Finite Group Databasebeta

← #34077137

Subgroups — #34077137

Order 19328.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#18012713C211yesmaximal, characteristic
2#23752147C212yesmaximal, complemented
3#23752142C212yesmaximal, complemented
4#23752129C212yesmaximal, complemented
5#23752127C214yesmaximal, complemented
6#23752123C212yesmaximal, complemented
7#23752075C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C151C211yesmaximal, characteristic, complemented
9C22×D604C2211yescharacteristic
10C22×C22⋊D151C2212yes—
11#18012713—24no—
12C2×C4⋊Dic151C2211yescharacteristic
13C2×Dic151⋊C4C2214yes—
14C2×C22⋊Dic151C2212yes—
15C2×(C2×D151)⋊C4C2212yes—
16C2×(C2×D151)⋊C4C2214yes—
17C2×(C2×D151)⋊C4C2214yes—
18C151⋊#95C2212yes—
19C151⋊#95C2212yes—
20C151⋊#95C2214yes—
21C22×Q8×C151C2211yescharacteristic
22C23×C4×C151C2211yescharacteristic
23C22⋊Q8×C151—24no—
24C2×C4⋊C4×C151C2212yes—
25C2×C4⋊C4×C151C2211yescharacteristic
26C2×C22⋊C4×C151C2212yes—
27C23×D151C2312yes—
28C2×D604—22no—
29C2×D604—42no—
30C2×C22⋊D151—24no—
31C2×C22⋊D151—44no—
32C22×Dic151C2314yes—
33C22×Dic151C2312yes—
34C4⋊Dic151—22no—
35Dic151⋊C4—44no—
36C22⋊Dic151—42no—
37(C2×D151)⋊C4—24no—
38(C2×D151)⋊C4—44no—
39(C2×D151)⋊C4—44no—
40C24×C151C2311yescharacteristic
41C2×Q8×C151—22no—
42C2×Q8×C151—42no—
43C22×C4×C151—42no—
44C22×C4×C151D414yes—
45C22×C4×C151C2312yes—
46C22×C4×C151C2312yes—
47C22×C4×C151C2311yescharacteristic
48C22×C4×C151C2311yescharacteristic
49C4⋊C4×C151—24no—
50C4⋊C4×C151—22no—
51C22⋊C4×C151—24no—
52C22×D151—22no—
53C22×D151—24no—
54C22×D151—24no—
55C22×D151—24no—
56D604—44no—
57C22⋊D151—48no—
58C2×Dic151—22no—
59C2×Dic151—24no—
60C2×Dic151—24no—
61C2×Dic151—24no—
62C2×Dic151—24no—
63C23×C151—22no—
64C23×C151—22no—
65C23×C151—22no—
66C23×C151C2×D412yes—
67C23×C151C2411yescharacteristic
68Q8×C151—44no—
69C2×C4×C151—22no—
70C2×C4×C151—22no—
71C2×C4×C151—22no—
72C2×C4×C151—22no—
73C2×C4×C151—24no—
74C2×C4×C151—22no—
75C2×C4×C151—22no—
76C2×C4×C151—44no—
77C2×C4×C151C2×D412yes—
78C2×C4×C151C2×D412yes—
79C22×C151—22no—
80C22×C151—22no—
81C22×C151—22no—
82C22×C151—44no—
83C22×C151C22×D411yescharacteristic
84C22×C151H(5,𝔽2)12yes—
85C22×C151H(5,𝔽2)12yes—
86C22×C1512−1+412yes—
87C2×D151—42no—
88C2×D151—42no—
89C2×D151—42no—
90C2×D151—48no—
91C4×C151—22no—
92C4×C151—22no—
93C4×C151—42no—
94C4×C151—42no—
95Dic151—42no—
96Dic151—44no—
97C2×C151—42no—
98C2×C151#53411yescharacteristic
99C2×C151#53511yescharacteristic
100C2×C151#53611yescharacteristic
101C2×C151#56214yes—
102D151—82no—
103C151#281611yescharacteristic, complemented, Sylow
104#2816—1511nomaximal, Sylow
105C2×C22⋊Q8D15111yescharacteristic, complemented
106C2×#121—1511no—
107#399—1512no—
108#398—1512no—
109#394—1512no—
110#394—1514no—
111#393—1512no—
112#381—1511no—
113C22×Q8C2×D15111yescharacteristic
114C22×D4—1512no—
115C22×D4—1511no—
116C23×C4C2×D15111yescharacteristic
117C22⋊Q8—24no—
118#121—3024no—
119C2×C4⋊C4—1511no—
120C2×C4⋊C4—1514no—
121C2×C4⋊C4C2×D15112yes—
122C2×C4⋊C4C2×D15111yescharacteristic
123C2×C22⋊C4—1512no—
124C2×C22⋊C4—1512no—
125C2×C22⋊C4—1514no—
126C2×C22⋊C4—1514no—
127C2×C22⋊C4C2×D15112yes—
128#95—1512no—
129#95—1512no—
130#95—1514no—
131C24—1512no—
132C24C22×D15111yescharacteristic
133C2×Q8—22no—
134C2×Q8—42no—
135C2×D4—3024no—
136C2×D4—3022no—
137C2×D4—6042no—
138C2×D4—6044no—
139C22×C4—42no—
140C22×C4—1514no—
141C22×C4—1512no—
142C22×C4C22⋊D15114yes—
143C22×C4C22×D15112yes—
144C22×C4C22×D15112yes—
145C22×C4C22×D15111yescharacteristic
146C22×C4C22×D15111yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—3022no—
150C4⋊C4—6044no—
151C22⋊C4—24no—
152C22⋊C4—3024no—
153C22⋊C4—6044no—
154C22⋊C4—6044no—
155C22⋊C4—6042no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—3022no—
160C23—3024no—
161C23—3024no—
162C23—3024no—
163C23C2×C22⋊D15112yes—
164C23C23×D15111yescharacteristic
165Q8—44no—
166D4—6044no—
167D4—6048no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—3022no—
177C2×C4—3024no—
178C2×C4—3024no—
179C2×C4—3024no—
180C2×C4—3024no—
181C2×C4C2×C22⋊D15112yes—
182C2×C4C2×C22⋊D15112yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—6042no—
188C22—6042no—
189C22—6042no—
190C22—6048no—
191C22C22×C22⋊D15111yescharacteristic
192C22#1801278412yes—
193C22#1801278512yes—
194C22#1801278712yes—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—6042no—
200C4—6044no—
201C2—42no—
202C2—12082no—
203C2#2375277111yescharacteristic
204C2#2375277611yescharacteristic
205C2#2375278611yescharacteristic
206C2#2375295814yes—
subgroupG/H
center, FrattiniC23C23×D151
commutator, socle, cosocleC23×C151C24
FittingC2×C22⋊Q8×C151C2
prime radicalC2×#18012713C2
hypercenterC2×C22⋊Q8D151
hypocenterC151#2816
LCSG ▷ C23×C151 ▷ C151
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #23752147C2semidirect
3. #23752142C2semidirect
4. #23752129C2semidirect
5. #23752127C2semidirect
6. #23752123C2semidirect
7. #23752075C2semidirect
8. C2×C22⋊Q8×C151C2semidirect
103. C151#2816semidirect
105. C2×C22⋊Q8D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127132
#237521472
#237521422
#237521292
#237521272
#237521232
#237520752
C2×C22⋊Q8×C1512
#2816151