Finite Group Databasebeta

← C73×#19358 #32928762

Subgroups — C73×#19358 #32928762

Order 18688.

608 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3218C212yesmaximal
2C2×#500×C73C211yesmaximal, characteristic, complemented
3C73×#2966C211yesmaximal, characteristic
4C73×#1694C212yesmaximal, complemented
5C73×#1693C212yesmaximal, complemented
6C73×#1674C212yesmaximal, complemented
7C73×#1673C212yesmaximal, complemented
8C73×#1568C211yesmaximal, characteristic, complemented
9C73×#1556C212yesmaximal, complemented
10C2×C4⋊Q8×C73C2212yes—
11#529×C73C2211yescharacteristic
12#500×C73—21no—
13#500×C73—21no—
14#500×C73C2212yes—
15#500×C73C2212yes—
16#497×C73—22no—
17#491×C73—22no—
18#462×C73C2214yes—
19#461×C73C2214yes—
20#451×C73—22no—
21#448×C73—22no—
22#446×C73C2211yescharacteristic
23C8×Q8×C73C2211yescharacteristic
24#436×C73C2211yescharacteristic
25#435×C73—21no—
26C8×D4×C73—21no—
27C8×D4×C73C2211yescharacteristic
28#433×C73—21no—
29#432×C73C2212yes—
30#432×C73C2211yescharacteristic
31C2×#107×C73—21no—
32C2×#107×C73—21no—
33#405×C73—21no—
34C2×C4×C8×C73C2211yescharacteristic
35#335×C73C2212yes—
36#334×C73C2212yes—
37#332×C73C2214yes—
38#332×C73C2214yes—
39C22×Q8×C73—22no—
40C2×Q16×C73—24no—
41C2×SD16×C73—24no—
42C2×M16×C73—21no—
43C22×C8×C73—21no—
44C22×C8×C73D411yescharacteristic
45C22×C8×C73D411yescharacteristic
46C4⋊Q8×C73—22no—
47C4⋊Q8×C73—22no—
48C4⋊Q8×C73—22no—
49C4⋊Q8×C73C2314yes—
50#126×C73—21no—
51#125×C73D411yescharacteristic
52#124×C73D411yescharacteristic
53#123×C73—21no—
54C22⋊Q8×C73D411yescharacteristic
55#121×C73D411yescharacteristic
56C4×Q8×C73C2311yescharacteristic
57C4×D4×C73—21no—
58C4×D4×C73C2311yescharacteristic
59#117×C73—21no—
60C2×C4⋊C4×C73—22no—
61C2×C4⋊C4×C73—22no—
62C2×C42×C73C2311yescharacteristic
63#107×C73—22no—
64#107×C73—22no—
65#107×C73—22no—
66#107×C73—22no—
67C4⋊C8×C73—21no—
68C4⋊C8×C73—21no—
69C4⋊C8×C73C2311yescharacteristic
70C4⋊C8×C73C2311yescharacteristic
71C4⋊C8×C73C2314yes—
72#103×C73—24no—
73#103×C73—24no—
74C22⋊C8×C73—21no—
75C22⋊C8×C73—22no—
76C22⋊C8×C73—21no—
77C22⋊C8×C73—21no—
78#97×C73—21no—
79#97×C73—21no—
80#97×C73—22no—
81C4×C8×C73—21no—
82C4×C8×C73D411yescharacteristic
83C4×C8×C73D411yescharacteristic
84C4×C8×C73C2311yescharacteristic
85C4×C8×C73C2311yescharacteristic
86C2×Q8×C73—24no—
87C2×Q8×C73—42no—
88C2×Q8×C73—84no—
89C2×Q8×C73C2×D411yescharacteristic
90C2×D4×C73—21no—
91C2×D4×C73C2×D411yescharacteristic
92C22×C4×C73—21no—
93C22×C4×C73—42no—
94C22×C4×C73C2×D411yescharacteristic
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73#4111yescharacteristic
97Q16×C73—84no—
98SD16×C73—84no—
99M16×C73—42no—
100C2×C8×C73—22no—
101C2×C8×C73—22no—
102C2×C8×C73—21no—
103C2×C8×C73—21no—
104C2×C8×C73—24no—
105C2×C8×C73—42no—
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73C2×D411yescharacteristic
109C2×C8×C73C2×D411yescharacteristic
110C4⋊C4×C73—21no—
111C4⋊C4×C73—21no—
112C4⋊C4×C73—21no—
113C4⋊C4×C73—21no—
114C4⋊C4×C73—24no—
115C4⋊C4×C73—24no—
116C4⋊C4×C73—42no—
117C4⋊C4×C73—42no—
118C4⋊C4×C73C2×D411yescharacteristic
119C4⋊C4×C73C2×D411yescharacteristic
120C22⋊C4×C73—21no—
121C22⋊C4×C73—21no—
122C22⋊C4×C73—21no—
123C22⋊C4×C73—21no—
124C22⋊C4×C73—21no—
125C42×C73—21no—
126C42×C73C2×D411yescharacteristic
127C42×C73C2×D411yescharacteristic
128C42×C73#4111yescharacteristic
129C42×C73C2411yescharacteristic
130C23×C73—21no—
131C23×C73#12111yescharacteristic
132Q8×C73—22no—
133Q8×C73—44no—
134Q8×C73—44no—
135Q8×C73—84no—
136D4×C73—22no—
137D4×C73—42no—
138C2×C4×C73—22no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—22no—
142C2×C4×C73—21no—
143C2×C4×C73—21no—
144C2×C4×C73—21no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—44no—
148C2×C4×C73—84no—
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73C22×D411yescharacteristic
153C2×C4×C73C22×D411yescharacteristic
154C2×C4×C73C2×#4111yescharacteristic
155C8×C73—21no—
156C8×C73—21no—
157C8×C73—41no—
158C8×C73—41no—
159C8×C73—44no—
160C8×C73#13712yes—
161C8×C73#13712yes—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—41no—
165C22×C73—41no—
166C22×C73—41no—
167C22×C73#47312yes—
168C22×C73C2×#12111yescharacteristic
169C4×C73—21no—
170C4×C73—21no—
171C4×C73—21no—
172C4×C73—21no—
173C4×C73—41no—
174C4×C73—41no—
175C4×C73—41no—
176C4×C73—84no—
177C4×C73C2×#13711yescharacteristic
178C4×C73C2×#13711yescharacteristic
179C4×C73#57612yes—
180#19358C7311yesmaximal, characteristic, direct product factor, Sylow
181C2×C73—22no—
182C2×C73—81no—
183C2×C73#303911yescharacteristic
184C2×C73#305811yescharacteristic
185C2×C73C2×#47311yescharacteristic
186#3218C2×C7312yes—
187C2×#500C2×C7311yescharacteristic, complemented
188#2966C2×C7311yescharacteristic
189#1694C2×C7312yescomplemented
190#1693C2×C7312yescomplemented
191#1674C2×C7312yescomplemented
192#1673C2×C7312yescomplemented
193#1568C2×C7311yescharacteristic, complemented
194#1556C2×C7312yescomplemented
195C73#1935811yescharacteristic, direct product factor, Sylow
196C2×C4⋊Q8C22×C7312yes—
197#529C22×C7311yescharacteristic
198#500—21no—
199#500—21no—
200#500C22×C7312yes—
201#500C22×C7312yes—
202#497—22no—
203#491—22no—
204#462C22×C7314yes—
205#461C22×C7314yes—
206#451—22no—
207#448—22no—
208#446C22×C7311yescharacteristic
209C8×Q8C22×C7311yescharacteristic
210#436C22×C7311yescharacteristic
211#435—21no—
212C8×D4—21no—
213C8×D4C22×C7311yescharacteristic
214#433—21no—
215#432C22×C7311yescharacteristic
216#432C22×C7312yes—
217C2×#107—21no—
218C2×#107—21no—
219#405—21no—
220C2×C4×C8C22×C7311yescharacteristic
221#335C22×C7312yes—
222#334C22×C7312yes—
223#332C22×C7314yes—
224#332C22×C7314yes—
225C22×Q8—22no—
226C2×Q16—24no—
227C2×SD16—24no—
228C2×M16—21no—
229C22×C8—21no—
230C22×C8D4×C7311yescharacteristic
231C22×C8D4×C7311yescharacteristic
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8C23×C7314yes—
236#126—21no—
237#125D4×C7311yescharacteristic
238#124D4×C7311yescharacteristic
239#123—21no—
240C22⋊Q8D4×C7311yescharacteristic
241#121D4×C7311yescharacteristic
242C4×Q8C23×C7311yescharacteristic
243C4×D4—21no—
244C4×D4C23×C7311yescharacteristic
245#117—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C42C23×C7311yescharacteristic
249#107—22no—
250#107—22no—
251#107—22no—
252#107—22no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8C23×C7311yescharacteristic
256C4⋊C8C23×C7311yescharacteristic
257C4⋊C8C23×C7314yes—
258#103—24no—
259#103—24no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264#97—21no—
265#97—21no—
266#97—22no—
267C4×C8—21no—
268C4×C8D4×C7311yescharacteristic
269C4×C8D4×C7311yescharacteristic
270C4×C8C23×C7311yescharacteristic
271C4×C8C23×C7311yescharacteristic
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—84no—
275C2×Q8C2×D4×C7311yescharacteristic
276C2×D4—21no—
277C2×D4C2×D4×C7311yescharacteristic
278C22×C4—21no—
279C22×C4—42no—
280C22×C4C2×D4×C7311yescharacteristic
281C22×C4C2×D4×C7311yescharacteristic
282C22×C4#41×C7311yescharacteristic
283Q16—84no—
284SD16—84no—
285M16—42no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—24no—
291C2×C8—42no—
292C2×C8C2×D4×C7311yescharacteristic
293C2×C8C2×D4×C7311yescharacteristic
294C2×C8C2×D4×C7311yescharacteristic
295C2×C8C2×D4×C7311yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4C2×D4×C7311yescharacteristic
305C4⋊C4C2×D4×C7311yescharacteristic
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C42—21no—
312C42C2×D4×C7311yescharacteristic
313C42C2×D4×C7311yescharacteristic
314C42#41×C7311yescharacteristic
315C42C24×C7311yescharacteristic
316C23—21no—
317C23#121×C7311yescharacteristic
318Q8—22no—
319Q8—44no—
320Q8—44no—
321Q8—84no—
322D4—22no—
323D4—42no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—84no—
335C2×C4#121×C7311yescharacteristic
336C2×C4#121×C7311yescharacteristic
337C2×C4#121×C7311yescharacteristic
338C2×C4C22×D4×C7311yescharacteristic
339C2×C4C22×D4×C7311yescharacteristic
340C2×C4C2×#41×C7311yescharacteristic
341C8—21no—
342C8—21no—
343C8—41no—
344C8—41no—
345C8—44no—
346C8#137×C7312yes—
347C8#137×C7312yes—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22#473×C7312yes—
354C22C2×#121×C7311yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—84no—
363C4C2×#137×C7311yescharacteristic
364C4C2×#137×C7311yescharacteristic
365C4#576×C7312yes—
366C2—22no—
367C2—81no—
368C2C73×#303911yescharacteristic
369C2C73×#305811yescharacteristic
370C2C2×#473×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC42C24×C73
prime radical#121×C73D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#500×C73C2semidirect
4. C73×#1694C2semidirect
5. C73×#1693C2semidirect
6. C73×#1674C2semidirect
7. C73×#1673C2semidirect
8. C73×#1568C2semidirect
9. C73×#1556C2semidirect
180. #19358C73direct (N × H)
187. C2×#500C2×C73semidirect
189. #1694C2×C73semidirect
190. #1693C2×C73semidirect
191. #1674C2×C73semidirect
192. #1673C2×C73semidirect
193. #1568C2×C73semidirect
194. #1556C2×C73semidirect
195. C73#19358direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#32182
C2×#500×C732
C73×#29662
C73×#16942
C73×#16932
C73×#16742
C73×#16732
C73×#15682
C73×#15562
#1935873