Finite Group Databasebeta

← C73×#8610 #32918014

Subgroups — C73×#8610 #32918014

Order 18688.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#435×C73C211yesmaximal, characteristic
2C73×#1808C211yesmaximal, characteristic, complemented
3C73×#1807C211yesmaximal, characteristic, complemented
4C73×#1799C212yesmaximal
5C73×#1289C212yesmaximal, complemented
6C22×M16×C73C411yescharacteristic, complemented
7C23×C8×C73C2211yescharacteristic
8C2×C4×D4×C73C2211yescharacteristic
9#435×C73—21no—
10#435×C73—21no—
11#435×C73—21no—
12#435×C73—21no—
13C2×#107×C73—21no—
14C2×#106×C73—21no—
15C2×C4⋊C8×C73C411yescharacteristic, complemented
16C2×#99×C73—22no—
17C2×C22⋊C8×C73C411yescharacteristic, complemented
18C2×C22⋊C8×C73C2211yescharacteristic
19C2×#97×C73C411yescharacteristic, complemented
20#380×C73C2212yes—
21#380×C73C2212yes—
22#342×C73—22no—
23#340×C73—21no—
24#340×C73—21no—
25C22×D4×C73D411yescharacteristic
26C23×C4×C73C2×C411yescharacteristic
27C23×C4×C73C2311yescharacteristic
28C2×M16×C73—21no—
29C2×M16×C73—21no—
30C2×M16×C73—42no—
31C22×C8×C73—42no—
32C22×C8×C73C2×C411yescharacteristic
33C22×C8×C73C2×C411yescharacteristic
34C22×C8×C73C2×C411yescharacteristic
35C22×C8×C73C2×C411yescharacteristic
36C22×C8×C73D411yescharacteristic
37C22×C8×C73D411yescharacteristic
38C22×C8×C73D411yescharacteristic
39C22×C8×C73D411yescharacteristic
40C4×D4×C73—21no—
41C4×D4×C73—21no—
42C4×D4×C73—21no—
43C4×D4×C73—21no—
44C2×C4⋊C4×C73—22no—
45C2×C4⋊C4×C73—22no—
46C2×C4⋊C4×C73Q811yescharacteristic
47C2×C22⋊C4×C73—22no—
48C2×C22⋊C4×C73—22no—
49C2×C22⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73Q811yescharacteristic
51C2×C42×C73C2×C411yescharacteristic
52#107×C73—42no—
53#106×C73—42no—
54C4⋊C8×C73—21no—
55C4⋊C8×C73—21no—
56#99×C73—44no—
57C22⋊C8×C73—21no—
58C22⋊C8×C73—21no—
59C22⋊C8×C73—21no—
60C22⋊C8×C73—21no—
61#97×C73—21no—
62#97×C73—21no—
63#95×C73—22no—
64#95×C73—22no—
65C24×C73C4⋊C411yescharacteristic
66C24×C73C2×D411yescharacteristic
67C2×D4×C73—21no—
68C2×D4×C73—21no—
69C2×D4×C73—22no—
70C2×D4×C73—22no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—41no—
74C22×C4×C73—41no—
75C22×C4×C73—41no—
76C22×C4×C73—41no—
77C22×C4×C73—42no—
78C22×C4×C73—42no—
79C22×C4×C73C4⋊C411yescharacteristic
80C22×C4×C73C4⋊C411yescharacteristic
81C22×C4×C73C4⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73C2×Q811yescharacteristic
86C22×C4×C73#4111yescharacteristic
87C22×C4×C73#4111yescharacteristic
88M16×C73—42no—
89M16×C73—42no—
90C2×C8×C73—22no—
91C2×C8×C73—22no—
92C2×C8×C73—22no—
93C2×C8×C73—21no—
94C2×C8×C73—21no—
95C2×C8×C73—41no—
96C2×C8×C73—41no—
97C2×C8×C73—41no—
98C2×C8×C73—41no—
99C2×C8×C73—41no—
100C2×C8×C73C22⋊C411yescharacteristic
101C2×C8×C73C22⋊C411yescharacteristic
102C2×C8×C73C22⋊C411yescharacteristic
103C2×C8×C73C22⋊C411yescharacteristic
104C4⋊C4×C73—21no—
105C4⋊C4×C73—21no—
106C4⋊C4×C73—42no—
107C4⋊C4×C73—42no—
108C4⋊C4×C73—42no—
109C4⋊C4×C73—42no—
110C22⋊C4×C73—21no—
111C22⋊C4×C73—21no—
112C22⋊C4×C73—21no—
113C22⋊C4×C73—21no—
114C22⋊C4×C73—42no—
115C22⋊C4×C73—42no—
116C22⋊C4×C73—42no—
117C22⋊C4×C73—42no—
118C42×C73—21no—
119C42×C73—21no—
120C23×C73—21no—
121C23×C73—22no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73C2×C4⋊C411yescharacteristic
127C23×C73#12111yescharacteristic
128C23×C73#12111yescharacteristic
129C23×C73AGL(1,ℤ/8)12yes—
130C23×C73#13712yes—
131D4×C73—42no—
132D4×C73—41no—
133D4×C73—41no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—22no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—42no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—82no—
153C2×C4×C73—82no—
154C2×C4×C73—82no—
155C2×C4×C73—82no—
156C2×C4×C73—82no—
157C2×C4×C73—82no—
158C2×C4×C73C2×C22⋊C411yescharacteristic
159C2×C4×C73#11711yescharacteristic
160C2×C4×C73C22⋊Q811yescharacteristic
161C2×C4×C73C22⋊Q811yescharacteristic
162C8×C73—21no—
163C8×C73—21no—
164C8×C73—21no—
165C8×C73—41no—
166C8×C73—41no—
167C8×C73#9912yescomplemented
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—42no—
175C22×C73—42no—
176C22×C73—41no—
177C22×C73—41no—
178C22×C73—41no—
179C22×C73—82no—
180C22×C73#38011yescharacteristic
181C22×C73#42812yes—
182C22×C73#46812yes—
183C22×C73#46911yescharacteristic
184C22×C73#47011yescharacteristic
185C4×C73—21no—
186C4×C73—21no—
187C4×C73—21no—
188C4×C73—41no—
189C4×C73—41no—
190C4×C73—41no—
191C4×C73—162no—
192C4×C73—162no—
193C4×C73C2×#9911yescharacteristic
194C4×C73#41011yescharacteristic
195#8610C7311yesmaximal, characteristic, direct product factor, Sylow
196C2×C73—21no—
197C2×C73—22no—
198C2×C73—22no—
199C2×C73—21no—
200C2×C73—81no—
201C2×C73#179911yescharacteristic
202C2×C73#180911yescharacteristic
203C2×C73#181211yescharacteristic
204C2×#435C2×C7311yescharacteristic
205#1808C2×C7311yescharacteristic, complemented
206#1807C2×C7311yescharacteristic, complemented
207#1799C2×C7312yes—
208#1289C2×C7312yescomplemented
209C73#861011yescharacteristic, direct product factor, Sylow
210C22×M16C4×C7311yescharacteristic, complemented
211C23×C8C22×C7311yescharacteristic
212C2×C4×D4C22×C7311yescharacteristic
213#435—21no—
214#435—21no—
215#435—21no—
216#435—21no—
217C2×#107—21no—
218C2×#106—21no—
219C2×C4⋊C8C4×C7311yescharacteristic, complemented
220C2×#99—22no—
221C2×C22⋊C8C4×C7311yescharacteristic, complemented
222C2×C22⋊C8C22×C7311yescharacteristic
223C2×#97C4×C7311yescharacteristic, complemented
224#380C22×C7312yes—
225#380C22×C7312yes—
226#342—22no—
227#340—21no—
228#340—21no—
229C22×D4D4×C7311yescharacteristic
230C23×C4C2×C4×C7311yescharacteristic
231C23×C4C23×C7311yescharacteristic
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—42no—
235C22×C8—42no—
236C22×C8C2×C4×C7311yescharacteristic
237C22×C8C2×C4×C7311yescharacteristic
238C22×C8C2×C4×C7311yescharacteristic
239C22×C8C2×C4×C7311yescharacteristic
240C22×C8D4×C7311yescharacteristic
241C22×C8D4×C7311yescharacteristic
242C22×C8D4×C7311yescharacteristic
243C22×C8D4×C7311yescharacteristic
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4Q8×C7311yescharacteristic
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4D4×C7311yescharacteristic
254C2×C22⋊C4Q8×C7311yescharacteristic
255C2×C42C2×C4×C7311yescharacteristic
256#107—42no—
257#106—42no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260#99—44no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265#97—21no—
266#97—21no—
267#95—22no—
268#95—22no—
269C24C4⋊C4×C7311yescharacteristic
270C24C2×D4×C7311yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4C4⋊C4×C7311yescharacteristic
284C22×C4C4⋊C4×C7311yescharacteristic
285C22×C4C4⋊C4×C7311yescharacteristic
286C22×C4C22×C4×C7311yescharacteristic
287C22×C4C2×D4×C7311yescharacteristic
288C22×C4C2×D4×C7311yescharacteristic
289C22×C4C2×Q8×C7311yescharacteristic
290C22×C4#41×C7311yescharacteristic
291C22×C4#41×C7311yescharacteristic
292M16—42no—
293M16—42no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8C22⋊C4×C7311yescharacteristic
305C2×C8C22⋊C4×C7311yescharacteristic
306C2×C8C22⋊C4×C7311yescharacteristic
307C2×C8C22⋊C4×C7311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23C2×C4⋊C4×C7311yescharacteristic
331C23#121×C7311yescharacteristic
332C23#121×C7311yescharacteristic
333C23AGL(1,ℤ/8)×C7312yes—
334C23#137×C7312yes—
335D4—42no—
336D4—41no—
337D4—41no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4C2×C22⋊C4×C7311yescharacteristic
363C2×C4#117×C7311yescharacteristic
364C2×C4C22⋊Q8×C7311yescharacteristic
365C2×C4C22⋊Q8×C7311yescharacteristic
366C8—21no—
367C8—21no—
368C8—21no—
369C8—41no—
370C8—41no—
371C8#99×C7312yescomplemented
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—82no—
384C22#380×C7311yescharacteristic
385C22#428×C7312yes—
386C22#468×C7312yes—
387C22#469×C7311yescharacteristic
388C22#470×C7311yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—162no—
396C4—162no—
397C4C2×#99×C7311yescharacteristic
398C4#410×C7311yescharacteristic
399C2—21no—
400C2—22no—
401C2—22no—
402C2—21no—
403C2—81no—
404C2C73×#179911yescharacteristic
405C2C73×#180911yescharacteristic
406C2C73×#181211yescharacteristic
subgroupG/H
center, socleC22×C73#380
commutatorC22×C4C22×C4×C73
Frattini, cosocleC23×C4C23×C73
prime radicalC22×D4×C73D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#1808C2semidirect
3. C73×#1807C2semidirect
5. C73×#1289C2semidirect
6. C22×M16×C73C4semidirect
15. C2×C4⋊C8×C73C4semidirect
17. C2×C22⋊C8×C73C4semidirect
19. C2×#97×C73C4semidirect
167. C8×C73#99semidirect
195. #8610C73direct (N × H)
205. #1808C2×C73semidirect
206. #1807C2×C73semidirect
208. #1289C2×C73semidirect
209. C73#8610direct (N × H)
210. C22×M16C4×C73semidirect
219. C2×C4⋊C8C4×C73semidirect
221. C2×C22⋊C8C4×C73semidirect
223. C2×#97C4×C73semidirect
371. C8#99×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#435×C732
C73×#18082
C73×#18072
C73×#17992
C73×#12892
#861073