Finite Group Databasebeta

← #34058401

Subgroups — #34058401

Order 19072.

1124 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750336C214yesmaximal, complemented
2#23750328C214yesmaximal, complemented
3#23750313C214yesmaximal, complemented
4#23750278C214yesmaximal, complemented
5#23750224C214yesmaximal, complemented
6#23750206C211yesmaximal, characteristic, complemented
7#23750138C214yesmaximal, complemented
8#23750137C212yesmaximal, complemented
9#23750105C212yesmaximal, complemented
10C2×C4⋊C4×D149C211yesmaximal, characteristic, complemented
11#544×C149C211yesmaximal, characteristic, complemented
12C2×#14558737C2214yes—
13C2×#14558736C2212yes—
14C22×C4×D149C2211yescharacteristic
15C22×C4×D149C2212yes—
16#18011906C2218yescomplemented
17#18011902C2218yescomplemented
18#18011896C2218yes—
19#18011896C2218yescomplemented
20#18011895C2214yes—
21#18011891C2212yes—
22#18011891C2212yes—
23#18011891C2214yescomplemented
24#18011887C2214yes—
25#18011887C2218yescomplemented
26#18011886C2212yes—
27#18011886C2212yes—
28#18011886C2214yes—
29#18011886C2214yescomplemented
30#18011885C2214yes—
31#18011881C2214yes—
32C4×Dic298C2214yescomplemented
33C4×D596C2214yes—
34C4×D596C2214yes—
35C4×D596C2214yescomplemented
36#18011873C2218yes—
37#18011871C2218yes—
38#18011869C2214yes—
39#18011869C2214yescomplemented
40C2×C4⋊Dic149C2211yescharacteristic
41C2×#14558715C2212yes—
42C4⋊C4×D149C2214yes—
43C4⋊C4×D149C2214yes—
44C4⋊C4×D149C2214yescomplemented
45C4⋊Q8×C149C2214yescomplemented
46#125×C149C2214yescomplemented
47C22⋊Q8×C149C2214yes—
48#117×C149C2212yes—
49C2×C4⋊C4×C149C2211yescharacteristic
50C23×D149C2311yescharacteristic
51#14558737—48no—
52#14558736—28no—
53C2×Dic298C2312yes—
54C2×D596C2312yes—
55C2×D596C2314yes—
56C2×D596C2314yes—
57C2×D596C2314yes—
58C2×C22⋊D149C2314yes—
59C22×Dic149C2311yescharacteristic
60C22×Dic149C2312yes—
61C2×C4×D149—24no—
62C2×C4×D149—44no—
63C2×C4×D149C2314yes—
64C2×C4×D149C2312yes—
65C2×C4×D149C2312yes—
66C2×C4×D149C2314yes—
67C2×C4×D149C2314yes—
68C2×C4×D149C2318yes—
69C4⋊Dic149C2314yes—
70C4⋊Dic149C2314yes—
71C4⋊Dic149C2314yes—
72C4⋊Dic149C2314yes—
73#14558715C2318yes—
74#14558715C2318yes—
75#14558713C2314yes—
76#14558713C2318yes—
77#14558713C2318yes—
78#14558713C2314yes—
79#14558713C2314yes—
80#14558713C2314yes—
81C4×Dic149C2318yes—
82C2×Q8×C149C2314yes—
83C22×C4×C149C2312yes—
84C22×C4×C149C2311yescharacteristic
85C4⋊C4×C149C2314yes—
86C4⋊C4×C149C2314yes—
87C4⋊C4×C149C2312yes—
88C4⋊C4×C149C2312yes—
89C4⋊C4×C149C2318yes—
90C22⋊C4×C149C2314yes—
91C42×C149C2314yes—
92C22×D149—22no—
93C22×D149—22no—
94C22×D149—22no—
95C22×D149C2414yes—
96C22×D149C2411yescharacteristic
97C22×D149C2411yescharacteristic
98Dic298—24no—
99D596—28no—
100D596—28no—
101D596—24no—
102D596—44no—
103C22⋊D149—28no—
104C2×Dic149—22no—
105C2×Dic149—42no—
106C2×Dic149C2414yes—
107C2×Dic149C2412yes—
108C2×Dic149C2414yes—
109C4×D149—28no—
110C4×D149—24no—
111C4×D149—24no—
112C4×D149—28no—
113C4×D149—44no—
114C4×D149—48no—
115C23×C149C2411yescharacteristic
116Q8×C149—44no—
117C2×C4×C149—42no—
118C2×C4×C149#4114yescomplemented
119C2×C4×C149C2414yes—
120C2×C4×C149C2414yes—
121C2×C4×C149C2414yes—
122C2×C4×C149C2412yes—
123C22×C149—21no—
124C22×C149—21no—
125C22×C149C2×#4112yes—
126C22×C149C2511yescharacteristic
127C2×D149—28no—
128C2×D149—21no—
129C2×D149—21no—
130C2×D149—21no—
131C2×D149—21no—
132C2×D149—24no—
133C2×D149—24no—
134C2×D149—24no—
135C2×D149H(5,𝔽2)12yes—
136C2×D1492−1+412yes—
137C4×C149—24no—
138C4×C149—24no—
139C4×C149—24no—
140C4×C149C2×#4114yes—
141Dic149—22no—
142Dic149—24no—
143Dic149—24no—
144C2×C149—22no—
145C2×C149C22×#4111yescharacteristic
146C2×C149C2×H(5,𝔽2)11yescharacteristic
147C2×C149C2×2−1+411yescharacteristic
148D149—22no—
149D149—22no—
150D149—44no—
151C149#349211yescharacteristic, complemented, Sylow
152#3492—1491nomaximal, Sylow
153#554—1494no—
154#553—1494no—
155#552—1494no—
156#547—1494no—
157#544D14911yescharacteristic, complemented
158#539—1494no—
159#537—1491no—
160#525—1494no—
161#525—1492no—
162#520—1492no—
163C22×C4⋊C4—1491no—
164C2×#41—1494no—
165C2×#41—1492no—
166C23×C4—1491no—
167C23×C4—1492no—
168C4⋊Q8C2×D14914yescomplemented
169#126—1498no—
170#125—1498no—
171#125C2×D14914yescomplemented
172#123—1498no—
173#123—1498no—
174#123—1494no—
175C22⋊Q8—1494no—
176C22⋊Q8—1498no—
177C22⋊Q8—1492no—
178C22⋊Q8—1492no—
179C22⋊Q8—1494no—
180C22⋊Q8C2×D14914yes—
181#121—1494no—
182#121—1492no—
183#121—1494no—
184#121—1492no—
185#121—1494no—
186#120—1494no—
187C4×Q8—1494no—
188C4×D4—1498no—
189C4×D4—1498no—
190C4×D4—1494no—
191C4×D4—1494no—
192C4×D4—1494no—
193#117—1494no—
194#117—1494no—
195#117C2×D14912yes—
196C2×C4⋊C4—1491no—
197C2×C4⋊C4—1492no—
198C2×C4⋊C4—1494no—
199C2×C4⋊C4—1494no—
200C2×C4⋊C4—1494no—
201C2×C4⋊C4C2×D14911yescharacteristic
202C24—1491no—
203#41—2988no—
204#41—5968no—
205C2×Q8—1492no—
206C2×Q8C22×D14914yes—
207C2×D4—1494no—
208C2×D4—1494no—
209C2×D4—1492no—
210C2×D4—1494no—
211C2×D4—1494no—
212C22×C4—2984no—
213C22×C4—5964no—
214C22×C4—1491no—
215C22×C4—1494no—
216C22×C4—1492no—
217C22×C4—1492no—
218C22×C4—1494no—
219C22×C4—1492no—
220C22×C4—1494no—
221C22×C4—1498no—
222C22×C4C22×D14912yes—
223C22×C4C22×D14911yescharacteristic
224C4⋊C4—1494no—
225C4⋊C4—1498no—
226C4⋊C4—1498no—
227C4⋊C4—1494no—
228C4⋊C4—1494no—
229C4⋊C4—1494no—
230C4⋊C4C22×D14914yes—
231C4⋊C4C22×D14912yes—
232C4⋊C4C22×D14912yes—
233C4⋊C4C22×D14914yes—
234C4⋊C4C22×D14918yes—
235C22⋊C4—1494no—
236C22⋊C4—1498no—
237C22⋊C4—1494no—
238C22⋊C4—1498no—
239C22⋊C4—1494no—
240C22⋊C4—1494no—
241C22⋊C4C22×D14914yes—
242C42—1498no—
243C42C22×D14914yes—
244C23—2982no—
245C23—2982no—
246C23—2982no—
247C23—1494no—
248C23—1491no—
249C23—1491no—
250C23C23×D14911yescharacteristic
251Q8—44no—
252Q8—2984no—
253D4—2988no—
254D4—2988no—
255D4—2988no—
256D4—2984no—
257D4—5964no—
258C2×C4—42no—
259C2×C4—2988no—
260C2×C4—2982no—
261C2×C4—2984no—
262C2×C4—2984no—
263C2×C4—2988no—
264C2×C4—5964no—
265C2×C4—5968no—
266C2×C4—5962no—
267C2×C4—1494no—
268C2×C4—1494no—
269C2×C4—1492no—
270C2×C4#1455873714yescomplemented
271C2×C4C23×D14914yes—
272C2×C4C23×D14914yes—
273C2×C4C23×D14914yes—
274C2×C4C23×D14912yes—
275C22—21no—
276C22—21no—
277C22—2988no—
278C22—2981no—
279C22—2981no—
280C22—2981no—
281C22—2981no—
282C22—2984no—
283C22—2984no—
284C22—2984no—
285C22—1492no—
286C22—1492no—
287C22C2×#1455873712yes—
288C22C24×D14911yescharacteristic
289C4—24no—
290C4—24no—
291C4—24no—
292C4—2982no—
293C4—2984no—
294C4—2984no—
295C4C2×#1455873714yes—
296C2—22no—
297C2—2982no—
298C2—2982no—
299C2—5964no—
300C2C22×#1455873711yescharacteristic
301C2H(5,𝔽2)×D14911yescharacteristic
302C22−1+4×D14911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#544×C149C2
hypercenter#544D149
hypocenterC149#3492
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #544

Complements

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

NH ≅ G/Ntype
1. #23750336C2semidirect
2. #23750328C2semidirect
3. #23750313C2semidirect
4. #23750278C2semidirect
5. #23750224C2semidirect
6. #23750206C2semidirect
7. #23750138C2semidirect
8. #23750137C2semidirect
9. #23750105C2semidirect
10. C2×C4⋊C4×D149C2semidirect
11. #544×C149C2semidirect
16. #18011906C22semidirect
17. #18011902C22semidirect
19. #18011896C22semidirect
23. #18011891C22semidirect
25. #18011887C22semidirect
29. #18011886C22semidirect
32. C4×Dic298C22semidirect
35. C4×D596C22semidirect
39. #18011869C22semidirect
44. C4⋊C4×D149C22semidirect
45. C4⋊Q8×C149C22semidirect
46. #125×C149C22semidirect
118. C2×C4×C149#41semidirect
151. C149#3492semidirect
157. #544D149semidirect
168. C4⋊Q8C2×D149semidirect
171. #125C2×D149semidirect
270. C2×C4#14558737semidirect

Maximal subgroups

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

subgroupindex
#237503362
#237503282
#237503132
#237502782
#237502242
#237502062
#237501382
#237501372
#237501052
C2×C4⋊C4×D1492
#544×C1492
#3492149