Finite Group Databasebeta

← #37546312

Subgroups — #37546312

Order 23168.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041992C211yesmaximal, characteristic, complemented
2#26041930C211yesmaximal, characteristic, complemented
3#26041676C211yesmaximal, characteristic, complemented
4#26041606C211yesmaximal, characteristic, complemented
5#26041603C211yesmaximal, characteristic, complemented
6#26041599C211yesmaximal, characteristic, complemented
7#26041597C211yesmaximal, characteristic, complemented
8#26041568C211yesmaximal, characteristic, complemented
9#26041558C211yesmaximal, characteristic, complemented
10#26041533C211yesmaximal, characteristic, complemented
11#26041532C211yesmaximal, characteristic, complemented
12#26041361C211yesmaximal, characteristic, complemented
13#26041355C211yesmaximal, characteristic, complemented
14#26041320C211yesmaximal, characteristic, complemented
15#428×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17C2×D1448C2211yescharacteristic
18C2×#15668432C2211yescharacteristic
19#19153815C2211yescharacteristic
20#19153815C2211yescharacteristic
21#19153804—21no—
22#19153802—21no—
23#19153801—21no—
24#19153799—21no—
25#19153794—21no—
26#19153791C2211yescharacteristic
27#19153791C2211yescharacteristic
28#19153790C2211yescharacteristic
29#19153790C2211yescharacteristic
30#19153789—21no—
31#19153778C2211yescharacteristic
32#19153778C2211yescharacteristic
33C4×C22⋊D181C2211yescharacteristic
34#19153774C2211yescharacteristic
35#19153773—21no—
36C4⋊C4×D181C2211yescharacteristic
37#19153749C2211yescharacteristic
38#19153748C2211yescharacteristic
39#19153747C2211yescharacteristic
40#19153747C2211yescharacteristic
41#19153747C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153738C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153736C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×M16×C181C2211yescharacteristic
53#117×C181C2211yescharacteristic
54C2×C4⋊C4×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#107×C181C2211yescharacteristic
57#106×C181C2211yescharacteristic
58#106×C181C2211yescharacteristic
59#15668449—44no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×D724—21no—
63C2×D724—21no—
64C2×D724C2311yescharacteristic
65C2×D724C2311yescharacteristic
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181D411yescharacteristic
69C2×C4×D181—21no—
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72C2×C4×D181C2311yescharacteristic
73D1448—22no—
74#15668432—22no—
75C2×#13089678C2311yescharacteristic
76C2×#13089678C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428D411yescharacteristic
83C22⋊Dic181D411yescharacteristic
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426D411yescharacteristic
92C4×Dic181—21no—
93C4×Dic181C2311yescharacteristic
94C22×C4×C181—21no—
95C22×C4×C181C2311yescharacteristic
96M16×C181—22no—
97C2×C8×C181C2311yescharacteristic
98C2×C8×C181C2311yescharacteristic
99C4⋊C4×C181—21no—
100C4⋊C4×C181C2311yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C22⋊C4×C181—21no—
105C42×C181—21no—
106C22×D181—21no—
107C22×D181—21no—
108C22×D181C2×D411yescharacteristic
109Dic362—42no—
110Dic362—81no—
111D724—21no—
112D724—21no—
113D724—21no—
114D724—21no—
115D724—42no—
116D724—81no—
117C22⋊D181—22no—
118C22⋊D181—42no—
119C22⋊D181—81no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181C2×D411yescharacteristic
123C2×Dic181C2×D411yescharacteristic
124C2×Dic181C2×D411yescharacteristic
125C4×D181—21no—
126C4×D181—21no—
127C4×D181—42no—
128C4×D181—42no—
129C4×D181—42no—
130#13089678—21no—
131#13089678—21no—
132C23×C181C2×D411yescharacteristic
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181—42no—
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2411yescharacteristic
142C8×C181—21no—
143C8×C181#4112yescomplemented
144C22×C181—21no—
145C22×C181—21no—
146C22×C181—21no—
147C22×C181C22×D411yescharacteristic
148C2×D181—21no—
149C2×D181—21no—
150C2×D181—21no—
151C2×D181—41no—
152C2×D181—41no—
153C2×D181—41no—
154C2×D181—41no—
155C2×D181—41no—
156C2×D181—41no—
157C4×C181—21no—
158C4×C181—41no—
159C4×C181—41no—
160C4×C181—41no—
161C4×C181—41no—
162C4×C181C2×#4111yescharacteristic
163C4×C1812−1+411yescharacteristic
164Dic181—21no—
165Dic181—21no—
166Dic181—41no—
167Dic181—41no—
168Dic181AGL(1,ℤ/8)12yes—
169C2×C181—41no—
170C2×C181#54711yescharacteristic
171C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C181#57711yescharacteristic
173D181—41no—
174D181—81no—
175D181—81no—
176C181#335411yescharacteristic, complemented, Sylow
177#3354—1811nomaximal, Sylow
178#555—1811no—
179#547—1811no—
180#498—1811no—
181#482—1811no—
182#482—1811no—
183#481—1811no—
184#481—1811no—
185#476—1811no—
186#474—1811no—
187#469—1811no—
188#469—1811no—
189C4×D8—1811no—
190#436—1811no—
191#428—1811no—
192#428D18111yescharacteristic, complemented
193C2×#41—3621no—
194C2×D8—1811no—
195C2×M16—1811no—
196C2×M16C2×D18111yescharacteristic
197C4⋊Q8—1811no—
198C4⋊Q8—1811no—
199#124—3621no—
200#124—3621no—
201#123—3621no—
202#123—3621no—
203C22⋊Q8—3621no—
204#121—3621no—
205#121—1811no—
206#121—1811no—
207#121—1811no—
208#121—1811no—
209C4×D4—1811no—
210C4×D4—1811no—
211C4×D4—1811no—
212#117—3621no—
213#117—1811no—
214#117C2×D18111yescharacteristic
215C2×C4⋊C4—1811no—
216C2×C4⋊C4C2×D18111yescharacteristic
217#107—1811no—
218#107C2×D18111yescharacteristic
219#107C2×D18111yescharacteristic
220#106—1811no—
221#106—1811no—
222#106—1811no—
223#106—1811no—
224#106C2×D18111yescharacteristic
225#106C2×D18111yescharacteristic
226C4⋊C8—1811no—
227#102—1811no—
228#102—1811no—
229#102—1811no—
230#102—1811no—
231#102—1811no—
232#102—1811no—
233C22⋊C8—1811no—
234C22⋊C8—1811no—
235C4×C8—1811no—
236#41—7244no—
237C2×Q8—3621no—
238C2×Q8—3621no—
239C2×D4—3621no—
240C2×D4—3621no—
241C2×D4—3621no—
242C2×D4—3621no—
243C2×D4—1811no—
244C2×D4—1811no—
245C2×D4—1811no—
246C22×C4—21no—
247C22×C4—3621no—
248C22×C4—3621no—
249C22×C4—3621no—
250C22×C4—1811no—
251C22×C4C22×D18111yescharacteristic
252D8—3622no—
253M16—22no—
254M16—3622no—
255C2×C8—1811no—
256C2×C8—1811no—
257C2×C8C22×D18111yescharacteristic
258C2×C8C22×D18111yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—3621no—
261C4⋊C4—3621no—
262C4⋊C4—3621no—
263C4⋊C4—1811no—
264C4⋊C4—1811no—
265C4⋊C4—1811no—
266C4⋊C4C22×D18111yescharacteristic
267C4⋊C4C22×D18111yescharacteristic
268C4⋊C4C22×D18111yescharacteristic
269C4⋊C4C22×D18111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3621no—
272C22⋊C4—3621no—
273C22⋊C4—3621no—
274C22⋊C4—3621no—
275C22⋊C4—3621no—
276C22⋊C4—3621no—
277C22⋊C4—3621no—
278C22⋊C4—1811no—
279C22⋊C4—1811no—
280C42—21no—
281C42—3621no—
282C42—1811no—
283C23—3621no—
284C23—3621no—
285C23—1811no—
286C23D4×D18111yescharacteristic
287Q8—14481no—
288Q8—7242no—
289D4—3622no—
290D4—3621no—
291D4—3621no—
292D4—3621no—
293D4—3621no—
294D4—14481no—
295D4—14481no—
296D4—7242no—
297D4—7242no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—3621no—
306C2×C4—3621no—
307C2×C4—3621no—
308C2×C4—3621no—
309C2×C4—1811no—
310C2×C4—1811no—
311C2×C4—1811no—
312C2×C4—7242no—
313C2×C4—7242no—
314C2×C4—7242no—
315C2×C4D4×D18111yescharacteristic
316C2×C4C23×D18111yescharacteristic
317C8—21no—
318C8—3621no—
319C8—3621no—
320C8#1566844912yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—3621no—
325C22—3621no—
326C22—3621no—
327C22—7241no—
328C22—7241no—
329C22—7241no—
330C22—7241no—
331C22—7241no—
332C22—7241no—
333C22C2×D4×D18111yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3621no—
340C4—3621no—
341C4—1812no—
342C4—7241no—
343C4—7241no—
344C4C2×#1566844911yescharacteristic
345C4#1915386411yescharacteristic
346C2—41no—
347C2—14481no—
348C2—14481no—
349C2—7241no—
350C2#2604193011yescharacteristic
351C2AGL(1,ℤ/8)×D18111yescharacteristic
352C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#428×C181C2
socleC22×C181C22×D4
hypercenter#428D181
hypocenterC181#3354
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #26041992C2semidirect
2. #26041930C2semidirect
3. #26041676C2semidirect
4. #26041606C2semidirect
5. #26041603C2semidirect
6. #26041599C2semidirect
7. #26041597C2semidirect
8. #26041568C2semidirect
9. #26041558C2semidirect
10. #26041533C2semidirect
11. #26041532C2semidirect
12. #26041361C2semidirect
13. #26041355C2semidirect
14. #26041320C2semidirect
15. #428×C181C2semidirect
143. C8×C181#41semidirect
176. C181#3354semidirect
192. #428D181semidirect
320. C8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260419922
#260419302
#260416762
#260416062
#260416032
#260415992
#260415972
#260415682
#260415582
#260415332
#260415322
#260413612
#260413552
#260413202
#428×C1812
#3354181