidirect
205. C2×C4⋊C16C2×C73semidirect
206. C73#33814direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#31702
C73×#31682
C73×#22292
C73×#22252
C73×#22242
C73×#22172
C73×#22162
C73×#22112
C73×#22092
C73×#22082
C2×C4⋊C16×C732
#3381473