Finite Group Databasebeta

← #37546676

Subgroups — #37546676

Order 23168.

348 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041990C211yesmaximal, characteristic, complemented
2#26041985C211yesmaximal, characteristic, complemented
3#26041687C211yesmaximal, characteristic, complemented
4#26041679C211yesmaximal, characteristic, complemented
5#26041582C211yesmaximal, characteristic, complemented
6#26041578C211yesmaximal, characteristic, complemented
7#26041578C211yesmaximal, characteristic, complemented
8#26041576C211yesmaximal, characteristic, complemented
9#26041558C211yesmaximal, characteristic, complemented
10#26041556C211yesmaximal, characteristic, complemented
11#26041407C211yesmaximal, characteristic, complemented
12#26041392C211yesmaximal, characteristic
13#26041390C211yesmaximal, characteristic
14#26041361C211yesmaximal, characteristic
15#501×C181C211yesmaximal, characteristic, complemented
16C2×D1448C2211yescharacteristic
17#19153815C2211yescharacteristic
18#19153807C2211yescharacteristic
19#19153807C2211yescharacteristic
20#19153807C2211yescharacteristic
21#19153796—21no—
22#19153796—21no—
23#19153794—21no—
24#19153794—21no—
25#19153792—21no—
26#19153792—21no—
27C4×Dic362C2211yescharacteristic
28C4×D724C2211yescharacteristic
29C4×D724C2211yescharacteristic
30#19153778C2211yescharacteristic
31#19153778C2211yescharacteristic
32#19153778C2211yescharacteristic
33#19153778C2211yescharacteristic
34C4⋊C4×D181—21no—
35C4⋊C4×D181—21no—
36#19153749C2211yescharacteristic
37#19153749C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153748C2211yescharacteristic
40#19153747C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic
53#125×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#106×C181C2211yescharacteristic
57#106×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362D411yescharacteristic
61C2×D724C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C4×D181—21no—
64C2×C4×D181—21no—
65C2×C4×D181—21no—
66C2×C4×D181—21no—
67C2×C4×D181—21no—
68C2×C4×D181—21no—
69D1448Q814yescomplemented
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C4⋊Dic181—21no—
73C4⋊Dic181—21no—
74C4⋊Dic181—21no—
75C4⋊Dic181D411yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428D411yescharacteristic
83#15668428D411yescharacteristic
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90C4×Dic181C2311yescharacteristic
91C4×Dic181C2311yescharacteristic
92C2×Q8×C181—21no—
93C2×C8×C181C2311yescharacteristic
94C2×C8×C181C2311yescharacteristic
95C4⋊C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181—21no—
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C42×C181C2311yescharacteristic
103C22×D181—21no—
104C22×D181—21no—
105Dic362—41no—
106Dic362—81no—
107D724—21no—
108D724—21no—
109D724C2×Q812yes—
110D724C2×Q812yes—
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181C2×D411yescharacteristic
114C2×Dic181C2×D411yescharacteristic
115C2×Dic181C2×D411yescharacteristic
116C2×Dic181C2×D411yescharacteristic
117C4×D181—42no—
118C4×D181—42no—
119C4×D181—81no—
120C4×D181—81no—
121C4×D181—81no—
122C4×D181—81no—
123#13089678—21no—
124#13089678—21no—
125Q8×C181—81no—
126C2×C4×C181—21no—
127C2×C4×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181C2×D411yescharacteristic
131C2×C4×C181C2×D411yescharacteristic
132C2×C4×C181C2411yescharacteristic
133C8×C181—21no—
134C8×C181C2×Q812yes—
135C22×C181C22×D411yescharacteristic
136C2×D181—22no—
137C2×D181—22no—
138C2×D181—41no—
139C2×D181—41no—
140C2×D181—41no—
141C2×D181—41no—
142C4×C181—21no—
143C4×C181—21no—
144C4×C181—41no—
145C4×C181—41no—
146C4×C181—41no—
147C4×C181—41no—
148C4×C181C22×Q811yescharacteristic
149C4×C1812−1+411yescharacteristic
150Dic181—21no—
151Dic181—21no—
152Dic181—21no—
153Dic181—21no—
154Dic181—41no—
155Dic181—41no—
156C2×C181D4×Q811yescharacteristic
157C2×C181#57511yescharacteristic
158C2×C181#57611yescharacteristic
159D181—42no—
160D181—42no—
161C181#338911yescharacteristic, complemented, Sylow
162#3389—1811nomaximal, Sylow
163#554—1811no—
164#554—1811no—
165#501—1811no—
166#501D18111yescharacteristic, complemented
167#499—1811no—
168#478—1811no—
169#478—1811no—
170#478—1811no—
171#478—1811no—
172#474—1811no—
173#474—1811no—
174#446—1811no—
175#442—1811no—
176#442—1811no—
177C4×D8—1811no—
178C2×D8—1811no—
179C4⋊Q8—1811no—
180C4⋊Q8C2×D18111yescharacteristic
181#125—1811no—
182#125—1811no—
183#125—1811no—
184#125C2×D18111yescharacteristic
185C22⋊Q8—3621no—
186C22⋊Q8—3621no—
187C22⋊Q8—3621no—
188C22⋊Q8—3621no—
189C22⋊Q8—3621no—
190C22⋊Q8—3621no—
191C4×Q8—1811no—
192C4×D4—1811no—
193C4×D4—1811no—
194C4×D4—1811no—
195C4×D4—1811no—
196C4×D4—1811no—
197C4×D4—1811no—
198C2×C4⋊C4—3621no—
199C2×C4⋊C4—3621no—
200#107—1811no—
201#107—1811no—
202#107—1811no—
203#107C2×D18111yescharacteristic
204#107C2×D18111yescharacteristic
205#106—1811no—
206#106—1811no—
207#106C2×D18111yescharacteristic
208#106C2×D18111yescharacteristic
209C4⋊C8—1811no—
210C4⋊C8—1811no—
211C4⋊C8—1811no—
212#102—1811no—
213#102—1811no—
214#102—1811no—
215#102—1811no—
216#102—1811no—
217#102—1811no—
218#97—1811no—
219#97C2×D18111yescharacteristic
220C4×C8—1811no—
221C2×Q8—21no—
222C2×Q8—3621no—
223C2×Q8—1811no—
224C2×D4—1811no—
225C2×D4—1811no—
226C22×C4—3621no—
227C22×C4—3621no—
228C22×C4—3621no—
229C22×C4—3621no—
230C22×C4—3621no—
231C22×C4—3621no—
232D8—1814no—
233C2×C8—1811no—
234C2×C8—1811no—
235C2×C8C22×D18111yescharacteristic
236C2×C8C22×D18111yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—3621no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—1811no—
248C4⋊C4—1811no—
249C4⋊C4—1811no—
250C4⋊C4—1811no—
251C4⋊C4—1811no—
252C4⋊C4C22×D18111yescharacteristic
253C4⋊C4C22×D18111yescharacteristic
254C4⋊C4C22×D18111yescharacteristic
255C4⋊C4C22×D18111yescharacteristic
256C22⋊C4—3621no—
257C22⋊C4—3621no—
258C22⋊C4—3621no—
259C22⋊C4—3621no—
260C22⋊C4—3621no—
261C22⋊C4—3621no—
262C42—1811no—
263C42—1811no—
264C42C22×D18111yescharacteristic
265C23—3621no—
266C23—3621no—
267Q8—81no—
268Q8—14481no—
269Q8—7241no—
270D4—3621no—
271D4—3621no—
272D4—1812no—
273D4—1812no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—3621no—
279C2×C4—3621no—
280C2×C4—14481no—
281C2×C4—14481no—
282C2×C4—14481no—
283C2×C4—14481no—
284C2×C4—1811no—
285C2×C4—1811no—
286C2×C4—1811no—
287C2×C4—1811no—
288C2×C4—7242no—
289C2×C4—7242no—
290C2×C4D4×D18111yescharacteristic
291C2×C4D4×D18111yescharacteristic
292C2×C4C23×D18111yescharacteristic
293C8—21no—
294C8—3621no—
295C8—3621no—
296C8Q8×D18112yes—
297C22—3622no—
298C22—3622no—
299C22—7241no—
300C22—7241no—
301C22—7241no—
302C22—7241no—
303C22C2×D4×D18111yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—3621no—
311C4—3621no—
312C4—3621no—
313C4—3621no—
314C4—7241no—
315C4—7241no—
316C4C2×Q8×D18111yescharacteristic
317C4#1915386411yescharacteristic
318C2—7242no—
319C2—7242no—
320C2#2604194411yescharacteristic
321C2#2604207811yescharacteristic
322C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#501×C181C2
socleC22×C181C22×D4
prime radicalC2×D1448C22
hypercenter#501D181
hypocenterC181#3389
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #26041990C2semidirect
2. #26041985C2semidirect
3. #26041687C2semidirect
4. #26041679C2semidirect
5. #26041582C2semidirect
6. #26041578C2semidirect
7. #26041578C2semidirect
8. #26041576C2semidirect
9. #26041558C2semidirect
10. #26041556C2semidirect
11. #26041407C2semidirect
15. #501×C181C2semidirect
69. D1448Q8semidirect
161. C181#3389semidirect
166. #501D181semidirect

Maximal subgroups

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

subgroupindex
#260419902
#260419852
#260416872
#260416792
#260415822
#260415782
#260415762
#260415582
#260415562
#260414072
#260413922
#260413902
#260413612
#501×C1812
#3389181