Finite Group Databasebeta

← #34031882

Subgroups — #34031882

Order 18688.

1702 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626844C214yesmaximal, complemented
2#22626842C214yesmaximal, complemented
3#22626835C218yesmaximal, complemented
4#22626733C214yesmaximal, complemented
5Q8×#11924038C214yesmaximal, complemented
6#22626612C214yesmaximal
7#22626608C212yesmaximal
8#533×D73C211yesmaximal, characteristic
9#17989639C22116yescomplemented
10#17989638C2218yescomplemented
11#17989637C2218yescomplemented
12#17989637C22116yescomplemented
13#17989636C2218yescomplemented
14Q8×#10596833C2218yescomplemented
15#17989612C22116yes—
16#17989611C22116yes—
17#17989609C2214yes—
18#17989608C2214yes—
19#17989606C2214yes—
20#17989603C2214yes—
21#17989592C2214yes—
22#17989591C2212yes—
23#17989591C2212yes—
24#17989591C2214yes—
25#17989558C2214yes—
26C4×#11924038C2214yes—
27C4×#11924038C2218yes—
28#17989144C411yescharacteristic
29#17989143C2212yes—
30#17989142C2212yes—
31#17989141C414yes—
32#17989140C412yes—
33#17989139C2214yes—
34C4⋊Q8×D73C412yes—
35#125×D73C412yes—
36C22⋊Q8×D73C414yes—
37C4×Q8×D73C2214yes—
38#117×D73C2212yes—
39C2×C42×D73C2211yescharacteristic
40#533×C73C411yescharacteristic
41C2×#11924038C2314yes—
42C2×#11924038C2314yes—
43#14556658C23116yes—
44C4⋊#10596833C2314yes—
45C4⋊#10596833C23116yes—
46C4⋊#10596833C2314yes—
47C4⋊#10596833C2314yes—
48C4⋊#10596833C2314yes—
49#14556648C2318yes—
50C22⋊#10596833C2318yes—
51#14556646C23116yes—
52#14556646C2318yes—
53C4×#10596833C2318yes—
54C4×#10596833C23116yes—
55C2×Q8×D73C2×C414yes—
56C22×C4×D73C2×C412yes—
57C22×C4×D73C2311yescharacteristic
58#14556591C2×C412yes—
59#14556590C2×C414yes—
60#14556590C2×C414yes—
61#14556589C2×C414yes—
62#14556583C2×C412yes—
63#14556582C2×C414yes—
64#14556581C2×C418yes—
65#14556571C2×C414yes—
66#14556570C2×C414yes—
67#14556569C2×C418yes—
68#14556568C2×C418yes—
69#14556567C2×C414yes—
70Q8×Dic73C2314yes—
71C4×Dic146C2×C418yes—
72C4×Dic146C2×C414yes—
73#14556556C2314yes—
74#14556556C2318yes—
75#14556550C2×C412yes—
76#14556549C2314yes—
77#14556548C2×C414yes—
78#14556547C2314yes—
79C4⋊C4×D73C2×C414yes—
80C4⋊C4×D73C2×C414yes—
81C4⋊C4×D73C2×C418yes—
82C22⋊C4×D73C2×C414yes—
83C2×C4×Dic73C2×C411yescharacteristic
84C2×C4×Dic73C2312yes—
85C42×D73C2×C411yescharacteristic
86C42×D73C2×C412yes—
87C42×D73C2×C411yescharacteristic
88C42×D73C2311yescharacteristic
89C42×D73C2311yescharacteristic
90C42×D73C2312yes—
91C42×D73C2314yes—
92C4⋊Q8×C73C2×C412yes—
93#125×C73C2×C412yes—
94C22⋊Q8×C73C2×C414yes—
95C4×Q8×C73C2×C414yes—
96#117×C73C2×C412yes—
97C2×C42×C73C2×C411yescharacteristic
98#11924038—28no—
99#11924038—28no—
100C2×#10596833C2418yes—
101C2×#10596833C2418yes—
102C23×D73C22×C411yescharacteristic
103C2×Dic146C22×C414yes—
104C2×Dic146C22×C418yes—
105Q8×D73—48no—
106C22×Dic73C22×C412yes—
107C22×Dic73C2411yescharacteristic
108C2×C4×D73—22no—
109C2×C4×D73—22no—
110C2×C4×D73—44no—
111C2×C4×D73C22×C411yescharacteristic
112C2×C4×D73C22×C411yescharacteristic
113C2×C4×D73C22×C412yes—
114C2×C4×D73C22×C412yes—
115C2×C4×D73C22×C412yes—
116C2×C4×D73C22×C412yes—
117C2×C4×D73C22×C418yes—
118C2×C4×D73C2411yescharacteristic
119C2×C4×D73C2411yescharacteristic
120C4⋊Dic73C22×C414yes—
121C4⋊Dic73C22×C414yes—
122C4⋊Dic73C22×C418yes—
123#11924007C22×C418yes—
124#11924007C22×C418yes—
125#11924007C22×C418yes—
126#11924007C22×C418yes—
127C22⋊Dic73C22×C414yes—
128#11924005C22×C418yes—
129C4×Dic73C22×C412yes—
130C4×Dic73C22×C412yes—
131C4×Dic73C22×C412yes—
132C4×Dic73C22×C411yescharacteristic
133C4×Dic73C22×C411yescharacteristic
134C4×Dic73C22×C414yes—
135C4×Dic73C2412yes—
136C4×Dic73C2412yes—
137C4×Dic73C2418yes—
138C2×Q8×C73C22×C414yes—
139C22×C4×C73C22×C411yescharacteristic
140C22×C4×C73C22×C412yes—
141C4⋊C4×C73C22×C414yes—
142C4⋊C4×C73C22×C418yes—
143C4⋊C4×C73C22×C414yes—
144C22⋊C4×C73C22×C414yes—
145C42×C73C22×C412yes—
146C42×C73C22×C411yescharacteristic
147C42×C73C22×C411yescharacteristic
148C42×C73C22×C414yes—
149C23×C73C23×C411yescharacteristic
150C22×D73—22no—
151C22×D73—42no—
152C22×D73C23×C411yescharacteristic
153C22×D73C23×C411yescharacteristic
154Q8×C73—28no—
155C2×C4×C73—22no—
156C2×C4×C73—24no—
157C2×C4×C73C23×C411yescharacteristic
158C2×C4×C73C23×C412yes—
159C2×C4×C73C23×C418yes—
160C2×C4×C73C23×C411yescharacteristic
161C2×C4×C73C23×C412yes—
162C2×Dic73—21no—
163C2×Dic73—21no—
164C2×Dic73—42no—
165C2×Dic73C23×C411yescharacteristic
166C2×Dic73C23×C412yes—
167C2×Dic73C23×C412yes—
168C2×Dic73C23×C418yes—
169C2×Dic73C2511yescharacteristic
170C4×D73—22no—
171C4×D73—22no—
172C4×D73—24no—
173C4×D73—24no—
174C4×D73—21no—
175C4×D73—21no—
176C4×D73—42no—
177C4×D73—42no—
178C4×D73—48no—
179C4×D732−1+412yes—
180C4×D732−1+412yes—
181Dic146—28no—
182Dic146—48no—
183#10596833—28no—
184#10596833—28no—
185C22×C73—21no—
186C22×C73—22no—
187C22×C73C24×C411yescharacteristic
188C2×D73—22no—
189C2×D73—21no—
190C2×D73—22no—
191C2×D73—44no—
192C2×D73#51912yes—
193C4×C73—21no—
194C4×C73—28no—
195C4×C73—22no—
196C4×C73#51912yes—
197C4×C73#56814yes—
198Dic73—21no—
199Dic73—28no—
200Dic73—22no—
201Dic73—22no—
202Dic73C2×2−1+411yescharacteristic
203Dic73C2×2−1+411yescharacteristic
204#60249—731nomaximal, Sylow
205C2×C73—41no—
206C2×C73C2×#51911yescharacteristic
207C2×C73C2×#56812yes—
208D73—22no—
209D73—41no—
210C2×#533—731no—
211#2987—734no—
212#2986—734no—
213#2985—738no—
214Q8×M16—734no—
215Q8×M16—734no—
216#2921—734no—
217#2920—732no—
218C73#6024911yescharacteristic, complemented, Sylow
219#533—732no—
220#533—732no—
221#533—734no—
222#533—734no—
223#533—732no—
224#533—731no—
225#533#6440811yescharacteristic
226C2×C4⋊Q8—732no—
227C2×#125—732no—
228C2×C22⋊Q8—734no—
229C2×C4×Q8—734no—
230C2×#117—732no—
231C22×C42—731no—
232#446—738no—
233#446—7316no—
234#446—738no—
235#446—7316no—
236C8×Q8—738no—
237C8×Q8—738no—
238#433—7316no—
239#433—7316no—
240#432—734no—
241#432—734no—
242#431—734no—
243#431—734no—
244#423—732no—
245#423—732no—
246#423—734no—
247#423—734no—
248#407—734no—
249C4×M16—734no—
250C4×M16—738no—
251C22×Q8—734no—
252C23×C4—732no—
253C23×C4—731no—
254C2×M16—734no—
255C2×M16—734no—
256C4⋊Q8—734no—
257C4⋊Q8—734no—
258C4⋊Q8—734no—
259C4⋊Q8—732no—
260C4⋊Q8C2×#6440812yes—
261#125—738no—
262#125—734no—
263#125—732no—
264#125C2×#6440812yes—
265C22⋊Q8—734no—
266C22⋊Q8—734no—
267C22⋊Q8—738no—
268C22⋊Q8—734no—
269C22⋊Q8—738no—
270C22⋊Q8C2×#6440814yes—
271C4×Q8—734no—
272C4×Q8—734no—
273C4×Q8—738no—
274C4×Q8—738no—
275C4×Q8—734no—
276C4×Q8C2×#6440814yes—
277#117—732no—
278#117—734no—
279#117—734no—
280#117—734no—
281#117C2×#6440812yes—
282C2×C4⋊C4—734no—
283C2×C4⋊C4—738no—
284C2×C4⋊C4—734no—
285C2×C22⋊C4—734no—
286C2×C42—731no—
287C2×C42—732no—
288C2×C42—734no—
289C2×C42—732no—
290C2×C42—731no—
291C2×C42—731no—
292C2×C42—732no—
293C2×C42—731no—
294C2×C42—731no—
295C2×C42C2×#6440811yescharacteristic
296C4⋊C8—734no—
297C4⋊C8—734no—
298C4⋊C8—734no—
299C4⋊C8—734no—
300C4⋊C8—7316no—
301C4⋊C8—7316no—
302C22⋊C8—738no—
303C22⋊C8—738no—
304#97—738no—
305#97—7316no—
306C4×C8—738no—
307C4×C8—7316no—
308C24—731no—
309C2×Q8—2928no—
310C2×Q8—738no—
311C2×Q8—734no—
312C2×Q8C22×#6440814yes—
313C22×C4—2924no—
314C22×C4—732no—
315C22×C4—732no—
316C22×C4—732no—
317C22×C4—738no—
318C22×C4—731no—
319C22×C4—731no—
320C22×C4—731no—
321C22×C4—732no—
322C22×C4—731no—
323C22×C4—732no—
324C22×C4—731no—
325C22×C4—1462no—
326C22×C4—1462no—
327C22×C4C22×#6440811yescharacteristic
328C22×C4C22×#6440812yes—
329M16—1468no—
330M16—1468no—
331C2×C8—738no—
332C2×C8—738no—
333C4⋊C4—738no—
334C4⋊C4—738no—
335C4⋊C4—738no—
336C4⋊C4—738no—
337C4⋊C4—734no—
338C4⋊C4—738no—
339C4⋊C4—734no—
340C4⋊C4C22×#6440814yes—
341C4⋊C4C22×#6440818yes—
342C4⋊C4C22×#6440814yes—
343C22⋊C4—734no—
344C22⋊C4—738no—
345C22⋊C4C22×#6440814yes—
346C42—732no—
347C42—732no—
348C42—731no—
349C42—732no—
350C42—731no—
351C42—734no—
352C42—732no—
353C42—738no—
354C42—732no—
355C42C22×#6440812yes—
356C42C22×#6440811yescharacteristic
357C42C22×#6440811yescharacteristic
358C42C22×#6440814yes—
359C23—2922no—
360C23—731no—
361C23—731no—
362C23—1462no—
363C23C23×#6440811yescharacteristic
364Q8—28no—
365Q8—2928no—
366Q8—1468no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—2922no—
370C2×C4—2922no—
371C2×C4—2928no—
372C2×C4—2922no—
373C2×C4—731no—
374C2×C4—732no—
375C2×C4—732no—
376C2×C4—731no—
377C2×C4—732no—
378C2×C4—732no—
379C2×C4—738no—
380C2×C4—1461no—
381C2×C4—1461no—
382C2×C4—1462no—
383C2×C4—1461no—
384C2×C4—1461no—
385C2×C4—1462no—
386C2×C4—1464no—
387C2×C4—1464no—
388C2×C4C23×#6440811yescharacteristic
389C2×C4C23×#6440812yes—
390C2×C4C23×#6440818yes—
391C2×C4C23×#6440811yescharacteristic
392C2×C4C23×#6440812yes—
393C8—1468no—
394C8—1468no—
395C22—21no—
396C22—22no—
397C22—2924no—
398C22—732no—
399C22—1461no—
400C22—1462no—
401C22—1462no—
402C22C24×#6440811yescharacteristic
403C4—21no—
404C4—22no—
405C4—28no—
406C4—731no—
407C4—731no—
408C4—1461no—
409C4—1462no—
410C4—1462no—
411C4—1468no—
412C4#1798965512yes—
413C4#1798966614yes—
414C2—41no—
415C2—2921no—
416C2—1462no—
417C2C2×#1798965511yescharacteristic
418C2C2×#1798966612yes—
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
Fitting#533×C73C4
prime radicalC23×D73C22×C4
cosocleC2×Dic73C25
hypercenter#533#64408
hypocenterC73#60249
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #22626844C2semidirect
2. #22626842C2semidirect
3. #22626835C2semidirect
4. #22626733C2semidirect
5. Q8×#11924038C2semidirect
9. #17989639C22semidirect
10. #17989638C22semidirect
11. #17989637C22semidirect
12. #17989637C22semidirect
13. #17989636C22semidirect
14. Q8×#10596833C22semidirect
218. C73#60249semidirect

Maximal subgroups

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

subgroupindex
#226268442
#226268422
#226268352
#226267332
Q8×#119240382
#226266122
#226266082
#533×D732
#6024973