Finite Group Databasebeta

← #34019760

Subgroups — #34019760

Order 18688.

564 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626840C212yesmaximal
2#22625576C212yesmaximal, complemented
3#22625544C212yesmaximal, complemented
4#22625543C212yesmaximal, complemented
5#22625525C212yesmaximal, complemented
6#22625480C211yesmaximal, characteristic
7#22625253C212yesmaximal
8#22625140C211yesmaximal, characteristic
9#22620512C211yesmaximal, characteristic
10#17989639—22no—
11#17989638C2212yes—
12#17989637C2212yes—
13#17989636—22no—
14#17989636C2212yes—
15Q8×#10596833C2212yes—
16#17989611—22no—
17#17989611—22no—
18#17989604C2211yescharacteristic
19#17989603C2212yes—
20C4×#11924037C2211yescharacteristic
21#17989451C2212yes—
22#17989449C2211yescharacteristic
23#17989449C2211yescharacteristic
24#17989437C2212yes—
25#17989437C2212yes—
26Q8⋊#10596833C2212yes—
27Q8⋊#10596833C2212yes—
28#17989429C2212yes—
29#17989428C2212yes—
30#17989139C2212yes—
31#17988991C411yescharacteristic
32#17988965C411yescharacteristic
33#17988936C412yes—
34#17988828C411yescharacteristic
35#17988826C412yes—
36Q16×Dic73C2211yescharacteristic
37#17988681C2211yescharacteristic
38#17988679C2212yes—
39C2×C8×Dic73C2211yescharacteristic
40#467×C73C411yescharacteristic
41C2×#11924037—21no—
42#14556658—22no—
43#14556658—22no—
44#14556658D412yes—
45C4⋊#10596833—22no—
46C4⋊#10596833—22no—
47C4⋊#10596833C2312yes—
48C4⋊#10596833C2312yes—
49C22⋊#10596833—22no—
50C22⋊#10596833D411yescharacteristic
51C22⋊#10596833D411yescharacteristic
52#14556646—22no—
53#14556646—22no—
54#14556646—21no—
55C4×#10596833—22no—
56C4×#10596833—22no—
57C4×#10596833—22no—
58C4×#10596833C2312yes—
59C4×#10596833C2312yes—
60C2×Dic292C2×C411yescharacteristic
61C2×#11924018C2×C412yes—
62C22×#10596831C2×C411yescharacteristic
63#14556590C2×C412yes—
64#14556589C2×C412yes—
65#14556581—22no—
66#14556570C2×C412yes—
67#14556568—22no—
68Q8×Dic73C2312yes—
69#14556556—22no—
70#14556556C2312yes—
71#14556547—22no—
72C2×C4×Dic73C2311yescharacteristic
73#14556525C2×C411yescharacteristic
74#14556524C2×C412yes—
75#14556516C2×C412yes—
76#14556515C2×C412yes—
77#14556514C2×C412yes—
78C8×Dic73C2×C411yescharacteristic
79C8×Dic73C2×C411yescharacteristic
80C8×Dic73C2311yescharacteristic
81C8×Dic73C2311yescharacteristic
82C2×Q16×C73C2×C411yescharacteristic
83C22×C8×C73C2×C411yescharacteristic
84C22⋊Q8×C73C2×C412yes—
85#107×C73C2×C411yescharacteristic
86#103×C73C2×C412yes—
87#11924037—42no—
88C2×#10596833—22no—
89C2×#10596833—22no—
90C2×#10596833—22no—
91C2×#10596833C2×D412yes—
92C2×#10596833C2×D412yes—
93C2×Dic146—22no—
94C2×Dic146C22×C412yes—
95C22×Dic73C2×D411yescharacteristic
96C22×Dic73#4111yescharacteristic
97Dic292—41no—
98#11924018—24no—
99C2×#10596831—22no—
100C2×#10596831C22×C411yescharacteristic
101C2×#10596831C22×C411yescharacteristic
102C4⋊Dic73—22no—
103C4⋊Dic73C22×C412yes—
104#11924007—22no—
105#11924007—22no—
106#11924007—22no—
107#11924007—22no—
108#11924007—22no—
109C22⋊Dic73—22no—
110C4×Dic73—22no—
111C4×Dic73—22no—
112C4×Dic73C22×C411yescharacteristic
113C4×Dic73C2×D411yescharacteristic
114C4×Dic73#4111yescharacteristic
115C4×Dic73C2411yescharacteristic
116C2×Q8×C73C22×C412yes—
117C22×C4×C73C22×C411yescharacteristic
118Q16×C73—41no—
119C2×C8×C73—41no—
120C2×C8×C73C22×C411yescharacteristic
121C2×C8×C73C22×C411yescharacteristic
122C4⋊C4×C73—22no—
123C4⋊C4×C73C22×C412yes—
124C22⋊C4×C73—22no—
125C23×C73C4×D411yescharacteristic
126Q8×C73—22no—
127Q8×C73—22no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—22no—
131C2×C4×C73—22no—
132C2×C4×C73C4×D411yescharacteristic
133C2×C4×C73C23×C411yescharacteristic
134C2×Dic73—22no—
135C2×Dic73—22no—
136C2×Dic73—22no—
137C2×Dic73—22no—
138C2×Dic73C4×D411yescharacteristic
139C2×Dic73C4×D411yescharacteristic
140C2×Dic73C22×D411yescharacteristic
141C2×Dic73C2×#4111yescharacteristic
142Dic146—22no—
143Dic146—22no—
144Dic146—44no—
145#10596833—22no—
146#10596833—22no—
147#10596833—42no—
148#10596833—42no—
149#10596833—42no—
150C8×C73—21no—
151C8×C73—21no—
152#10596831—21no—
153#10596831#13112yes—
154C22×C73—21no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73C2×C4×D411yescharacteristic
158C4×C73—21no—
159C4×C73—42no—
160C4×C73—42no—
161C4×C73C2×#13111yescharacteristic
162C4×C73#56811yescharacteristic
163Dic73—21no—
164Dic73—21no—
165Dic73—42no—
166Dic73—42no—
167Dic73#44412yes—
168Dic73#57812yes—
169#18090—731nomaximal, Sylow
170C2×C73—41no—
171C2×C73#293511yescharacteristic
172C2×C73#295011yescharacteristic
173C2×C73C2×#44411yescharacteristic
174#3170—731no—
175#2986—732no—
176#1593—732no—
177#1584—732no—
178#1584—732no—
179#1579—732no—
180#1564—731no—
181#1485—732no—
182#1451—731no—
183C73#1809011yescharacteristic, complemented, Sylow
184#533—732no—
185#500—731no—
186#494—731no—
187#487—732no—
188#467—731no—
189#467—732no—
190#467#6440811yescharacteristic
191#446—732no—
192#446—732no—
193#446—1462no—
194C8×Q8—732no—
195C8×Q8—732no—
196C8×Q8—1462no—
197C4×Q16—731no—
198C4×Q16—731no—
199C4×Q16—732no—
200#433—1462no—
201#433—1462no—
202#431—731no—
203#431—732no—
204C4×M16—731no—
205C2×C4×C8—731no—
206#335—732no—
207#334—731no—
208#334—731no—
209Q8⋊C8—732no—
210Q8⋊C8—732no—
211Q8⋊C8—732no—
212Q8⋊C8—732no—
213#322—732no—
214#322—732no—
215C2×Q16—732no—
216C2×Q16—731no—
217C2×Q16C2×#6440811yescharacteristic
218C2×M16—1461no—
219C22×C8—731no—
220C22×C8C2×#6440811yescharacteristic
221C4⋊Q8—732no—
222C4⋊Q8—732no—
223#125—1462no—
224C22⋊Q8—732no—
225C22⋊Q8—1462no—
226C22⋊Q8C2×#6440812yes—
227C4×Q8—732no—
228C4×Q8—732no—
229C4×Q8—1462no—
230#117—1462no—
231C2×C42—731no—
232#107—731no—
233#107—732no—
234#107C2×#6440811yescharacteristic
235C4⋊C8—732no—
236C4⋊C8—732no—
237C4⋊C8—732no—
238C4⋊C8—1462no—
239C4⋊C8—1462no—
240C4⋊C8—1462no—
241C4⋊C8—1462no—
242#103—732no—
243#103—732no—
244#103—732no—
245#103C2×#6440812yes—
246C22⋊C8—731no—
247C22⋊C8—731no—
248C22⋊C8—1462no—
249#97—1461no—
250#97—1462no—
251#97—1462no—
252C4×C8—732no—
253C4×C8—731no—
254C4×C8—731no—
255C4×C8—731no—
256C4×C8—731no—
257C4×C8—732no—
258C4×C8—1462no—
259C4×C8—1462no—
260C4×C8—1462no—
261C2×Q8—732no—
262C2×Q8—1462no—
263C2×Q8C22×#6440812yes—
264C22×C4—731no—
265C22×C4—731no—
266C22×C4C22×#6440811yescharacteristic
267Q16—41no—
268Q16—2921no—
269Q16—1464no—
270M16—2922no—
271C2×C8—41no—
272C2×C8—731no—
273C2×C8—732no—
274C2×C8—732no—
275C2×C8—731no—
276C2×C8—1462no—
277C2×C8—1462no—
278C2×C8—1462no—
279C2×C8—1462no—
280C2×C8C22×#6440811yescharacteristic
281C2×C8C22×#6440811yescharacteristic
282C4⋊C4—22no—
283C4⋊C4—732no—
284C4⋊C4—1462no—
285C4⋊C4—1462no—
286C4⋊C4—1462no—
287C4⋊C4—1462no—
288C4⋊C4—1462no—
289C4⋊C4—1462no—
290C4⋊C4C22×#6440812yes—
291C22⋊C4—22no—
292C22⋊C4—1462no—
293C42—731no—
294C42—731no—
295C42—731no—
296C42—731no—
297C42—1462no—
298C42—1462no—
299C23D4×#6440811yescharacteristic
300Q8—22no—
301Q8—22no—
302Q8—2924no—
303Q8—1462no—
304Q8—1462no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—731no—
310C2×C4—731no—
311C2×C4—731no—
312C2×C4—731no—
313C2×C4—1462no—
314C2×C4—1462no—
315C2×C4—1462no—
316C2×C4—1462no—
317C2×C4D4×#6440811yescharacteristic
318C2×C4C23×#6440811yescharacteristic
319C8—21no—
320C8—21no—
321C8—2922no—
322C8—2922no—
323C8—2922no—
324C8—732no—
325C8—1462no—
326C8—1462no—
327C8—1461no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22C2×D4×#6440811yescharacteristic
332C4—21no—
333C4—42no—
334C4—42no—
335C4—2922no—
336C4—2922no—
337C4—732no—
338C4—732no—
339C4—1461no—
340C4—1461no—
341C4D73⋊#13111yescharacteristic
342C4#1798966511yescharacteristic
343C2—41no—
344C2#2262667511yescharacteristic
345C2#2262669211yescharacteristic
346C2D73⋊#44411yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#467×C73C4
socleC22×C73C2×C4×D4
prime radicalC23×C73C4×D4
cosocleC4×Dic73C24
hypercenter#467#64408
hypocenterC73#18090
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #22625576C2semidirect
3. #22625544C2semidirect
4. #22625543C2semidirect
5. #22625525C2semidirect
183. C73#18090semidirect

Maximal subgroups

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

subgroupindex
#226268402
#226255762
#226255442
#226255432
#226255252
#226254802
#226252532
#226251402
#226205122
#1809073