Finite Group Databasebeta

← #34056060

Subgroups — #34056060

Order 19072.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#23750180C211yesmaximal, characteristic, complemented
2#23750157C211yesmaximal, characteristic
3#23749934C212yesmaximal, complemented
4#23749921C211yesmaximal, characteristic, complemented
5#23749915C211yesmaximal, characteristic, complemented
6#23749882C212yesmaximal, complemented
7#23749868C212yesmaximal
8#23749845C212yesmaximal, complemented
9#23749832C211yesmaximal, characteristic, complemented
10#23749647C211yesmaximal, characteristic, complemented
11#461×C149C211yesmaximal, characteristic, complemented
12C2×#14558726C2212yes—
13C2×#14558719C2212yes—
14#18011911C2211yescharacteristic
15#18011904—21no—
16#18011902—21no—
17#18011902C2211yescharacteristic
18#18011898C2211yescharacteristic
19#18011893—21no—
20#18011891—21no—
21#18011891C2211yescharacteristic
22#18011887C2212yes—
23#18011882C2211yescharacteristic
24Q8×Dic149C2212yes—
25C4×Dic298C2211yescharacteristic
26#18011876—21no—
27C4×C22⋊D149—21no—
28#18011870—21no—
29#18011869—21no—
30C42×D149C2211yescharacteristic
31#18011845C2212yes—
32#18011842C2212yes—
33C4⋊#11924746C2211yescharacteristic
34#18011840C2212yes—
35#18011836C2212yes—
36#18011835C2212yes—
37#18011832C2212yes—
38C2×SD16×C149C2212yes—
39C4⋊Q8×C149C2211yescharacteristic
40C4×D4×C149C2211yescharacteristic
41C4⋊C8×C149C2211yescharacteristic
42#103×C149C2212yes—
43C2×Dic298—21no—
44C2×Dic298C2311yescharacteristic
45C2×C22⋊D149—21no—
46C2×C4×D149D411yescharacteristic
47C2×C4×D149D411yescharacteristic
48C2×C4×D149C2311yescharacteristic
49#14558726—42no—
50#14558719—42no—
51C2×#11924746C2312yes—
52C4⋊Dic149—21no—
53C4⋊Dic149—21no—
54C4⋊Dic149—22no—
55C4⋊Dic149—21no—
56C4⋊Dic149C2312yes—
57C4⋊Dic149C2311yescharacteristic
58#14558715—21no—
59#14558715—22no—
60#14558715—21no—
61C22⋊Dic149—21no—
62C22⋊Dic149—21no—
63C22⋊Dic149—21no—
64#14558713—22no—
65#14558713—21no—
66C4×Dic149—22no—
67C4×Dic149—21no—
68C4×Dic149D411yescharacteristic
69C4×Dic149D411yescharacteristic
70C4×Dic149C2311yescharacteristic
71C2×Q8×C149C2312yes—
72C2×D4×C149C2311yescharacteristic
73C22×C4×C149—21no—
74SD16×C149—42no—
75C2×C8×C149C2312yes—
76C4⋊C4×C149—21no—
77C4⋊C4×C149—21no—
78C4⋊C4×C149C2311yescharacteristic
79C22⋊C4×C149—21no—
80C42×C149C2311yescharacteristic
81C22×D149C2×D411yescharacteristic
82Dic298—21no—
83Dic298—21no—
84Dic298—42no—
85C22⋊D149—42no—
86C2×Dic149—21no—
87C2×Dic149—22no—
88C2×Dic149—21no—
89C2×Dic149C2×D411yescharacteristic
90C2×Dic149C2×D411yescharacteristic
91C2×Dic149C2×D411yescharacteristic
92C4×D149—22no—
93C4×D149—21no—
94C4×D149—21no—
95C4×D149—41no—
96#11924746—22no—
97C23×C149—21no—
98Q8×C149—22no—
99Q8×C149—22no—
100D4×C149—21no—
101D4×C149—21no—
102C2×C4×C149—21no—
103C2×C4×C149—22no—
104C2×C4×C149—42no—
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2×D411yescharacteristic
107C2×C4×C149C2411yescharacteristic
108C8×C149—22no—
109C22×C149—41no—
110C22×C149—41no—
111C22×C149—41no—
112C22×C149C22×D411yescharacteristic
113C2×D149—21no—
114C2×D149—21no—
115C2×D149—21no—
116C4×C149—21no—
117C4×C149—42no—
118C4×C149—41no—
119C4×C149#13712yes—
120C4×C149H(5,𝔽2)11yescharacteristic
121C4×C1492−1+411yescharacteristic
122Dic149—21no—
123Dic149—21no—
124Dic149—21no—
125Dic149—41no—
126Dic149—41no—
127Dic149—42no—
128C2×C149—81no—
129C2×C149#53711yescharacteristic
130C2×C149C2×#13711yescharacteristic
131C2×C149#57511yescharacteristic
132D149—41no—
133C149#325211yescharacteristic, complemented, Sylow
134#3252—1491nomaximal, Sylow
135#533—1491no—
136#529—1491no—
137#479—1492no—
138#477—1491no—
139#476—1491no—
140#470—1492no—
141#468—1492no—
142#464—1492no—
143#462—1491no—
144#461D14911yescharacteristic, complemented
145#423—1491no—
146C2×Q16—1492no—
147C2×SD16C2×D14912yes—
148C2×M16—1492no—
149C4⋊Q8—1491no—
150C4⋊Q8C2×D14911yescharacteristic
151#126—2981no—
152#125—2981no—
153#125—1491no—
154#124—1491no—
155#123—2981no—
156C22⋊Q8—2981no—
157C22⋊Q8—1492no—
158C22⋊Q8—1491no—
159#121—1491no—
160C4×Q8—2981no—
161C4×Q8—1492no—
162C4×Q8—1491no—
163C4×D4—2981no—
164C4×D4C2×D14911yescharacteristic
165#117—2981no—
166#117—2981no—
167C2×C42—1491no—
168#107—1492no—
169#106—1492no—
170C4⋊C8—1492no—
171C4⋊C8—1491no—
172C4⋊C8C2×D14911yescharacteristic
173#103—1492no—
174#103—1492no—
175#103C2×D14912yes—
176#102—1492no—
177C2×Q8—2981no—
178C2×Q8—1491no—
179C2×Q8C22×D14912yes—
180C2×D4—2981no—
181C2×D4C22×D14911yescharacteristic
182C22×C4—21no—
183C22×C4—1491no—
184C22×C4—1491no—
185C22×C4—1491no—
186Q16—5962no—
187SD16—42no—
188M16—5962no—
189C2×C8—1492no—
190C2×C8C22×D14912yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—2981no—
194C4⋊C4—2981no—
195C4⋊C4—2981no—
196C4⋊C4—2982no—
197C4⋊C4—2982no—
198C4⋊C4—2981no—
199C4⋊C4—2981no—
200C4⋊C4—1492no—
201C4⋊C4—1491no—
202C4⋊C4C22×D14911yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—2982no—
205C22⋊C4—2981no—
206C22⋊C4—2981no—
207C22⋊C4—2981no—
208C22⋊C4—2981no—
209C42—2982no—
210C42—2981no—
211C42—1491no—
212C42—1491no—
213C42—1491no—
214C42C22×D14911yescharacteristic
215C23—21no—
216C23—1491no—
217Q8—22no—
218Q8—22no—
219Q8—2981no—
220Q8—2981no—
221Q8—5962no—
222D4—21no—
223D4—21no—
224D4—5962no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—2982no—
229C2×C4—2981no—
230C2×C4—2981no—
231C2×C4—2981no—
232C2×C4—2981no—
233C2×C4—2982no—
234C2×C4—5961no—
235C2×C4—1491no—
236C2×C4—1491no—
237C2×C4—1491no—
238C2×C4D4×D14911yescharacteristic
239C2×C4D4×D14911yescharacteristic
240C2×C4C23×D14911yescharacteristic
241C8—22no—
242C8—2982no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—2981no—
247C22—2981no—
248C22—2981no—
249C22C2×D4×D14911yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—2981no—
254C4—2981no—
255C4—2981no—
256C4—5961no—
257C4—5962no—
258C4—5961no—
259C4#1801194012yes—
260C4#1801195611yescharacteristic
261C4#1801195811yescharacteristic
262C2—81no—
263C2—5961no—
264C2#2375020411yescharacteristic
265C2C2×#1801194011yescharacteristic
266C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#461×C149C2
socleC22×C149C22×D4
prime radical#18011882C22
hypercenter#461D149
hypocenterC149#3252
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #23750180C2semidirect
3. #23749934C2semidirect
4. #23749921C2semidirect
5. #23749915C2semidirect
6. #23749882C2semidirect
8. #23749845C2semidirect
9. #23749832C2semidirect
10. #23749647C2semidirect
11. #461×C149C2semidirect
133. C149#3252semidirect
144. #461D149semidirect

Maximal subgroups

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

subgroupindex
#237501802
#237501572
#237499342
#237499212
#237499152
#237498822
#237498682
#237498452
#237498322
#237496472
#461×C1492
#3252149