Finite Group Databasebeta

← #37543409

Subgroups — #37543409

Order 23168.

472 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153848C211yesmaximal, characteristic, complemented
2C2×#19153824C211yesmaximal, characteristic, complemented
3#26041859C211yesmaximal, characteristic
4#26041852C211yesmaximal, characteristic, complemented
5#26041536C211yesmaximal, characteristic, complemented
6#26041528C211yesmaximal, characteristic, complemented
7#26041525C211yesmaximal, characteristic, complemented
8#26041518C211yesmaximal, characteristic, complemented
9#26041517C211yesmaximal, characteristic, complemented
10#26041510C211yesmaximal, characteristic, complemented
11#26041509C211yesmaximal, characteristic, complemented
12#26041318C211yesmaximal, characteristic, complemented
13#26041274C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#470×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668449C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19#19153848—24no—
20C2×#15668439C2211yescharacteristic
21C2×#15668438C2211yescharacteristic
22C2×#15668436C2211yescharacteristic
23#19153824—24no—
24C2×#15668432C2211yescharacteristic
25C2×#15668432C2211yescharacteristic
26C22×#13089678C2211yescharacteristic
27C2×C8×D181C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic
29#19153815—21no—
30#19153802—21no—
31#19153801—21no—
32#19153798—21no—
33#19153796C2211yescharacteristic
34#19153795C2211yescharacteristic
35#19153794—21no—
36#19153792C2211yescharacteristic
37#19153792C2211yescharacteristic
38#19153791—21no—
39#19153791C2211yescharacteristic
40#19153790C2211yescharacteristic
41#19153789—21no—
42#19153789—21no—
43#19153774C2211yescharacteristic
44C4⋊C4×D181C2211yescharacteristic
45#19153750C2211yescharacteristic
46#19153749C2211yescharacteristic
47#19153747C2211yescharacteristic
48#19153741C2211yescharacteristic
49#19153740C2211yescharacteristic
50#19153740C2211yescharacteristic
51#19153739C2211yescharacteristic
52#19153738C2211yescharacteristic
53#19153736C2211yescharacteristic
54C2×Q16×C181C2211yescharacteristic
55C2×M16×C181C2211yescharacteristic
56C22⋊Q8×C181C2211yescharacteristic
57C22⋊Q8×C181C2211yescharacteristic
58#106×C181C2211yescharacteristic
59#103×C181C2211yescharacteristic
60#103×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—22no—
63#15668449—22no—
64#15668449—22no—
65C2×Dic362—21no—
66C2×Dic362D411yescharacteristic
67C2×Dic362C2311yescharacteristic
68Q8×D181—22no—
69Q8×D181—22no—
70C2×D724—21no—
71C2×D724—21no—
72C2×D724D411yescharacteristic
73C2×D724C2311yescharacteristic
74C2×C22⋊D181—21no—
75C2×C22⋊D181D411yescharacteristic
76C2×C22⋊D181D411yescharacteristic
77C2×C4×D181—21no—
78C2×C4×D181—21no—
79C2×C4×D181—21no—
80C2×C4×D181C2311yescharacteristic
81C2×C4×D181C2311yescharacteristic
82#15668439—22no—
83#15668438—22no—
84#15668436—22no—
85#15668432—22no—
86#15668432D414yescomplemented
87C2×#13089678—22no—
88C2×#13089678C2311yescharacteristic
89C2×#13089678C2311yescharacteristic
90C8×D181—22no—
91C8×D181—22no—
92C4⋊Dic181—21no—
93C4⋊Dic181C2311yescharacteristic
94C4⋊Dic181C2311yescharacteristic
95#15668428—21no—
96#15668428—21no—
97#15668428—21no—
98#15668428—21no—
99C22⋊Dic181—21no—
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106C4×Dic181—21no—
107C2×Q8×C181C2311yescharacteristic
108C2×Q8×C181C2311yescharacteristic
109C22×C4×C181C2311yescharacteristic
110Q16×C181—22no—
111M16×C181—22no—
112C2×C8×C181C2311yescharacteristic
113C2×C8×C181C2311yescharacteristic
114C4⋊C4×C181—21no—
115C4⋊C4×C181—21no—
116C4⋊C4×C181C2311yescharacteristic
117C4⋊C4×C181C2311yescharacteristic
118C22⋊C4×C181—21no—
119C22⋊C4×C181—21no—
120C22×D181—21no—
121C22×D181C2×D411yescharacteristic
122C22×D181C2×D411yescharacteristic
123Dic362—22no—
124Dic362—21no—
125Dic362—21no—
126Dic362—42no—
127D724—22no—
128D724—21no—
129D724—21no—
130D724—42no—
131D724—81no—
132C22⋊D181—22no—
133C22⋊D181—22no—
134C22⋊D181—81no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181C2×D411yescharacteristic
139C2×Dic181C2×D411yescharacteristic
140C4×D181—21no—
141C4×D181—21no—
142C4×D181—42no—
143C4×D181—42no—
144C4×D181—81no—
145C4×D181C2×D412yes—
146C4×D181#4112yes—
147#13089678—21no—
148#13089678C2×D412yes—
149C23×C181C2×D411yescharacteristic
150Q8×C181—21no—
151Q8×C181—21no—
152Q8×C181—21no—
153Q8×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181—21no—
157C2×C4×C181—21no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181C2×D411yescharacteristic
161C2×C4×C181C2411yescharacteristic
162C8×C181—21no—
163C8×C181C2×D412yes—
164C22×C181—21no—
165C22×C181—21no—
166C22×C181—21no—
167C22×C181C22×D411yescharacteristic
168C2×D181—21no—
169C2×D181—21no—
170C2×D181—21no—
171C2×D181—21no—
172C2×D181—21no—
173C2×D181—41no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181#12112yes—
177C4×C181—21no—
178C4×C181—41no—
179C4×C181—41no—
180C4×C181—41no—
181C4×C181—41no—
182C4×C181C22×D411yescharacteristic
183C4×C181C2×#4111yescharacteristic
184Dic181—21no—
185Dic181—41no—
186Dic181—41no—
187Dic181—41no—
188Dic181#12112yes—
189C2×C181—41no—
190C2×C181C2×#12111yescharacteristic
191C2×C181#57711yescharacteristic
192C2×C181#57811yescharacteristic
193D181—22no—
194D181—41no—
195D181—81no—
196C181#306311yescharacteristic, complemented, Sylow
197#3063—1811nomaximal, Sylow
198C2×#137—1811no—
199C2×#131—1811no—
200#537—1811no—
201#536—1811no—
202#470—1811no—
203#470D18111yescharacteristic, complemented
204#468—1811no—
205#468—1811no—
206#467—1811no—
207#467—1811no—
208#465—1811no—
209#465—1811no—
210#428—1811no—
211#416—1811no—
212C2×#103—1811no—
213C2×#41—1811no—
214C2×#41—1811no—
215C22×Q8—1811no—
216#137—3624no—
217C2×Q16—1811no—
218C2×Q16C2×D18111yescharacteristic
219C2×SD16—1811no—
220C2×SD16—1811no—
221#131—3624no—
222C2×M16—1811no—
223C2×M16—1811no—
224C2×M16C2×D18111yescharacteristic
225C22×C8—1811no—
226C22×C8—1811no—
227C22×C8—1811no—
228C4⋊Q8—3621no—
229#124—3621no—
230#123—3621no—
231#123—3621no—
232C22⋊Q8—3621no—
233C22⋊Q8—1811no—
234C22⋊Q8—1811no—
235C22⋊Q8—1811no—
236C22⋊Q8—1811no—
237C22⋊Q8C2×D18111yescharacteristic
238C22⋊Q8C2×D18111yescharacteristic
239#121—3621no—
240#121—3621no—
241#121—3621no—
242#121—1811no—
243#121—1811no—
244#117—1811no—
245C2×C4⋊C4—1811no—
246#107—1811no—
247#107—1811no—
248#106—1811no—
249#106C2×D18111yescharacteristic
250#103—1811no—
251#103—1811no—
252#103—1811no—
253#103—1811no—
254#103C2×D18111yescharacteristic
255#103C2×D18111yescharacteristic
256#102—1811no—
257#102—1811no—
258#41—3622no—
259#41—3622no—
260#41—3622no—
261#41—3622no—
262C2×Q8—3621no—
263C2×Q8—3622no—
264C2×Q8—3622no—
265C2×Q8—1811no—
266C2×Q8—1811no—
267C2×Q8C22×D18111yescharacteristic
268C2×Q8C22×D18111yescharacteristic
269C2×D4—3621no—
270C2×D4—3621no—
271C2×D4—3621no—
272C2×D4—1811no—
273C2×D4—1811no—
274C2×D4—1811no—
275C2×D4—1811no—
276C22×C4—3621no—
277C22×C4—3621no—
278C22×C4—3621no—
279C22×C4—1811no—
280C22×C4—1811no—
281C22×C4C22×D18111yescharacteristic
282Q16—22no—
283Q16—3622no—
284SD16—3622no—
285SD16—3622no—
286M16—22no—
287M16—3622no—
288M16—1814no—
289C2×C8—3622no—
290C2×C8—3622no—
291C2×C8—3622no—
292C2×C8—1811no—
293C2×C8—1811no—
294C2×C8C22×D18111yescharacteristic
295C2×C8C22×D18111yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—3621no—
299C4⋊C4—3621no—
300C4⋊C4—3621no—
301C4⋊C4—3621no—
302C4⋊C4—3621no—
303C4⋊C4—1811no—
304C4⋊C4—1811no—
305C4⋊C4C22×D18111yescharacteristic
306C4⋊C4C22×D18111yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—3621no—
310C22⋊C4—3621no—
311C22⋊C4—3621no—
312C22⋊C4—3621no—
313C22⋊C4—3621no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C42—3621no—
317C23—3621no—
318C23—1811no—
319C23—1811no—
320C23D4×D18111yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—3622no—
326Q8—3621no—
327Q8—3621no—
328Q8—7242no—
329D4—3622no—
330D4—3622no—
331D4—3622no—
332D4—3621no—
333D4—3621no—
334D4—14481no—
335D4—14481no—
336D4—7242no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3621no—
344C2×C4—3621no—
345C2×C4—3621no—
346C2×C4—3621no—
347C2×C4—3621no—
348C2×C4—14481no—
349C2×C4—1811no—
350C2×C4—1811no—
351C2×C4—1812no—
352C2×C4—1812no—
353C2×C4—7242no—
354C2×C4—7242no—
355C2×C4D4×D18111yescharacteristic
356C2×C4C23×D18111yescharacteristic
357C8—21no—
358C8—3621no—
359C8—1812no—
360C8D4×D18112yes—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—3621no—
365C22—3621no—
366C22—3621no—
367C22—3621no—
368C22—3621no—
369C22—1812no—
370C22—7241no—
371C22—7241no—
372C22—7241no—
373C22C2×D4×D18111yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—3621no—
380C4—1812no—
381C4—7241no—
382C4—7241no—
383C4—7241no—
384C4C2×D4×D18111yescharacteristic
385C4#41×D18111yescharacteristic
386C2—41no—
387C2—3622no—
388C2—14481no—
389C2—7241no—
390C2#121×D18111yescharacteristic
391C2#2604208811yescharacteristic
392C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#470×C181C2
socleC22×C181C22×D4
prime radical#26041859C2
hypercenter#470D181
hypocenterC181#3063
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#19153848C2semidirect
2. C2×#19153824C2semidirect
4. #26041852C2semidirect
5. #26041536C2semidirect
6. #26041528C2semidirect
7. #26041525C2semidirect
8. #26041518C2semidirect
9. #26041517C2semidirect
10. #26041510C2semidirect
11. #26041509C2semidirect
12. #26041318C2semidirect
13. #26041274C2semidirect
14. #103×D181C2semidirect
15. #470×C181C2semidirect
86. #15668432D4semidirect
196. C181#3063semidirect
203. #470D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538482
C2×#191538242
#260418592
#260418522
#260415362
#260415282
#260415252
#260415182
#260415172
#260415102
#260415092
#260413182
#260412742
#103×D1812
#470×C1812
#3063181