Finite Group Databasebeta

← #32978886

Subgroups — #32978886

Order 18688.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609960C211yesmaximal, characteristic, complemented
2#22609956C211yesmaximal, characteristic, complemented
3#22609919C212yesmaximal
4#22607799C212yesmaximal, complemented
5C2×#435×C73C211yesmaximal, characteristic
6C2×#14556525—21no—
7C2×#14556523—21no—
8C2×#14556506—22no—
9#17988447C2212yes—
10#17988447C2212yes—
11#17988354—22no—
12#17988350—21no—
13#17988350—21no—
14C22×M16×C73C411yescharacteristic, complemented
15C23×C8×C73C2211yescharacteristic
16C2×C4×D4×C73C2211yescharacteristic
17#435×C73—21no—
18#435×C73—21no—
19#435×C73—21no—
20#435×C73—21no—
21C2×C4⋊C8×C73C411yescharacteristic, complemented
22C2×C22⋊C8×C73C411yescharacteristic, complemented
23C2×C22⋊C8×C73C2211yescharacteristic
24C2×#97×C73C411yescharacteristic, complemented
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73—22no—
27C2×C22⋊Dic73—22no—
28C2×C22⋊Dic73—22no—
29#14556525—42no—
30#14556523—42no—
31#14556506—44no—
32#14556498—22no—
33#14556498—22no—
34C22×D4×C73D411yescharacteristic
35C23×C4×C73C2×C411yescharacteristic
36C23×C4×C73C2311yescharacteristic
37C2×M16×C73—21no—
38C2×M16×C73—21no—
39C2×M16×C73—42no—
40C22×C8×C73—42no—
41C22×C8×C73C2×C411yescharacteristic
42C22×C8×C73C2×C411yescharacteristic
43C22×C8×C73C2×C411yescharacteristic
44C22×C8×C73C2×C411yescharacteristic
45C22×C8×C73D411yescharacteristic
46C22×C8×C73D411yescharacteristic
47C22×C8×C73D411yescharacteristic
48C22×C8×C73D411yescharacteristic
49C4×D4×C73—21no—
50C4×D4×C73—21no—
51C4×D4×C73—21no—
52C4×D4×C73—21no—
53C2×C4⋊C4×C73Q811yescharacteristic
54C2×C22⋊C4×C73D411yescharacteristic
55C2×C22⋊C4×C73Q811yescharacteristic
56C2×C42×C73C2×C411yescharacteristic
57C4⋊C8×C73—21no—
58C4⋊C8×C73—21no—
59C22⋊C8×C73—21no—
60C22⋊C8×C73—21no—
61C22⋊C8×C73—21no—
62C22⋊C8×C73—21no—
63#97×C73—21no—
64#97×C73—21no—
65C22×Dic73—42no—
66C22×Dic73—42no—
67C4⋊Dic73—42no—
68C4⋊Dic73—42no—
69C4⋊Dic73—42no—
70C4⋊Dic73—42no—
71C22⋊Dic73—42no—
72C22⋊Dic73—42no—
73C22⋊Dic73—42no—
74C22⋊Dic73—42no—
75C24×C73C4⋊C411yescharacteristic
76C24×C73C2×D411yescharacteristic
77C2×D4×C73—21no—
78C2×D4×C73—21no—
79C2×D4×C73—22no—
80C2×D4×C73—22no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—41no—
84C22×C4×C73—41no—
85C22×C4×C73—41no—
86C22×C4×C73—41no—
87C22×C4×C73C4⋊C411yescharacteristic
88C22×C4×C73C4⋊C411yescharacteristic
89C22×C4×C73C4⋊C411yescharacteristic
90C22×C4×C73C22×C411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93C22×C4×C73C2×Q811yescharacteristic
94C22×C4×C73#4111yescharacteristic
95C22×C4×C73#4111yescharacteristic
96M16×C73—42no—
97M16×C73—42no—
98C2×C8×C73—22no—
99C2×C8×C73—22no—
100C2×C8×C73—22no—
101C2×C8×C73—21no—
102C2×C8×C73—21no—
103C2×C8×C73—41no—
104C2×C8×C73—41no—
105C2×C8×C73—41no—
106C2×C8×C73—41no—
107C2×C8×C73—41no—
108C2×C8×C73C22⋊C411yescharacteristic
109C2×C8×C73C22⋊C411yescharacteristic
110C2×C8×C73C22⋊C411yescharacteristic
111C2×C8×C73C22⋊C411yescharacteristic
112C4⋊C4×C73—21no—
113C4⋊C4×C73—21no—
114C22⋊C4×C73—21no—
115C22⋊C4×C73—21no—
116C22⋊C4×C73—21no—
117C22⋊C4×C73—21no—
118C42×C73—21no—
119C42×C73—21no—
120C23×C73—21no—
121C23×C73—22no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73C2×C4⋊C411yescharacteristic
127C23×C73#12111yescharacteristic
128C23×C73#12111yescharacteristic
129C23×C73AGL(1,ℤ/8)12yes—
130C23×C73#13712yes—
131D4×C73—42no—
132D4×C73—41no—
133D4×C73—41no—
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—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—22no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—42no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73C2×C22⋊C411yescharacteristic
153C2×C4×C73#11711yescharacteristic
154C2×C4×C73C22⋊Q811yescharacteristic
155C2×C4×C73C22⋊Q811yescharacteristic
156C2×Dic73—82no—
157C2×Dic73—82no—
158C2×Dic73—82no—
159C2×Dic73—82no—
160C2×Dic73—82no—
161C2×Dic73—82no—
162C8×C73—21no—
163C8×C73—21no—
164C8×C73—21no—
165C8×C73—41no—
166C8×C73—41no—
167C8×C73#9912yescomplemented
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—42no—
175C22×C73—42no—
176C22×C73—41no—
177C22×C73—41no—
178C22×C73—41no—
179C22×C73—82no—
180C22×C73#38011yescharacteristic
181C22×C73#42812yes—
182C22×C73#46812yes—
183C22×C73#46911yescharacteristic
184C22×C73#47011yescharacteristic
185C4×C73—21no—
186C4×C73—21no—
187C4×C73—21no—
188C4×C73—41no—
189C4×C73—41no—
190C4×C73—41no—
191C4×C73C2×#9911yescharacteristic
192C4×C73#41011yescharacteristic
193Dic73—162no—
194Dic73—162no—
195#8610—731nomaximal, Sylow
196C2×C73—21no—
197C2×C73—22no—
198C2×C73—22no—
199C2×C73—21no—
200C2×C73—81no—
201C2×C73#179911yescharacteristic
202C2×C73#180911yescharacteristic
203C2×C73#181211yescharacteristic
204C2×#435D7311yescharacteristic
205#1808—731no—
206#1807—731no—
207#1799—732no—
208#1289—732no—
209C73#861011yescharacteristic, complemented, Sylow
210C22×M16Dic7311yescharacteristic, complemented
211C23×C8C2×D7311yescharacteristic
212C2×C4×D4C2×D7311yescharacteristic
213#435—21no—
214#435—21no—
215#435—21no—
216#435—21no—
217C2×#107—1461no—
218C2×#106—1461no—
219C2×C4⋊C8Dic7311yescharacteristic, complemented
220C2×#99—1462no—
221C2×C22⋊C8Dic7311yescharacteristic, complemented
222C2×C22⋊C8C2×D7311yescharacteristic
223C2×#97Dic7311yescharacteristic, complemented
224#380—732no—
225#380—732no—
226#342—1462no—
227#340—1461no—
228#340—1461no—
229C22×D4D29211yescharacteristic
230C23×C4C2×Dic7311yescharacteristic
231C23×C4C22×D7311yescharacteristic
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—42no—
235C22×C8—42no—
236C22×C8C2×Dic7311yescharacteristic
237C22×C8C2×Dic7311yescharacteristic
238C22×C8C2×Dic7311yescharacteristic
239C22×C8C2×Dic7311yescharacteristic
240C22×C8C22⋊D7311yescharacteristic
241C22×C8C22⋊D7311yescharacteristic
242C22×C8C22⋊D7311yescharacteristic
243C22×C8C22⋊D7311yescharacteristic
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C2×C4⋊C4—1462no—
249C2×C4⋊C4—1462no—
250C2×C4⋊C4Dic14611yescharacteristic
251C2×C22⋊C4—1462no—
252C2×C22⋊C4—1462no—
253C2×C22⋊C4Dic14611yescharacteristic
254C2×C22⋊C4D29211yescharacteristic
255C2×C42C2×Dic7311yescharacteristic
256#107—2922no—
257#106—2922no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260#99—2924no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265#97—21no—
266#97—21no—
267#95—1462no—
268#95—1462no—
269C24C4⋊Dic7311yescharacteristic
270C24C2×D29211yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—2922no—
282C22×C4—2922no—
283C22×C4C4⋊Dic7311yescharacteristic
284C22×C4C4⋊Dic7311yescharacteristic
285C22×C4C4⋊Dic7311yescharacteristic
286C22×C4C22×Dic7311yescharacteristic
287C22×C4C2×C22⋊D7311yescharacteristic
288C22×C4C2×C22⋊D7311yescharacteristic
289C22×C4C2×Dic14611yescharacteristic
290C22×C4#1192402811yescharacteristic
291C22×C4#1192402811yescharacteristic
292M16—42no—
293M16—42no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8C22⋊Dic7311yescharacteristic
305C2×C8C22⋊Dic7311yescharacteristic
306C2×C8C22⋊Dic7311yescharacteristic
307C2×C8C22⋊Dic7311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—2922no—
311C4⋊C4—2922no—
312C4⋊C4—2922no—
313C4⋊C4—2922no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—2922no—
319C22⋊C4—2922no—
320C22⋊C4—2922no—
321C22⋊C4—2922no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23C2×C4⋊Dic7311yescharacteristic
331C23#1455656511yescharacteristic
332C23#1455656511yescharacteristic
333C23#1455661912yes—
334C23#1455662412yes—
335D4—42no—
336D4—41no—
337D4—41no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—5842no—
357C2×C4—5842no—
358C2×C4—5842no—
359C2×C4—5842no—
360C2×C4—5842no—
361C2×C4—5842no—
362C2×C4C2×C22⋊Dic7311yescharacteristic
363C2×C4#1455655011yescharacteristic
364C2×C4#1455657011yescharacteristic
365C2×C4#1455657011yescharacteristic
366C8—21no—
367C8—21no—
368C8—21no—
369C8—41no—
370C8—41no—
371C8#1455650612yescomplemented
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—82no—
384C22#1798844711yescharacteristic
385C22#1798862912yes—
386C22#1798883312yes—
387C22#1798884011yescharacteristic
388C22#1798884511yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—11682no—
396C4—11682no—
397C4C2×#1455650611yescharacteristic
398C4#1798856111yescharacteristic
399C2—21no—
400C2—22no—
401C2—22no—
402C2—21no—
403C2—81no—
404C2#2260991911yescharacteristic
405C2#2260996411yescharacteristic
406C2#2260997711yescharacteristic
subgroupG/H
centerC22#17988447
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#435×C73C2
socleC22×C73#380
prime radicalC22×D4×C73D4
cosocleC23×C4×C73C23
hypercenterC2×#435D73
hypocenterC73#8610
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. #22609960C2semidirect
2. #22609956C2semidirect
4. #22607799C2semidirect
14. C22×M16×C73C4semidirect
21. C2×C4⋊C8×C73C4semidirect
22. C2×C22⋊C8×C73C4semidirect
24. C2×#97×C73C4semidirect
167. C8×C73#99semidirect
209. C73#8610semidirect
210. C22×M16Dic73semidirect
219. C2×C4⋊C8Dic73semidirect
221. C2×C22⋊C8Dic73semidirect
223. C2×#97Dic73semidirect
371. C8#14556506semidirect

Maximal subgroups

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

subgroupindex
#226099602
#226099562
#226099192
#226077992
C2×#435×C732
#861073