Finite Group Databasebeta

← C73×#33805 #32943209

Subgroups — C73×#33805 #32943209

Order 18688.

496 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
1C73×#3168C211yesmaximal, characteristic
2C2×#500×C73C211yesmaximal, characteristic, complemented
3C73×#2255C212yesmaximal, complemented
4C73×#2243C212yesmaximal, complemented
5C73×#2236C212yesmaximal, complemented
6C73×#2233C212yesmaximal
7C73×#2225C212yesmaximal, complemented
8C73×#2224C212yesmaximal, complemented
9C73×#2163C211yesmaximal, characteristic, complemented
10C2×C4⋊Q8×C73—21no—
11#529×C73—21no—
12#500×C73—21no—
13#500×C73—21no—
14#500×C73C2212yes—
15#500×C73C2212yes—
16#499×C73C2211yescharacteristic
17#495×C73C2211yescharacteristic
18#488×C73—21no—
19#467×C73C2211yescharacteristic
20#466×C73C2211yescharacteristic
21#465×C73—21no—
22C4×Q16×C73C2211yescharacteristic
23C4×SD16×C73—21no—
24C4×D8×C73C2211yescharacteristic
25C2×#107×C73—21no—
26C2×#107×C73C2212yes—
27C2×C4×C8×C73C2211yescharacteristic
28#367×C73C2214yes—
29#366×C73C2214yes—
30C4⋊C16×C73C2212yes—
31#358×C73C2214yes—
32#357×C73C2214yes—
33C22⋊C16×C73C2212yes—
34#346×C73C2212yes—
35C22×Q8×C73—41no—
36C2×Q16×C73C2311yescharacteristic
37C2×SD16×C73—21no—
38C2×D8×C73C2311yescharacteristic
39C22×C8×C73D411yescharacteristic
40C22×C8×C73C2311yescharacteristic
41C4⋊Q8×C73—22no—
42C4⋊Q8×C73—22no—
43C4⋊Q8×C73—41no—
44C4⋊Q8×C73—41no—
45#126×C73—41no—
46#125×C73—21no—
47#124×C73—21no—
48#123×C73—41no—
49C22⋊Q8×C73—21no—
50#121×C73—21no—
51C4×Q8×C73—21no—
52C4×D4×C73—21no—
53C4×D4×C73—41no—
54#117×C73—41no—
55C2×C4⋊C4×C73—22no—
56C2×C4⋊C4×C73—41no—
57C2×C42×C73D411yescharacteristic
58C2×C16×C73C2314yes—
59#107×C73—22no—
60#107×C73—22no—
61#107×C73—22no—
62#107×C73C2314yes—
63#107×C73C2311yescharacteristic
64#107×C73C2311yescharacteristic
65#106×C73—21no—
66#103×C73—21no—
67#103×C73—21no—
68#102×C73—21no—
69#102×C73—21no—
70C4×C8×C73D411yescharacteristic
71C4×C8×C73D411yescharacteristic
72C4×C8×C73C2311yescharacteristic
73C4×C8×C73C2311yescharacteristic
74C2×Q8×C73—21no—
75C2×Q8×C73—42no—
76C2×Q8×C73—42no—
77C2×Q8×C73—84no—
78C2×D4×C73—21no—
79C2×D4×C73—41no—
80C22×C4×C73—42no—
81C22×C4×C73—41no—
82C22×C4×C73D811yescharacteristic
83C22×C4×C73D811yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85Q16×C73—22no—
86SD16×C73—42no—
87D8×C73—22no—
88C2×C8×C73—22no—
89C2×C8×C73—22no—
90C2×C8×C73C2×D411yescharacteristic
91C2×C8×C73C2×D411yescharacteristic
92C2×C8×C73C2×D411yescharacteristic
93C2×C8×C73C2411yescharacteristic
94C4⋊C4×C73—21no—
95C4⋊C4×C73—21no—
96C4⋊C4×C73—24no—
97C4⋊C4×C73—42no—
98C4⋊C4×C73—42no—
99C4⋊C4×C73—42no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—41no—
104C22⋊C4×C73—41no—
105C22⋊C4×C73—41no—
106C22⋊C4×C73—41no—
107C22⋊C4×C73—41no—
108C22⋊C4×C73—41no—
109C42×C73—41no—
110C42×C73D811yescharacteristic
111C42×C73D811yescharacteristic
112C42×C73C2×D411yescharacteristic
113C42×C73C2×D411yescharacteristic
114C16×C73—24no—
115C23×C73—41no—
116C23×C73C2×D811yescharacteristic
117Q8×C73—41no—
118Q8×C73—41no—
119Q8×C73—84no—
120Q8×C73—84no—
121D4×C73—41no—
122D4×C73—41no—
123D4×C73—82no—
124C2×C4×C73—22no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—22no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—44no—
132C2×C4×C73—84no—
133C2×C4×C73—82no—
134C2×C4×C73C2×D811yescharacteristic
135C2×C4×C73C2×D811yescharacteristic
136C2×C4×C73C2×D811yescharacteristic
137C2×C4×C73C2×D811yescharacteristic
138C2×C4×C73C2×D811yescharacteristic
139C2×C4×C73C22×D411yescharacteristic
140C8×C73—21no—
141C8×C73—21no—
142C8×C73#13712yes—
143C8×C732−1+412yes—
144C22×C73—21no—
145C22×C73—21no—
146C22×C73—81no—
147C22×C73—81no—
148C22×C73—81no—
149C22×C73#51012yes—
150C22×C73C22×D811yescharacteristic
151C4×C73—21no—
152C4×C73—21no—
153C4×C73—21no—
154C4×C73—21no—
155C4×C73—81no—
156C4×C73—84no—
157C4×C73—81no—
158C4×C73—81no—
159C4×C73#50912yes—
160C4×C73#53611yescharacteristic
161C4×C73C2×#13711yescharacteristic
162#33805C7311yesmaximal, characteristic, direct product factor, Sylow
163C2×C73—22no—
164C2×C73—161no—
165C2×C73#322611yescharacteristic
166C2×C73C2×#50911yescharacteristic
167C2×C73C2×#51011yescharacteristic
168#3168C2×C7311yescharacteristic
169C2×#500C2×C7311yescharacteristic, complemented
170#2255C2×C7312yescomplemented
171#2243C2×C7312yescomplemented
172#2236C2×C7312yescomplemented
173#2233C2×C7312yes—
174#2225C2×C7312yescomplemented
175#2224C2×C7312yescomplemented
176#2163C2×C7311yescharacteristic, complemented
177C73#3380511yescharacteristic, direct product factor, Sylow
178C2×C4⋊Q8—21no—
179#529—21no—
180#500—21no—
181#500—21no—
182#500C22×C7312yes—
183#500C22×C7312yes—
184#499C22×C7311yescharacteristic
185#495C22×C7311yescharacteristic
186#488—21no—
187#467C22×C7311yescharacteristic
188#466C22×C7311yescharacteristic
189#465—21no—
190C4×Q16C22×C7311yescharacteristic
191C4×SD16—21no—
192C4×D8C22×C7311yescharacteristic
193C2×#107—21no—
194C2×#107C22×C7312yes—
195C2×C4×C8C22×C7311yescharacteristic
196#367C22×C7314yes—
197#366C22×C7314yes—
198C4⋊C16C22×C7312yes—
199#358C22×C7314yes—
200#357C22×C7314yes—
201C22⋊C16C22×C7312yes—
202#346C22×C7312yes—
203C22×Q8—41no—
204C2×Q16C23×C7311yescharacteristic
205C2×SD16—21no—
206C2×D8C23×C7311yescharacteristic
207C22×C8D4×C7311yescharacteristic
208C22×C8C23×C7311yescharacteristic
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211C4⋊Q8—41no—
212C4⋊Q8—41no—
213#126—41no—
214#125—21no—
215#124—21no—
216#123—41no—
217C22⋊Q8—21no—
218#121—21no—
219C4×Q8—21no—
220C4×D4—21no—
221C4×D4—41no—
222#117—41no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—41no—
225C2×C42D4×C7311yescharacteristic
226C2×C16C23×C7314yes—
227#107—22no—
228#107—22no—
229#107—22no—
230#107C23×C7314yes—
231#107C23×C7311yescharacteristic
232#107C23×C7311yescharacteristic
233#106—21no—
234#103—21no—
235#103—21no—
236#102—21no—
237#102—21no—
238C4×C8D4×C7311yescharacteristic
239C4×C8D4×C7311yescharacteristic
240C4×C8C23×C7311yescharacteristic
241C4×C8C23×C7311yescharacteristic
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—84no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4D8×C7311yescharacteristic
251C22×C4D8×C7311yescharacteristic
252C22×C4C2×D4×C7311yescharacteristic
253Q16—22no—
254SD16—42no—
255D8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8C2×D4×C7311yescharacteristic
259C2×C8C2×D4×C7311yescharacteristic
260C2×C8C2×D4×C7311yescharacteristic
261C2×C8C24×C7311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—24no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—41no—
278C42D8×C7311yescharacteristic
279C42D8×C7311yescharacteristic
280C42C2×D4×C7311yescharacteristic
281C42C2×D4×C7311yescharacteristic
282C16—24no—
283C23—41no—
284C23C2×D8×C7311yescharacteristic
285Q8—41no—
286Q8—41no—
287Q8—84no—
288Q8—84no—
289D4—41no—
290D4—41no—
291D4—82no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4C2×D8×C7311yescharacteristic
303C2×C4C2×D8×C7311yescharacteristic
304C2×C4C2×D8×C7311yescharacteristic
305C2×C4C2×D8×C7311yescharacteristic
306C2×C4C2×D8×C7311yescharacteristic
307C2×C4C22×D4×C7311yescharacteristic
308C8—21no—
309C8—21no—
310C8#137×C7312yes—
311C82−1+4×C7312yes—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22#510×C7312yes—
318C22C22×D8×C7311yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—81no—
324C4—84no—
325C4—81no—
326C4—81no—
327C4#509×C7312yes—
328C4#536×C7311yescharacteristic
329C4C2×#137×C7311yescharacteristic
330C2—22no—
331C2—161no—
332C2C73×#322611yescharacteristic
333C2C2×#509×C7311yescharacteristic
334C2C2×#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
2. C2×#500×C73C2semidirect
3. C73×#2255C2semidirect
4. C73×#2243C2semidirect
5. C73×#2236C2semidirect
7. C73×#2225C2semidirect
8. C73×#2224C2semidirect
9. C73×#2163C2semidirect
162. #33805C73direct (N × H)
169. C2×#500C2×C73semidirect
170. #2255C2×C73semidirect
171. #2243C2×C73semidirect
172. #2236C2×C73semidirect
174. #2225C2×C73semidirect
175. #2224C2×C73semidirect
176. #2163C2×C73semidirect
177. C73#33805direct (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×#500×C732
C73×#22552
C73×#22432
C73×#22362
C73×#22332
C73×#22252
C73×#22242
C73×#21632
#3380573