Finite Group Databasebeta

← #34056065

Subgroups — #34056065

Order 19072.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#23750182C211yesmaximal, characteristic, complemented
2#23750159C211yesmaximal, characteristic
3#23749933C212yesmaximal, complemented
4#23749921C211yesmaximal, characteristic, complemented
5#23749916C211yesmaximal, characteristic, complemented
6#23749883C212yesmaximal, complemented
7#23749871C212yesmaximal
8#23749848C212yesmaximal, complemented
9#23749836C211yesmaximal, characteristic, complemented
10#23749831C211yesmaximal, characteristic, complemented
11#423×C149C211yesmaximal, characteristic, complemented
12C2×Dic596C2212yes—
13C2×#14558725C2212yes—
14#18011911C2211yescharacteristic
15#18011911C2211yescharacteristic
16#18011905—21no—
17#18011903—21no—
18#18011903C2211yescharacteristic
19#18011899C2211yescharacteristic
20#18011894—21no—
21#18011890—21no—
22#18011890C2211yescharacteristic
23#18011890C2212yes—
24#18011885C2211yescharacteristic
25C4×Dic298—21no—
26C4×Dic298C2211yescharacteristic
27C4×Dic298C2212yes—
28C4×D596C2211yescharacteristic
29C4×C22⋊D149—21no—
30#18011870—21no—
31#18011868—21no—
32#18011845C2212yes—
33#18011843C2212yes—
34#18011836C2212yes—
35#18011836C2212yes—
36#18011836C2212yes—
37#18011833C2212yes—
38C2×M16×C149C2212yes—
39C2×C42×C149C2211yescharacteristic
40C4⋊C8×C149C2211yescharacteristic
41C4⋊C8×C149C2211yescharacteristic
42C4⋊C8×C149C2212yes—
43C2×Dic298—21no—
44C2×Dic298C2312yes—
45C2×Dic298C2311yescharacteristic
46C2×D596C2311yescharacteristic
47C2×C22⋊D149—21no—
48C2×C4×D149—21no—
49Dic596—42no—
50#14558725—42no—
51C4⋊Dic149—21no—
52C4⋊Dic149—21no—
53C4⋊Dic149—21no—
54C4⋊Dic149C2312yes—
55C4⋊Dic149C2311yescharacteristic
56C4⋊Dic149C2311yescharacteristic
57#14558715—21no—
58#14558715—21no—
59#14558715—21no—
60#14558715—21no—
61#14558715—22no—
62#14558715—22no—
63C22⋊Dic149—22no—
64C22⋊Dic149—21no—
65#14558713—21no—
66#14558713—21no—
67#14558713—21no—
68#14558713—21no—
69C4×Dic149—21no—
70C4×Dic149—22no—
71C22×C4×C149D411yescharacteristic
72C22×C4×C149D411yescharacteristic
73C22×C4×C149C2311yescharacteristic
74M16×C149—42no—
75C2×C8×C149C2312yes—
76C2×C8×C149C2312yes—
77C42×C149D411yescharacteristic
78C42×C149D411yescharacteristic
79C42×C149C2311yescharacteristic
80C42×C149C2311yescharacteristic
81C22×D149—21no—
82Dic298—22no—
83Dic298—21no—
84Dic298—22no—
85Dic298—21no—
86Dic298—42no—
87D596—21no—
88D596—21no—
89C22⋊D149—42no—
90C2×Dic149—21no—
91C2×Dic149—21no—
92C2×Dic149—22no—
93C2×Dic149—21no—
94C2×Dic149—22no—
95C4×D149—42no—
96C23×C149C2×D411yescharacteristic
97C2×C4×C149—22no—
98C2×C4×C149—21no—
99C2×C4×C149—21no—
100C2×C4×C149—41no—
101C2×C4×C149C2×D411yescharacteristic
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2×D411yescharacteristic
104C2×C4×C149C2×D411yescharacteristic
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2411yescharacteristic
107C8×C149—22no—
108C8×C149—22no—
109C22×C149—21no—
110C22×C149—21no—
111C22×C149—21no—
112C22×C149C22×D411yescharacteristic
113C2×D149—41no—
114C2×D149—41no—
115C2×D149—41no—
116C4×C149—21no—
117C4×C149—21no—
118C4×C149—21no—
119C4×C149—21no—
120C4×C149#13712yes—
121C4×C149H(5,𝔽2)11yescharacteristic
122C4×C1492−1+411yescharacteristic
123Dic149—41no—
124Dic149—42no—
125Dic149—41no—
126Dic149—41no—
127Dic149—42no—
128C2×C149—41no—
129C2×C149#53711yescharacteristic
130C2×C149C2×#13711yescharacteristic
131C2×C149#57511yescharacteristic
132D149—81no—
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#461—1491no—
145#423D14911yescharacteristic, complemented
146C2×Q16—1492no—
147C2×SD16—1492no—
148C2×M16C2×D14912yes—
149C4⋊Q8—1491no—
150C4⋊Q8—1491no—
151#126—2981no—
152#125—2981no—
153#125—1491no—
154#124—1491no—
155#123—2981no—
156C22⋊Q8—2981no—
157C22⋊Q8—1491no—
158C22⋊Q8—1492no—
159#121—1491no—
160C4×Q8—2981no—
161C4×Q8—1491no—
162C4×Q8—1492no—
163C4×D4—2981no—
164C4×D4—1491no—
165#117—2981no—
166#117—2981no—
167C2×C42C2×D14911yescharacteristic
168#107—1492no—
169#106—1492no—
170C4⋊C8C2×D14911yescharacteristic
171C4⋊C8C2×D14911yescharacteristic
172C4⋊C8C2×D14912yes—
173#103—1492no—
174#103—1492no—
175#103—1492no—
176#102—1492no—
177C2×Q8—2981no—
178C2×Q8—1491no—
179C2×Q8—1492no—
180C2×D4—2981no—
181C2×D4—1491no—
182C22×C4—2981no—
183C22×C4D59611yescharacteristic
184C22×C4D59611yescharacteristic
185C22×C4C22×D14911yescharacteristic
186Q16—5962no—
187SD16—5962no—
188M16—42no—
189C2×C8C22×D14912yes—
190C2×C8C22×D14912yes—
191C4⋊C4—2981no—
192C4⋊C4—2981no—
193C4⋊C4—2981no—
194C4⋊C4—2981no—
195C4⋊C4—2981no—
196C4⋊C4—2981no—
197C4⋊C4—2982no—
198C4⋊C4—2982no—
199C4⋊C4—2981no—
200C4⋊C4—1492no—
201C4⋊C4—1491no—
202C4⋊C4—1491no—
203C22⋊C4—2982no—
204C22⋊C4—2981no—
205C22⋊C4—2981no—
206C22⋊C4—2981no—
207C22⋊C4—2981no—
208C22⋊C4—2981no—
209C42—2981no—
210C42—2982no—
211C42D59611yescharacteristic
212C42D59611yescharacteristic
213C42C22×D14911yescharacteristic
214C42C22×D14911yescharacteristic
215C23—2981no—
216C23C2×D59611yescharacteristic
217Q8—2982no—
218Q8—2981no—
219Q8—2981no—
220Q8—2982no—
221Q8—5962no—
222D4—2981no—
223D4—2981no—
224D4—5962no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—2981no—
230C2×C4—2982no—
231C2×C4—2981no—
232C2×C4—2982no—
233C2×C4—2981no—
234C2×C4—5962no—
235C2×C4C2×D59611yescharacteristic
236C2×C4C2×D59611yescharacteristic
237C2×C4C2×D59611yescharacteristic
238C2×C4C2×D59611yescharacteristic
239C2×C4C2×D59611yescharacteristic
240C2×C4C23×D14911yescharacteristic
241C8—22no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—5961no—
247C22—5961no—
248C22—5961no—
249C22C22×D59611yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—5961no—
255C4—5961no—
256C4—5962no—
257C4—5961no—
258C4—5962no—
259C4#1801194412yes—
260C4#1801195611yescharacteristic
261C4#1801195911yescharacteristic
262C2—41no—
263C2—11921no—
264C2#2375020611yescharacteristic
265C2C2×#1801194411yescharacteristic
266C2#2375042511yescharacteristic
subgroupG/H
centerC22C22×D596
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#423×C149C2
socleC22×C149C22×D4
prime radical#18011885C22
hypercenter#423D149
hypocenterC149#3252
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #23750182C2semidirect
3. #23749933C2semidirect
4. #23749921C2semidirect
5. #23749916C2semidirect
6. #23749883C2semidirect
8. #23749848C2semidirect
9. #23749836C2semidirect
10. #23749831C2semidirect
11. #423×C149C2semidirect
133. C149#3252semidirect
145. #423D149semidirect

Maximal subgroups

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

subgroupindex
#237501822
#237501592
#237499332
#237499212
#237499162
#237498832
#237498712
#237498482
#237498362
#237498312
#423×C1492
#3252149