Finite Group Databasebeta

← #33056533

Subgroups — #33056533

Order 18688.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620490C211yesmaximal, characteristic
2#22618861C211yesmaximal, characteristic, complemented
3#22618642C211yesmaximal, characteristic
4#22608832C214yesmaximal, complemented
5#22608742C214yesmaximal, complemented
6#22608385C211yesmaximal, characteristic, complemented
7#22608381C211yesmaximal, characteristic
8C2×#17988308C211yesmaximal, characteristic, complemented
9C73×#1567C211yesmaximal, characteristic, complemented
10C2×#14556599—21no—
11#17989144C2211yescharacteristic
12#17989137C2211yescharacteristic
13#17988985C411yescharacteristic
14#17988960C411yescharacteristic
15#17988938C411yescharacteristic
16#17988934C411yescharacteristic
17#17988816C414yes—
18#17988714—21no—
19#17988714C2212yes—
20#17988711—21no—
21#17988711C2212yes—
22C4×#11924017C2214yes—
23#17988664—21no—
24#17988664C2212yes—
25#17988652—21no—
26#17988652C2212yes—
27#17988615—21no—
28#17988555—21no—
29#17988545—21no—
30C2×C4×#10596831C2211yescharacteristic
31C2×C4×#10596831C2211yescharacteristic
32#17988317C2214yes—
33#17988314C2214yes—
34#17988308C2212yes—
35#17988308C2212yes—
36#423×C73C2211yescharacteristic
37#423×C73C2211yescharacteristic
38C2×C4×C8×C73C2211yescharacteristic
39#335×C73C2214yes—
40C2×#11924028—21no—
41C2×#11924017C2×C414yes—
42#14556599—44no—
43C2×#11924011—22no—
44C22×#10596831—21no—
45C22×#10596831D411yescharacteristic
46C22×#10596831D411yescharacteristic
47C2×C8×D73—22no—
48#14556591C2×C411yescharacteristic
49#14556591C2×C411yescharacteristic
50#14556591C2×C411yescharacteristic
51#14556588C2×C411yescharacteristic
52#14556583—21no—
53#14556579—21no—
54#14556570—21no—
55#14556570C2×C412yes—
56#14556565—21no—
57#14556565C2×C412yes—
58C4×Dic146—21no—
59C4×Dic146C2312yes—
60C4×D292—21no—
61C4×D292C2312yes—
62#14556550—21no—
63#14556523C2×C414yes—
64#14556520—22no—
65#14556520—22no—
66#14556516C2×C414yes—
67#14556513C2×C414yes—
68#14556503—22no—
69#14556503—22no—
70#14556501—22no—
71C4×#10596831—21no—
72C4×#10596831D411yescharacteristic
73C4×#10596831D411yescharacteristic
74C4×#10596831C2312yes—
75C4×#10596831C2311yescharacteristic
76C4×#10596831C2311yescharacteristic
77C8×Dic73—22no—
78C2×M16×C73—21no—
79C2×M16×C73—21no—
80C22×C8×C73C2×C412yes—
81C2×C42×C73C2311yescharacteristic
82C4⋊C8×C73—21no—
83C4⋊C8×C73—21no—
84C4⋊C8×C73C2312yes—
85C4⋊C8×C73C2312yes—
86C4×C8×C73C2×C411yescharacteristic
87C4×C8×C73C2×C411yescharacteristic
88C4×C8×C73C2312yes—
89#11924028—42no—
90#11924028—42no—
91C2×Dic146—21no—
92C2×Dic146—21no—
93C2×Dic146C22×C412yes—
94C2×D292—21no—
95C2×D292—21no—
96C2×D292C22×C412yes—
97C2×C22⋊D73—22no—
98C2×C4×D73—22no—
99#11924017—28no—
100#11924011—44no—
101C2×#10596831—22no—
102C2×#10596831—22no—
103C2×#10596831—22no—
104C2×#10596831—42no—
105C2×#10596831C2×D411yescharacteristic
106C2×#10596831C2×D411yescharacteristic
107C2×#10596831C2×D411yescharacteristic
108C2×#10596831C2×D411yescharacteristic
109C8×D73—44no—
110C4⋊Dic73—21no—
111C4⋊Dic73—21no—
112C4⋊Dic73—21no—
113C4⋊Dic73—21no—
114C4⋊Dic73C22×C412yes—
115C4⋊Dic73C22×C412yes—
116#11924007—22no—
117#11924007—22no—
118C22⋊Dic73—22no—
119#11924005—22no—
120#11924005—22no—
121C4×Dic73—22no—
122C22×C4×C73C22×C411yescharacteristic
123C22×C4×C73C2×D411yescharacteristic
124C22×C4×C73#4111yescharacteristic
125M16×C73—42no—
126M16×C73—42no—
127C2×C8×C73—24no—
128C2×C8×C73—22no—
129C2×C8×C73—22no—
130C2×C8×C73C22×C412yes—
131C2×C8×C73C22×C412yes—
132C42×C73C22×C411yescharacteristic
133C42×C73C2×D411yescharacteristic
134C42×C73#4111yescharacteristic
135C42×C73C2411yescharacteristic
136C23×C73C4×D411yescharacteristic
137C22×D73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73C4×D411yescharacteristic
141C2×C4×C73C4×D411yescharacteristic
142C2×C4×C73C4×D411yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)12yes—
144C2×C4×C73#13712yes—
145C2×C4×C73C23×C411yescharacteristic
146C2×C4×C73C22×D411yescharacteristic
147C2×C4×C73C2×#4111yescharacteristic
148C22⋊D73—44no—
149C2×Dic73—22no—
150C2×Dic73—22no—
151C2×Dic73—22no—
152D292—24no—
153D292—42no—
154D292—42no—
155C4×D73—42no—
156C4×D73—42no—
157Dic146—24no—
158Dic146—42no—
159Dic146—42no—
160C8×C73—22no—
161C8×C73—42no—
162C8×C73—42no—
163C8×C73#13114yescomplemented
164#10596831—21no—
165#10596831—21no—
166#10596831—21no—
167#10596831—21no—
168#10596831—42no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73#44012yes—
172C22×C73C2×C4×D411yescharacteristic
173C2×D73—42no—
174C2×D73—42no—
175C2×D73—42no—
176C4×C73—21no—
177C4×C73—21no—
178C4×C73#44012yes—
179C4×C73#44412yes—
180C4×C73C2×#13112yes—
181C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
182C4×C73C2×#13711yescharacteristic
183Dic73—42no—
184Dic73—42no—
185Dic73—42no—
186#17960—731nomaximal, Sylow
187C2×C73—22no—
188C2×C73#293111yescharacteristic
189C2×C73C2×#44011yescharacteristic
190C2×C73C2×#44411yescharacteristic
191D73—82no—
192#3167—731no—
193#2960—731no—
194#2931—731no—
195#1592—734no—
196SD16⋊C8—734no—
197#1567D7311yescharacteristic, complemented
198#1493—731no—
199#1492—731no—
200C2×#322—731no—
201C73#1796011yescharacteristic, complemented, Sylow
202C2×#131—1461no—
203#533—731no—
204#532—731no—
205#498—731no—
206#492—731no—
207#487—731no—
208#486—731no—
209#465—734no—
210#446—732no—
211#446—1461no—
212C8×Q8—732no—
213C8×Q8—1461no—
214C4×SD16—734no—
215#436—732no—
216#436—1461no—
217C8×D4—732no—
218C8×D4—1461no—
219#424—1461no—
220#423C2×D7311yescharacteristic
221#423C2×D7311yescharacteristic
222#408—1461no—
223#405—1461no—
224C2×C4×C8—731no—
225C2×C4×C8—731no—
226C2×C4×C8C2×D7311yescharacteristic
227#335C2×D7314yes—
228Q8⋊C8—734no—
229D4⋊C8—734no—
230#322—732no—
231#322—732no—
232C2×#41—1461no—
233C2×SD16—734no—
234#131—2924no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—1462no—
238C22×C8—731no—
239C22×C8—731no—
240C22×C8—1462no—
241C22×C8—1461no—
242C22×C8C4×D7312yes—
243C4⋊Q8—731no—
244C4⋊Q8—731no—
245C4⋊Q8—731no—
246#127—731no—
247#125—1461no—
248#124—1461no—
249C22⋊Q8—732no—
250C22⋊Q8—1461no—
251#121—732no—
252#121—1461no—
253C4×Q8—732no—
254C4×Q8—1461no—
255C4×D4—732no—
256C4×D4—1461no—
257#117—1461no—
258C2×C42C22×D7311yescharacteristic
259#106—734no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—1462no—
263C4⋊C8—1462no—
264C4⋊C8C22×D7312yes—
265C4⋊C8C22×D7312yes—
266#103—734no—
267#102—734no—
268C22⋊C8—1462no—
269C22⋊C8—1462no—
270#97—1462no—
271C4×C8—731no—
272C4×C8—731no—
273C4×C8—731no—
274C4×C8—731no—
275C4×C8—732no—
276C4×C8—1462no—
277C4×C8—1461no—
278C4×C8C4×D7311yescharacteristic
279C4×C8C4×D7311yescharacteristic
280C4×C8C22×D7312yes—
281#41—2922no—
282#41—2922no—
283C2×Q8—732no—
284C2×Q8—1461no—
285C2×Q8—1461no—
286C2×D4—732no—
287C2×D4—1462no—
288C2×D4—1461no—
289C2×D4—1461no—
290C22×C4—1462no—
291C22×C4C2×C4×D7311yescharacteristic
292C22×C4D4×D7311yescharacteristic
293C22×C4#1192402911yescharacteristic
294SD16—1468no—
295M16—42no—
296M16—42no—
297M16—2924no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—2924no—
302C2×C8—2922no—
303C2×C8—731no—
304C2×C8—731no—
305C2×C8—731no—
306C2×C8—731no—
307C2×C8—1462no—
308C2×C8—1462no—
309C2×C8—1462no—
310C2×C8C2×C4×D7312yes—
311C2×C8C2×C4×D7312yes—
312C4⋊C4—732no—
313C4⋊C4—732no—
314C4⋊C4—1462no—
315C4⋊C4—1461no—
316C4⋊C4—1461no—
317C4⋊C4—1461no—
318C4⋊C4—1461no—
319C4⋊C4—1462no—
320C22⋊C4—1462no—
321C22⋊C4—1462no—
322C22⋊C4—1462no—
323C42—1462no—
324C42C2×C4×D7311yescharacteristic
325C42D4×D7311yescharacteristic
326C42#1192402911yescharacteristic
327C42C23×D7311yescharacteristic
328C23—1462no—
329C23#1455655311yescharacteristic
330Q8—2922no—
331Q8—2922no—
332Q8—1464no—
333D4—2924no—
334D4—2922no—
335D4—2922no—
336D4—1464no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—2922no—
340C2×C4—2922no—
341C2×C4—1462no—
342C2×C4—1462no—
343C2×C4—1462no—
344C2×C4#1455655311yescharacteristic
345C2×C4#1455655311yescharacteristic
346C2×C4#1455655311yescharacteristic
347C2×C4#1455661512yes—
348C2×C4#1455662312yes—
349C2×C4C22×C4×D7311yescharacteristic
350C2×C4C2×D4×D7311yescharacteristic
351C2×C4C2×#1192402911yescharacteristic
352C8—22no—
353C8—42no—
354C8—42no—
355C8—2922no—
356C8—1461no—
357C8—1461no—
358C8—1461no—
359C8—1461no—
360C8#1455659914yescomplemented
361C22—21no—
362C22—21no—
363C22—2922no—
364C22—2922no—
365C22—2922no—
366C22#1798868612yes—
367C22C2×#1455655311yescharacteristic
368C4—21no—
369C4—21no—
370C4—2922no—
371C4—2922no—
372C4—2922no—
373C4#1798868612yes—
374C4#1798870812yes—
375C4C2×#1455659912yes—
376C4C2×#1455661511yescharacteristic
377C4C2×#1455662311yescharacteristic
378C2—22no—
379C2—5842no—
380C2#2261864211yescharacteristic
381C2C2×#1798868611yescharacteristic
382C2C2×#1798870811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1567C2
socleC22×C73C2×C4×D4
prime radical#17989137C22
cosocleC42×C73C24
hypercenter#1567D73
hypocenterC73#17960
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1567

Complements

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

NH ≅ G/Ntype
2. #22618861C2semidirect
4. #22608832C2semidirect
5. #22608742C2semidirect
6. #22608385C2semidirect
8. C2×#17988308C2semidirect
9. C73×#1567C2semidirect
163. C8×C73#131semidirect
197. #1567D73semidirect
201. C73#17960semidirect
360. C8#14556599semidirect

Maximal subgroups

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

subgroupindex
#226204902
#226188612
#226186422
#226088322
#226087422
#226083852
#226083812
C2×#179883082
C73×#15672
#1796073