Finite Group Databasebeta

← #34056019

Subgroups — #34056019

Order 19072.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#23750176C211yesmaximal, characteristic, complemented
2#23750157C211yesmaximal, characteristic, complemented
3#23749924C212yesmaximal, complemented
4#23749917C211yesmaximal, characteristic, complemented
5#23749860C212yesmaximal, complemented
6#23749855C212yesmaximal, complemented
7#23749839C212yesmaximal, complemented
8#23749826C211yesmaximal, characteristic, complemented
9#23749820C211yesmaximal, characteristic, complemented
10C4⋊C8×D149C211yesmaximal, characteristic, complemented
11#476×C149C211yesmaximal, characteristic, complemented
12C2×#14558737—21no—
13C2×#14558725C2212yes—
14C2×#14558721C2212yes—
15C2×C8×D149C2212yes—
16#18011908C2211yescharacteristic
17#18011904—21no—
18#18011902C2211yescharacteristic
19#18011900—21no—
20#18011898C2211yescharacteristic
21#18011893—21no—
22#18011891C2211yescharacteristic
23#18011886—21no—
24#18011886C2212yes—
25#18011882C2211yescharacteristic
26C4×Dic298C2211yescharacteristic
27#18011873—21no—
28#18011873C2212yes—
29C4×C22⋊D149—21no—
30#18011870—21no—
31C42×D149C2211yescharacteristic
32#18011844C2212yes—
33C4⋊#11924746C2211yescharacteristic
34#18011840C2212yes—
35#18011834C2212yes—
36#18011833C2212yes—
37#18011831C2212yes—
38C4⋊Q8×C149C2211yescharacteristic
39C4×D4×C149C2211yescharacteristic
40#106×C149C2212yes—
41C4⋊C8×C149C2211yescharacteristic
42#102×C149C2212yes—
43#14558737—44no—
44C2×Dic298C2311yescharacteristic
45C2×D596—22no—
46C2×D596—21no—
47C2×D596—21no—
48C2×D596C2312yes—
49C2×C22⋊D149—21no—
50C2×C4×D149—22no—
51C2×C4×D149D411yescharacteristic
52C2×C4×D149D411yescharacteristic
53C2×C4×D149C2311yescharacteristic
54#14558725—42no—
55#14558721—42no—
56C2×#11924746C2312yes—
57C8×D149—42no—
58C4⋊Dic149—21no—
59C4⋊Dic149—21no—
60C4⋊Dic149C2311yescharacteristic
61#14558715—21no—
62#14558715—21no—
63C22⋊Dic149—21no—
64C22⋊Dic149—21no—
65C22⋊Dic149—21no—
66#14558713—21no—
67#14558713—22no—
68#14558713—22no—
69C4×Dic149—21no—
70C4×Dic149D411yescharacteristic
71C4×Dic149D411yescharacteristic
72C4×Dic149C2311yescharacteristic
73C2×Q8×C149—21no—
74C2×D4×C149C2311yescharacteristic
75C22×C4×C149—21no—
76C2×C8×C149C2312yes—
77C4⋊C4×C149—21no—
78C4⋊C4×C149C2312yes—
79C4⋊C4×C149C2311yescharacteristic
80C22⋊C4×C149—21no—
81C42×C149C2311yescharacteristic
82C22×D149—22no—
83C22×D149C2×D411yescharacteristic
84Dic298—21no—
85Dic298—21no—
86D596—22no—
87D596—22no—
88D596—42no—
89D596—44no—
90D596—42no—
91C22⋊D149—42no—
92C2×Dic149—21no—
93C2×Dic149—21no—
94C2×Dic149C2×D411yescharacteristic
95C2×Dic149C2×D411yescharacteristic
96C2×Dic149C2×D411yescharacteristic
97C4×D149—22no—
98C4×D149—22no—
99C4×D149—21no—
100C4×D149—21no—
101C4×D149—44no—
102#11924746—22no—
103C23×C149—21no—
104Q8×C149—42no—
105D4×C149—21no—
106D4×C149—21no—
107C2×C4×C149—22no—
108C2×C4×C149—21no—
109C2×C4×C149—42no—
110C2×C4×C149C2×D411yescharacteristic
111C2×C4×C149C2×D411yescharacteristic
112C2×C4×C149C2411yescharacteristic
113C8×C149—22no—
114C22×C149—41no—
115C22×C149—41no—
116C22×C149—41no—
117C22×C149C22×D411yescharacteristic
118C2×D149—21no—
119C2×D149—21no—
120C2×D149—42no—
121C2×D149—42no—
122C2×D149—42no—
123C2×D149AGL(1,ℤ/8)12yes—
124C4×C149—21no—
125C4×C149—41no—
126C4×C149—42no—
127C4×C149#13512yes—
128C4×C149H(5,𝔽2)11yescharacteristic
129C4×C1492−1+411yescharacteristic
130Dic149—21no—
131Dic149—21no—
132Dic149—21no—
133Dic149—41no—
134Dic149—41no—
135C2×C149—81no—
136C2×C149#53711yescharacteristic
137C2×C149C2×#13511yescharacteristic
138C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
139D149—22no—
140D149—82no—
141C149#324711yescharacteristic, complemented, Sylow
142#3247—1491nomaximal, Sylow
143#532—1491no—
144#529—1491no—
145#478—1492no—
146#477—1491no—
147#476D14911yescharacteristic, complemented
148#466—1492no—
149#465—1492no—
150#463—1492no—
151#460—1491no—
152#459—1491no—
153C2×C4⋊C8—1491no—
154C2×#41—2981no—
155C2×SD16—1492no—
156C2×D8—1492no—
157C22×C8—1492no—
158C4⋊Q8C2×D14911yescharacteristic
159#127—1491no—
160#126—2981no—
161#125—1491no—
162#124—2981no—
163#124—1491no—
164#123—2981no—
165C22⋊Q8—1491no—
166#121—2981no—
167#121—1491no—
168#121—1492no—
169C4×Q8—1491no—
170C4×D4—2981no—
171C4×D4—2981no—
172C4×D4—1492no—
173C4×D4C2×D14911yescharacteristic
174#117—2981no—
175C2×C42—1491no—
176#107—1492no—
177#106C2×D14912yes—
178C4⋊C8—1492no—
179C4⋊C8—1491no—
180C4⋊C8C2×D14911yescharacteristic
181#103—1492no—
182#102—1492no—
183#102—1492no—
184#102C2×D14912yes—
185#41—5964no—
186C2×Q8—21no—
187C2×Q8—1491no—
188C2×D4—2982no—
189C2×D4—2981no—
190C2×D4—2981no—
191C2×D4—2981no—
192C2×D4—1492no—
193C2×D4C22×D14911yescharacteristic
194C22×C4—21no—
195C22×C4—2982no—
196C22×C4—1491no—
197C22×C4—1491no—
198C22×C4—1491no—
199SD16—5962no—
200D8—5962no—
201C2×C8—5962no—
202C2×C8—1492no—
203C2×C8C22×D14912yes—
204C4⋊C4—21no—
205C4⋊C4—2981no—
206C4⋊C4—2981no—
207C4⋊C4—2981no—
208C4⋊C4—2981no—
209C4⋊C4—1491no—
210C4⋊C4C22×D14911yescharacteristic
211C4⋊C4C22×D14912yes—
212C22⋊C4—21no—
213C22⋊C4—2981no—
214C22⋊C4—2982no—
215C22⋊C4—2982no—
216C22⋊C4—2981no—
217C22⋊C4—2981no—
218C22⋊C4—2981no—
219C42—2981no—
220C42—1491no—
221C42—1491no—
222C42—1491no—
223C42C22×D14911yescharacteristic
224C23—21no—
225C23—2982no—
226C23—1491no—
227Q8—42no—
228Q8—2981no—
229Q8—2981no—
230D4—21no—
231D4—21no—
232D4—2982no—
233D4—2982no—
234D4—5962no—
235D4—5964no—
236D4—5962no—
237D4—5962no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—2982no—
242C2×C4—2982no—
243C2×C4—2981no—
244C2×C4—2981no—
245C2×C4—2981no—
246C2×C4—2981no—
247C2×C4—5964no—
248C2×C4—1491no—
249C2×C4—1491no—
250C2×C4—1491no—
251C2×C4D4×D14911yescharacteristic
252C2×C4D4×D14911yescharacteristic
253C2×C4C23×D14911yescharacteristic
254C8—22no—
255C8—2982no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—2981no—
260C22—2981no—
261C22—5962no—
262C22—5962no—
263C22—5962no—
264C22—1492no—
265C22C2×D4×D14911yescharacteristic
266C4—21no—
267C4—41no—
268C4—42no—
269C4—2981no—
270C4—2981no—
271C4—2981no—
272C4—5961no—
273C4—5961no—
274C4#1801193212yes—
275C4#1801195711yescharacteristic
276C4#1801195911yescharacteristic
277C2—81no—
278C2—2982no—
279C2—11922no—
280C2#2375020511yescharacteristic
281C2C2×#1801193211yescharacteristic
282C2AGL(1,ℤ/8)×D14911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#476×C149C2
socleC22×C149C22×D4
hypercenter#476D149
hypocenterC149#3247
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #23750176C2semidirect
2. #23750157C2semidirect
3. #23749924C2semidirect
4. #23749917C2semidirect
5. #23749860C2semidirect
6. #23749855C2semidirect
7. #23749839C2semidirect
8. #23749826C2semidirect
9. #23749820C2semidirect
10. C4⋊C8×D149C2semidirect
11. #476×C149C2semidirect
141. C149#3247semidirect
147. #476D149semidirect

Maximal subgroups

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

subgroupindex
#237501762
#237501572
#237499242
#237499172
#237498602
#237498552
#237498392
#237498262
#237498202
C4⋊C8×D1492
#476×C1492
#3247149