Finite Group Databasebeta

← #14544292

Subgroups — #14544292

Order 2176.

344 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 70 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#11763827C211yesmaximal, characteristic, complemented
2#11763814C211yesmaximal, characteristic, complemented
3#11763667C214yesmaximal, complemented
4#127×D17C211yesmaximal, characteristic
5C2×D4×D17—21no—
6C2×D4×D17—21no—
7C2×D4×D17C412yes—
8C2×#63243—21no—
9C17×#127C411yescharacteristic
10#10587626—21no—
11#10587626—21no—
12#10587626C2212yes—
13#10587560C411yescharacteristic
14C42×D17C2211yescharacteristic
15#10587544—21no—
16#10587544C2212yescomplemented
17#10587544C2212yescomplemented
18#10587543—21no—
19C4×#3624—21no—
20C23×D17—22no—
21C2×D4×C17—21no—
22C2×D4×C17—21no—
23C2×D4×C17C2×C412yes—
24C2×C22⋊D17—22no—
25C2×C22⋊D17—22no—
26C2×C22⋊D17—22no—
27D4×D17—44no—
28D4×D17—44no—
29D4×D17—42no—
30D4×D17—42no—
31C2×D68—21no—
32C2×D68—21no—
33C2×D68C2×C412yes—
34C2×C4×D17C2×C411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×#3624—22no—
38C2×#3624—22no—
39#63243—42no—
40C42×C17C2×C411yescharacteristic
41C4×Dic17D411yescharacteristic
42C4×Dic17D411yescharacteristic
43C4×Dic17C2311yescharacteristic
44C23×C17—22no—
45C22×D17—22no—
46C22×D17—44no—
47C22×D17—44no—
48C22×D17—44no—
49C22×D17C22⋊C411yescharacteristic
50D4×C17—42no—
51D4×C17—42no—
52D4×C17—42no—
53C2×C4×C17C22⋊C411yescharacteristic
54C2×C4×C17C22⋊C411yescharacteristic
55C2×C4×C17C22×C411yescharacteristic
56C22⋊D17—44no—
57C22⋊D17—42no—
58C22⋊D17—42no—
59C22⋊D17—44no—
60C2×Dic17C22⋊C411yescharacteristic
61C2×Dic17C2×D411yescharacteristic
62C2×Dic17C2×D411yescharacteristic
63D68—42no—
64D68—42no—
65D68—42no—
66C4×D17—22no—
67C4×D17—22no—
68C4×D17D812yes—
69C4×D17SD1612yes—
70#3624—42no—
71#3624—42no—
72#1545—171nomaximal, Sylow
73C22×C17—42no—
74C22×C17—42no—
75C22×C17—42no—
76C22×C17C2×C22⋊C411yescharacteristic
77C2×D17—21no—
78C2×D17—21no—
79C2×D17—42no—
80C2×D17—42no—
81C2×D17—42no—
82C2×D17—88no—
83C2×D17#10212yes—
84C4×C17—21no—
85C4×C17—21no—
86C4×C17#10212yes—
87Dic17#10012yes—
88Dic17C2×D811yescharacteristic
89Dic17C2×SD1611yescharacteristic
90Dic17AGL(1,ℤ/8)12yes—
91C2×#127—171no—
92#431—171no—
93#423—171no—
94#331—174no—
95C2×C17—82no—
96C2×C17C2×#10011yescharacteristic
97C2×C17C2×#10211yescharacteristic
98C2×C17#41811yescharacteristic
99D17—22no—
100D17—82no—
101C22×D4—172no—
102C22×D4—341no—
103C22×D4—341no—
104C2×M16—341no—
105#127—172no—
106#127—171no—
107#127—341no—
108#127—341no—
109#127#59511yescharacteristic
110C2×C42—171no—
111C4⋊C8—172no—
112C4⋊C8—172no—
113C4⋊C8—341no—
114C22⋊C8—341no—
115C4×C8—341no—
116C17#154511yescharacteristic, complemented, Sylow
117C24—342no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—172no—
121C2×D4—342no—
122C2×D4—342no—
123C2×D4—342no—
124C2×D4—341no—
125C2×D4—341no—
126C2×D4—682no—
127C2×D4—682no—
128C2×D4—684no—
129C2×D4—684no—
130C2×D4C2×#59512yes—
131C22×C4—171no—
132C22×C4—171no—
133C22×C4—171no—
134M16—682no—
135C2×C8—342no—
136C2×C8—342no—
137C42—171no—
138C42—171no—
139C42—171no—
140C42C2×#59511yescharacteristic
141C23—22no—
142C23—171no—
143C23—342no—
144C23—684no—
145C23—684no—
146C23—684no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—682no—
151D4—682no—
152D4—682no—
153D4—682no—
154D4—684no—
155D4—684no—
156D4—682no—
157C2×C4—171no—
158C2×C4—171no—
159C2×C4—171no—
160C2×C4—172no—
161C2×C4—172no—
162C2×C4—342no—
163C2×C4—342no—
164C2×C4C22⋊#59511yescharacteristic
165C2×C4C22⋊#59511yescharacteristic
166C2×C4C22×#59511yescharacteristic
167C8—682no—
168C8—682no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—172no—
173C22—341no—
174C22—341no—
175C22—682no—
176C22—682no—
177C22—682no—
178C22—1368no—
179C22C2×C22⋊#59511yescharacteristic
180C4—21no—
181C4—21no—
182C4—172no—
183C4—171no—
184C4—171no—
185C4—172no—
186C4#1058754712yes—
187C2—82no—
188C2—342no—
189C2—1362no—
190C2D17⋊#10011yescharacteristic
191C2C2×#1058754711yescharacteristic
192C2#1176380111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#127C4
socleC22×C17C2×C22⋊C4
prime radical#127×D17C2
cosocleC4×Dic17C23
hypercenter#127#595
hypocenterC17#1545
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #11763827C2semidirect
2. #11763814C2semidirect
3. #11763667C2semidirect
16. #10587544C22semidirect
17. #10587544C22semidirect
116. C17#1545semidirect

Maximal subgroups

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

subgroupindex
#117638272
#117638142
#117636672
#127×D172
#154517