Finite Group Databasebeta

← #34058398

Subgroups — #34058398

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#23750333C214yesmaximal, complemented
2#23750322C214yesmaximal, complemented
3#23750309C214yesmaximal, complemented
4#23750275C214yesmaximal, complemented
5#23750256C211yesmaximal, characteristic, complemented
6#23750220C214yesmaximal, complemented
7#23750135C214yesmaximal, complemented
8#23750132C212yesmaximal, complemented
9#23750108C212yesmaximal, complemented
10C2×C4⋊C4×D149C211yesmaximal, characteristic, complemented
11#537×C149C211yesmaximal, characteristic, complemented
12C2×#14558735C2214yes—
13C22×C4×D149C2211yescharacteristic
14C22×C4×D149C2212yes—
15#18011910C2214yescomplemented
16#18011904C2218yescomplemented
17#18011902C2214yescomplemented
18#18011901C2218yescomplemented
19#18011893C2218yes—
20#18011892C2214yes—
21#18011892C2218yescomplemented
22#18011891C2214yes—
23#18011890C2214yes—
24#18011888C2218yescomplemented
25#18011887C2212yes—
26#18011887C2212yes—
27#18011882C2212yes—
28#18011882C2212yes—
29#18011882C2214yes—
30#18011879C2214yes—
31Q8×Dic149C2214yescomplemented
32D4×Dic149C2218yes—
33D4×Dic149C2214yescomplemented
34C4×C22⋊D149C2214yes—
35C4×C22⋊D149C2214yes—
36#18011871C2218yes—
37#18011870C2212yes—
38#18011870C2214yescomplemented
39#18011869C2214yes—
40C2×C4⋊Dic149C2211yescharacteristic
41C2×C4⋊Dic149C2214yescomplemented
42C2×#14558715C2212yes—
43C4⋊C4×D149C2214yes—
44C4⋊C4×D149C2214yes—
45C2×#41×C149C2212yes—
46C22⋊Q8×C149C2214yescomplemented
47#121×C149C2214yes—
48#121×C149C2214yescomplemented
49C2×C4⋊C4×C149C2211yescharacteristic
50C23×D149C2311yescharacteristic
51#14558735—48no—
52C2×Dic298C2314yes—
53C2×C22⋊D149C2314yes—
54C2×C22⋊D149C2314yes—
55C22×Dic149C2311yescharacteristic
56C22×Dic149C2312yes—
57C22×Dic149C2318yes—
58C2×C4×D149—24no—
59C2×C4×D149—44no—
60C2×C4×D149C2314yes—
61C2×C4×D149C2312yes—
62C2×C4×D149C2312yes—
63C2×C4×D149C2314yes—
64C4⋊Dic149C2314yes—
65C4⋊Dic149C2312yes—
66C4⋊Dic149C2314yes—
67C4⋊Dic149C2318yes—
68C4⋊Dic149C2312yes—
69#14558715C2314yes—
70#14558715C2318yes—
71#14558715C2318yes—
72#14558715C2314yes—
73C22⋊Dic149C2314yes—
74C22⋊Dic149C2318yes—
75C22⋊Dic149C2314yes—
76C22⋊Dic149C2314yes—
77#14558713C2314yes—
78#14558713C2314yes—
79C4×Dic149C2318yes—
80C4×Dic149C2314yes—
81#41×C149—28no—
82C2×Q8×C149C2312yes—
83C2×D4×C149C2314yes—
84C2×D4×C149C2312yes—
85C2×D4×C149C2314yes—
86C22×C4×C149C2314yes—
87C22×C4×C149C2312yes—
88C22×C4×C149C2311yescharacteristic
89C4⋊C4×C149C2314yes—
90C4⋊C4×C149C2314yes—
91C22⋊C4×C149C2318yes—
92C22×D149—22no—
93C22×D149—22no—
94C22×D149—22no—
95C22×D149C2411yescharacteristic
96C22×D149C2411yescharacteristic
97Dic298—44no—
98C22⋊D149—28no—
99C22⋊D149—28no—
100C2×Dic149—22no—
101C2×Dic149—42no—
102C2×Dic149—48no—
103C2×Dic149C2414yes—
104C2×Dic149C2412yes—
105C2×Dic149C2414yes—
106C2×Dic149C2414yes—
107C4×D149—28no—
108C4×D149—24no—
109C4×D149—24no—
110C4×D149—44no—
111C23×C149C2414yes—
112C23×C149C2411yescharacteristic
113Q8×C149—24no—
114D4×C149—28no—
115D4×C149—24no—
116D4×C149—44no—
117C2×C4×C149—28no—
118C2×C4×C149—42no—
119C2×C4×C149#4114yescomplemented
120C2×C4×C149C2414yes—
121C2×C4×C149C2414yes—
122C2×C4×C149C2412yes—
123C22×C149—21no—
124C22×C149—21no—
125C22×C149—24no—
126C22×C149—24no—
127C22×C149—24no—
128C22×C149C2×#4112yes—
129C22×C149C2511yescharacteristic
130C2×D149—28no—
131C2×D149—21no—
132C2×D149—21no—
133C2×D149—21no—
134C2×D149—21no—
135C2×D149H(5,𝔽2)12yes—
136C2×D1492−1+412yes—
137C4×C149—24no—
138C4×C149—24no—
139C4×C149C2×#4114yes—
140Dic149—22no—
141Dic149—24no—
142Dic149—24no—
143Dic149—24no—
144C2×C149—22no—
145C2×C149—44no—
146C2×C149C22×#4111yescharacteristic
147C2×C149C2×H(5,𝔽2)11yescharacteristic
148C2×C149C2×2−1+411yescharacteristic
149D149—22no—
150D149—22no—
151C149#349211yescharacteristic, complemented, Sylow
152#3492—1491nomaximal, Sylow
153#554—1494no—
154#553—1494no—
155#552—1494no—
156#547—1494no—
157#544—1491no—
158#539—1494no—
159#537D14911yescharacteristic, complemented
160#525—1492no—
161#525—1494no—
162#520—1492no—
163C22×C4⋊C4—1491no—
164C2×#41—1494no—
165C2×#41C2×D14912yes—
166C23×C4—1491no—
167C23×C4—1492no—
168C4⋊Q8—1494no—
169#126—1498no—
170#125—1498no—
171#125—1494no—
172#123—1494no—
173#123—1498no—
174#123—1498no—
175C22⋊Q8—1492no—
176C22⋊Q8—1492no—
177C22⋊Q8—1494no—
178C22⋊Q8—1494no—
179C22⋊Q8—1498no—
180C22⋊Q8C2×D14914yescomplemented
181#121—1492no—
182#121—1494no—
183#121—1492no—
184#121C2×D14914yes—
185#121C2×D14914yescomplemented
186#120—1494no—
187C4×Q8—1494no—
188C4×D4—1498no—
189C4×D4—1494no—
190C4×D4—1494no—
191C4×D4—1494no—
192C4×D4—1498no—
193#117—1492no—
194#117—1494no—
195#117—1494no—
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—28no—
204#41—5968no—
205C2×Q8—1494no—
206C2×Q8C22×D14912yes—
207C2×D4—1494no—
208C2×D4—1494no—
209C2×D4C22×D14914yes—
210C2×D4C22×D14912yes—
211C2×D4C22×D14914yes—
212C22×C4—2984no—
213C22×C4—5964no—
214C22×C4—1491no—
215C22×C4—1494no—
216C22×C4—1492no—
217C22×C4—1492no—
218C22×C4—1492no—
219C22×C4—1498no—
220C22×C4—1494no—
221C22×C4C22×D14914yes—
222C22×C4C22×D14912yes—
223C22×C4C22×D14911yescharacteristic
224C4⋊C4—1494no—
225C4⋊C4—1498no—
226C4⋊C4—1492no—
227C4⋊C4—1498no—
228C4⋊C4—1492no—
229C4⋊C4—1494no—
230C4⋊C4—1494no—
231C4⋊C4—1498no—
232C4⋊C4—1494no—
233C4⋊C4C22×D14914yes—
234C4⋊C4C22×D14914yes—
235C22⋊C4—1494no—
236C22⋊C4—1498no—
237C22⋊C4—1494no—
238C22⋊C4—1494no—
239C22⋊C4—1494no—
240C22⋊C4—1494no—
241C22⋊C4C22×D14918yes—
242C42—1498no—
243C42—1494no—
244C23—2982no—
245C23—2982no—
246C23—2982no—
247C23—1491no—
248C23—1491no—
249C23C23×D14914yes—
250C23C23×D14911yescharacteristic
251Q8—24no—
252Q8—5964no—
253D4—28no—
254D4—24no—
255D4—44no—
256D4—2988no—
257D4—2988no—
258C2×C4—28no—
259C2×C4—42no—
260C2×C4—2988no—
261C2×C4—2982no—
262C2×C4—2984no—
263C2×C4—2984no—
264C2×C4—5964no—
265C2×C4—5962no—
266C2×C4—5968no—
267C2×C4—1494no—
268C2×C4—1494no—
269C2×C4—1494no—
270C2×C4—1492no—
271C2×C4#1455873514yescomplemented
272C2×C4C23×D14914yes—
273C2×C4C23×D14914yes—
274C2×C4C23×D14912yes—
275C22—21no—
276C22—21no—
277C22—24no—
278C22—24no—
279C22—24no—
280C22—2988no—
281C22—2981no—
282C22—2981no—
283C22—2981no—
284C22—2981no—
285C22—1492no—
286C22—1492no—
287C22C2×#1455873512yes—
288C22C24×D14911yescharacteristic
289C4—24no—
290C4—24no—
291C4—2982no—
292C4—2984no—
293C4—2984no—
294C4—2984no—
295C4C2×#1455873514yes—
296C2—22no—
297C2—44no—
298C2—2982no—
299C2—2982no—
300C2C22×#1455873511yescharacteristic
301C2H(5,𝔽2)×D14911yescharacteristic
302C22−1+4×D14911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#537×C149C2
hypercenter#537D149
hypocenterC149#3492
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #23750333C2semidirect
2. #23750322C2semidirect
3. #23750309C2semidirect
4. #23750275C2semidirect
5. #23750256C2semidirect
6. #23750220C2semidirect
7. #23750135C2semidirect
8. #23750132C2semidirect
9. #23750108C2semidirect
10. C2×C4⋊C4×D149C2semidirect
11. #537×C149C2semidirect
15. #18011910C22semidirect
16. #18011904C22semidirect
17. #18011902C22semidirect
18. #18011901C22semidirect
21. #18011892C22semidirect
24. #18011888C22semidirect
31. Q8×Dic149C22semidirect
33. D4×Dic149C22semidirect
38. #18011870C22semidirect
41. C2×C4⋊Dic149C22semidirect
46. C22⋊Q8×C149C22semidirect
48. #121×C149C22semidirect
119. C2×C4×C149#41semidirect
151. C149#3492semidirect
159. #537D149semidirect
180. C22⋊Q8C2×D149semidirect
185. #121C2×D149semidirect
271. C2×C4#14558735semidirect

Maximal subgroups

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

subgroupindex
#237503332
#237503222
#237503092
#237502752
#237502562
#237502202
#237501352
#237501322
#237501082
C2×C4⋊C4×D1492
#537×C1492
#3492149