Finite Group Databasebeta

← #31779232

Subgroups — #31779232

Order 17536.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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#22577460C211yesmaximal, characteristic, complemented
2C4×#14555494C211yesmaximal, characteristic, complemented
3D4⋊#11764040C214yesmaximal, complemented
4C2×C4×D4×C137C211yesmaximal, characteristic
5C2×#14555494—21no—
6C4⋊#11764040C2212yescomplemented
7C22⋊#11764040—21no—
8#17983999—21no—
9#17983999—21no—
10C4×#11764040C2212yescomplemented
11C22×D4×C137C411yescharacteristic
12C23×C4×C137—21no—
13C4×D4×C137—21no—
14C4×D4×C137—21no—
15C4×D4×C137C412yes—
16C4×D4×C137C2212yes—
17C2×C4⋊C4×C137C411yescharacteristic
18C2×C22⋊C4×C137—21no—
19C2×C42×C137C2211yescharacteristic
20#14555494—24no—
21C2×#11764040—22no—
22C2×#11764040—22no—
23C24×C137—21no—
24C2×D4×C137—24no—
25C2×D4×C137—21no—
26C2×D4×C137C2×C412yes—
27C22×C4×C137—21no—
28C22×C4×C137—22no—
29C22×C4×C137—22no—
30C22×C4×C137—44no—
31C22×C4×C137C2×C411yescharacteristic
32C22×C4×C137D411yescharacteristic
33C22×C4×C137D411yescharacteristic
34C4⋊C4×C137—21no—
35C4⋊C4×C137C2×C412yes—
36C22⋊C4×C137—22no—
37C22⋊C4×C137—22no—
38C42×C137C2×C411yescharacteristic
39C42×C137D411yescharacteristic
40C42×C137D411yescharacteristic
41C42×C137C2311yescharacteristic
42#11764040—24no—
43#11764040—42no—
44C23×C137—24no—
45C23×C137—22no—
46C23×C137—42no—
47C23×C137—42no—
48C23×C137C22⋊C411yescharacteristic
49D4×C137—24no—
50D4×C137—22no—
51D4×C137M1614yescomplemented
52C2×C4×C137—22no—
53C2×C4×C137—22no—
54C2×C4×C137—22no—
55C2×C4×C137—22no—
56C2×C4×C137—44no—
57C2×C4×C137—42no—
58C2×C4×C137—44no—
59C2×C4×C137C22⋊C411yescharacteristic
60C2×C4×C137C22⋊C411yescharacteristic
61C2×C4×C137C22⋊C411yescharacteristic
62C2×C4×C137C22×C411yescharacteristic
63C2×C4×C137C2×D411yescharacteristic
64C2×C4×C137C2×D411yescharacteristic
65C22×C137—21no—
66C22×C137—21no—
67C22×C137—24no—
68C22×C137—48no—
69C22×C137—42no—
70C22×C137—42no—
71C22×C137#10412yes—
72C22×C137C2×C22⋊C411yescharacteristic
73C4×C137—21no—
74C4×C137—21no—
75C4×C137—21no—
76C4×C137—21no—
77C4×C137—42no—
78C4×C137C2×M1612yes—
79C4×C137AGL(1,ℤ/8)12yes—
80C2×C137—22no—
81C2×C137—44no—
82C2×C137#40711yescharacteristic
83C2×C137#41811yescharacteristic
84C2×C137C2×#10411yescharacteristic
85C137#149011yescharacteristic, complemented, Sylow
86#1490—1371nomaximal, Sylow
87C2×C4×D4D13711yescharacteristic
88#432—1371no—
89C4×M16—1371no—
90D4⋊C8—1374no—
91C22×D4Dic13711yescharacteristic
92C23×C4—21no—
93C2×M16—2741no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4Dic13712yes—
97C4×D4C2×D13712yes—
98C2×C4⋊C4Dic13711yescharacteristic
99C2×C22⋊C4—21no—
100C2×C42C2×D13711yescharacteristic
101C4⋊C8—1372no—
102C22⋊C8—2741no—
103#97—2741no—
104#97—2741no—
105C4×C8—1372no—
106C24—21no—
107C2×D4—24no—
108C2×D4—21no—
109C2×D4C2×Dic13712yes—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—44no—
114C22×C4C2×Dic13711yescharacteristic
115C22×C4C22⋊D13711yescharacteristic
116C22×C4C22⋊D13711yescharacteristic
117M16—2744no—
118C2×C8—2742no—
119C2×C8—2742no—
120C4⋊C4—21no—
121C4⋊C4C2×Dic13712yes—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C42C2×Dic13711yescharacteristic
125C42C22⋊D13711yescharacteristic
126C42C22⋊D13711yescharacteristic
127C42C22×D13711yescharacteristic
128C23—24no—
129C23—22no—
130C23—42no—
131C23—42no—
132C23C22⋊Dic13711yescharacteristic
133D4—24no—
134D4—22no—
135D4#1455549414yescomplemented
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—44no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4C22⋊Dic13711yescharacteristic
144C2×C4C22⋊Dic13711yescharacteristic
145C2×C4C22⋊Dic13711yescharacteristic
146C2×C4C22×Dic13711yescharacteristic
147C2×C4C2×C22⋊D13711yescharacteristic
148C2×C4C2×C22⋊D13711yescharacteristic
149C8—2744no—
150C8—5482no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—48no—
155C22—42no—
156C22—42no—
157C22#1798401512yes—
158C22C2×C22⋊Dic13711yescharacteristic
159C4—21no—
160C4—21no—
161C4—21no—
162C4—21no—
163C4—42no—
164C4C2×#1455549412yes—
165C4#1798411412yes—
166C2—22no—
167C2—44no—
168C2#2257736711yescharacteristic
169C2#2257740811yescharacteristic
170C2C2×#1798401511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
FittingC2×C4×D4×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC22×D4×C137C4
cosocleC42×C137C23
hypercenterC2×C4×D4D137
hypocenterC137#1490
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22577460C2semidirect
2. C4×#14555494C2semidirect
3. D4⋊#11764040C2semidirect
6. C4⋊#11764040C22semidirect
10. C4×#11764040C22semidirect
51. D4×C137M16semidirect
85. C137#1490semidirect
135. D4#14555494semidirect

Maximal subgroups

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

subgroupindex
#225774602
C4×#145554942
D4⋊#117640402
C2×C4×D4×C1372
#1490137