Finite Group Databasebeta

← #14544022

Subgroups — #14544022

Order 2176.

332 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 84 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×C22⋊#3624C212yesmaximal, complemented
2#11763655C212yesmaximal, complemented
3#11763653C212yesmaximal, complemented
4C2×#10587567C211yesmaximal, characteristic
5C22×C22⋊D17C411yescharacteristic
6C23×Dic17C2211yescharacteristic
7C22×C4×D17C411yescharacteristic
8C22×#3624—22no—
9C2×C22⋊C4×C17C411yescharacteristic
10C2×#63227C2211yescharacteristic
11C2×#63225C2211yescharacteristic
12C2×C4×Dic17C411yescharacteristic
13#10587567—22no—
14#10587567—22no—
15C22⋊#3624—22no—
16C22⋊#3624C2214yes—
17C24×C17C2×C411yescharacteristic
18C23×D17C2×C411yescharacteristic
19C22×C4×C17C2×C411yescharacteristic
20C22×C4×C17C2×C411yescharacteristic
21C2×C22⋊D17—22no—
22C2×C22⋊D17—22no—
23C2×C22⋊D17C814yescomplemented
24C22×Dic17—42no—
25C22×Dic17C2×C411yescharacteristic
26C22×Dic17C2×C411yescharacteristic
27C22×Dic17D412yes—
28C22×Dic17D412yes—
29C22×Dic17C2311yescharacteristic
30C2×C4×D17—21no—
31C2×C4×D17—21no—
32C2×C4×D17—42no—
33C2×#3624—24no—
34C2×#3624—44no—
35C22⋊C4×C17—22no—
36#63227—22no—
37#63225—22no—
38C4×Dic17—21no—
39C4×Dic17—21no—
40C23×C17—41no—
41C23×C17—41no—
42C23×C17C22⋊C412yes—
43C23×C17C2×C812yes—
44C23×C17M1612yes—
45C23×C17C22×C411yescharacteristic
46C22×D17—21no—
47C22×D17—21no—
48C22×D17—41no—
49C22×D17—41no—
50C22×D17C2×C812yescomplemented
51C2×C4×C17—21no—
52C2×C4×C17—21no—
53C2×C4×C17—41no—
54C2×C4×C17—41no—
55C22⋊D17—28no—
56C2×Dic17—21no—
57C2×Dic17—21no—
58C2×Dic17—24no—
59C2×Dic17—22no—
60C2×Dic17—42no—
61C2×Dic17—41no—
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C22⋊C411yescharacteristic
64C2×Dic17C2×C812yescomplemented
65C2×Dic17C2×D412yes—
66C4×D17—42no—
67C4×D17—42no—
68#3624—44no—
69#1459—171nomaximal, Sylow
70C22×C17—21no—
71C22×C17—22no—
72C22×C17—21no—
73C22×C17—22no—
74C22×C17—42no—
75C22×C17—42no—
76C22×C17C22⋊C814yes—
77C22×C17C2×C22⋊C411yescharacteristic
78C22×C17C22×C811yescharacteristic
79C22×C17C2×M1611yescharacteristic
80C2×D17—22no—
81C2×D17—42no—
82C2×D17—41no—
83C2×D17—41no—
84C2×D17—42no—
85C4×C17—41no—
86C4×C17—41no—
87Dic17—22no—
88Dic17—22no—
89Dic17—21no—
90Dic17#9912yes—
91C2×C4×D4—171no—
92C2×C22⋊C8—172no—
93#324—172no—
94#323—172no—
95C2×C17—21no—
96C2×C17—21no—
97C2×C17—24no—
98C2×C17C2×C22⋊C811yescharacteristic
99C2×C17C2×#9911yescharacteristic
100C2×C17#41311yescharacteristic
101D17—42no—
102C22×D4—171no—
103C23×C4—171no—
104C23×C4—171no—
105C22×C8—342no—
106C4×D4—342no—
107C4×D4—342no—
108C2×C4⋊C4—171no—
109C2×C22⋊C4—171no—
110C2×C22⋊C4#59511yescharacteristic
111C2×C42—171no—
112C22⋊C8—174no—
113C22⋊C8—342no—
114C17#145911yescharacteristic, complemented, Sylow
115C24—171no—
116C24C2×#59511yescharacteristic
117C2×D4—174no—
118C2×D4—342no—
119C2×D4—342no—
120C22×C4—172no—
121C22×C4—171no—
122C22×C4—171no—
123C22×C4—172no—
124C22×C4—171no—
125C22×C4—341no—
126C22×C4—341no—
127C22×C4—682no—
128C22×C4—682no—
129C22×C4C2×#59511yescharacteristic
130C22×C4C2×#59511yescharacteristic
131C2×C8—344no—
132C2×C8—684no—
133C4⋊C4—342no—
134C22⋊C4—22no—
135C22⋊C4—342no—
136C42—341no—
137C42—341no—
138C23—41no—
139C23—41no—
140C23—172no—
141C23—341no—
142C23—341no—
143C23—681no—
144C23—681no—
145C23#6324212yes—
146C23#6324312yes—
147C23C22⋊#59512yes—
148C23C22×#59511yescharacteristic
149D4—348no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—171no—
155C2×C4—172no—
156C2×C4—171no—
157C2×C4—172no—
158C2×C4—344no—
159C2×C4—342no—
160C2×C4—341no—
161C2×C4—341no—
162C2×C4—682no—
163C2×C4—682no—
164C2×C4—681no—
165C2×C4—682no—
166C8—684no—
167C22—21no—
168C22—22no—
169C22—21no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22—342no—
174C22—682no—
175C22—682no—
176C22—681no—
177C22—681no—
178C22#1058754314yes—
179C22C2×#6324211yescharacteristic
180C22C2×#6324311yescharacteristic
181C22C2×C22⋊#59511yescharacteristic
182C4—41no—
183C4—41no—
184C4—172no—
185C4—342no—
186C4—342no—
187C4—341no—
188C2—21no—
189C2—21no—
190C2—24no—
191C2—682no—
192C2C2×#1058754311yescharacteristic
193C2D17⋊#9911yescharacteristic
194C2#1176379611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC22×C17C22×C8
FrattiniC23C22×#595
FittingC2×C22⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×C22⋊D17C4
cosocleC22×Dic17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#1459
LCSG ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#3624C2semidirect
2. #11763655C2semidirect
3. #11763653C2semidirect
23. C2×C22⋊D17C8semidirect
50. C22×D17C2×C8semidirect
64. C2×Dic17C2×C8semidirect
114. C17#1459semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#36242
#117636552
#117636532
C2×#105875672
#145917