Finite Group Databasebeta

← C73×#11848 #32921252

Subgroups — C73×#11848 #32921252

Order 18688.

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

normal subgroups only

This group has 572 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×#2406C211yesmaximal, characteristic
2C73×#1821C211yesmaximal, characteristic, complemented
3C2×#324×C73C211yesmaximal, characteristic, complemented
4C73×#1354C212yesmaximal, complemented
5C73×#1345C212yesmaximal, complemented
6C22×#41×C73C411yescharacteristic
7C2×#123×C73—21no—
8C22×C4⋊C4×C73C2211yescharacteristic
9#407×C73—21no—
10C2×C22⋊C8×C73—21no—
11#395×C73C411yescharacteristic
12#395×C73C2211yescharacteristic
13#394×C73C411yescharacteristic
14#394×C73C2211yescharacteristic
15#386×C73—21no—
16#385×C73—21no—
17#385×C73—21no—
18#381×C73C411yescharacteristic
19#343×C73—21no—
20#343×C73C2212yes—
21#324×C73—21no—
22#324×C73C2212yes—
23C2×#41×C73—41no—
24C2×#41×C73—41no—
25C2×#41×C73—44no—
26C22×Q8×C73C2×C411yescharacteristic
27C22×D4×C73—21no—
28C22×D4×C73C2×C411yescharacteristic
29C23×C4×C73—21no—
30C23×C4×C73C2×C411yescharacteristic
31C23×C4×C73D411yescharacteristic
32C23×C4×C73D411yescharacteristic
33C2×M16×C73—22no—
34C22×C8×C73—41no—
35#123×C73—44no—
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—41no—
39C2×C4⋊C4×C73—41no—
40C2×C4⋊C4×C73C2×C411yescharacteristic
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—
49C22⋊C8×C73—22no—
50C22⋊C8×C73—41no—
51C22⋊C8×C73—41no—
52C22⋊C8×C73—41no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73C2×C411yescharacteristic
57#95×C73C2×C411yescharacteristic
58C24×C73—21no—
59C24×C73C22⋊C411yescharacteristic
60#41×C73—44no—
61#41×C73—44no—
62#41×C73—44no—
63#41×C73—44no—
64C2×Q8×C73—42no—
65C2×Q8×C73—41no—
66C2×D4×C73—24no—
67C2×D4×C73—41no—
68C2×D4×C73—42no—
69C2×D4×C73—41no—
70C2×D4×C73—41no—
71C2×D4×C73—42no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—22no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—41no—
83C22×C4×C73—42no—
84C22×C4×C73—42no—
85C22×C4×C73—42no—
86C22×C4×C73—42no—
87C22×C4×C73—42no—
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C22⋊C411yescharacteristic
91C22×C4×C73C22×C411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93C22×C4×C73C2×D411yescharacteristic
94M16×C73—42no—
95M16×C73—42no—
96C2×C8×C73—42no—
97C2×C8×C73—42no—
98C2×C8×C73—82no—
99C4⋊C4×C73—41no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—42no—
104C4⋊C4×C73—42no—
105C22⋊C4×C73—42no—
106C22⋊C4×C73—42no—
107C22⋊C4×C73—42no—
108C22⋊C4×C73—42no—
109C22⋊C4×C73—42no—
110C23×C73—22no—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73—22no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73#9911yescharacteristic
122C23×C73#9911yescharacteristic
123C23×C73C2×C22⋊C411yescharacteristic
124Q8×C73—42no—
125Q8×C73—42no—
126D4×C73—44no—
127D4×C73—42no—
128D4×C73—42no—
129D4×C73—44no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—22no—
133C2×C4×C73—21no—
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—42no—
141C2×C4×C73—42no—
142C2×C4×C73—41no—
143C2×C4×C73—42no—
144C2×C4×C73—48no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—81no—
159C2×C4×C73#13512yes—
160C8×C73—82no—
161C8×C73—82no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—42no—
170C22×C73—42no—
171C22×C73—42no—
172C22×C73—42no—
173C22×C73—42no—
174C22×C73—81no—
175C22×C73—81no—
176C22×C73—81no—
177C22×C73—81no—
178C22×C73AΣL(1,𝔽16)12yes—
179C22×C73C2×#9911yescharacteristic
180C22×C73#41611yescharacteristic
181C22×C73#42111yescharacteristic
182C4×C73—22no—
183C4×C73—41no—
184C4×C73—41no—
185C4×C73—42no—
186C4×C73—41no—
187C4×C73—41no—
188C4×C73—41no—
189C4×C73—81no—
190C4×C73—81no—
191#11848C7311yesmaximal, characteristic, direct product factor, Sylow
192C2×C73—21no—
193C2×C73—21no—
194C2×C73—22no—
195C2×C73—41no—
196C2×C73—82no—
197C2×C73#151611yescharacteristic
198C2×C73C2×AΣL(1,𝔽16)11yescharacteristic
199C2×C73#213311yescharacteristic
200#2406C2×C7311yescharacteristic
201#1821C2×C7311yescharacteristic, complemented
202C2×#324C2×C7311yescharacteristic, complemented
203#1354C2×C7312yescomplemented
204#1345C2×C7312yescomplemented
205C73#1184811yescharacteristic, direct product factor, Sylow
206C22×#41C4×C7311yescharacteristic
207C2×#123—21no—
208C22×C4⋊C4C22×C7311yescharacteristic
209#407—21no—
210C2×C22⋊C8—21no—
211#395C4×C7311yescharacteristic
212#395C22×C7311yescharacteristic
213#394C4×C7311yescharacteristic
214#394C22×C7311yescharacteristic
215#386—21no—
216#385—21no—
217#385—21no—
218#381C4×C7311yescharacteristic
219#343—21no—
220#343C22×C7312yes—
221#324—21no—
222#324C22×C7312yes—
223C2×#41—41no—
224C2×#41—41no—
225C2×#41—44no—
226C22×Q8C2×C4×C7311yescharacteristic
227C22×D4—21no—
228C22×D4C2×C4×C7311yescharacteristic
229C23×C4—21no—
230C23×C4C2×C4×C7311yescharacteristic
231C23×C4D4×C7311yescharacteristic
232C23×C4D4×C7311yescharacteristic
233C2×M16—22no—
234C22×C8—41no—
235#123—44no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C2×C4×C7311yescharacteristic
241C2×C4⋊C4D4×C7311yescharacteristic
242C2×C4⋊C4D4×C7311yescharacteristic
243C2×C4⋊C4C23×C7311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C22⋊C8—22no—
250C22⋊C8—41no—
251C22⋊C8—41no—
252C22⋊C8—41no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95C2×C4×C7311yescharacteristic
257#95C2×C4×C7311yescharacteristic
258C24—21no—
259C24C22⋊C4×C7311yescharacteristic
260#41—44no—
261#41—44no—
262#41—44no—
263#41—44no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×D4—24no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C22⋊C4×C7311yescharacteristic
289C22×C4C22⋊C4×C7311yescharacteristic
290C22×C4C22⋊C4×C7311yescharacteristic
291C22×C4C22×C4×C7311yescharacteristic
292C22×C4C2×D4×C7311yescharacteristic
293C22×C4C2×D4×C7311yescharacteristic
294M16—42no—
295M16—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—82no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#99×C7311yescharacteristic
322C23#99×C7311yescharacteristic
323C23C2×C22⋊C4×C7311yescharacteristic
324Q8—42no—
325Q8—42no—
326D4—44no—
327D4—42no—
328D4—42no—
329D4—44no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4#135×C7312yes—
360C8—82no—
361C8—82no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22AΣL(1,𝔽16)×C7312yes—
379C22C2×#99×C7311yescharacteristic
380C22#416×C7311yescharacteristic
381C22#421×C7311yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—82no—
396C2C73×#151611yescharacteristic
397C2C2×AΣL(1,𝔽16)×C7311yescharacteristic
398C2C73×#213311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#99
commutatorC22×C4C22×C4×C73
Frattini, cosocleC2×C4⋊C4C23×C73
prime radicalC22×#41×C73C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C24×C73 ◁ C22×C4⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#1821C2semidirect
3. C2×#324×C73C2semidirect
4. C73×#1354C2semidirect
5. C73×#1345C2semidirect
191. #11848C73direct (N × H)
201. #1821C2×C73semidirect
202. C2×#324C2×C73semidirect
203. #1354C2×C73semidirect
204. #1345C2×C73semidirect
205. C73#11848direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#24062
C73×#18212
C2×#324×C732
C73×#13542
C73×#13452
#1184873