Finite Group Databasebeta

← #34055876

Subgroups — #34055876

Order 19072.

368 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 111 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#23750173C211yesmaximal, characteristic
2#125×D149C211yesmaximal, characteristic, complemented
3#23750035C211yesmaximal, characteristic, complemented
4#23750034C211yesmaximal, characteristic, complemented
5#23749989C211yesmaximal, characteristic, complemented
6#23749942C212yesmaximal, complemented
7#23749937C212yesmaximal, complemented
8#23749927C212yesmaximal, complemented
9#23749923C212yesmaximal, complemented
10#23749680C211yesmaximal, characteristic, complemented
11#490×C149C211yesmaximal, characteristic, complemented
12C2×#14558735—21no—
13#18011911C2211yescharacteristic
14#18011903C2211yescharacteristic
15#18011902C2211yescharacteristic
16#18011902C2211yescharacteristic
17#18011901—21no—
18#18011901—21no—
19#18011898—21no—
20#18011882—21no—
21#18011882C2212yes—
22D4×Dic149—21no—
23D4×Dic149C2212yes—
24C4⋊C4×D149—21no—
25C4⋊C4×D149—21no—
26C4⋊C4×D149C2211yescharacteristic
27C4⋊C4×D149C2211yescharacteristic
28C42×D149C2211yescharacteristic
29#18011845C2212yes—
30#18011844C2212yes—
31#18011843C2212yes—
32#18011842C2212yes—
33#18011840C2212yes—
34#18011832C2212yes—
35#18011832C2212yes—
36#18011823C2212yes—
37#18011822C2211yescharacteristic
38#127×C149C2211yescharacteristic
39#125×C149C2211yescharacteristic
40#102×C149C2212yes—
41#102×C149C2212yes—
42#97×C149C2211yescharacteristic
43#14558735—44no—
44C2×Dic298—21no—
45C2×C22⋊D149—22no—
46C22×Dic149—22no—
47C2×C4×D149—21no—
48C2×C4×D149—21no—
49C2×C4×D149D411yescharacteristic
50C2×C4×D149D411yescharacteristic
51C2×C4×D149C2311yescharacteristic
52C2×#11924746C2312yes—
53C4⋊Dic149—21no—
54C4⋊Dic149—21no—
55C4⋊Dic149—21no—
56C4⋊Dic149C2312yes—
57C4⋊Dic149C2311yescharacteristic
58C4⋊Dic149C2311yescharacteristic
59#14558715—21no—
60#14558715—21no—
61#14558715—21no—
62#14558715—21no—
63#14558715—21no—
64#14558715—21no—
65C22⋊Dic149—22no—
66C22⋊Dic149—22no—
67C4×Dic149D411yescharacteristic
68C4×Dic149D411yescharacteristic
69C4×Dic149C2311yescharacteristic
70C2×D4×C149—21no—
71C2×D4×C149—21no—
72C2×D4×C149C2312yes—
73C2×C8×C149C2312yes—
74C4⋊C4×C149—21no—
75C4⋊C4×C149—21no—
76C4⋊C4×C149C2311yescharacteristic
77C4⋊C4×C149C2311yescharacteristic
78C42×C149C2311yescharacteristic
79C22×D149C2×D411yescharacteristic
80Dic298—42no—
81C22⋊D149—44no—
82C2×Dic149—21no—
83C2×Dic149—21no—
84C2×Dic149—22no—
85C2×Dic149—44no—
86C2×Dic149C2×D411yescharacteristic
87C2×Dic149C2×D411yescharacteristic
88C2×Dic149C2×D411yescharacteristic
89C4×D149—22no—
90C4×D149—22no—
91C4×D149—41no—
92C4×D149—42no—
93C4×D149—42no—
94#11924746—22no—
95C23×C149—22no—
96D4×C149—24no—
97D4×C149—42no—
98D4×C149—42no—
99C2×C4×C149—21no—
100C2×C4×C149—21no—
101C2×C4×C149C2×D411yescharacteristic
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2411yescharacteristic
104C8×C149—22no—
105C22×C149—42no—
106C22×C149—42no—
107C22×C149—42no—
108C22×C149C22×D411yescharacteristic
109C2×D149—21no—
110C2×D149—21no—
111C2×D149AGL(1,ℤ/8)12yes—
112C4×C149—21no—
113C4×C149—21no—
114C4×C149—41no—
115C4×C149—41no—
116C4×C1492−1+412yes—
117Dic149—21no—
118Dic149—21no—
119Dic149—21no—
120Dic149—41no—
121Dic149—42no—
122Dic149—41no—
123C2×C149—82no—
124C2×C149#53611yescharacteristic
125C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C149#57511yescharacteristic
127D149—22no—
128C149#323411yescharacteristic, complemented, Sylow
129#3234—1491nomaximal, Sylow
130#532—1491no—
131C2×#125—1491no—
132#501—1491no—
133#501—1491no—
134#490—1491no—
135#490D14911yescharacteristic, complemented
136#481—1492no—
137#480—1492no—
138#478—1492no—
139#478—1492no—
140#432—1491no—
141C2×#41—2981no—
142C4⋊Q8—1491no—
143#127C2×D14911yescharacteristic
144#125—2981no—
145#125—2981no—
146#125—1491no—
147#125—1491no—
148#125—1491no—
149#125C2×D14911yescharacteristic
150#124—2981no—
151#121—2981no—
152#121—1492no—
153C4×D4—2981no—
154C4×D4—1492no—
155C2×C4⋊C4—2981no—
156C2×C4⋊C4—2981no—
157C2×C4⋊C4—1491no—
158C2×C4⋊C4—1491no—
159C2×C42—1491no—
160#107—1492no—
161#107—1492no—
162#106—1492no—
163#106—1492no—
164C4⋊C8—1492no—
165#102—1492no—
166#102—1492no—
167#102C2×D14912yes—
168#102C2×D14912yes—
169C22⋊C8—1492no—
170#97—1491no—
171#97C2×D14911yescharacteristic
172#41—5964no—
173C2×Q8—2981no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—2982no—
177C2×D4C22×D14912yes—
178C22×C4—2981no—
179C22×C4—2981no—
180C22×C4—2982no—
181C22×C4—1491no—
182C22×C4—1491no—
183C22×C4—1491no—
184C2×C8—1492no—
185C2×C8C22×D14912yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—2981no—
189C4⋊C4—2981no—
190C4⋊C4—2981no—
191C4⋊C4—2981no—
192C4⋊C4—2981no—
193C4⋊C4—2981no—
194C4⋊C4—2981no—
195C4⋊C4—2981no—
196C4⋊C4—2981no—
197C4⋊C4—1491no—
198C4⋊C4—1491no—
199C4⋊C4—1492no—
200C4⋊C4C22×D14911yescharacteristic
201C4⋊C4C22×D14911yescharacteristic
202C22⋊C4—2982no—
203C22⋊C4—2982no—
204C42—1491no—
205C42—1491no—
206C42—1491no—
207C42C22×D14911yescharacteristic
208C23—22no—
209C23—1491no—
210Q8—5962no—
211D4—24no—
212D4—42no—
213D4—42no—
214D4—5964no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—2982no—
218C2×C4—2982no—
219C2×C4—2982no—
220C2×C4—2981no—
221C2×C4—2981no—
222C2×C4—5961no—
223C2×C4—5962no—
224C2×C4—5962no—
225C2×C4—5964no—
226C2×C4—1491no—
227C2×C4—1491no—
228C2×C4—1491no—
229C2×C4D4×D14911yescharacteristic
230C2×C4D4×D14911yescharacteristic
231C2×C4C23×D14911yescharacteristic
232C8—22no—
233C8—2982no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—2981no—
238C22—2981no—
239C22—1492no—
240C22C2×D4×D14911yescharacteristic
241C4—21no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—2981no—
246C4—2981no—
247C4—2981no—
248C4—5962no—
249C4—5961no—
250C4—5961no—
251C4#1801195812yes—
252C2—82no—
253C2—2982no—
254C2#2375019511yescharacteristic
255C2AGL(1,ℤ/8)×D14911yescharacteristic
256C2#2375042111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#490×C149C2
socleC22×C149C22×D4
prime radical#23750173C2
hypercenter#490D149
hypocenterC149#3234
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #125×D149C2semidirect
3. #23750035C2semidirect
4. #23750034C2semidirect
5. #23749989C2semidirect
6. #23749942C2semidirect
7. #23749937C2semidirect
8. #23749927C2semidirect
9. #23749923C2semidirect
10. #23749680C2semidirect
11. #490×C149C2semidirect
128. C149#3234semidirect
135. #490D149semidirect

Maximal subgroups

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

subgroupindex
#237501732
#125×D1492
#237500352
#237500342
#237499892
#237499422
#237499372
#237499272
#237499232
#237496802
#490×C1492
#3234149