Finite Group Databasebeta

← #37545134

Subgroups — #37545134

Order 23168.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041900C211yesmaximal, characteristic, complemented
2#26041884C211yesmaximal, characteristic, complemented
3#26041587C211yesmaximal, characteristic, complemented
4#26041580C211yesmaximal, characteristic, complemented
5#26041570C211yesmaximal, characteristic, complemented
6#26041560C211yesmaximal, characteristic, complemented
7#26041532C211yesmaximal, characteristic, complemented
8#26041525C211yesmaximal, characteristic, complemented
9#26041515C211yesmaximal, characteristic, complemented
10#26041508C211yesmaximal, characteristic, complemented
11#26041495C211yesmaximal, characteristic, complemented
12#26041495C211yesmaximal, characteristic, complemented
13#26041480C211yesmaximal, characteristic, complemented
14#26041475C211yesmaximal, characteristic, complemented
15#424×C181C211yesmaximal, characteristic, complemented
16C2×#15668438C2211yescharacteristic
17C2×#15668438C2211yescharacteristic
18C2×D1448C2211yescharacteristic
19C2×D1448C2211yescharacteristic
20#19153816C2211yescharacteristic
21#19153813C2211yescharacteristic
22#19153811—21no—
23#19153808C2211yescharacteristic
24#19153804C2211yescharacteristic
25#19153802—21no—
26#19153801—21no—
27#19153801—21no—
28#19153798—21no—
29#19153795C2211yescharacteristic
30#19153794—21no—
31#19153790C2211yescharacteristic
32#19153790C2211yescharacteristic
33#19153790C2211yescharacteristic
34#19153789—21no—
35#19153789—21no—
36C4×Dic362C2211yescharacteristic
37C4×D724C2211yescharacteristic
38C4×D724C2211yescharacteristic
39C4×D724C2211yescharacteristic
40#19153750C2211yescharacteristic
41#19153750C2211yescharacteristic
42#19153748C2211yescharacteristic
43#19153748C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153738C2211yescharacteristic
49#19153738C2211yescharacteristic
50#19153738C2211yescharacteristic
51#19153738C2211yescharacteristic
52C2×M16×C181C2211yescharacteristic
53C22×C8×C181C2211yescharacteristic
54#117×C181C2211yescharacteristic
55C4⋊C8×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57C4⋊C8×C181C2211yescharacteristic
58C4⋊C8×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C2×D724—21no—
62C2×D724—21no—
63C2×D724C2311yescharacteristic
64C2×D724C2311yescharacteristic
65C2×D724C2311yescharacteristic
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C4×D181—21no—
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72#15668438—41no—
73#15668438—41no—
74D1448—41no—
75D1448—41no—
76C4⋊Dic181—21no—
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85C22⋊Dic181—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94C4×Dic181—21no—
95C22×C4×C181C2311yescharacteristic
96M16×C181—41no—
97C2×C8×C181—41no—
98C2×C8×C181C2311yescharacteristic
99C2×C8×C181C2311yescharacteristic
100C2×C8×C181C2311yescharacteristic
101C2×C8×C181C2311yescharacteristic
102C4⋊C4×C181D411yescharacteristic
103C4⋊C4×C181D411yescharacteristic
104C22⋊C4×C181D411yescharacteristic
105C22⋊C4×C181D411yescharacteristic
106C42×C181C2311yescharacteristic
107C42×C181C2311yescharacteristic
108C22×D181—21no—
109C22×D181—21no—
110C22×D181—21no—
111Dic362—21no—
112Dic362—21no—
113Dic362—81no—
114D724—21no—
115D724—21no—
116D724—21no—
117D724—21no—
118D724—21no—
119D724—21no—
120D724—81no—
121D724—81no—
122C22⋊D181—81no—
123C22⋊D181—81no—
124C22⋊D181—81no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C2×Dic181—21no—
128C2×Dic181—21no—
129C2×Dic181—21no—
130C4×D181—81no—
131C4×D181—81no—
132C4×D181—81no—
133C23×C181C2×D411yescharacteristic
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2×D411yescharacteristic
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2411yescharacteristic
142C8×C181—21no—
143C8×C181—21no—
144C8×C181—21no—
145C8×C181—21no—
146C22×C181—21no—
147C22×C181—21no—
148C22×C181—21no—
149C22×C181C22×D411yescharacteristic
150C2×D181—41no—
151C2×D181—41no—
152C2×D181—41no—
153C2×D181—41no—
154C2×D181—41no—
155C2×D181—41no—
156C2×D181—41no—
157C2×D181—41no—
158C2×D181—41no—
159C4×C181—21no—
160C4×C181—21no—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—21no—
164C4×C181H(5,𝔽2)11yescharacteristic
165C4×C1812−1+411yescharacteristic
166Dic181—41no—
167Dic181—41no—
168Dic181—41no—
169Dic181—41no—
170Dic181—41no—
171C2×C181—41no—
172C2×C181#53711yescharacteristic
173C2×C181#57611yescharacteristic
174C2×C181#57711yescharacteristic
175D181—81no—
176D181—81no—
177D181—81no—
178C181#325711yescharacteristic, complemented, Sylow
179#3257—1811nomaximal, Sylow
180#542—1811no—
181#540—1811no—
182#479—1811no—
183#478—1811no—
184#476—1811no—
185#474—1811no—
186#469—1811no—
187#468—1811no—
188#466—1811no—
189#465—1811no—
190#463—1811no—
191#463—1811no—
192#460—1811no—
193#459—1811no—
194#424D18111yescharacteristic, complemented
195C2×SD16—1811no—
196C2×SD16—1811no—
197C2×D8—1811no—
198C2×D8—1811no—
199C2×M16C2×D18111yescharacteristic
200C22×C8C2×D18111yescharacteristic
201C4⋊Q8—1811no—
202#127—1811no—
203#126—3621no—
204#125—1811no—
205#124—3621no—
206#124—1811no—
207#123—3621no—
208#123—3621no—
209#123—3621no—
210C22⋊Q8—3621no—
211C22⋊Q8—1811no—
212#121—3621no—
213#121—3621no—
214#121—1811no—
215#121—1811no—
216#121—1811no—
217C4×Q8—1811no—
218C4×D4—1811no—
219C4×D4—1811no—
220C4×D4—1811no—
221#117C2×D18111yescharacteristic
222#107—1811no—
223#107—1811no—
224#106—1811no—
225#106—1811no—
226C4⋊C8C2×D18111yescharacteristic
227C4⋊C8C2×D18111yescharacteristic
228C4⋊C8C2×D18111yescharacteristic
229C4⋊C8C2×D18111yescharacteristic
230#103—1811no—
231#103—1811no—
232#102—1811no—
233#102—1811no—
234#102—1811no—
235#102—1811no—
236#102—1811no—
237#102—1811no—
238C2×Q8—3621no—
239C2×Q8—1811no—
240C2×D4—3621no—
241C2×D4—3621no—
242C2×D4—3621no—
243C2×D4—3621no—
244C2×D4—3621no—
245C2×D4—1811no—
246C2×D4—1811no—
247C2×D4—1811no—
248C22×C4—3621no—
249C22×C4—3621no—
250C22×C4—3621no—
251C22×C4C22×D18111yescharacteristic
252SD16—7241no—
253SD16—7241no—
254D8—7241no—
255D8—7241no—
256M16—41no—
257C2×C8—41no—
258C2×C8C22×D18111yescharacteristic
259C2×C8C22×D18111yescharacteristic
260C2×C8C22×D18111yescharacteristic
261C2×C8C22×D18111yescharacteristic
262C4⋊C4—3621no—
263C4⋊C4—3621no—
264C4⋊C4—3621no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
267C4⋊C4—1811no—
268C4⋊C4—1811no—
269C4⋊C4—1811no—
270C4⋊C4—1811no—
271C4⋊C4D72411yescharacteristic
272C4⋊C4D72411yescharacteristic
273C22⋊C4—3621no—
274C22⋊C4—3621no—
275C22⋊C4—3621no—
276C22⋊C4—3621no—
277C22⋊C4—3621no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4D72411yescharacteristic
283C22⋊C4D72411yescharacteristic
284C42—3621no—
285C42C22×D18111yescharacteristic
286C42C22×D18111yescharacteristic
287C23—3621no—
288C23—3621no—
289C23—3621no—
290C23C2×D72411yescharacteristic
291Q8—3621no—
292Q8—3621no—
293Q8—14481no—
294D4—3621no—
295D4—3621no—
296D4—3621no—
297D4—3621no—
298D4—3621no—
299D4—3621no—
300D4—14481no—
301D4—14481no—
302D4—14481no—
303D4—14481no—
304D4—14481no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—3621no—
308C2×C4—3621no—
309C2×C4—3621no—
310C2×C4—3621no—
311C2×C4—3621no—
312C2×C4—14481no—
313C2×C4—14481no—
314C2×C4—14481no—
315C2×C4C2×D72411yescharacteristic
316C2×C4C2×D72411yescharacteristic
317C2×C4C2×D72411yescharacteristic
318C2×C4C2×D72411yescharacteristic
319C2×C4C2×D72411yescharacteristic
320C2×C4C23×D18111yescharacteristic
321C8—21no—
322C8—21no—
323C8—21no—
324C8—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—7241no—
329C22—7241no—
330C22—7241no—
331C22—7241no—
332C22—7241no—
333C22—7241no—
334C22—7241no—
335C22—7241no—
336C22—7241no—
337C22C22×D72411yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—7241no—
344C4—7241no—
345C4—7241no—
346C4—7241no—
347C4—7241no—
348C4#1915386111yescharacteristic
349C4#1915386411yescharacteristic
350C2—41no—
351C2—14481no—
352C2—14481no—
353C2—14481no—
354C2#2604186011yescharacteristic
355C2#2604208411yescharacteristic
356C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#424×C181C2
socleC22×C181C22×D4
hypercenter#424D181
hypocenterC181#3257
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #26041900C2semidirect
2. #26041884C2semidirect
3. #26041587C2semidirect
4. #26041580C2semidirect
5. #26041570C2semidirect
6. #26041560C2semidirect
7. #26041532C2semidirect
8. #26041525C2semidirect
9. #26041515C2semidirect
10. #26041508C2semidirect
11. #26041495C2semidirect
12. #26041495C2semidirect
13. #26041480C2semidirect
14. #26041475C2semidirect
15. #424×C181C2semidirect
178. C181#3257semidirect
194. #424D181semidirect

Maximal subgroups

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

subgroupindex
#260419002
#260418842
#260415872
#260415802
#260415702
#260415602
#260415322
#260415252
#260415152
#260415082
#260414952
#260414802
#260414752
#424×C1812
#3257181