Finite Group Databasebeta

← #34047622

Subgroups — #34047622

Order 19072.

660 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D149C211yesmaximal, characteristic
2C2×#18011869C211yesmaximal, characteristic, complemented
3C22×C4⋊Dic149C211yesmaximal, characteristic, complemented
4#23749517C212yesmaximal, complemented
5#23749512C212yesmaximal, complemented
6#23749502C212yesmaximal
7#23749490C211yesmaximal, characteristic, complemented
8#23749482C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic149C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic149C212yesmaximal, complemented
11#380×C149C211yesmaximal, characteristic, complemented
12C22×C22⋊D149C2211yescharacteristic
13C23×Dic149C2212yes—
14C22×C4×D149C2211yescharacteristic
15C4×C22⋊D149C418yescomplemented
16#18011869—24no—
17C2×C4⋊Dic149—24no—
18C2×C4⋊Dic149C2211yescharacteristic
19C2×C4⋊Dic149C2212yes—
20C2×C4⋊Dic149C2211yescharacteristic
21C2×#14558715C2211yescharacteristic
22C2×C22⋊Dic149C2211yescharacteristic
23C2×#14558713C2212yes—
24C2×#14558713C2212yes—
25C2×#14558713C2211yescharacteristic
26C2×C4×Dic149C2212yes—
27C2×C4×Dic149C2212yes—
28C2×C4×Dic149C2211yescharacteristic
29#18011818C2212yes—
30#18011818C2212yes—
31#18011818C2212yes—
32#18011818C2212yes—
33C23×C4×C149C2211yescharacteristic
34C2×C4⋊C4×C149C2211yescharacteristic
35C2×C4⋊C4×C149C2211yescharacteristic
36C2×C22⋊C4×C149C2212yes—
37#95×C149C2212yes—
38C23×D149C2311yescharacteristic
39C2×C22⋊D149—22no—
40C2×C22⋊D149—22no—
41C2×C22⋊D149C2×C414yes—
42C22×Dic149—24no—
43C22×Dic149—24no—
44C22×Dic149—42no—
45C22×Dic149C2312yes—
46C22×Dic149C2311yescharacteristic
47C22×Dic149C2311yescharacteristic
48C22×Dic149C2311yescharacteristic
49C22×Dic149C2312yes—
50C2×C4×D149—22no—
51C2×C4×D149—22no—
52C2×C4×D149C2×C414yescomplemented
53C4⋊Dic149—24no—
54C4⋊Dic149—22no—
55C4⋊Dic149D414yescomplemented
56#14558715C2×C414yescomplemented
57C22⋊Dic149C2×C414yes—
58#14558713—24no—
59#14558713—42no—
60#14558713C2×C414yescomplemented
61C4×Dic149—24no—
62C4×Dic149—42no—
63C4×Dic149C2×C414yescomplemented
64C24×C149C2311yescharacteristic
65C22×C4×C149—22no—
66C22×C4×C149—22no—
67C22×C4×C149C2×C414yes—
68C22×C4×C149C2312yes—
69C22×C4×C149C2312yes—
70C22×C4×C149C2311yescharacteristic
71C22×C4×C149C2311yescharacteristic
72C4⋊C4×C149—22no—
73C4⋊C4×C149—22no—
74C22⋊C4×C149—24no—
75C22×D149—21no—
76C22×D149—21no—
77C22×D149—21no—
78C22×D149—21no—
79C22×D149—21no—
80C22×D149—21no—
81C22×D149C22×C412yes—
82C22⋊D149—28no—
83C2×Dic149—21no—
84C2×Dic149—21no—
85C2×Dic149—22no—
86C2×Dic149—21no—
87C2×Dic149—21no—
88C2×Dic149—22no—
89C2×Dic149—21no—
90C2×Dic149—21no—
91C2×Dic149—22no—
92C2×Dic149—22no—
93C2×Dic149—22no—
94C2×Dic149—48no—
95C2×Dic149C22×C412yes—
96C2×Dic149C22×C412yes—
97C2×Dic149C22×C412yes—
98C2×Dic149C2×D414yes—
99C4×D149—28no—
100C23×C149—21no—
101C23×C149—21no—
102C23×C149—21no—
103C23×C149—21no—
104C23×C149C22×C412yes—
105C23×C149#4112yes—
106C23×C149#4112yes—
107C23×C149C2411yescharacteristic
108C2×C4×C149—21no—
109C2×C4×C149—21no—
110C2×C4×C149—28no—
111C2×C4×C149—22no—
112C2×C4×C149—22no—
113C2×C4×C149—22no—
114C2×C4×C149—22no—
115C2×C4×C149—22no—
116C2×C4×C149—22no—
117C2×C4×C149C22×C412yes—
118C2×C4×C149C22×C412yes—
119C2×C4×C149C2×D412yes—
120C2×C4×C149#4112yes—
121C22×C149—22no—
122C22×C149—22no—
123C22×C149—22no—
124C22×C149—22no—
125C22×C149—22no—
126C22×C149—22no—
127C22×C149#11714yescomplemented
128C22×C149C23×C411yescharacteristic
129C22×C149C22×D411yescharacteristic
130C22×C149C2×#4111yescharacteristic
131C22×C149C2×#4111yescharacteristic
132C22×C149C2×#4111yescharacteristic
133C22×C149H(5,𝔽2)11yescharacteristic
134C22×C1492−1+411yescharacteristic
135C2×D149—22no—
136C2×D149—22no—
137C2×D149—22no—
138C2×D149—41no—
139C2×D149—41no—
140C2×D149—41no—
141C2×D149—41no—
142C4×C149—22no—
143C4×C149—42no—
144C4×C149—42no—
145C4×C149C4×D414yes—
146Dic149—22no—
147Dic149—22no—
148Dic149—22no—
149Dic149—24no—
150Dic149—42no—
151C2×C149—24no—
152C2×C149C2×#11711yescharacteristic
153C2×C149C2×C4×D411yescharacteristic
154C2×C149#52011yescharacteristic
155C2×C149#54611yescharacteristic
156C2×C149#54811yescharacteristic
157C2×C149#55211yescharacteristic
158C2×C149#55311yescharacteristic
159D149—42no—
160C149#235211yescharacteristic, complemented, Sylow
161#2352—1491nomaximal, Sylow
162C2×C4×D4—1491no—
163C2×#117—1491no—
164C22×C4⋊C4—1491no—
165#388—1492no—
166#387—1492no—
167#386—1492no—
168#382—1491no—
169#380—1491no—
170#380D14911yescharacteristic, complemented
171C4×C4⋊C4—1491no—
172C4×C22⋊C4—1492no—
173C22×D4—1491no—
174C23×C4—1492no—
175C23×C4—1491no—
176C23×C4C2×D14911yescharacteristic
177C4×D4—1498no—
178#117—2984no—
179C2×C4⋊C4—2984no—
180C2×C4⋊C4—1491no—
181C2×C4⋊C4—1492no—
182C2×C4⋊C4—1491no—
183C2×C4⋊C4—1491no—
184C2×C4⋊C4C2×D14911yescharacteristic
185C2×C4⋊C4C2×D14911yescharacteristic
186C2×C22⋊C4—1492no—
187C2×C22⋊C4—1492no—
188C2×C22⋊C4—1491no—
189C2×C22⋊C4—1491no—
190C2×C22⋊C4C2×D14912yes—
191C2×C42—1492no—
192C2×C42—1492no—
193C2×C42—1491no—
194#95—1492no—
195#95—1492no—
196#95—1492no—
197#95—1492no—
198#95C2×D14912yes—
199C24—1491no—
200C24C22×D14911yescharacteristic
201C2×D4—2982no—
202C2×D4—2982no—
203C2×D4—1494no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—2982no—
207C22×C4—2982no—
208C22×C4—2984no—
209C22×C4—2984no—
210C22×C4—5962no—
211C22×C4—1492no—
212C22×C4—1491no—
213C22×C4—1491no—
214C22×C4—1491no—
215C22×C4—1492no—
216C22×C4—1494no—
217C22×C4C4×D14914yes—
218C22×C4C22×D14912yes—
219C22×C4C22×D14912yes—
220C22×C4C22×D14911yescharacteristic
221C22×C4C22×D14911yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—2984no—
225C4⋊C4—2982no—
226C4⋊C4—1494no—
227C4⋊C4—1494no—
228C22⋊C4—24no—
229C22⋊C4—2984no—
230C22⋊C4—5962no—
231C22⋊C4—1494no—
232C22⋊C4—1494no—
233C42—2984no—
234C42—5962no—
235C42—1494no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—2981no—
241C23—2981no—
242C23—2981no—
243C23—2981no—
244C23—2981no—
245C23—2981no—
246C23—1492no—
247C23C2×C4×D14912yes—
248C23#1455873512yes—
249C23#1455873712yes—
250C23C23×D14911yescharacteristic
251D4—2988no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—28no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—2981no—
262C2×C4—2981no—
263C2×C4—2981no—
264C2×C4—2982no—
265C2×C4—2981no—
266C2×C4—2981no—
267C2×C4—2982no—
268C2×C4—2981no—
269C2×C4—2988no—
270C2×C4—2982no—
271C2×C4—2982no—
272C2×C4—2982no—
273C2×C4—5968no—
274C2×C4—1492no—
275C2×C4—1492no—
276C2×C4—1492no—
277C2×C4—1494no—
278C2×C4C2×C4×D14912yes—
279C2×C4C2×C4×D14912yes—
280C2×C4D4×D14912yes—
281C2×C4#1455873512yes—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—2982no—
289C22—2982no—
290C22—2982no—
291C22—5961no—
292C22—5961no—
293C22—5961no—
294C22—5961no—
295C22#1801186914yescomplemented
296C22C22×C4×D14911yescharacteristic
297C22C2×D4×D14911yescharacteristic
298C22C2×#1455873511yescharacteristic
299C22C2×#1455873511yescharacteristic
300C22C2×#1455873711yescharacteristic
301C22#1801195711yescharacteristic
302C22#1801195811yescharacteristic
303C4—22no—
304C4—42no—
305C4—42no—
306C4—2984no—
307C4—2982no—
308C4—2982no—
309C4—2982no—
310C4—5962no—
311C4#1801187114yes—
312C2—24no—
313C2—5962no—
314C2C2×#1801186911yescharacteristic
315C2C2×#1801187111yescharacteristic
316C2#2375010511yescharacteristic
317C2#2375026611yescharacteristic
318C2#2375028111yescharacteristic
319C2#2375031711yescharacteristic
320C2#2375032711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
Fitting#380×C149C2
socle, cosocleC23×C149C24
prime radicalC22×C22⋊D149C22
hypercenter#380D149
hypocenterC149#2352
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#18011869C2semidirect
3. C22×C4⋊Dic149C2semidirect
4. #23749517C2semidirect
5. #23749512C2semidirect
7. #23749490C2semidirect
8. #23749482C2semidirect
9. C4⋊C4×Dic149C2semidirect
10. C22⋊C4×Dic149C2semidirect
11. #380×C149C2semidirect
15. C4×C22⋊D149C4semidirect
52. C2×C4×D149C2×C4semidirect
55. C4⋊Dic149D4semidirect
56. #14558715C2×C4semidirect
60. #14558713C2×C4semidirect
63. C4×Dic149C2×C4semidirect
127. C22×C149#117semidirect
160. C149#2352semidirect
170. #380D149semidirect
295. C22#18011869semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1492
C2×#180118692
C22×C4⋊Dic1492
#237495172
#237495122
#237495022
#237494902
#237494822
C4⋊C4×Dic1492
C22⋊C4×Dic1492
#380×C1492
#2352149