Finite Group Databasebeta

← #14544041

Subgroups — #14544041

Order 2176.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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#11763780C212yesmaximal, complemented
2#11763654C212yesmaximal, complemented
3#11763653C212yesmaximal, complemented
4C2×C4×D68C211yesmaximal, characteristic
5C22×D68C411yescharacteristic
6C22×C4×D17C411yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×#63243—22no—
9C2×C42×C17C411yescharacteristic
10C2×#63227C411yescharacteristic
11C2×#63227C2211yescharacteristic
12C2×C4⋊Dic17C2211yescharacteristic
13C4×D68—22no—
14C4×D68—22no—
15C22⋊#3624C2214yes—
16#10587543—22no—
17C23×D17C2×C411yescharacteristic
18C23×D17C2×C411yescharacteristic
19C22×C4×C17C2×C411yescharacteristic
20C22×C4×C17C2×C411yescharacteristic
21C22×C4×C17C2×C411yescharacteristic
22C22×Dic17C2×C411yescharacteristic
23C22×Dic17C2311yescharacteristic
24C2×D68—22no—
25C2×D68—22no—
26C2×D68—22no—
27C2×C4×D17—21no—
28C2×C4×D17—21no—
29C2×C4×D17—42no—
30C2×C4×D17—42no—
31C2×C4×D17D412yes—
32C2×C4×D17D412yes—
33C2×#3624—24no—
34#63243—44no—
35C42×C17—21no—
36C42×C17—21no—
37#63227—22no—
38#63227—22no—
39C4⋊Dic17—22no—
40C23×C17C22×C411yescharacteristic
41C22×D17—21no—
42C22×D17—21no—
43C22×D17—21no—
44C22×D17—21no—
45C22×D17—41no—
46C22×D17—41no—
47C22×D17—41no—
48C22×D17—41no—
49C22×D17C22⋊C412yes—
50C22×D17M1612yescomplemented
51C2×C4×C17—21no—
52C2×C4×C17—21no—
53C2×C4×C17—22no—
54C2×C4×C17—21no—
55C2×C4×C17—41no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17M1612yescomplemented
59C2×Dic17—21no—
60C2×Dic17—41no—
61C2×Dic17—41no—
62C2×Dic17C2×D412yes—
63D68—28no—
64C4×D17—24no—
65C4×D17—42no—
66C4×D17—42no—
67C4×D17—42no—
68#3624—44no—
69#1465—171nomaximal, Sylow
70C22×C17—21no—
71C22×C17—21no—
72C22×C17C2×C22⋊C411yescharacteristic
73C22×C17C2×M1611yescharacteristic
74C22×C17C2×M1611yescharacteristic
75C2×D17—22no—
76C2×D17—22no—
77C2×D17—22no—
78C2×D17—22no—
79C2×D17—42no—
80C2×D17—42no—
81C2×D17—42no—
82C2×D17—41no—
83C2×D17—41no—
84C2×D17—42no—
85C4×C17—22no—
86C4×C17—21no—
87C4×C17—41no—
88C4×C17#9912yes—
89Dic17—22no—
90Dic17—41no—
91C2×C4×D4—171no—
92#407—172no—
93#324—172no—
94#323—172no—
95C2×C17—21no—
96C2×C17—21no—
97C2×C17#40711yescharacteristic
98C2×C17C2×#9911yescharacteristic
99C2×C17#41311yescharacteristic
100D17—42no—
101D17—42no—
102C22×D4—171no—
103C23×C4—171no—
104C23×C4—171no—
105C2×M16—342no—
106C4×D4—342no—
107C4×D4—342no—
108C2×C4⋊C4—171no—
109C2×C22⋊C4—171no—
110C2×C22⋊C4—171no—
111C2×C42#59511yescharacteristic
112C22⋊C8—174no—
113C22⋊C8—342no—
114C17#146511yescharacteristic, complemented, Sylow
115C24—171no—
116C24—171no—
117C2×D4—342no—
118C2×D4—342no—
119C2×D4—342no—
120C22×C4—171no—
121C22×C4—172no—
122C22×C4—172no—
123C22×C4—171no—
124C22×C4—341no—
125C22×C4—341no—
126C22×C4—682no—
127C22×C4—682no—
128C22×C4C2×#59511yescharacteristic
129C22×C4C2×#59511yescharacteristic
130C22×C4C2×#59511yescharacteristic
131M16—684no—
132C2×C8—344no—
133C4⋊C4—342no—
134C22⋊C4—342no—
135C22⋊C4—342no—
136C42—21no—
137C42—21no—
138C23—172no—
139C23—172no—
140C23—341no—
141C23—341no—
142C23—341no—
143C23—341no—
144C23—681no—
145C23—681no—
146C23—681no—
147C23—681no—
148C23C22×#59511yescharacteristic
149D4—348no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—172no—
156C2×C4—344no—
157C2×C4—341no—
158C2×C4—682no—
159C2×C4—682no—
160C2×C4—681no—
161C2×C4—682no—
162C2×C4—681no—
163C2×C4#6324312yescomplemented
164C2×C4C22⋊#59511yescharacteristic
165C2×C4C22⋊#59511yescharacteristic
166C8—684no—
167C22—21no—
168C22—21no—
169C22—342no—
170C22—342no—
171C22—342no—
172C22—342no—
173C22—682no—
174C22—682no—
175C22—682no—
176C22—682no—
177C22—681no—
178C22—681no—
179C22C2×#6324311yescharacteristic
180C22D17⋊M1611yescharacteristic
181C22C2×C22⋊#59511yescharacteristic
182C4—22no—
183C4—21no—
184C4—41no—
185C4—342no—
186C4—681no—
187C4#1058753912yes—
188C2—21no—
189C2—21no—
190C2—682no—
191C2—682no—
192C2#1176378011yescharacteristic
193C2C2×#1058753911yescharacteristic
194C2#1176379511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C42×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D68C4
cosocleC22×Dic17C23
hypercenterC2×C42#595
hypocenterC17#1465
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. #11763780C2semidirect
2. #11763654C2semidirect
3. #11763653C2semidirect
50. C22×D17M16semidirect
58. C2×C4×C17M16semidirect
114. C17#1465semidirect
163. C2×C4#63243semidirect

Maximal subgroups

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

subgroupindex
#117637802
#117636542
#117636532
C2×C4×D682
#146517