Finite Group Databasebeta

← #34056005

Subgroups — #34056005

Order 19072.

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

normal subgroups only

This group has 392 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#23750179C211yesmaximal, characteristic, complemented
2#23750173C211yesmaximal, characteristic
3#23749913C212yesmaximal, complemented
4#23749911C211yesmaximal, characteristic, complemented
5#23749910C211yesmaximal, characteristic, complemented
6#23749853C212yesmaximal, complemented
7#23749851C212yesmaximal, complemented
8#23749828C212yesmaximal, complemented
9#23749822C211yesmaximal, characteristic, complemented
10C4⋊C8×D149C211yesmaximal, characteristic, complemented
11#460×C149C211yesmaximal, characteristic, complemented
12C2×#14558735—21no—
13C2×#14558724C2212yes—
14C2×C8×D149C2212yes—
15#18011911C2211yescharacteristic
16#18011910C2211yescharacteristic
17#18011909C2211yescharacteristic
18#18011901—21no—
19#18011898—21no—
20#18011891C2211yescharacteristic
21#18011889—21no—
22#18011887C2211yescharacteristic
23#18011882—21no—
24#18011882C2212yes—
25C4×Dic298—21no—
26C4×Dic298C2211yescharacteristic
27D4×Dic149—21no—
28D4×Dic149C2212yes—
29#18011868—21no—
30C42×D149C2211yescharacteristic
31#18011843C2212yes—
32#18011842C2212yes—
33C4⋊#11924746C2211yescharacteristic
34#18011840C2212yes—
35#18011836C2212yes—
36#18011835C2212yes—
37#18011832C2212yes—
38C2×SD16×C149C2212yes—
39#127×C149C2211yescharacteristic
40C4×Q8×C149C2211yescharacteristic
41C4⋊C8×C149C2211yescharacteristic
42#102×C149C2212yes—
43#14558735—44no—
44C2×Dic298—21no—
45C2×Dic298—21no—
46C2×Dic298C2311yescharacteristic
47C2×C22⋊D149—22no—
48C22×Dic149—22no—
49C2×C4×D149D411yescharacteristic
50C2×C4×D149D411yescharacteristic
51C2×C4×D149C2311yescharacteristic
52#14558724—42no—
53C2×#11924746C2312yes—
54C8×D149—42no—
55C4⋊Dic149—21no—
56C4⋊Dic149—21no—
57C4⋊Dic149C2311yescharacteristic
58C4⋊Dic149C2312yes—
59#14558715—21no—
60#14558715—21no—
61#14558715—21no—
62#14558715—21no—
63#14558715—21no—
64C22⋊Dic149—22no—
65C22⋊Dic149—22no—
66#14558713—21no—
67#14558713—21no—
68C4×Dic149—21no—
69C4×Dic149D411yescharacteristic
70C4×Dic149D411yescharacteristic
71C4×Dic149C2311yescharacteristic
72C2×Q8×C149C2311yescharacteristic
73C2×D4×C149—21no—
74C2×D4×C149—21no—
75C2×D4×C149C2312yes—
76SD16×C149—42no—
77C2×C8×C149C2312yes—
78C4⋊C4×C149—21no—
79C4⋊C4×C149C2311yescharacteristic
80C42×C149—21no—
81C42×C149C2311yescharacteristic
82C22×D149C2×D411yescharacteristic
83Dic298—21no—
84Dic298—21no—
85Dic298—42no—
86Dic298—42no—
87C22⋊D149—44no—
88C2×Dic149—22no—
89C2×Dic149—21no—
90C2×Dic149—21no—
91C2×Dic149—44no—
92C2×Dic149C2×D411yescharacteristic
93C2×Dic149C2×D411yescharacteristic
94C2×Dic149C2×D411yescharacteristic
95C4×D149—21no—
96C4×D149—21no—
97C4×D149—41no—
98C4×D149SD1614yescomplemented
99#11924746—22no—
100C23×C149—22no—
101Q8×C149—21no—
102Q8×C149—21no—
103D4×C149—22no—
104D4×C149—22no—
105D4×C149—42no—
106D4×C149—42no—
107C2×C4×C149—21no—
108C2×C4×C149—21no—
109C2×C4×C149C2×D411yescharacteristic
110C2×C4×C149C2×D411yescharacteristic
111C2×C4×C149C2411yescharacteristic
112C8×C149—22no—
113C22×C149—42no—
114C22×C149—42no—
115C22×C149—42no—
116C22×C149C22×D411yescharacteristic
117C2×D149—21no—
118C2×D149—21no—
119C2×D149C2×SD1612yes—
120C4×C149—21no—
121C4×C149—41no—
122C4×C149—41no—
123C4×C149C2×SD1612yes—
124C4×C149H(5,𝔽2)11yescharacteristic
125C4×C1492−1+411yescharacteristic
126Dic149—21no—
127Dic149—21no—
128Dic149—41no—
129Dic149—42no—
130Dic149—41no—
131Dic149C2×SD1612yes—
132C2×C149—82no—
133C2×C149#53711yescharacteristic
134C2×C149C22×SD1611yescharacteristic
135C2×C149#57511yescharacteristic
136D149—22no—
137C149#324611yescharacteristic, complemented, Sylow
138#3246—1491nomaximal, Sylow
139#533—1491no—
140#532—1491no—
141#476—1492no—
142#475—1491no—
143#475—1491no—
144#465—1492no—
145#465—1492no—
146#461—1492no—
147#460—1491no—
148#460D14911yescharacteristic, complemented
149C2×C4⋊C8—1491no—
150C2×#41—2981no—
151C2×SD16—1492no—
152C2×SD16C2×D14912yes—
153C22×C8—1492no—
154C4⋊Q8—1491no—
155C4⋊Q8—1491no—
156C4⋊Q8—1491no—
157#127C2×D14911yescharacteristic
158#125—2981no—
159#124—2981no—
160C22⋊Q8—2981no—
161C22⋊Q8—1491no—
162C22⋊Q8—1491no—
163#121—2981no—
164#121—1492no—
165C4×Q8—2981no—
166C4×Q8—1491no—
167C4×Q8C2×D14911yescharacteristic
168C4×D4—2981no—
169C4×D4—1492no—
170#117—2981no—
171C2×C42—1491no—
172#106—1492no—
173#106—1492no—
174C4⋊C8—1492no—
175C4⋊C8—1491no—
176C4⋊C8C2×D14911yescharacteristic
177#103—1492no—
178#103—1492no—
179#102—1492no—
180#102C2×D14912yes—
181#41—5964no—
182C2×Q8—2981no—
183C2×Q8—2981no—
184C2×Q8—1491no—
185C2×Q8C22×D14911yescharacteristic
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—2982no—
189C2×D4C22×D14912yes—
190C22×C4—2982no—
191C22×C4—1491no—
192C22×C4—1491no—
193C22×C4—1491no—
194SD16—42no—
195SD16—5962no—
196C2×C8—5962no—
197C2×C8—1492no—
198C2×C8C22×D14912yes—
199C4⋊C4—21no—
200C4⋊C4—2981no—
201C4⋊C4—2981no—
202C4⋊C4—2981no—
203C4⋊C4—2981no—
204C4⋊C4—2981no—
205C4⋊C4—2981no—
206C4⋊C4—2981no—
207C4⋊C4—1491no—
208C4⋊C4—1492no—
209C4⋊C4C22×D14911yescharacteristic
210C22⋊C4—2981no—
211C22⋊C4—2981no—
212C22⋊C4—2982no—
213C22⋊C4—2982no—
214C42—21no—
215C42—2981no—
216C42—1491no—
217C42—1491no—
218C42—1491no—
219C42C22×D14911yescharacteristic
220C23—22no—
221C23—1491no—
222Q8—21no—
223Q8—21no—
224Q8—2981no—
225Q8—2981no—
226Q8—5962no—
227Q8—5962no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—42no—
232D4—5964no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—2981no—
236C2×C4—2981no—
237C2×C4—2981no—
238C2×C4—2981no—
239C2×C4—2982no—
240C2×C4—5961no—
241C2×C4—5964no—
242C2×C4—1491no—
243C2×C4—1491no—
244C2×C4—1491no—
245C2×C4—1494no—
246C2×C4D4×D14911yescharacteristic
247C2×C4D4×D14911yescharacteristic
248C2×C4C23×D14911yescharacteristic
249C8—22no—
250C8—2982no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—2981no—
255C22—2981no—
256C22—1492no—
257C22C2×D4×D14911yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—2981no—
262C4—2981no—
263C4—5961no—
264C4—5961no—
265C4—5962no—
266C4—1492no—
267C4SD16×D14912yes—
268C4#1801195611yescharacteristic
269C4#1801195811yescharacteristic
270C2—82no—
271C2—2982no—
272C2#2375020411yescharacteristic
273C2C2×SD16×D14911yescharacteristic
274C2#2375042111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#460×C149C2
socleC22×C149C22×D4
prime radical#23750173C2
hypercenter#460D149
hypocenterC149#3246
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #23750179C2semidirect
3. #23749913C2semidirect
4. #23749911C2semidirect
5. #23749910C2semidirect
6. #23749853C2semidirect
7. #23749851C2semidirect
8. #23749828C2semidirect
9. #23749822C2semidirect
10. C4⋊C8×D149C2semidirect
11. #460×C149C2semidirect
98. C4×D149SD16semidirect
137. C149#3246semidirect
148. #460D149semidirect

Maximal subgroups

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

subgroupindex
#237501792
#237501732
#237499132
#237499112
#237499102
#237498532
#237498512
#237498282
#237498222
C4⋊C8×D1492
#460×C1492
#3246149