Finite Group Databasebeta

← C73×#10071 #32919475

Subgroups — C73×#10071 #32919475

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486×C73C211yesmaximal, characteristic
2C2×#475×C73C211yesmaximal, characteristic, complemented
3C73×#2548C211yesmaximal, characteristic
4C73×#1981C211yesmaximal, characteristic, complemented
5C2×#331×C73C211yesmaximal, characteristic
6C2×Q8⋊C8×C73C211yesmaximal, characteristic, complemented
7C73×#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C73C2211yescharacteristic
9C2×#127×C73C2211yescharacteristic
10C2×#124×C73—21no—
11C2×C4×Q8×C73C2211yescharacteristic
12#486×C73—22no—
13#486×C73—22no—
14#475×C73—22no—
15#475×C73—22no—
16C2×#106×C73—21no—
17C2×C4⋊C8×C73C2211yescharacteristic
18C2×C4⋊C8×C73C2211yescharacteristic
19C2×#103×C73—21no—
20C2×#103×C73—22no—
21C2×#102×C73—22no—
22C2×C4×C8×C73C2211yescharacteristic
23#394×C73—22no—
24#390×C73—21no—
25#390×C73—22no—
26C4×C4⋊C4×C73C2211yescharacteristic
27#331×C73—22no—
28Q8⋊C8×C73—22no—
29C22×Q8×C73—21no—
30C22×Q8×C73D411yescharacteristic
31C22×D4×C73—21no—
32C22×D4×C73—21no—
33C22×D4×C73D412yes—
34C22×C8×C73—21no—
35C22×C8×C73—21no—
36C22×C8×C73—21no—
37C4⋊Q8×C73—21no—
38C4⋊Q8×C73—21no—
39C4⋊Q8×C73—21no—
40C4⋊Q8×C73D412yes—
41#127×C73—21no—
42#127×C73—21no—
43#127×C73—21no—
44#127×C73—21no—
45#124×C73—42no—
46#124×C73—42no—
47C4×Q8×C73—21no—
48C4×Q8×C73—21no—
49C4×Q8×C73—21no—
50C4×Q8×C73—21no—
51C2×C4⋊C4×C73—21no—
52C2×C4⋊C4×C73—21no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73D412yes—
55C2×C4⋊C4×C73D411yescharacteristic
56C2×C22⋊C4×C73—24no—
57C2×C22⋊C4×C73—24no—
58C2×C42×C73—21no—
59C2×C42×C73—21no—
60C2×C42×C73C2311yescharacteristic
61#106×C73—24no—
62C4⋊C8×C73—21no—
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65C4⋊C8×C73D412yes—
66#103×C73—24no—
67#103×C73—44no—
68#102×C73—28no—
69C4×C8×C73—21no—
70C4×C8×C73D412yes—
71#95×C73—21no—
72C24×C73—22no—
73C2×Q8×C73—21no—
74C2×Q8×C73—21no—
75C2×Q8×C73—21no—
76C2×Q8×C73—22no—
77C2×Q8×C73—21no—
78C2×Q8×C73—21no—
79C2×Q8×C73—41no—
80C2×Q8×C73—42no—
81C2×Q8×C73—42no—
82C2×Q8×C73SD1612yes—
83C2×D4×C73—22no—
84C2×D4×C73—22no—
85C2×D4×C73—44no—
86C2×D4×C73—41no—
87C2×D4×C73—41no—
88C2×D4×C73—41no—
89C2×D4×C73—41no—
90C2×D4×C73—41no—
91C2×D4×C73—41no—
92C2×D4×C73—42no—
93C2×D4×C73—41no—
94C2×D4×C73—42no—
95C2×D4×C73—41no—
96C22×C4×C73—21no—
97C22×C4×C73—22no—
98C22×C4×C73—21no—
99C22×C4×C73—41no—
100C22×C4×C73C2×D411yescharacteristic
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73C2×D411yescharacteristic
103C2×C8×C73—22no—
104C2×C8×C73—22no—
105C2×C8×C73—42no—
106C2×C8×C73—41no—
107C2×C8×C73—41no—
108C2×C8×C73—41no—
109C2×C8×C73—42no—
110C4⋊C4×C73—22no—
111C4⋊C4×C73—22no—
112C4⋊C4×C73—21no—
113C4⋊C4×C73—21no—
114C4⋊C4×C73—22no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—41no—
120C22⋊C4×C73—48no—
121C22⋊C4×C73—84no—
122C42×C73—41no—
123C42×C73—41no—
124C42×C73—41no—
125C42×C73—41no—
126C42×C73C2×D411yescharacteristic
127C42×C73C2×D411yescharacteristic
128C42×C73C2×D411yescharacteristic
129C42×C73#4111yescharacteristic
130C23×C73—42no—
131C23×C73—44no—
132C23×C73—42no—
133C23×C73—42no—
134C23×C73—44no—
135C23×C73#12011yescharacteristic
136Q8×C73—22no—
137Q8×C73—22no—
138Q8×C73—41no—
139Q8×C73—41no—
140Q8×C73—44no—
141Q8×C73—81no—
142Q8×C73—81no—
143D4×C73—44no—
144D4×C73—82no—
145D4×C73—81no—
146D4×C73—81no—
147D4×C73—82no—
148D4×C73—81no—
149D4×C73—81no—
150D4×C73—81no—
151D4×C73—81no—
152C2×C4×C73—22no—
153C2×C4×C73—22no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—42no—
158C2×C4×C73—42no—
159C2×C4×C73—42no—
160C2×C4×C73—42no—
161C2×C4×C73—42no—
162C2×C4×C73—82no—
163C2×C4×C73#12011yescharacteristic
164C2×C4×C73#12011yescharacteristic
165C2×C4×C73#12111yescharacteristic
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12711yescharacteristic
169C2×C4×C73C2×SD1611yescharacteristic
170C2×C4×C73#13511yescharacteristic
171C2×C4×C73#13512yes—
172C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C73AGL(1,ℤ/8)12yes—
174C2×C4×C73#13711yescharacteristic
175C8×C73—42no—
176C8×C73—42no—
177C8×C73—81no—
178C22×C73—84no—
179C22×C73—84no—
180C22×C73—82no—
181C22×C73—82no—
182C22×C73—82no—
183C22×C73#39211yescharacteristic
184C22×C73#44912yes—
185C22×C73#45011yescharacteristic
186C22×C73#45211yescharacteristic
187C22×C73#45512yes—
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—42no—
191C4×C73—82no—
192C4×C73—81no—
193C4×C73—82no—
194C4×C73#44811yescharacteristic
195C4×C73#44911yescharacteristic
196C4×C73#46312yes—
197C4×C73#46511yescharacteristic
198C4×C73#46811yescharacteristic
199C4×C73#49511yescharacteristic
200C4×C73#49611yescharacteristic
201#10071C7311yesmaximal, characteristic, direct product factor, Sylow
202C2×C73—162no—
203C2×C73#164012yes—
204C2×C73#164212yes—
205C2×C73#200011yescharacteristic
206C2×C73#200411yescharacteristic
207C2×C73#201211yescharacteristic
208C2×#486C2×C7311yescharacteristic
209C2×#475C2×C7311yescharacteristic, complemented
210#2548C2×C7311yescharacteristic
211#1981C2×C7311yescharacteristic, complemented
212C2×#331C2×C7311yescharacteristic
213C2×Q8⋊C8C2×C7311yescharacteristic, complemented
214#1279C2×C7311yescharacteristic, complemented
215C73#1007111yescharacteristic, direct product factor, Sylow
216C2×C4⋊Q8C22×C7311yescharacteristic
217C2×#127C22×C7311yescharacteristic
218C2×#124—21no—
219C2×C4×Q8C22×C7311yescharacteristic
220#486—22no—
221#486—22no—
222#475—22no—
223#475—22no—
224C2×#106—21no—
225C2×C4⋊C8C22×C7311yescharacteristic
226C2×C4⋊C8C22×C7311yescharacteristic
227C2×#103—21no—
228C2×#103—22no—
229C2×#102—22no—
230C2×C4×C8C22×C7311yescharacteristic
231#394—22no—
232#390—22no—
233#390—21no—
234C4×C4⋊C4C22×C7311yescharacteristic
235#331—22no—
236Q8⋊C8—22no—
237C22×Q8—21no—
238C22×Q8D4×C7311yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4D4×C7312yes—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8D4×C7312yes—
249#127—21no—
250#127—21no—
251#127—21no—
252#127—21no—
253#124—42no—
254#124—42no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4D4×C7311yescharacteristic
263C2×C4⋊C4D4×C7312yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42C23×C7311yescharacteristic
269#106—24no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8D4×C7312yes—
274#103—24no—
275#103—44no—
276#102—28no—
277C4×C8—21no—
278C4×C8D4×C7312yes—
279#95—21no—
280C24—22no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8SD16×C7312yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—44no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4C2×D4×C7311yescharacteristic
309C22×C4C2×D4×C7311yescharacteristic
310C22×C4C2×D4×C7311yescharacteristic
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C22⋊C4—48no—
329C22⋊C4—84no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42C2×D4×C7311yescharacteristic
335C42C2×D4×C7311yescharacteristic
336C42C2×D4×C7311yescharacteristic
337C42#41×C7311yescharacteristic
338C23—42no—
339C23—44no—
340C23—42no—
341C23—42no—
342C23—44no—
343C23#120×C7311yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—41no—
347Q8—41no—
348Q8—44no—
349Q8—81no—
350Q8—81no—
351D4—44no—
352D4—82no—
353D4—81no—
354D4—81no—
355D4—82no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4#120×C7311yescharacteristic
372C2×C4#120×C7311yescharacteristic
373C2×C4#121×C7311yescharacteristic
374C2×C4#121×C7311yescharacteristic
375C2×C4#121×C7311yescharacteristic
376C2×C4#127×C7311yescharacteristic
377C2×C4C2×SD16×C7311yescharacteristic
378C2×C4#135×C7311yescharacteristic
379C2×C4#135×C7312yes—
380C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
381C2×C4AGL(1,ℤ/8)×C7312yes—
382C2×C4#137×C7311yescharacteristic
383C8—42no—
384C8—42no—
385C8—81no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—84no—
390C22—84no—
391C22#392×C7311yescharacteristic
392C22#449×C7312yes—
393C22#450×C7311yescharacteristic
394C22#452×C7311yescharacteristic
395C22#455×C7312yes—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—82no—
401C4—82no—
402C4#448×C7311yescharacteristic
403C4#449×C7311yescharacteristic
404C4#463×C7312yes—
405C4#465×C7311yescharacteristic
406C4#468×C7311yescharacteristic
407C4#495×C7311yescharacteristic
408C4#496×C7311yescharacteristic
409C2—162no—
410C2C73×#164012yes—
411C2C73×#164212yes—
412C2C73×#200011yescharacteristic
413C2C73×#200411yescharacteristic
414C2C73×#201211yescharacteristic
subgroupG/H
center, socleC23×C73#120
commutator, Frattini, cosocleC2×C42C23×C73
prime radicalC2×#127×C73C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#475×C73C2semidirect
4. C73×#1981C2semidirect
6. C2×Q8⋊C8×C73C2semidirect
7. C73×#1279C2semidirect
201. #10071C73direct (N × H)
209. C2×#475C2×C73semidirect
211. #1981C2×C73semidirect
213. C2×Q8⋊C8C2×C73semidirect
214. #1279C2×C73semidirect
215. C73#10071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#486×C732
C2×#475×C732
C73×#25482
C73×#19812
C2×#331×C732
C2×Q8⋊C8×C732
C73×#12792
#1007173