Finite Group Databasebeta

← #32983769

Subgroups — #32983769

Order 18688.

912 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 123 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#22618960C211yesmaximal, characteristic, complemented
2#22609919C211yesmaximal, characteristic, complemented
3#22609862C212yesmaximal, complemented
4#22607798C212yesmaximal, complemented
5C73×#2408C211yesmaximal, characteristic
6#17988659—24no—
7#17988644—21no—
8C2×#14556512—24no—
9C2×#14556506—21no—
10#17988551—21no—
11C2×C22⋊#10596831C2212yes—
12#17988447C2212yes—
13#17988354—21no—
14C23×D4×C73C2211yescharacteristic
15C2×#121×C73—21no—
16C2×#120×C73—21no—
17C2×C4×D4×C73C2211yescharacteristic
18#394×C73C412yescomplemented
19#392×C73C412yescomplemented
20#390×C73—21no—
21#380×C73C2211yescharacteristic
22#379×C73—21no—
23C2×#11924012—22no—
24C22×#10596831D414yescomplemented
25C2×C4⋊Dic73—22no—
26C2×C22⋊Dic73—22no—
27C4⋊#10596831—22no—
28#14556512—48no—
29#14556506—42no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33#14556502—22no—
34#14556498—22no—
35C25×C73—21no—
36C22×D4×C73—22no—
37C22×D4×C73—22no—
38C22×D4×C73—21no—
39C22×D4×C73—22no—
40C22×D4×C73—44no—
41C22×D4×C73C2×C412yes—
42C22×D4×C73D411yescharacteristic
43C23×C4×C73D411yescharacteristic
44C23×C4×C73C2311yescharacteristic
45#121×C73—44no—
46#120×C73—44no—
47C4×D4×C73—22no—
48C4×D4×C73—22no—
49C2×C4⋊C4×C73—21no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—22no—
53C2×C22⋊C4×C73—22no—
54C2×C22⋊C4×C73—22no—
55C2×C22⋊C4×C73C2×C411yescharacteristic
56C2×C22⋊C4×C73D411yescharacteristic
57C2×C42×C73D411yescharacteristic
58#95×C73C2×C412yes—
59C22×Dic73—42no—
60#11924012—82no—
61C2×#10596831—24no—
62C2×#10596831—44no—
63C4⋊Dic73—44no—
64C22⋊Dic73—42no—
65C22⋊Dic73—42no—
66C24×C73—22no—
67C24×C73—44no—
68C24×C73—42no—
69C24×C73—42no—
70C24×C73—41no—
71C24×C73—41no—
72C24×C73—82no—
73C24×C73C22⋊C411yescharacteristic
74C24×C73C2×D411yescharacteristic
75C2×D4×C73—28no—
76C2×D4×C73—24no—
77C2×D4×C73—22no—
78C2×D4×C73—22no—
79C2×D4×C73—21no—
80C2×D4×C73—21no—
81C2×D4×C73—24no—
82C2×D4×C73—42no—
83C2×D4×C73—44no—
84C2×D4×C73—44no—
85C2×D4×C73—48no—
86C2×D4×C73—44no—
87C2×D4×C73—44no—
88C2×D4×C73—41no—
89C2×D4×C73—41no—
90C2×D4×C73—82no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—22no—
94C22×C4×C73—42no—
95C22×C4×C73—42no—
96C22×C4×C73C22⋊C411yescharacteristic
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C22⋊C411yescharacteristic
99C22×C4×C73C22×C411yescharacteristic
100C22×C4×C73C2×D412yes—
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73#4112yes—
103C4⋊C4×C73—22no—
104C4⋊C4×C73—42no—
105C22⋊C4×C73—22no—
106C22⋊C4×C73—22no—
107C22⋊C4×C73—44no—
108C22⋊C4×C73—44no—
109C22⋊C4×C73—42no—
110C22⋊C4×C73—82no—
111C42×C73—21no—
112C42×C73—21no—
113C23×C73—24no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73—42no—
122C23×C73—48no—
123C23×C73—42no—
124C23×C73—42no—
125C23×C73—42no—
126C23×C73—42no—
127C23×C73—42no—
128C23×C73—84no—
129C23×C73—88no—
130C23×C73—84no—
131C23×C73—84no—
132C23×C73—82no—
133C23×C73—82no—
134C23×C73C2×C22⋊C411yescharacteristic
135C23×C73#12112yes—
136C23×C73AGL(1,ℤ/8)11yescharacteristic
137C23×C73AGL(1,ℤ/8)12yes—
138C23×C73#13711yescharacteristic
139D4×C73—28no—
140D4×C73—44no—
141D4×C73—44no—
142D4×C73—42no—
143D4×C73—42no—
144D4×C73—44no—
145D4×C73—88no—
146C2×C4×C73—22no—
147C2×C4×C73—24no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—21no—
153C2×C4×C73—41no—
154C2×C4×C73—42no—
155C2×C4×C73—42no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—42no—
159C2×C4×C73—42no—
160C2×C4×C73—42no—
161C2×C4×C73—81no—
162C2×C4×C73C4×D412yes—
163C2×C4×C73#12011yescharacteristic
164C2×C4×C73#12311yescharacteristic
165C2×Dic73—82no—
166C2×Dic73—82no—
167C2×Dic73—82no—
168#10596831—44no—
169C22×C73—22no—
170C22×C73—21no—
171C22×C73—22no—
172C22×C73—21no—
173C22×C73—41no—
174C22×C73—42no—
175C22×C73—41no—
176C22×C73—42no—
177C22×C73—44no—
178C22×C73—44no—
179C22×C73—41no—
180C22×C73—44no—
181C22×C73—41no—
182C22×C73—41no—
183C22×C73—81no—
184C22×C73—82no—
185C22×C73—82no—
186C22×C73—88no—
187C22×C73—88no—
188C22×C73—82no—
189C22×C73—82no—
190C22×C73#38611yescharacteristic
191C22×C73#41711yescharacteristic
192C22×C73#41811yescharacteristic
193C22×C73#46812yes—
194C22×C73#46912yes—
195C4×C73—21no—
196C4×C73—22no—
197C4×C73—41no—
198C4×C73—41no—
199C4×C73—41no—
200C4×C73—82no—
201C4×C73B(4,𝔽2)12yes—
202Dic73—162no—
203#9472—731nomaximal, Sylow
204C2×C73—22no—
205C2×C73—21no—
206C2×C73—21no—
207C2×C73—22no—
208C2×C73—81no—
209C2×C73—84no—
210C2×C73#188911yescharacteristic
211C2×C73#189511yescharacteristic
212C2×C73#191011yescharacteristic
213#2973—731no—
214#2408D7311yescharacteristic
215#1799—731no—
216#1786—732no—
217#1289—732no—
218C73#947211yescharacteristic, complemented, Sylow
219C23×D4C2×D7311yescharacteristic
220C2×#121—21no—
221C2×#120—21no—
222C2×C4×D4C2×D7311yescharacteristic
223#435—1464no—
224#432—1461no—
225C2×#102—1464no—
226C2×#99—1461no—
227#407—1461no—
228C2×C22⋊C8—732no—
229#394Dic7312yescomplemented
230#392Dic7312yescomplemented
231#390—21no—
232#380—732no—
233#380C2×D7311yescharacteristic
234#379—21no—
235#342—1461no—
236C25—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—44no—
242C22×D4C2×Dic7312yes—
243C22×D4C22⋊D7311yescharacteristic
244C23×C4C22⋊D7311yescharacteristic
245C23×C4C22×D7311yescharacteristic
246C2×M16—1462no—
247C22×C8—734no—
248#121—44no—
249#120—44no—
250C4×D4—22no—
251C4×D4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—1462no—
254C2×C4⋊C4C2×Dic7311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—1462no—
260C2×C22⋊C4C2×Dic7311yescharacteristic
261C2×C22⋊C4C22⋊D7311yescharacteristic
262C2×C42C22⋊D7311yescharacteristic
263C4⋊C8—1462no—
264#102—2928no—
265#99—2922no—
266C22⋊C8—1462no—
267C22⋊C8—1462no—
268C22⋊C8—1462no—
269#97—1462no—
270#95—1462no—
271#95C2×Dic7312yes—
272C24—22no—
273C24—44no—
274C24—42no—
275C24—41no—
276C24—42no—
277C24—41no—
278C24—82no—
279C24C22⋊Dic7311yescharacteristic
280C24C2×C22⋊D7311yescharacteristic
281C2×D4—28no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—24no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—48no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—82no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—2922no—
303C22×C4C22⋊Dic7311yescharacteristic
304C22×C4C22⋊Dic7311yescharacteristic
305C22×C4C22⋊Dic7311yescharacteristic
306C22×C4C22×Dic7311yescharacteristic
307C22×C4D4×D7312yes—
308C22×C4C2×C22⋊D7311yescharacteristic
309C22×C4#1192402712yes—
310M16—5842no—
311C2×C8—2924no—
312C2×C8—1464no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—2924no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—82no—
322C22⋊C4—2922no—
323C22⋊C4—2922no—
324C42—21no—
325C42—21no—
326C23—24no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—48no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—84no—
342C23—84no—
343C23—88no—
344C23—84no—
345C23—82no—
346C23—82no—
347C23C2×C22⋊Dic7311yescharacteristic
348C23#1455656312yes—
349C23#1455661611yescharacteristic
350C23#1455661712yes—
351C23#1455662111yescharacteristic
352D4—28no—
353D4—44no—
354D4—44no—
355D4—42no—
356D4—42no—
357D4—44no—
358D4—88no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—5842no—
376C2×C4—5842no—
377C2×C4—5842no—
378C2×C4D4×Dic7312yes—
379C2×C4#1455655911yescharacteristic
380C2×C4#1455657211yescharacteristic
381C8—2924no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—22no—
386C22—41no—
387C22—42no—
388C22—41no—
389C22—42no—
390C22—44no—
391C22—44no—
392C22—41no—
393C22—41no—
394C22—44no—
395C22—41no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—88no—
400C22—88no—
401C22—82no—
402C22—82no—
403C22#1798846711yescharacteristic
404C22#1798858811yescharacteristic
405C22#1798859211yescharacteristic
406C22#1798883112yes—
407C22#1798883812yes—
408C4—21no—
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—82no—
414C4—11682no—
415C4#1798877312yes—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—22no—
420C2—81no—
421C2—84no—
422C2#2261034011yescharacteristic
423C2#2261037011yescharacteristic
424C2#2261045911yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
Fitting, prime radicalC73×#2408C2
socleC22×C73#386
cosocleC23×C4×C73C23
hypercenter#2408D73
hypocenterC73#9472
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #22618960C2semidirect
2. #22609919C2semidirect
3. #22609862C2semidirect
4. #22607798C2semidirect
18. #394×C73C4semidirect
19. #392×C73C4semidirect
24. C22×#10596831D4semidirect
218. C73#9472semidirect
229. #394Dic73semidirect
230. #392Dic73semidirect

Maximal subgroups

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

subgroupindex
#226189602
#226099192
#226098622
#226077982
C73×#24082
#947273