Finite Group Databasebeta

← #34080332

Subgroups — #34080332

Order 19328.

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

normal subgroups only

This group has 362 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
1C4⋊Q8×D151C211yesmaximal, characteristic, complemented
2#23752747C211yesmaximal, characteristic
3#23752622C211yesmaximal, characteristic, complemented
4#23752574C211yesmaximal, characteristic, complemented
5#23752572C211yesmaximal, characteristic, complemented
6#23752544C212yesmaximal, complemented
7#23752527C212yesmaximal
8#23752506C212yesmaximal, complemented
9#23752501C212yesmaximal, complemented
10#23752267C211yesmaximal, characteristic, complemented
11#501×C151C211yesmaximal, characteristic, complemented
12C2×Q8×D151—21no—
13#18012739C2211yescharacteristic
14#18012738—21no—
15#18012738C2212yes—
16#18012737—21no—
17#18012734—21no—
18#18012727C2211yescharacteristic
19#18012724—21no—
20#18012719C2211yescharacteristic
21#18012714C2211yescharacteristic
22#18012704C2211yescharacteristic
23#18012701—21no—
24#18012701C2211yescharacteristic
25#18012697—21no—
26C4⋊C4×D151—21no—
27C4⋊C4×D151C2212yes—
28C42×D151C2211yescharacteristic
29#18012672C2212yes—
30#18012670C2212yes—
31Dic151⋊C8C2212yes—
32#18012664C2212yes—
33#18012662C2212yes—
34#18012661C2212yes—
35#18012659C2212yes—
36(C2×D151)⋊C8C2212yes—
37#18012650C2211yescharacteristic
38C4⋊Q8×C151C2211yescharacteristic
39#125×C151C2211yescharacteristic
40#107×C151C2212yes—
41#106×C151C2212yes—
42#97×C151C2211yescharacteristic
43C2×Dic302—22no—
44C2×Dic302—21no—
45C2×Dic302C2311yescharacteristic
46Q8×D151—44no—
47C2×D604—21no—
48C2×D604C2311yescharacteristic
49C2×C4×D151—22no—
50C2×C4×D151—21no—
51C2×C4×D151D411yescharacteristic
52C2×C4×D151D411yescharacteristic
53C2×C4×D151C2311yescharacteristic
54C2×C151⋊C8C2312yes—
55C4⋊Dic151—21no—
56C4⋊Dic151—21no—
57C4⋊Dic151C2312yes—
58Dic151⋊C4—22no—
59Dic151⋊C4—22no—
60Dic151⋊C4—21no—
61(C2×D151)⋊C4—21no—
62(C2×D151)⋊C4—21no—
63(C2×D151)⋊C4—21no—
64(C2×D151)⋊C4—21no—
65(C2×D151)⋊C4—21no—
66C4×Dic151—21no—
67C4×Dic151D411yescharacteristic
68C4×Dic151D411yescharacteristic
69C4×Dic151C2311yescharacteristic
70C2×Q8×C151—21no—
71C2×C8×C151C2312yes—
72C4⋊C4×C151—21no—
73C4⋊C4×C151—21no—
74C4⋊C4×C151—21no—
75C4⋊C4×C151C2311yescharacteristic
76C4⋊C4×C151C2312yes—
77C4⋊C4×C151C2311yescharacteristic
78C42×C151C2311yescharacteristic
79C22×D151—21no—
80C22×D151C2×D411yescharacteristic
81Dic302—22no—
82Dic302—44no—
83Dic302—42no—
84D604—22no—
85D604—42no—
86C2×Dic151—21no—
87C2×Dic151—22no—
88C2×Dic151C2×D411yescharacteristic
89C2×Dic151C2×D411yescharacteristic
90C2×Dic151C2×D411yescharacteristic
91C4×D151—22no—
92C4×D151—22no—
93C4×D151—22no—
94C4×D151—44no—
95C4×D151—42no—
96C151⋊C8—22no—
97Q8×C151—42no—
98C2×C4×C151—22no—
99C2×C4×C151—21no—
100C2×C4×C151—21no—
101C2×C4×C151C2×D411yescharacteristic
102C2×C4×C151C2×D411yescharacteristic
103C2×C4×C151C2411yescharacteristic
104C8×C151—22no—
105C22×C151C22×D411yescharacteristic
106C2×D151—21no—
107C2×D151—21no—
108C2×D151—41no—
109C2×D151—41no—
110C2×D151—41no—
111C2×D151AGL(1,ℤ/8)12yes—
112C4×C151—21no—
113C4×C151—21no—
114C4×C151—42no—
115C4×C151—41no—
116C4×C151—41no—
117C4×C1512−1+412yes—
118Dic151—21no—
119Dic151—21no—
120Dic151—41no—
121Dic151—42no—
122Dic151#13712yes—
123C2×C151#53611yescharacteristic
124C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C151C2×#13711yescharacteristic
126D151—22no—
127D151—81no—
128C151#323811yescharacteristic, complemented, Sylow
129#3238—1511nomaximal, Sylow
130C2×C4⋊Q8—1511no—
131#529—1511no—
132#501—1511no—
133#501D15111yescharacteristic, complemented
134#489—1511no—
135#489—1511no—
136#483—1512no—
137#480—1512no—
138#477—1512no—
139#476—1512no—
140#432—1511no—
141C22×Q8—3021no—
142C4⋊Q8—3021no—
143C4⋊Q8—3021no—
144C4⋊Q8—1512no—
145C4⋊Q8—1511no—
146C4⋊Q8C2×D15111yescharacteristic
147#126—3021no—
148#125C2×D15111yescharacteristic
149#124—1511no—
150#123—3021no—
151C22⋊Q8—1511no—
152#121—1511no—
153C4×Q8—1511no—
154C4×D4—3021no—
155C4×D4—1511no—
156#117—3021no—
157C2×C4⋊C4—3021no—
158C2×C4⋊C4—1512no—
159C2×C42—1511no—
160#107—1512no—
161#107C2×D15112yes—
162#106—1512no—
163#106C2×D15112yes—
164C4⋊C8—1512no—
165#103—1512no—
166#103—1512no—
167#102—1512no—
168#102—1512no—
169C22⋊C8—1512no—
170#97—1511no—
171#97C2×D15111yescharacteristic
172C2×Q8—21no—
173C2×Q8—3022no—
174C2×Q8—3021no—
175C2×Q8—6044no—
176C2×Q8—1511no—
177C2×D4—3021no—
178C2×D4—1511no—
179C22×C4—3022no—
180C22×C4—3021no—
181C22×C4—1511no—
182C22×C4—1511no—
183C22×C4—1511no—
184C2×C8—1512no—
185C2×C8C22×D15112yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—3022no—
190C4⋊C4—3022no—
191C4⋊C4—3021no—
192C4⋊C4—3021no—
193C4⋊C4—3021no—
194C4⋊C4—1512no—
195C4⋊C4C22×D15112yes—
196C4⋊C4C22×D15111yescharacteristic
197C4⋊C4C22×D15111yescharacteristic
198C22⋊C4—3021no—
199C22⋊C4—3021no—
200C22⋊C4—3021no—
201C22⋊C4—3021no—
202C22⋊C4—3021no—
203C42—3021no—
204C42—1511no—
205C42—1511no—
206C42—1511no—
207C42C22×D15111yescharacteristic
208C23—3021no—
209C23—1511no—
210Q8—42no—
211Q8—3022no—
212Q8—6042no—
213Q8—6044no—
214D4—3022no—
215D4—6042no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—3022no—
220C2×C4—3022no—
221C2×C4—3022no—
222C2×C4—3021no—
223C2×C4—3022no—
224C2×C4—6044no—
225C2×C4—6042no—
226C2×C4—1511no—
227C2×C4—1511no—
228C2×C4—1511no—
229C2×C4D4×D15111yescharacteristic
230C2×C4D4×D15111yescharacteristic
231C2×C4C23×D15111yescharacteristic
232C8—22no—
233C8—3022no—
234C22—3021no—
235C22—3021no—
236C22—6041no—
237C22—6041no—
238C22—6041no—
239C22—1512no—
240C22C2×D4×D15111yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—3021no—
247C4—3021no—
248C4—6042no—
249C4—6041no—
250C4—1512no—
251C4#1801278712yes—
252C2—3022no—
253C2—12081no—
254C2#2375278711yescharacteristic
255C2AGL(1,ℤ/8)×D15111yescharacteristic
256C2#137×D15111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#501×C151C2
socleC22×C151C22×D4
prime radical#18012714C22
hypercenter#501D151
hypocenterC151#3238
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D151C2semidirect
3. #23752622C2semidirect
4. #23752574C2semidirect
5. #23752572C2semidirect
6. #23752544C2semidirect
8. #23752506C2semidirect
9. #23752501C2semidirect
10. #23752267C2semidirect
11. #501×C151C2semidirect
128. C151#3238semidirect
133. #501D151semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1512
#237527472
#237526222
#237525742
#237525722
#237525442
#237525272
#237525062
#237525012
#237522672
#501×C1512
#3238151