Finite Group Databasebeta

← #34081759

Subgroups — #34081759

Order 19328.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#23752935C211yesmaximal, characteristic, complemented
2#23752569C212yesmaximal, complemented
3#23752564C211yesmaximal, characteristic, complemented
4#23752516C212yesmaximal, complemented
5#23752499C211yesmaximal, characteristic, complemented
6#23752427C212yesmaximal
7#23752417C211yesmaximal, characteristic, complemented
8Q8×C151⋊C8C211yesmaximal, characteristic, complemented
9C4×#14558992C212yesmaximal, complemented
10C4×#14558989C211yesmaximal, characteristic
11#554×C151C211yesmaximal, characteristic, complemented
12C2×#14558992C2212yes—
13C2×#14558989C2211yescharacteristic
14#18012739C2211yescharacteristic
15#18012734—22no—
16#18012727C2212yes—
17#18012715—21no—
18C4×Dic302C2212yes—
19C4×D604C2211yescharacteristic
20#18012697—21no—
21#18012672C2211yescharacteristic
22#18012670C2212yes—
23C4⋊(C151⋊C8)C2211yescharacteristic
24C4⋊(C151⋊C8)C2212yes—
25#18012662C2212yes—
26#18012662C2212yes—
27#18012662C2212yes—
28#18012660C2211yescharacteristic
29#18012659C2212yes—
30#18012659C2211yescharacteristic
31C4×C151⋊C8C2211yescharacteristic
32C4×C151⋊C8C2212yes—
33C4⋊Q8×C151C2211yescharacteristic
34#125×C151C2212yes—
35C22⋊Q8×C151—22no—
36C22⋊Q8×C151—21no—
37C4×Q8×C151C2211yescharacteristic
38C4×D4×C151C2211yescharacteristic
39C4×D4×C151C2212yes—
40C2×C4⋊C4×C151—21no—
41C2×Dic302C2312yes—
42C2×D604C2311yescharacteristic
43C2×C4×D151—21no—
44#14558992—24no—
45#14558989—22no—
46C2×C151⋊C8C2311yescharacteristic
47C2×C151⋊C8C2312yes—
48C2×C151⋊C8C2311yescharacteristic
49C4⋊Dic151—21no—
50C4⋊Dic151—21no—
51C4⋊Dic151C2311yescharacteristic
52Dic151⋊C4—22no—
53(C2×D151)⋊C4—22no—
54(C2×D151)⋊C4—21no—
55(C2×D151)⋊C4—22no—
56C4×Dic151—22no—
57C2×Q8×C151—21no—
58C2×Q8×C151D411yescharacteristic
59C2×D4×C151C2311yescharacteristic
60C22×C4×C151—21no—
61C22×C4×C151—22no—
62C4⋊C4×C151—22no—
63C4⋊C4×C151—21no—
64C4⋊C4×C151—22no—
65C4⋊C4×C151D411yescharacteristic
66C4⋊C4×C151D412yes—
67C4⋊C4×C151C2311yescharacteristic
68C4⋊C4×C151C2312yes—
69C22⋊C4×C151—22no—
70C22⋊C4×C151—21no—
71C42×C151C2311yescharacteristic
72C42×C151C2312yes—
73C22×D151—21no—
74Dic302—22no—
75Dic302—22no—
76D604—21no—
77D604—21no—
78C2×Dic151—21no—
79C2×Dic151—22no—
80C4×D151—42no—
81C151⋊C8—21no—
82C151⋊C8—22no—
83C151⋊C8—21no—
84C23×C151—21no—
85Q8×C151—22no—
86Q8×C151—81no—
87D4×C151—21no—
88D4×C151#4112yes—
89C2×C4×C151—22no—
90C2×C4×C151—21no—
91C2×C4×C151—42no—
92C2×C4×C151—82no—
93C2×C4×C151C2×D411yescharacteristic
94C2×C4×C151C2×D412yes—
95C2×C4×C151C2×D412yes—
96C2×C4×C151C2×D411yescharacteristic
97C2×C4×C151C2411yescharacteristic
98C22×C151—22no—
99C22×C151—41no—
100C22×C151—41no—
101C22×C151C22×D411yescharacteristic
102C2×D151—41no—
103C2×D151—41no—
104C2×D151—41no—
105C4×C151—22no—
106C4×C151—22no—
107C4×C151—21no—
108C4×C151—41no—
109C4×C151—42no—
110C4×C151#13512yes—
111C4×C151C2×#4111yescharacteristic
112C4×C1512−1+411yescharacteristic
113Dic151—42no—
114Dic151—41no—
115C2×C151—42no—
116C2×C151#54811yescharacteristic
117C2×C151C2×#13511yescharacteristic
118C2×C151#57711yescharacteristic
119D151—81no—
120C151#336011yescharacteristic, complemented, Sylow
121#3360—1511nomaximal, Sylow
122#556—1511no—
123#554D15111yescharacteristic, complemented
124#488—1512no—
125#487—1511no—
126#479—1512no—
127#476—1511no—
128#463—1512no—
129#461—1511no—
130C8×Q8—1511no—
131C4×SD16—1512no—
132C4×D8—1511no—
133C2×SD16—1512no—
134C2×D8—1511no—
135C4⋊Q8—1511no—
136C4⋊Q8C2×D15111yescharacteristic
137#126—3022no—
138#125C2×D15112yes—
139#124—1512no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—3021no—
143C4×Q8—1512no—
144C4×Q8C2×D15111yescharacteristic
145C4×D4—1511no—
146C4×D4C2×D15111yescharacteristic
147C4×D4C2×D15112yes—
148#117—3021no—
149C2×C4⋊C4—21no—
150#107—1511no—
151#106—1512no—
152C4⋊C8—1511no—
153C4⋊C8—1512no—
154#103—1512no—
155#103—1512no—
156#103—1512no—
157#102—1511no—
158#102—1511no—
159#102—1512no—
160C4×C8—1511no—
161C4×C8—1512no—
162C2×Q8—21no—
163C2×Q8—1512no—
164C2×Q8C22⋊D15111yescharacteristic
165C2×D4—1511no—
166C2×D4C22×D15111yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—3021no—
170SD16—3024no—
171D8—3022no—
172C2×C8—1511no—
173C2×C8—1512no—
174C2×C8—1511no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—3022no—
179C4⋊C4—3021no—
180C4⋊C4—3021no—
181C4⋊C4—1511no—
182C4⋊C4C22⋊D15111yescharacteristic
183C4⋊C4C22⋊D15112yes—
184C4⋊C4C22×D15112yes—
185C4⋊C4C22×D15111yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—3022no—
189C22⋊C4—3021no—
190C22⋊C4—3022no—
191C42—3022no—
192C42C22×D15111yescharacteristic
193C42C22×D15112yes—
194C23—21no—
195C23—3021no—
196Q8—22no—
197Q8—81no—
198Q8—3022no—
199Q8—3022no—
200D4—21no—
201D4—3021no—
202D4—3021no—
203D4#1455900512yes—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—3021no—
209C2×C4—3022no—
210C2×C4—6042no—
211C2×C4C2×C22⋊D15112yes—
212C2×C4C2×C22⋊D15112yes—
213C2×C4C2×C22⋊D15111yescharacteristic
214C2×C4C2×C22⋊D15111yescharacteristic
215C2×C4C23×D15111yescharacteristic
216C8—3021no—
217C8—3022no—
218C8—3021no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—6041no—
223C22—6041no—
224C22—6041no—
225C22C22×C22⋊D15111yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—6041no—
232C4—6042no—
233C4#1801275912yes—
234C4C2×#1455900511yescharacteristic
235C4#1801278711yescharacteristic
236C2—42no—
237C2—12081no—
238C2#2375287311yescharacteristic
239C2C2×#1801275911yescharacteristic
240C2#2375302311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#554×C151C2
socleC22×C151C22×D4
prime radicalC2×#14558989C22
hypercenter#554D151
hypocenterC151#3360
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #23752935C2semidirect
2. #23752569C2semidirect
3. #23752564C2semidirect
4. #23752516C2semidirect
5. #23752499C2semidirect
7. #23752417C2semidirect
8. Q8×C151⋊C8C2semidirect
9. C4×#14558992C2semidirect
11. #554×C151C2semidirect
120. C151#3360semidirect
123. #554D151semidirect

Maximal subgroups

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

subgroupindex
#237529352
#237525692
#237525642
#237525162
#237524992
#237524272
#237524172
Q8×C151⋊C82
C4×#145589922
C4×#145589892
#554×C1512
#3360151