Finite Group Databasebeta

← C73×#7770 #32917174

Subgroups — C73×#7770 #32917174

Order 18688.

460 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2724C211yesmaximal, characteristic
2C73×#2058C211yesmaximal, characteristic, complemented
3C73×#2043C211yesmaximal, characteristic, complemented
4C73×#2023C211yesmaximal, characteristic
5C73×#1271C211yesmaximal, characteristic, complemented
6C73×#1271C211yesmaximal, characteristic, complemented
7C73×#1270C211yesmaximal, characteristic
8C2×C4⋊Q8×C73C2211yescharacteristic
9C2×#123×C73—21no—
10C2×#106×C73—21no—
11C2×#103×C73—21no—
12C2×#103×C73—21no—
13C2×#102×C73—21no—
14#398×C73—21no—
15#397×C73—21no—
16#397×C73—21no—
17#390×C73C2211yescharacteristic
18#389×C73C2211yescharacteristic
19#389×C73C2211yescharacteristic
20#385×C73—21no—
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×C73D411yescharacteristic
29C23×C4×C73—21no—
30C22×C8×C73—21no—
31C22×C8×C73—21no—
32C22×C8×C73—21no—
33C4⋊Q8×C73—42no—
34#123×C73—42no—
35#123×C73—42no—
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73D411yescharacteristic
42C2×C4⋊C4×C73D411yescharacteristic
43C2×C4⋊C4×C73D411yescharacteristic
44C2×C4⋊C4×C73D411yescharacteristic
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C42×C73C2311yescharacteristic
52#106×C73—42no—
53#103×C73—42no—
54#103×C73—42no—
55#102×C73—42no—
56#95×C73—21no—
57#95×C73—21no—
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61C24×C73—21no—
62C2×Q8×C73—22no—
63C2×Q8×C73—41no—
64C2×Q8×C73—41no—
65C2×Q8×C73—41no—
66C2×Q8×C73—41no—
67C2×Q8×C73—41no—
68C2×D4×C73—21no—
69C2×D4×C73—41no—
70C2×D4×C73—41no—
71C2×D4×C73SD1612yes—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—42no—
78C22×C4×C73—41no—
79C22×C4×C73—81no—
80C22×C4×C73—81no—
81C22×C4×C73C2×D411yescharacteristic
82C22×C4×C73C2×D411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C2×C8×C73—22no—
85C2×C8×C73—41no—
86C2×C8×C73—41no—
87C2×C8×C73—41no—
88C2×C8×C73—41no—
89C2×C8×C73—41no—
90C2×C8×C73—41no—
91C2×C8×C73—41no—
92C2×C8×C73—41no—
93C4⋊C4×C73—22no—
94C4⋊C4×C73—22no—
95C4⋊C4×C73—21no—
96C4⋊C4×C73—21no—
97C4⋊C4×C73—21no—
98C4⋊C4×C73—21no—
99C4⋊C4×C73—41no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—41no—
104C4⋊C4×C73—41no—
105C4⋊C4×C73—41no—
106C4⋊C4×C73—41no—
107C4⋊C4×C73—41no—
108C4⋊C4×C73—81no—
109C22⋊C4×C73—42no—
110C22⋊C4×C73—42no—
111C22⋊C4×C73—42no—
112C22⋊C4×C73—41no—
113C22⋊C4×C73—41no—
114C22⋊C4×C73—41no—
115C22⋊C4×C73—41no—
116C42×C73—41no—
117C23×C73—22no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73#12011yescharacteristic
125Q8×C73—42no—
126Q8×C73—42no—
127Q8×C73—81no—
128Q8×C73—81no—
129D4×C73—42no—
130D4×C73—81no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—42no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—84no—
154C2×C4×C73—81no—
155C2×C4×C73—81no—
156C2×C4×C73C2×SD1611yescharacteristic
157C2×C4×C73#13511yescharacteristic
158C2×C4×C73#13511yescharacteristic
159C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C73#13711yescharacteristic
162C8×C73—42no—
163C8×C73—81no—
164C8×C73—81no—
165C22×C73—42no—
166C22×C73—42no—
167C22×C73—42no—
168C22×C73—81no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73#44911yescharacteristic
173C22×C73#45011yescharacteristic
174C22×C73#45211yescharacteristic
175C22×C73#45411yescharacteristic
176C22×C73#45511yescharacteristic
177C22×C73#45611yescharacteristic
178C22×C73B(4,𝔽2)11yescharacteristic
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—82no—
186C4×C73—81no—
187C4×C73—81no—
188C4×C73—81no—
189C4×C73—81no—
190C4×C73—81no—
191#7770C7311yesmaximal, characteristic, direct product factor, Sylow
192C2×C73—82no—
193C2×C73#159711yescharacteristic
194C2×C73#160111yescharacteristic
195C2×C73#160311yescharacteristic
196C2×C73#163511yescharacteristic
197C2×C73#164011yescharacteristic
198C2×C73#165411yescharacteristic
199C2×C73#166411yescharacteristic
200#2724C2×C7311yescharacteristic
201#2058C2×C7311yescharacteristic, complemented
202#2043C2×C7311yescharacteristic, complemented
203#2023C2×C7311yescharacteristic
204#1271C2×C7311yescharacteristic, complemented
205#1271C2×C7311yescharacteristic, complemented
206#1270C2×C7311yescharacteristic
207C73#777011yescharacteristic, direct product factor, Sylow
208C2×C4⋊Q8C22×C7311yescharacteristic
209C2×#123—21no—
210C2×#106—21no—
211C2×#103—21no—
212C2×#103—21no—
213C2×#102—21no—
214#398—21no—
215#397—21no—
216#397—21no—
217#390C22×C7311yescharacteristic
218#389C22×C7311yescharacteristic
219#389C22×C7311yescharacteristic
220#385—21no—
221#385—21no—
222#340—21no—
223#340—21no—
224#336C22×C7311yescharacteristic
225#336C22×C7311yescharacteristic
226#336C22×C7311yescharacteristic
227C22×Q8—21no—
228C22×D4D4×C7311yescharacteristic
229C23×C4—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C4⋊Q8—42no—
234#123—42no—
235#123—42no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4D4×C7311yescharacteristic
242C2×C4⋊C4D4×C7311yescharacteristic
243C2×C4⋊C4D4×C7311yescharacteristic
244C2×C4⋊C4D4×C7311yescharacteristic
245C2×C4⋊C4D4×C7311yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42C23×C7311yescharacteristic
252#106—42no—
253#103—42no—
254#103—42no—
255#102—42no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4SD16×C7312yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4C2×D4×C7311yescharacteristic
282C22×C4C2×D4×C7311yescharacteristic
283C22×C4C2×D4×C7311yescharacteristic
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—81no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—41no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#120×C7311yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329D4—42no—
330D4—81no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—84no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4C2×SD16×C7311yescharacteristic
357C2×C4#135×C7311yescharacteristic
358C2×C4#135×C7311yescharacteristic
359C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
360C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
361C2×C4#137×C7311yescharacteristic
362C8—42no—
363C8—81no—
364C8—81no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#449×C7311yescharacteristic
373C22#450×C7311yescharacteristic
374C22#452×C7311yescharacteristic
375C22#454×C7311yescharacteristic
376C22#455×C7311yescharacteristic
377C22#456×C7311yescharacteristic
378C22B(4,𝔽2)×C7311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C2—82no—
392C2C73×#159711yescharacteristic
393C2C73×#160111yescharacteristic
394C2C73×#160311yescharacteristic
395C2C73×#163511yescharacteristic
396C2C73×#164011yescharacteristic
397C2C73×#165411yescharacteristic
398C2C73×#166411yescharacteristic
subgroupG/H
center, socleC23×C73#120
commutator, Frattini, cosocleC2×C42C23×C73
prime radicalC22×D4×C73D4
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×#2058C2semidirect
3. C73×#2043C2semidirect
5. C73×#1271C2semidirect
6. C73×#1271C2semidirect
191. #7770C73direct (N × H)
201. #2058C2×C73semidirect
202. #2043C2×C73semidirect
204. #1271C2×C73semidirect
205. #1271C2×C73semidirect
207. C73#7770direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#27242
C73×#20582
C73×#20432
C73×#20232
C73×#12712
C73×#12702
#777073