Finite Group Databasebeta

← C73×#12248 #32921652

Subgroups — C73×#12248 #32921652

Order 18688.

428 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2609C211yesmaximal, characteristic
2C73×#2061C211yesmaximal, characteristic, complemented
3C73×#1611C211yesmaximal, characteristic, complemented
4C73×#1599C211yesmaximal, characteristic
5C73×#1346C211yesmaximal, characteristic, complemented
6C73×#1345C211yesmaximal, characteristic
7C73×#1324C211yesmaximal, characteristic, complemented
8#544×C73C2211yescharacteristic
9C2×#117×C73—21no—
10#483×C73—21no—
11#450×C73—21no—
12#428×C73—21no—
13#398×C73C2211yescharacteristic
14#396×C73—21no—
15#394×C73C2211yescharacteristic
16#388×C73—21no—
17#387×C73—21no—
18#386×C73—21no—
19#385×C73—21no—
20#382×C73C2211yescharacteristic
21#343×C73C2211yescharacteristic
22#339×C73—21no—
23#328×C73—21no—
24#328×C73—21no—
25#324×C73C2211yescharacteristic
26#324×C73C2211yescharacteristic
27C22×D4×C73D411yescharacteristic
28C23×C4×C73—21no—
29C2×SD16×C73—41no—
30C2×M16×C73—21no—
31C4⋊Q8×C73—41no—
32#125×C73—41no—
33C22⋊Q8×C73—21no—
34C22⋊Q8×C73—21no—
35#117×C73—21no—
36#117×C73—42no—
37#117×C73—42no—
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×C73C2311yescharacteristic
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C42×C73—21no—
50C2×C42×C73D411yescharacteristic
51#107×C73—41no—
52#106×C73—41no—
53#106×C73—41no—
54#103×C73—41no—
55#102×C73—41no—
56C22⋊C8×C73—21no—
57C22⋊C8×C73—21no—
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61#95×C73—21no—
62#95×C73D411yescharacteristic
63#95×C73D411yescharacteristic
64C24×C73—21no—
65C2×Q8×C73—21no—
66C2×Q8×C73—21no—
67C2×D4×C73—41no—
68C2×D4×C73—42no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—42no—
75C22×C4×C73—42no—
76C22×C4×C73—42no—
77C22×C4×C73C2×D411yescharacteristic
78C22×C4×C73C2×D411yescharacteristic
79C22×C4×C73C2×D411yescharacteristic
80SD16×C73—82no—
81M16×C73—22no—
82M16×C73—41no—
83C2×C8×C73—41no—
84C2×C8×C73—41no—
85C2×C8×C73—41no—
86C4⋊C4×C73—21no—
87C4⋊C4×C73—21no—
88C4⋊C4×C73—21no—
89C4⋊C4×C73—21no—
90C4⋊C4×C73—41no—
91C4⋊C4×C73—41no—
92C4⋊C4×C73—41no—
93C4⋊C4×C73—42no—
94C4⋊C4×C73—42no—
95C4⋊C4×C73—41no—
96C4⋊C4×C73—41no—
97C4⋊C4×C73—41no—
98C4⋊C4×C73—41no—
99C22⋊C4×C73—42no—
100C22⋊C4×C73—42no—
101C22⋊C4×C73—42no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—41no—
104C22⋊C4×C73—41no—
105C22⋊C4×C73—81no—
106C42×C73—21no—
107C42×C73—42no—
108C42×C73—41no—
109C42×C73SD1612yescomplemented
110C23×C73—22no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73#12011yescharacteristic
118Q8×C73—81no—
119Q8×C73—81no—
120D4×C73—42no—
121D4×C73—81no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—22no—
125C2×C4×C73—22no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—48no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
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×C73C2×SD1611yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C8×C73—42no—
149C8×C73—81no—
150C8×C73—81no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—42no—
154C22×C73—42no—
155C22×C73—42no—
156C22×C73—81no—
157C22×C73—81no—
158C22×C73—81no—
159C22×C73—81no—
160C22×C73#45011yescharacteristic
161C22×C73#45611yescharacteristic
162C22×C73B(4,𝔽2)11yescharacteristic
163C4×C73—21no—
164C4×C73—21no—
165C4×C73—42no—
166C4×C73—42no—
167C4×C73—41no—
168C4×C73—41no—
169C4×C73—81no—
170C4×C73—81no—
171C4×C73—81no—
172C4×C73—81no—
173C4×C73—81no—
174#12248C7311yesmaximal, characteristic, direct product factor, Sylow
175C2×C73—21no—
176C2×C73—21no—
177C2×C73—82no—
178C2×C73#160411yescharacteristic
179C2×C73#219711yescharacteristic
180C2×C73#220411yescharacteristic
181#2609C2×C7311yescharacteristic
182#2061C2×C7311yescharacteristic, complemented
183#1611C2×C7311yescharacteristic, complemented
184#1599C2×C7311yescharacteristic
185#1346C2×C7311yescharacteristic, complemented
186#1345C2×C7311yescharacteristic
187#1324C2×C7311yescharacteristic, complemented
188C73#1224811yescharacteristic, direct product factor, Sylow
189#544C22×C7311yescharacteristic
190C2×#117—21no—
191#483—21no—
192#450—21no—
193#428—21no—
194#398C22×C7311yescharacteristic
195#396—21no—
196#394C22×C7311yescharacteristic
197#388—21no—
198#387—21no—
199#386—21no—
200#385—21no—
201#382C22×C7311yescharacteristic
202#343C22×C7311yescharacteristic
203#339—21no—
204#328—21no—
205#328—21no—
206#324C22×C7311yescharacteristic
207#324C22×C7311yescharacteristic
208C22×D4D4×C7311yescharacteristic
209C23×C4—21no—
210C2×SD16—41no—
211C2×M16—21no—
212C4⋊Q8—41no—
213#125—41no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216#117—21no—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4D4×C7311yescharacteristic
223C2×C4⋊C4D4×C7311yescharacteristic
224C2×C4⋊C4C23×C7311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42D4×C7311yescharacteristic
232#107—41no—
233#106—41no—
234#106—41no—
235#103—41no—
236#102—41no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95D4×C7311yescharacteristic
244#95D4×C7311yescharacteristic
245C24—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×D4×C7311yescharacteristic
259C22×C4C2×D4×C7311yescharacteristic
260C22×C4C2×D4×C7311yescharacteristic
261SD16—82no—
262M16—22no—
263M16—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—81no—
287C42—21no—
288C42—42no—
289C42—41no—
290C42SD16×C7312yescomplemented
291C23—22no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#120×C7311yescharacteristic
299Q8—81no—
300Q8—81no—
301D4—42no—
302D4—81no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—48no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4C2×SD16×C7311yescharacteristic
328C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
329C8—42no—
330C8—81no—
331C8—81no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#450×C7311yescharacteristic
342C22#456×C7311yescharacteristic
343C22B(4,𝔽2)×C7311yescharacteristic
344C4—21no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—81no—
355C2—21no—
356C2—21no—
357C2—82no—
358C2C73×#160411yescharacteristic
359C2C73×#219711yescharacteristic
360C2C73×#220411yescharacteristic
subgroupG/H
center, socleC22×C73B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C73
prime radicalC22×D4×C73D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C73 ◁ C2×C4⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2061C2semidirect
3. C73×#1611C2semidirect
5. C73×#1346C2semidirect
7. C73×#1324C2semidirect
109. C42×C73SD16semidirect
174. #12248C73direct (N × H)
182. #2061C2×C73semidirect
183. #1611C2×C73semidirect
185. #1346C2×C73semidirect
187. #1324C2×C73semidirect
188. C73#12248direct (N × H)
290. C42SD16×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#26092
C73×#20612
C73×#16112
C73×#15992
C73×#13462
C73×#13452
C73×#13242
#1224873