Finite Group Databasebeta

← #32973009

Subgroups — #32973009

Order 18688.

560 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#22610005C211yesmaximal, characteristic
2#22608258C211yesmaximal, characteristic
3C73×#1437C211yesmaximal, characteristic
4#17988551—21no—
5#17988443—21no—
6#17988354—21no—
7#17988354C412yescomplemented
8#17988311—21no—
9#17988311C412yescomplemented
10C22×C22⋊C4×C73—21no—
11C22×C22⋊C4×C73C2211yescharacteristic
12C22×C42×C73C411yescharacteristic, complemented
13C2×#95×C73—21no—
14C2×#95×C73C411yescharacteristic, complemented
15C2×#11924012—41no—
16C2×C22⋊Dic73—22no—
17C2×C22⋊Dic73—41no—
18C2×C22⋊Dic73—41no—
19C22⋊#10596831—22no—
20C22⋊#10596831—41no—
21C25×C73D411yescharacteristic
22C23×C4×C73—21no—
23C23×C4×C73—21no—
24C23×C4×C73C2×C411yescharacteristic
25C23×C4×C73D411yescharacteristic
26C2×C22⋊C4×C73—24no—
27C2×C22⋊C4×C73—41no—
28C2×C22⋊C4×C73—41no—
29C2×C22⋊C4×C73—42no—
30C2×C22⋊C4×C73—42no—
31C2×C22⋊C4×C73C2×C411yescharacteristic
32C2×C22⋊C4×C73C2×C411yescharacteristic
33C2×C22⋊C4×C73D411yescharacteristic
34C2×C22⋊C4×C73Q811yescharacteristic
35C2×C42×C73—41no—
36C2×C42×C73—42no—
37#95×C73—42no—
38#95×C73—41no—
39C22×Dic73—42no—
40#11924012—81no—
41#11924012—81no—
42C2×#10596831—42no—
43C22⋊Dic73—42no—
44C22⋊Dic73—42no—
45C22⋊Dic73—81no—
46C22⋊Dic73—81no—
47C22⋊Dic73—81no—
48C22⋊Dic73—81no—
49C24×C73—22no—
50C24×C73—21no—
51C24×C73—21no—
52C24×C73—41no—
53C24×C73—41no—
54C24×C73—41no—
55C24×C73—41no—
56C24×C73—41no—
57C24×C73C22⋊C411yescharacteristic
58C24×C73C22⋊C411yescharacteristic
59C24×C73C4⋊C411yescharacteristic
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73—21no—
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—22no—
68C22×C4×C73—41no—
69C22×C4×C73—41no—
70C22×C4×C73—41no—
71C22×C4×C73—41no—
72C22×C4×C73—41no—
73C22×C4×C73—41no—
74C22×C4×C73—41no—
75C22×C4×C73—41no—
76C22×C4×C73—41no—
77C22×C4×C73—41no—
78C22×C4×C73—41no—
79C22×C4×C73—41no—
80C22×C4×C73—41no—
81C22×C4×C73—41no—
82C22×C4×C73C4211yescharacteristic
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C4⋊C411yescharacteristic
85C22×C4×C73C4⋊C411yescharacteristic
86C22⋊C4×C73—41no—
87C22⋊C4×C73—41no—
88C22⋊C4×C73—41no—
89C22⋊C4×C73—41no—
90C22⋊C4×C73—48no—
91C42×C73—42no—
92C42×C73—42no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—21no—
104C23×C73—42no—
105C23×C73—42no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—41no—
110C23×C73—42no—
111C23×C73—42no—
112C23×C73—42no—
113C23×C73—42no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73—41no—
118C23×C73—81no—
119C23×C73—81no—
120C23×C73—81no—
121C23×C73—81no—
122C23×C73—81no—
123C23×C73—81no—
124C23×C73#9511yescharacteristic
125C23×C73#9911yescharacteristic
126C23×C73#9911yescharacteristic
127C23×C73#10412yes—
128C2×C4×C73—24no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—42no—
134C2×C4×C73—42no—
135C2×C4×C73—42no—
136C2×C4×C73—42no—
137C2×C4×C73—42no—
138C2×C4×C73—42no—
139C2×C4×C73—41no—
140C2×C4×C73—42no—
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—42no—
149C2×C4×C73—42no—
150C2×C4×C73—42no—
151C2×C4×C73—81no—
152C2×C4×C73—81no—
153C2×C4×C73—81no—
154C2×C4×C73—81no—
155C2×C4×C73—81no—
156C2×Dic73—82no—
157C2×Dic73—82no—
158C2×Dic73—82no—
159#10596831—82no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—22no—
175C22×C73—42no—
176C22×C73—42no—
177C22×C73—41no—
178C22×C73—41no—
179C22×C73—41no—
180C22×C73—41no—
181C22×C73—41no—
182C22×C73—81no—
183C22×C73—82no—
184C22×C73—82no—
185C22×C73—82no—
186C22×C73—82no—
187C22×C73—81no—
188C22×C73H(3,ℤ/4)11yescharacteristic
189C22×C73#33911yescharacteristic
190C22×C73#34211yescharacteristic
191C22×C73AΣL(1,𝔽16)11yescharacteristic
192C22×C73#35211yescharacteristic
193C22×C73#35311yescharacteristic
194C22×C73#35411yescharacteristic
195C4×C73—44no—
196C4×C73—41no—
197C4×C73—41no—
198C4×C73—41no—
199C4×C73—41no—
200C4×C73—82no—
201Dic73—162no—
202#7213—731nomaximal, Sylow
203C2×C73—21no—
204C2×C73—21no—
205C2×C73—21no—
206C2×C73—21no—
207C2×C73—21no—
208C2×C73—21no—
209C2×C73—42no—
210C2×C73—81no—
211C2×C73#128411yescharacteristic
212C2×C73#139011yescharacteristic
213C2×C73#139311yescharacteristic
214#1819—731no—
215#1462—731no—
216#1437D7311yescharacteristic
217C73#721311yescharacteristic, complemented, Sylow
218C22×C22⋊C4—21no—
219C22×C22⋊C4C2×D7311yescharacteristic
220C22×C42Dic7311yescharacteristic, complemented
221#407—1461no—
222#379—1461no—
223C2×#95—21no—
224C2×#95Dic7311yescharacteristic, complemented
225#342—732no—
226#342—1461no—
227#323—732no—
228#323—1461no—
229C25C22⋊D7311yescharacteristic
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C2×Dic7311yescharacteristic
233C23×C4C22⋊D7311yescharacteristic
234C2×M16—2921no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—2921no—
241C2×C22⋊C4—2921no—
242C2×C22⋊C4—1462no—
243C2×C22⋊C4Dic14611yescharacteristic
244C2×C22⋊C4C4×D7311yescharacteristic
245C2×C22⋊C4C4×D7311yescharacteristic
246C2×C22⋊C4D29211yescharacteristic
247C2×C42—41no—
248C2×C42—42no—
249C22⋊C8—2921no—
250C22⋊C8—1462no—
251#95—41no—
252#95—42no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24#1192400511yescharacteristic
262C24C22⋊Dic7311yescharacteristic
263C24#1192400711yescharacteristic
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—2922no—
287C22×C4C4×Dic7311yescharacteristic
288C22×C4#1192400511yescharacteristic
289C22×C4#1192400711yescharacteristic
290C22×C4C4⋊Dic7311yescharacteristic
291M16—5841no—
292M16—5841no—
293C2×C8—2922no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—48no—
299C22⋊C4—2922no—
300C22⋊C4—2922no—
301C22⋊C4—5841no—
302C22⋊C4—5841no—
303C22⋊C4—5841no—
304C22⋊C4—5841no—
305C42—42no—
306C42—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23#1455649811yescharacteristic
339C23#1455650611yescharacteristic
340C23#1455650611yescharacteristic
341C23#1455651912yes—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—5842no—
371C2×C4—5842no—
372C2×C4—5842no—
373C8—5842no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—81no—
402C22#1798834311yescharacteristic
403C22#1798834711yescharacteristic
404C22#1798835411yescharacteristic
405C22#1798837411yescharacteristic
406C22#1798837711yescharacteristic
407C22#1798838011yescharacteristic
408C22#1798838211yescharacteristic
409C4—44no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—11682no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—81no—
424C2#2260778411yescharacteristic
425C2#2260804911yescharacteristic
426C2#2260805611yescharacteristic
subgroupG/H
centerC22#17988354
commutatorC22×C4×C73C42
FrattiniC22×C22⋊C4C2×D73
FittingC73×#1437C2
socleC22×C73#342
prime radicalC25×C73D4
cosocleC22×C22⋊C4×C73C22
hypercenter#1437D73
hypocenterC73#7213
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1437

Complements

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

NH ≅ G/Ntype
7. #17988354C4semidirect
9. #17988311C4semidirect
12. C22×C42×C73C4semidirect
14. C2×#95×C73C4semidirect
217. C73#7213semidirect
220. C22×C42Dic73semidirect
224. C2×#95Dic73semidirect

Maximal subgroups

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

subgroupindex
#226100052
#226082582
C73×#14372
#721373