Finite Group Databasebeta

← #14544111

Subgroups — #14544111

Order 2176.

312 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 74 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#11763826C211yesmaximal, characteristic, complemented
2#11763812C211yesmaximal, characteristic, complemented
3#11763657C214yesmaximal, complemented
4C2×#10587579C211yesmaximal, characteristic
5C22×D68C411yescharacteristic
6C22×C4×D17—21no—
7C2×#63247—21no—
8C2×C4⋊C4×C17C411yescharacteristic
9C2×#63227—21no—
10C2×C4×Dic17C2211yescharacteristic
11#10587579—21no—
12#10587579—21no—
13#10587579C412yes—
14#10587579C2212yes—
15C22⋊#3624—21no—
16#10587544—21no—
17#10587544—21no—
18C4⋊#3624C2212yescomplemented
19C4×#3624C2212yescomplemented
20C23×D17—21no—
21C22×C4×C17—21no—
22C22×C4×C17C2×C411yescharacteristic
23C22×Dic17D411yescharacteristic
24C22×Dic17D411yescharacteristic
25C2×D68—21no—
26C2×D68—24no—
27C2×D68C2×C412yes—
28C2×C4×D17—22no—
29C2×C4×D17—22no—
30C2×C4×D17—42no—
31C2×C4×D17—42no—
32#63247—42no—
33C2×#3624—22no—
34C2×#3624—22no—
35C4⋊C4×C17—21no—
36C4⋊C4×C17C2×C412yes—
37#63227—22no—
38#63227—22no—
39C4×Dic17C2×C411yescharacteristic
40C4×Dic17D411yescharacteristic
41C4×Dic17D411yescharacteristic
42C4×Dic17C2311yescharacteristic
43C23×C17C22⋊C411yescharacteristic
44C22×D17—24no—
45C22×D17—22no—
46C22×D17—42no—
47C22×D17—42no—
48C2×C4×C17—22no—
49C2×C4×C17—22no—
50C2×C4×C17—42no—
51C2×C4×C17C22⋊C411yescharacteristic
52C2×C4×C17C22×C411yescharacteristic
53C2×Dic17—41no—
54C2×Dic17C22⋊C411yescharacteristic
55C2×Dic17C22⋊C411yescharacteristic
56C2×Dic17D812yes—
57C2×Dic17SD1612yes—
58C2×Dic17C2×D411yescharacteristic
59C2×Dic17C2×D411yescharacteristic
60D68—24no—
61D68—22no—
62D68M1614yescomplemented
63C4×D17—44no—
64C4×D17—42no—
65C4×D17—42no—
66#3624—42no—
67#3624—42no—
68#1486—171nomaximal, Sylow
69C22×C17—21no—
70C22×C17—21no—
71C22×C17#10212yes—
72C22×C17C2×C22⋊C411yescharacteristic
73C2×D17—24no—
74C2×D17—48no—
75C2×D17—42no—
76C2×D17—42no—
77C4×C17—21no—
78C4×C17—42no—
79C4×C17C2×M1612yes—
80Dic17—21no—
81Dic17—21no—
82Dic17#10212yes—
83Dic17C2×D811yescharacteristic
84Dic17C2×SD1611yescharacteristic
85C2×C4×D4—171no—
86#431—171no—
87#423—171no—
88D4⋊C8—174no—
89C2×C17—22no—
90C2×C17#40711yescharacteristic
91C2×C17C2×#10211yescharacteristic
92C2×C17#42111yescharacteristic
93D17—44no—
94C22×D4—171no—
95C23×C4—341no—
96C2×M16—341no—
97C4×D4—172no—
98C4×D4—172no—
99C4×D4—341no—
100C4×D4—341no—
101C2×C4⋊C4#59511yescharacteristic
102C2×C22⋊C4—341no—
103C2×C42—171no—
104C4⋊C8—172no—
105C4⋊C8—341no—
106C4⋊C8—341no—
107C22⋊C8—341no—
108C4×C8—172no—
109C17#148611yescharacteristic, complemented, Sylow
110C24—341no—
111C2×D4—172no—
112C2×D4—344no—
113C2×D4—341no—
114C22×C4—21no—
115C22×C4—171no—
116C22×C4—171no—
117C22×C4—342no—
118C22×C4—342no—
119C22×C4—682no—
120C22×C4—682no—
121C22×C4C2×#59511yescharacteristic
122M16—682no—
123C2×C8—342no—
124C2×C8—342no—
125C4⋊C4—21no—
126C4⋊C4C2×#59512yes—
127C22⋊C4—342no—
128C22⋊C4—342no—
129C42—171no—
130C42—171no—
131C42—171no—
132C42—171no—
133C23—344no—
134C23—342no—
135C23—682no—
136C23—682no—
137C23C22⋊#59511yescharacteristic
138D4—174no—
139D4—344no—
140D4—342no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—171no—
145C2×C4—171no—
146C2×C4—172no—
147C2×C4—172no—
148C2×C4—171no—
149C2×C4—171no—
150C2×C4—682no—
151C2×C4—684no—
152C2×C4—682no—
153C2×C4—681no—
154C2×C4C22⋊#59511yescharacteristic
155C2×C4C22×#59511yescharacteristic
156C8—682no—
157C8—682no—
158C22—21no—
159C22—21no—
160C22—344no—
161C22—688no—
162C22—682no—
163C22—682no—
164C22#1058754712yes—
165C22C2×C22⋊#59511yescharacteristic
166C4—21no—
167C4—42no—
168C4—171no—
169C4—172no—
170C4—171no—
171C4—341no—
172C4—341no—
173C4D17⋊M1612yes—
174C2—22no—
175C2—684no—
176C2#1176378111yescharacteristic
177C2C2×#1058754711yescharacteristic
178C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D68C4
cosocleC4×Dic17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1486
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11763826C2semidirect
2. #11763812C2semidirect
3. #11763657C2semidirect
18. C4⋊#3624C22semidirect
19. C4×#3624C22semidirect
62. D68M16semidirect
109. C17#1486semidirect

Maximal subgroups

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

subgroupindex
#117638262
#117638122
#117636572
C2×#105875792
#148617