Finite Group Databasebeta

← #34053984

Subgroups — #34053984

Order 19072.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011958C211yesmaximal, characteristic
2C2×#18011942C211yesmaximal, characteristic, complemented
3C22×#14558726C211yesmaximal, characteristic, complemented
4#23749785C214yesmaximal, complemented
5#23749780C214yesmaximal, complemented
6#23749633C211yesmaximal, characteristic, complemented
7#23749619C211yesmaximal, characteristic, complemented
8#23749592C211yesmaximal, characteristic, complemented
9#536×C149C211yesmaximal, characteristic, complemented
10#18011958—24no—
11#18011958—24no—
12C2×#14558736C2212yes—
13C2×#14558736C2211yescharacteristic
14C2×#14558735—22no—
15C2×#14558735—21no—
16C22×Dic298C2212yes—
17C22×Dic298C2211yescharacteristic
18C2×Q8×D149—21no—
19#18011942—24no—
20C2×#14558726—24no—
21C2×#14558726C2214yes—
22C2×#14558726C2212yes—
23C2×#14558723C2212yes—
24C2×#14558720C2211yescharacteristic
25C22×#11924746C2211yescharacteristic
26#18011834C2214yes—
27#18011834C2212yes—
28#18011831C2212yes—
29C22⋊#11924746C2214yes—
30C2×#41×C149C2211yescharacteristic
31C22×Q8×C149C2211yescharacteristic
32C4⋊Q8×C149—21no—
33#124×C149—21no—
34#123×C149—22no—
35C22⋊Q8×C149C2214yes—
36#117×C149C2211yescharacteristic
37#14558736—24no—
38#14558736—24no—
39#14558736—22no—
40#14558736D414yes—
41#14558735—216no—
42#14558735—28no—
43C2×Dic298—24no—
44C2×Dic298—24no—
45C2×Dic298—22no—
46C2×Dic298—21no—
47C2×Dic298—22no—
48C2×Dic298D414yes—
49C2×Dic298C2311yescharacteristic
50C2×Dic298C2312yes—
51Q8×D149—28no—
52C2×D596C2311yescharacteristic
53C2×C22⋊D149—22no—
54C2×C22⋊D149—21no—
55C22×Dic149—22no—
56C22×Dic149—21no—
57C2×C4×D149—22no—
58C2×C4×D149—21no—
59#14558726—28no—
60#14558726—24no—
61#14558723—24no—
62#14558720—22no—
63C2×#11924746—22no—
64C2×#11924746C2312yes—
65C2×#11924746C2312yes—
66#41×C149—22no—
67#41×C149—22no—
68C2×Q8×C149—21no—
69C2×Q8×C149—22no—
70C2×Q8×C149—22no—
71C2×Q8×C149D411yescharacteristic
72C2×Q8×C149C2312yes—
73C2×D4×C149D412yes—
74C2×D4×C149D411yescharacteristic
75C22×C4×C149—21no—
76C22×C4×C149C2312yes—
77C22×C4×C149C2311yescharacteristic
78C4⋊C4×C149—24no—
79C4⋊C4×C149C2312yes—
80C22⋊C4×C149—21no—
81C22⋊C4×C149—24no—
82C42×C149—21no—
83C22×D149—21no—
84Dic298—28no—
85Dic298—24no—
86Dic298—21no—
87Dic298—22no—
88Dic298C2×D412yes—
89Dic298C2×D414yes—
90D596—21no—
91D596C2×D412yes—
92C22⋊D149—28no—
93C22⋊D149—24no—
94C2×Dic149—28no—
95C2×Dic149—24no—
96C2×Dic149—22no—
97C2×Dic149—21no—
98C4×D149—28no—
99C4×D149—24no—
100#11924746—22no—
101#11924746—22no—
102C23×C149C2×D412yes—
103C23×C149C2×D411yescharacteristic
104Q8×C149—22no—
105Q8×C149—22no—
106Q8×C149—22no—
107Q8×C149—42no—
108D4×C149—24no—
109D4×C149—22no—
110C2×C4×C149—22no—
111C2×C4×C149—22no—
112C2×C4×C149—22no—
113C2×C4×C149—22no—
114C2×C4×C149—42no—
115C2×C4×C149C2×D412yes—
116C2×C4×C149C2×D412yes—
117C2×C4×C149C2×D412yes—
118C2×C4×C149C2×D411yescharacteristic
119C2×C4×C149C2411yescharacteristic
120C22×C149—22no—
121C22×C149—22no—
122C22×C149—24no—
123C22×C149#12012yes—
124C22×C149C22×D411yescharacteristic
125C2×D149—22no—
126C2×D149—42no—
127C4×C149—22no—
128C4×C149—42no—
129C4×C149—42no—
130C4×C149#12012yes—
131C4×C149C22×D411yescharacteristic
132C4×C149C22×D411yescharacteristic
133Dic149—24no—
134Dic149—22no—
135C2×C149—22no—
136C2×C149—42no—
137C2×C149C2×#12011yescharacteristic
138C2×C149#57812yes—
139D149—42no—
140C149#301111yescharacteristic, complemented, Sylow
141#3011—1491nomaximal, Sylow
142C2×2−1+4—1491no—
143C2×#137—1491no—
144C22×Q16—1491no—
145#536D14911yescharacteristic, complemented
146#452—1494no—
147#451—1494no—
148#419—1491no—
149#416—1491no—
150#408—1491no—
1512−1+4—2984no—
1522−1+4—2984no—
153C2×#41—2982no—
154C2×#41—2981no—
155C2×#41—1492no—
156C2×#41—1491no—
157C2×#41C2×D14911yescharacteristic
158C22×Q8—2981no—
159C22×Q8—1492no—
160C22×Q8—1491no—
161C22×Q8C2×D14911yescharacteristic
162#137—2984no—
163C2×Q16—2984no—
164C2×Q16—1494no—
165C2×Q16—1492no—
166C2×SD16—1492no—
167C2×M16—1491no—
168C22×C8—1491no—
169C4⋊Q8—21no—
170#124—21no—
171#123—22no—
172C22⋊Q8C2×D14914yes—
173#117C2×D14911yescharacteristic
174#103—1492no—
175#103—1494no—
176#102—1492no—
177C22⋊C8—1494no—
178#41—22no—
179#41—22no—
180#41—2984no—
181#41—2984no—
182#41—29816no—
183#41—2988no—
184#41—2982no—
185#41—1494no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—2984no—
190C2×Q8—2984no—
191C2×Q8—2988no—
192C2×Q8—2982no—
193C2×Q8—2982no—
194C2×Q8—2981no—
195C2×Q8—1494no—
196C2×Q8—1491no—
197C2×Q8—1492no—
198C2×Q8C22⋊D14911yescharacteristic
199C2×Q8C22×D14912yes—
200C2×D4—2982no—
201C2×D4—2981no—
202C2×D4—1491no—
203C2×D4C22⋊D14912yes—
204C2×D4C22⋊D14911yescharacteristic
205C22×C4—21no—
206C22×C4—2982no—
207C22×C4—2981no—
208C22×C4—2982no—
209C22×C4—2981no—
210C22×C4C22×D14912yes—
211C22×C4C22×D14911yescharacteristic
212Q16—2988no—
213Q16—2984no—
214SD16—2984no—
215M16—2982no—
216C2×C8—2982no—
217C2×C8—1492no—
218C2×C8—1492no—
219C4⋊C4—24no—
220C4⋊C4C22×D14912yes—
221C22⋊C4—21no—
222C22⋊C4—24no—
223C42—21no—
224C23—2981no—
225C23C2×C22⋊D14912yes—
226C23C2×C22⋊D14911yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—22no—
230Q8—42no—
231Q8—2988no—
232Q8—2981no—
233Q8—2982no—
234Q8—2984no—
235Q8—1492no—
236Q8—1494no—
237D4—24no—
238D4—22no—
239D4—2988no—
240D4—2984no—
241D4—2981no—
242D4—1492no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—2988no—
249C2×C4—2988no—
250C2×C4—2982no—
251C2×C4—2981no—
252C2×C4—2984no—
253C2×C4—2984no—
254C2×C4D4×D14912yes—
255C2×C4D4×D14912yes—
256C2×C4C2×C22⋊D14912yes—
257C2×C4C2×C22⋊D14911yescharacteristic
258C2×C4C23×D14911yescharacteristic
259C8—2982no—
260C8—2982no—
261C22—22no—
262C22—24no—
263C22—22no—
264C22—2982no—
265C22—5962no—
266C22#1801187912yes—
267C22C22×C22⋊D14911yescharacteristic
268C4—22no—
269C4—42no—
270C4—42no—
271C4—2984no—
272C4—2982no—
273C4#1801187912yes—
274C4C2×D4×D14911yescharacteristic
275C4C2×D4×D14911yescharacteristic
276C2—22no—
277C2—42no—
278C2—5962no—
279C2C2×#1801187911yescharacteristic
280C2#2375044012yes—
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#536×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011958C2
hypercenter#536D149
hypocenterC149#3011
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#18011942C2semidirect
3. C22×#14558726C2semidirect
4. #23749785C2semidirect
5. #23749780C2semidirect
6. #23749633C2semidirect
7. #23749619C2semidirect
8. #23749592C2semidirect
9. #536×C149C2semidirect
140. C149#3011semidirect
145. #536D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119582
C2×#180119422
C22×#145587262
#237497852
#237497802
#237496332
#237496192
#237495922
#536×C1492
#3011149