Finite Group Databasebeta

← #32978611

Subgroups — #32978611

Order 18688.

1478 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1478 conjugacy classes of subgroups — too many to draw. Only its 155 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#22609852C212yesmaximal, complemented
2#22608261C212yesmaximal, complemented
3#22608258C212yesmaximal, complemented
4C22×#121×C73C211yesmaximal, characteristic
5C2×#14556508—24no—
6#17988551C2214yes—
7#17988313—24no—
8#17988311—24no—
9C23×D4×C73C411yescharacteristic
10C23×D4×C73C411yescharacteristic
11C23×D4×C73C2211yescharacteristic
12C24×C4×C73C411yescharacteristic
13C2×#121×C73—24no—
14C2×#121×C73—24no—
15C2×#121×C73—24no—
16C22×C4⋊C4×C73C2211yescharacteristic
17C22×C22⋊C4×C73C411yescharacteristic
18C22×C22⋊C4×C73C2211yescharacteristic
19C2×#11924012—24no—
20#14556508—48no—
21C22⋊#10596831—24no—
22C22⋊#10596831—24no—
23C25×C73C2×C411yescharacteristic
24C25×C73C2×C411yescharacteristic
25C25×C73C2×C411yescharacteristic
26C22×D4×C73—22no—
27C22×D4×C73—22no—
28C22×D4×C73—22no—
29C22×D4×C73—22no—
30C22×D4×C73—22no—
31C22×D4×C73—22no—
32C22×D4×C73—22no—
33C22×D4×C73—22no—
34C22×D4×C73—44no—
35C22×D4×C73—44no—
36C22×D4×C73—44no—
37C22×D4×C73D414yes—
38C23×C4×C73—22no—
39C23×C4×C73—22no—
40C23×C4×C73—22no—
41C23×C4×C73—44no—
42C23×C4×C73C2×C411yescharacteristic
43C23×C4×C73C2×C411yescharacteristic
44C23×C4×C73C2×C411yescharacteristic
45C23×C4×C73C2311yescharacteristic
46#121×C73—232no—
47C2×C4⋊C4×C73—22no—
48C2×C4⋊C4×C73—22no—
49C2×C4⋊C4×C73D414yes—
50C2×C22⋊C4×C73—22no—
51C2×C22⋊C4×C73—22no—
52C2×C22⋊C4×C73—22no—
53C2×C22⋊C4×C73—22no—
54C2×C22⋊C4×C73—22no—
55C2×C22⋊C4×C73D414yes—
56#11924012—48no—
57C2×#10596831—44no—
58C24×C73—24no—
59C24×C73—21no—
60C24×C73—21no—
61C24×C73—21no—
62C24×C73—21no—
63C24×C73—21no—
64C24×C73—21no—
65C24×C73—42no—
66C24×C73—42no—
67C24×C73—42no—
68C24×C73—42no—
69C24×C73—42no—
70C24×C73—42no—
71C24×C73—42no—
72C24×C73—42no—
73C24×C73C22⋊C412yes—
74C24×C73C22⋊C412yes—
75C24×C73C22⋊C412yes—
76C24×C73C22×C411yescharacteristic
77C2×D4×C73—28no—
78C2×D4×C73—28no—
79C2×D4×C73—28no—
80C2×D4×C73—28no—
81C2×D4×C73—44no—
82C2×D4×C73—48no—
83C2×D4×C73—48no—
84C2×D4×C73—44no—
85C2×D4×C73—48no—
86C2×D4×C73—48no—
87C2×D4×C73—48no—
88C2×D4×C73—44no—
89C2×D4×C73—48no—
90C2×D4×C73—44no—
91C2×D4×C73—44no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73—24no—
95C22×C4×C73—28no—
96C22×C4×C73—21no—
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—42no—
101C22×C4×C73—48no—
102C22×C4×C73—48no—
103C22×C4×C73—44no—
104C22×C4×C73—44no—
105C22×C4×C73—42no—
106C22×C4×C73—42no—
107C22×C4×C73C22⋊C412yes—
108C22×C4×C73C22⋊C412yes—
109C22×C4×C73C22⋊C412yes—
110C22×C4×C73C2×D412yes—
111C22×C4×C73C2×D412yes—
112C22×C4×C73C2×D412yes—
113C4⋊C4×C73—28no—
114C22⋊C4×C73—28no—
115C22⋊C4×C73—28no—
116C23×C73—22no—
117C23×C73—22no—
118C23×C73—22no—
119C23×C73—24no—
120C23×C73—24no—
121C23×C73—22no—
122C23×C73—22no—
123C23×C73—22no—
124C23×C73—22no—
125C23×C73—24no—
126C23×C73—24no—
127C23×C73—42no—
128C23×C73—42no—
129C23×C73—42no—
130C23×C73—416no—
131C23×C73—42no—
132C23×C73—42no—
133C23×C73—42no—
134C23×C73—42no—
135C23×C73—42no—
136C23×C73—42no—
137C23×C73—42no—
138C23×C73—42no—
139C23×C73—42no—
140C23×C73—42no—
141C23×C73—42no—
142C23×C73—42no—
143C23×C73—42no—
144C23×C73—42no—
145C23×C73—42no—
146C23×C73—42no—
147C23×C73—42no—
148C23×C73—42no—
149C23×C73—42no—
150C23×C73—88no—
151C23×C73—88no—
152C23×C73#9914yes—
153C23×C73C2×C22⋊C411yescharacteristic
154C23×C73C2×C22⋊C411yescharacteristic
155C23×C73C2×C22⋊C411yescharacteristic
156D4×C73—416no—
157D4×C73—416no—
158D4×C73—416no—
159C2×C4×C73—24no—
160C2×C4×C73—24no—
161C2×C4×C73—22no—
162C2×C4×C73—24no—
163C2×C4×C73—22no—
164C2×C4×C73—22no—
165C2×C4×C73—22no—
166C2×C4×C73—42no—
167C2×C4×C73—42no—
168C2×C4×C73—416no—
169C2×C4×C73—42no—
170C2×C4×C73—42no—
171C2×C4×C73—42no—
172C2×C4×C73—42no—
173C2×C4×C73—42no—
174C2×C4×C73—42no—
175C2×C4×C73#10014yescomplemented
176C2×C4×C73#12014yes—
177#10596831—84no—
178C22×C73—28no—
179C22×C73—21no—
180C22×C73—21no—
181C22×C73—21no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—24no—
186C22×C73—44no—
187C22×C73—44no—
188C22×C73—48no—
189C22×C73—48no—
190C22×C73—42no—
191C22×C73—42no—
192C22×C73—42no—
193C22×C73—42no—
194C22×C73—42no—
195C22×C73—42no—
196C22×C73—42no—
197C22×C73—42no—
198C22×C73—84no—
199C22×C73—84no—
200C22×C73—84no—
201C22×C73—84no—
202C22×C73—84no—
203C22×C73—84no—
204C22×C73#37911yescharacteristic
205C22×C73C2×#9912yes—
206C22×C73C2×#10012yes—
207C22×C73#41312yes—
208C4×C73—24no—
209C4×C73—24no—
210C4×C73—42no—
211C4×C73—42no—
212#8549—731nomaximal, Sylow
213C2×C73—22no—
214C2×C73—22no—
215C2×C73—22no—
216C2×C73—44no—
217C2×C73—82no—
218C2×C73—82no—
219C2×C73#178111yescharacteristic
220C2×C73#178311yescharacteristic
221C2×C73#178511yescharacteristic
222C22×#121D7311yescharacteristic
223#1783—732no—
224#1463—732no—
225#1462—732no—
226C73#854911yescharacteristic, complemented, Sylow
227C23×D4Dic7311yescharacteristic
228C23×D4Dic7311yescharacteristic
229C23×D4C2×D7311yescharacteristic
230C24×C4Dic7311yescharacteristic
231C2×#121—24no—
232C2×#121—24no—
233C2×#121—24no—
234C22×C4⋊C4C2×D7311yescharacteristic
235C22×C22⋊C4Dic7311yescharacteristic
236C22×C22⋊C4C2×D7311yescharacteristic
237C2×#100—1464no—
238#407—734no—
239#324—1464no—
240#323—1464no—
241C25C2×Dic7311yescharacteristic
242C25C2×Dic7311yescharacteristic
243C25C2×Dic7311yescharacteristic
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4—44no—
255C22×D4C22⋊D7314yes—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4C2×Dic7311yescharacteristic
261C23×C4C2×Dic7311yescharacteristic
262C23×C4C2×Dic7311yescharacteristic
263C23×C4C22×D7311yescharacteristic
264C2×M16—1464no—
265#121—232no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C22⋊D7314yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4C22⋊D7314yes—
275#100—2928no—
276C22⋊C8—1464no—
277C22⋊C8—1464no—
278C24—24no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24C22⋊Dic7312yes—
294C24C22⋊Dic7312yes—
295C24C22⋊Dic7312yes—
296C24C22×Dic7311yescharacteristic
297C2×D4—28no—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—44no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—44no—
309C2×D4—48no—
310C2×D4—44no—
311C2×D4—44no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4C22⋊Dic7312yes—
328C22×C4C22⋊Dic7312yes—
329C22×C4C22⋊Dic7312yes—
330C22×C4C2×C22⋊D7312yes—
331C22×C4C2×C22⋊D7312yes—
332C22×C4C2×C22⋊D7312yes—
333M16—2928no—
334C2×C8—2924no—
335C4⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—28no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—24no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—416no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—88no—
373C23—88no—
374C23#1455650614yes—
375C23C2×C22⋊Dic7311yescharacteristic
376C23C2×C22⋊Dic7311yescharacteristic
377C23C2×C22⋊Dic7311yescharacteristic
378D4—416no—
379D4—416no—
380D4—416no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—416no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4#1455650814yescomplemented
398C2×C4#1455656014yes—
399C8—5844no—
400C22—28no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—24no—
408C22—44no—
409C22—44no—
410C22—48no—
411C22—48no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—84no—
421C22—84no—
422C22—84no—
423C22—84no—
424C22—84no—
425C22—84no—
426C22#1798844311yescharacteristic
427C22C2×#1455650612yes—
428C22C2×#1455650812yes—
429C22#1798857112yes—
430C4—24no—
431C4—24no—
432C4—42no—
433C4—42no—
434C2—22no—
435C2—22no—
436C2—22no—
437C2—44no—
438C2—82no—
439C2—82no—
440C2#2260984611yescharacteristic
441C2#2260985211yescharacteristic
442C2#2260985911yescharacteristic
subgroupG/H
centerC22#17988443
commutatorC24×C73C22×C4
FrattiniC23×C4C22×D73
Fitting, prime radicalC22×#121×C73C2
socleC22×C73#379
cosocleC23×C4×C73C23
hypercenterC22×#121D73
hypocenterC73#8549
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. #22609852C2semidirect
2. #22608261C2semidirect
3. #22608258C2semidirect
175. C2×C4×C73#100semidirect
226. C73#8549semidirect
397. C2×C4#14556508semidirect

Maximal subgroups

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

subgroupindex
#226098522
#226082612
#226082582
C22×#121×C732
#854973