Finite Group Databasebeta

← C73×#7762 #32917166

Subgroups — C73×#7762 #32917166

Order 18688.

542 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2682C211yesmaximal, characteristic
2C73×#2043C211yesmaximal, characteristic, complemented
3C73×#2023C211yesmaximal, characteristic, complemented
4C73×#2023C211yesmaximal, characteristic, complemented
5C73×#1271C211yesmaximal, characteristic, complemented
6C73×#1270C211yesmaximal, characteristic, complemented
7C73×#1270C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C73C2211yescharacteristic
9C2×#123×C73—21no—
10C2×#121×C73—21no—
11C2×#103×C73—21no—
12C2×#103×C73—21no—
13C2×#102×C73—21no—
14C2×#102×C73—21no—
15#396×C73—21no—
16#394×C73—21no—
17#392×C73—21no—
18#390×C73C2211yescharacteristic
19#390×C73C2211yescharacteristic
20#389×C73C2211yescharacteristic
21#385×C73—21no—
22#340×C73—21no—
23#340×C73—21no—
24#336×C73C2211yescharacteristic
25#336×C73C2211yescharacteristic
26#336×C73C2211yescharacteristic
27C22×Q8×C73—21no—
28C22×D4×C73—21no—
29C22×D4×C73—21no—
30C22×D4×C73D411yescharacteristic
31C22×D4×C73D411yescharacteristic
32C23×C4×C73—21no—
33C22×C8×C73—21no—
34C22×C8×C73—21no—
35C22×C8×C73—21no—
36C4⋊Q8×C73—42no—
37#123×C73—42no—
38#123×C73—42no—
39#121×C73—42no—
40#121×C73—42no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C4⋊C4×C73D411yescharacteristic
47C2×C4⋊C4×C73D411yescharacteristic
48C2×C4⋊C4×C73D411yescharacteristic
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C42×C73C2311yescharacteristic
57#103×C73—42no—
58#103×C73—42no—
59#102×C73—42no—
60#102×C73—42no—
61#95×C73—21no—
62#95×C73—21no—
63C24×C73—21no—
64C24×C73—21no—
65C2×Q8×C73—22no—
66C2×Q8×C73—41no—
67C2×Q8×C73—41no—
68C2×Q8×C73—41no—
69C2×Q8×C73—41no—
70C2×Q8×C73—41no—
71C2×D4×C73—21no—
72C2×D4×C73—21no—
73C2×D4×C73—21no—
74C2×D4×C73—41no—
75C2×D4×C73—41no—
76C2×D4×C73—41no—
77C2×D4×C73—41no—
78C2×D4×C73—42no—
79C2×D4×C73—42no—
80C2×D4×C73—42no—
81C2×D4×C73—81no—
82C2×D4×C73—81no—
83C2×D4×C73—81no—
84C2×D4×C73SD1612yes—
85C22×C4×C73—21no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—42no—
90C22×C4×C73—42no—
91C22×C4×C73—41no—
92C22×C4×C73—81no—
93C22×C4×C73C2×D411yescharacteristic
94C22×C4×C73C2×D411yescharacteristic
95C22×C4×C73C2×D411yescharacteristic
96C2×C8×C73—41no—
97C2×C8×C73—41no—
98C2×C8×C73—41no—
99C2×C8×C73—41no—
100C2×C8×C73—41no—
101C2×C8×C73—41no—
102C2×C8×C73—41no—
103C2×C8×C73—41no—
104C2×C8×C73—41no—
105C4⋊C4×C73—21no—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—22no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C4⋊C4×C73—41no—
111C4⋊C4×C73—42no—
112C4⋊C4×C73—41no—
113C4⋊C4×C73—41no—
114C4⋊C4×C73—41no—
115C4⋊C4×C73—41no—
116C4⋊C4×C73D812yes—
117C22⋊C4×C73—42no—
118C22⋊C4×C73—42no—
119C22⋊C4×C73—41no—
120C22⋊C4×C73—41no—
121C22⋊C4×C73—41no—
122C22⋊C4×C73—41no—
123C22⋊C4×C73—41no—
124C22⋊C4×C73—41no—
125C22⋊C4×C73—41no—
126C22⋊C4×C73—41no—
127C22⋊C4×C73—81no—
128C42×C73—41no—
129C23×C73—22no—
130C23×C73—41no—
131C23×C73—41no—
132C23×C73—41no—
133C23×C73—41no—
134C23×C73—41no—
135C23×C73—41no—
136C23×C73—41no—
137C23×C73—41no—
138C23×C73—41no—
139C23×C73—41no—
140C23×C73—41no—
141C23×C73—41no—
142C23×C73—41no—
143C23×C73#12011yescharacteristic
144Q8×C73—42no—
145Q8×C73—42no—
146Q8×C73—81no—
147Q8×C73—81no—
148D4×C73—42no—
149D4×C73—42no—
150D4×C73—81no—
151D4×C73—81no—
152D4×C73—84no—
153D4×C73—84no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—22no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73—41no—
170C2×C4×C73—41no—
171C2×C4×C73—42no—
172C2×C4×C73—41no—
173C2×C4×C73—84no—
174C2×C4×C73—81no—
175C2×C4×C73—81no—
176C2×C4×C73C2×D811yescharacteristic
177C2×C4×C73C2×SD1611yescharacteristic
178C2×C4×C73#13511yescharacteristic
179C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C73#13711yescharacteristic
182C8×C73—81no—
183C8×C73—81no—
184C8×C73—81no—
185C22×C73—42no—
186C22×C73—42no—
187C22×C73—42no—
188C22×C73—81no—
189C22×C73—81no—
190C22×C73—81no—
191C22×C73—81no—
192C22×C73—81no—
193C22×C73—81no—
194C22×C73—81no—
195C22×C73—81no—
196C22×C73—81no—
197C22×C73—81no—
198C22×C73—81no—
199C22×C73#44711yescharacteristic
200C22×C73#45011yescharacteristic
201C22×C73#45211yescharacteristic
202C22×C73#45411yescharacteristic
203C22×C73#45511yescharacteristic
204C22×C73#45511yescharacteristic
205C22×C73B(4,𝔽2)11yescharacteristic
206C4×C73—21no—
207C4×C73—21no—
208C4×C73—21no—
209C4×C73—21no—
210C4×C73—21no—
211C4×C73—21no—
212C4×C73—42no—
213C4×C73—82no—
214C4×C73—81no—
215C4×C73—81no—
216C4×C73—81no—
217#7762C7311yesmaximal, characteristic, direct product factor, Sylow
218C2×C73—82no—
219C2×C73—161no—
220C2×C73#159811yescharacteristic
221C2×C73#160111yescharacteristic
222C2×C73#160611yescharacteristic
223C2×C73#163511yescharacteristic
224C2×C73#163911yescharacteristic
225C2×C73#164111yescharacteristic
226C2×C73#166211yescharacteristic
227#2682C2×C7311yescharacteristic
228#2043C2×C7311yescharacteristic, complemented
229#2023C2×C7311yescharacteristic, complemented
230#2023C2×C7311yescharacteristic, complemented
231#1271C2×C7311yescharacteristic, complemented
232#1270C2×C7311yescharacteristic, complemented
233#1270C2×C7311yescharacteristic, complemented
234C73#776211yescharacteristic, direct product factor, Sylow
235C2×C4⋊Q8C22×C7311yescharacteristic
236C2×#123—21no—
237C2×#121—21no—
238C2×#103—21no—
239C2×#103—21no—
240C2×#102—21no—
241C2×#102—21no—
242#396—21no—
243#394—21no—
244#392—21no—
245#390C22×C7311yescharacteristic
246#390C22×C7311yescharacteristic
247#389C22×C7311yescharacteristic
248#385—21no—
249#340—21no—
250#340—21no—
251#336C22×C7311yescharacteristic
252#336C22×C7311yescharacteristic
253#336C22×C7311yescharacteristic
254C22×Q8—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4D4×C7311yescharacteristic
258C22×D4D4×C7311yescharacteristic
259C23×C4—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C4⋊Q8—42no—
264#123—42no—
265#123—42no—
266#121—42no—
267#121—42no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4D4×C7311yescharacteristic
273C2×C4⋊C4D4×C7311yescharacteristic
274C2×C4⋊C4D4×C7311yescharacteristic
275C2×C4⋊C4D4×C7311yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42C23×C7311yescharacteristic
284#103—42no—
285#103—42no—
286#102—42no—
287#102—42no—
288#95—21no—
289#95—21no—
290C24—21no—
291C24—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4—81no—
311C2×D4SD16×C7312yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—81no—
320C22×C4C2×D4×C7311yescharacteristic
321C22×C4C2×D4×C7311yescharacteristic
322C22×C4C2×D4×C7311yescharacteristic
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4D8×C7312yes—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—81no—
355C42—41no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23#120×C7311yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
379D4—84no—
380D4—84no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—84no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4C2×D8×C7311yescharacteristic
404C2×C4C2×SD16×C7311yescharacteristic
405C2×C4#135×C7311yescharacteristic
406C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
407C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
408C2×C4#137×C7311yescharacteristic
409C8—81no—
410C8—81no—
411C8—81no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#447×C7311yescharacteristic
427C22#450×C7311yescharacteristic
428C22#452×C7311yescharacteristic
429C22#454×C7311yescharacteristic
430C22#455×C7311yescharacteristic
431C22#455×C7311yescharacteristic
432C22B(4,𝔽2)×C7311yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—82no—
441C4—81no—
442C4—81no—
443C4—81no—
444C2—82no—
445C2—161no—
446C2C73×#159811yescharacteristic
447C2C73×#160111yescharacteristic
448C2C73×#160611yescharacteristic
449C2C73×#163511yescharacteristic
450C2C73×#163911yescharacteristic
451C2C73×#164111yescharacteristic
452C2C73×#166211yescharacteristic
subgroupG/H
center, socleC23×C73#120
commutator, Frattini, cosocleC2×C42C23×C73
prime radicalC73×#2682C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2043C2semidirect
3. C73×#2023C2semidirect
4. C73×#2023C2semidirect
5. C73×#1271C2semidirect
6. C73×#1270C2semidirect
7. C73×#1270C2semidirect
217. #7762C73direct (N × H)
228. #2043C2×C73semidirect
229. #2023C2×C73semidirect
230. #2023C2×C73semidirect
231. #1271C2×C73semidirect
232. #1270C2×C73semidirect
233. #1270C2×C73semidirect
234. C73#7762direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#26822
C73×#20432
C73×#20232
C73×#12712
C73×#12702
#776273