Finite Group Databasebeta

← #32981739

Subgroups — #32981739

Order 18688.

946 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618218C211yesmaximal, characteristic
2#22609841C211yesmaximal, characteristic
3C2×#17988323C212yesmaximal, complemented
4#22607810C212yesmaximal, complemented
5C2×#380×C73C211yesmaximal, characteristic
6C2×#14556570C412yescomplemented
7C2×#14556570C2211yescharacteristic
8C2×C22⋊#10596831—22no—
9C2×C22⋊#10596831C2212yes—
10#17988460—21no—
11#17988460—22no—
12#17988447C412yescomplemented
13#17988447C2211yescharacteristic
14#17988443—21no—
15#17988323—24no—
16C24×C4×C73C2211yescharacteristic
17C22×C4⋊C4×C73—21no—
18C22×C22⋊C4×C73—21no—
19#380×C73—24no—
20C2×#95×C73C2212yes—
21C22×Dic146C2×C411yescharacteristic
22C22×#10596831—22no—
23#14556570—22no—
24#14556570—22no—
25#14556570—44no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C4⋊Dic73C2×C412yes—
28C2×#11924007—22no—
29C2×#11924007—21no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—21no—
33C2×C22⋊Dic73—21no—
34C22⋊#10596831—22no—
35C22⋊#10596831—22no—
36C22⋊#10596831—44no—
37#14556498—22no—
38#14556498—21no—
39C25×C73D411yescharacteristic
40C23×C4×C73—22no—
41C23×C4×C73—22no—
42C23×C4×C73—22no—
43C23×C4×C73—44no—
44C23×C4×C73C2×C412yes—
45C23×C4×C73D411yescharacteristic
46C23×C4×C73D412yes—
47C23×C4×C73C2311yescharacteristic
48C2×C4⋊C4×C73—24no—
49C2×C4⋊C4×C73—44no—
50C2×C22⋊C4×C73—24no—
51C2×C22⋊C4×C73—42no—
52C2×C22⋊C4×C73—42no—
53#95×C73—22no—
54#95×C73D414yes—
55C2×Dic146—22no—
56C2×Dic146—42no—
57C22×Dic73—21no—
58C22×Dic73—22no—
59C22×Dic73—21no—
60C2×#10596831—44no—
61C2×#10596831—42no—
62C4⋊Dic73—22no—
63C4⋊Dic73—42no—
64#11924007—44no—
65#11924007—42no—
66C22⋊Dic73—44no—
67C22⋊Dic73—44no—
68C22⋊Dic73—41no—
69C22⋊Dic73—41no—
70C22⋊Dic73—41no—
71C22⋊Dic73—41no—
72C24×C73—21no—
73C24×C73—21no—
74C24×C73—21no—
75C24×C73—21no—
76C24×C73—42no—
77C24×C73—42no—
78C24×C73C22⋊C412yes—
79C24×C73SD1612yes—
80C24×C73Q1612yes—
81C24×C73C2×D411yescharacteristic
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—24no—
85C22×C4×C73—24no—
86C22×C4×C73—24no—
87C22×C4×C73—24no—
88C22×C4×C73—24no—
89C22×C4×C73—24no—
90C22×C4×C73—42no—
91C22×C4×C73—42no—
92C22×C4×C73—42no—
93C22×C4×C73—48no—
94C22×C4×C73—48no—
95C22×C4×C73—44no—
96C22×C4×C73—44no—
97C22×C4×C73—42no—
98C22×C4×C73—42no—
99C22×C4×C73C22⋊C412yes—
100C22×C4×C73C22×C411yescharacteristic
101C22×C4×C73C2×D412yes—
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73#4112yes—
104C4⋊C4×C73—48no—
105C22⋊C4×C73—48no—
106C23×C73—22no—
107C23×C73—22no—
108C23×C73—22no—
109C23×C73—22no—
110C23×C73—22no—
111C23×C73—22no—
112C23×C73—22no—
113C23×C73—22no—
114C23×C73—22no—
115C23×C73—22no—
116C23×C73—22no—
117C23×C73—22no—
118C23×C73—44no—
119C23×C73—44no—
120C23×C73—42no—
121C23×C73—42no—
122C23×C73—42no—
123C23×C73—42no—
124C23×C73—44no—
125C23×C73—44no—
126C23×C73#10314yes—
127C23×C73C2×C22⋊C411yescharacteristic
128C23×C73#12011yescharacteristic
129C23×C73#12311yescharacteristic
130C23×C73C2×SD1611yescharacteristic
131C23×C73C2×Q1611yescharacteristic
132C23×C73AGL(1,ℤ/8)11yescharacteristic
133C23×C73#13711yescharacteristic
134C2×C4×C73—24no—
135C2×C4×C73—24no—
136C2×C4×C73—22no—
137C2×C4×C73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—48no—
143C2×C4×C73—48no—
144C2×C4×C73—44no—
145C2×C4×C73—44no—
146C2×C4×C73—44no—
147C2×C4×C73—84no—
148C2×C4×C73—84no—
149C2×C4×C73#9914yes—
150C2×C4×C73C4×D412yes—
151C2×C4×C73#12112yes—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—42no—
155C2×Dic73—42no—
156C2×Dic73—42no—
157C2×Dic73—42no—
158C2×Dic73—42no—
159Dic146—82no—
160#10596831—82no—
161C22×C73—21no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—22no—
166C22×C73—22no—
167C22×C73—22no—
168C22×C73—22no—
169C22×C73—24no—
170C22×C73—24no—
171C22×C73—24no—
172C22×C73—44no—
173C22×C73—44no—
174C22×C73—44no—
175C22×C73—44no—
176C22×C73—42no—
177C22×C73—42no—
178C22×C73—42no—
179C22×C73—42no—
180C22×C73#38611yescharacteristic
181C22×C73C2×#9912yes—
182C22×C73C2×#10311yescharacteristic
183C22×C73#41711yescharacteristic
184C22×C73#42112yes—
185C22×C73#45011yescharacteristic
186C22×C73#45111yescharacteristic
187C22×C73#48311yescharacteristic
188C22×C73#48411yescharacteristic
189C4×C73—22no—
190C4×C73—22no—
191C4×C73—24no—
192C4×C73—84no—
193Dic73—81no—
194Dic73—81no—
195Dic73—82no—
196#9151—731nomaximal, Sylow
197C2×C73—21no—
198C2×C73—21no—
199C2×C73—21no—
200C2×C73—21no—
201C2×C73—44no—
202C2×C73#151312yes—
203C2×C73#152012yes—
204C2×C73#187211yescharacteristic
205C2×C73#189711yescharacteristic
206C2×C73#189811yescharacteristic
207#2848—731no—
208C2×#380D7311yescharacteristic
209#1779—731no—
210C2×#328—732no—
211#1294—732no—
212C73#915111yescharacteristic, complemented, Sylow
213C24×C4C2×D7311yescharacteristic
214C2×C22⋊Q8—732no—
215C2×C22⋊Q8—731no—
216C22×C4⋊C4—21no—
217C22×C22⋊C4—21no—
218C2×C22⋊C8—732no—
219C2×C22⋊C8—1462no—
220#385—1462no—
221#385—1461no—
222#380—24no—
223#380—732no—
224#380—731no—
225#379—1461no—
226C2×#95C2×D7312yes—
227#328—1464no—
228C25C22⋊D7311yescharacteristic
229C22×Q8—731no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—44no—
234C23×C4C4×D7312yes—
235C23×C4D29212yes—
236C23×C4C22⋊D7311yescharacteristic
237C23×C4C22×D7311yescharacteristic
238C22×C8—1462no—
239C22⋊Q8—2924no—
240C22⋊Q8—1462no—
241C22⋊Q8—1462no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—732no—
245C2×C4⋊C4—731no—
246C2×C4⋊C4—1462no—
247C2×C4⋊C4—1461no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—1462no—
252C2×C22⋊C4—1462no—
253C2×C22⋊C4—1461no—
254C2×C22⋊C4—1461no—
255C22⋊C8—2924no—
256C22⋊C8—1462no—
257C22⋊C8—1462no—
258#95—22no—
259#95—1462no—
260#95—1461no—
261#95C22⋊D7314yes—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—42no—
267C24—42no—
268C24#1192400512yes—
269C24#1192401612yes—
270C24#1192401812yes—
271C24C2×C22⋊D7311yescharacteristic
272C2×Q8—2922no—
273C2×Q8—1462no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—1461no—
292C22×C4—1462no—
293C22×C4—1461no—
294C22×C4#1192400512yes—
295C22×C4C2×C4×D7311yescharacteristic
296C22×C4C2×C22⋊D7312yes—
297C22×C4C2×D29211yescharacteristic
298C22×C4#1192402812yes—
299C2×C8—2924no—
300C2×C8—2922no—
301C4⋊C4—48no—
302C4⋊C4—2922no—
303C4⋊C4—2924no—
304C4⋊C4—2922no—
305C4⋊C4—1462no—
306C22⋊C4—48no—
307C22⋊C4—2924no—
308C22⋊C4—2924no—
309C22⋊C4—2921no—
310C22⋊C4—2921no—
311C22⋊C4—2921no—
312C22⋊C4—2921no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—44no—
326C23—44no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23#1455651414yes—
334C23C2×#1192400511yescharacteristic
335C23#1455656011yescharacteristic
336C23#1455657411yescharacteristic
337C23C2×#1192401611yescharacteristic
338C23C2×#1192401811yescharacteristic
339C23#1455661711yescharacteristic
340C23#1455662211yescharacteristic
341Q8—5842no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—2921no—
358C2×C4—2921no—
359C2×C4—2922no—
360C2×C4—2922no—
361C2×C4—2922no—
362C2×C4—2922no—
363C2×C4—2922no—
364C2×C4#1455650514yes—
365C2×C4C4×C22⋊D7312yes—
366C2×C4#1455656512yes—
367C8—5842no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22#1798846611yescharacteristic
388C22C2×#1455650512yes—
389C22C2×#1455651411yescharacteristic
390C22#1798858611yescharacteristic
391C22#1798860512yes—
392C22#1798873811yescharacteristic
393C22#1798874311yescharacteristic
394C22#1798891711yescharacteristic
395C22#1798892211yescharacteristic
396C4—22no—
397C4—24no—
398C4—22no—
399C4—84no—
400C4—5842no—
401C4—5841no—
402C4—5841no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—44no—
408C2#2260847212yes—
409C2#2260850212yes—
410C2#2261023811yescharacteristic
411C2#2261037911yescharacteristic
412C2#2261038411yescharacteristic
subgroupG/H
centerC23C2×#11924005
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#380×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC25×C73D4
cosocleC23×C4×C73C23
hypercenterC2×#380D73
hypocenterC73#9151
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. C2×#17988323C2semidirect
4. #22607810C2semidirect
6. C2×#14556570C4semidirect
12. #17988447C4semidirect
212. C73#9151semidirect

Maximal subgroups

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

subgroupindex
#226182182
#226098412
C2×#179883232
#226078102
C2×#380×C732
#915173