Finite Group Databasebeta

← #14544244

Subgroups — #14544244

Order 2176.

264 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 58 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#11763832C211yesmaximal, characteristic, complemented
2C4×#63243C211yesmaximal, characteristic, complemented
3#11763665C212yesmaximal, complemented
4#11763664C212yesmaximal, complemented
5#124×D17C211yesmaximal, characteristic
6C2×Q8×D17C411yescharacteristic
7C2×D4×D17C411yescharacteristic
8C2×#63243—21no—
9C17×#124C411yescharacteristic
10#10587633C2211yescharacteristic
11#10587622C2211yescharacteristic
12#10587571—21no—
13#10587571—21no—
14C22⋊C4×D17—21no—
15C22⋊C4×D17—21no—
16#10587561C411yescharacteristic
17C42×D17C2211yescharacteristic
18#10587544—21no—
19#10587543—21no—
20#10587542C2212yescomplemented
21#10587542C2212yescomplemented
22C4×#3624—21no—
23C23×D17—21no—
24C2×Q8×C17C2×C411yescharacteristic
25C2×D4×C17C2×C411yescharacteristic
26C2×C22⋊D17—21no—
27Q8×D17—42no—
28D4×D17—42no—
29C2×D68C2×C411yescharacteristic
30C2×C4×D17—21no—
31C2×C4×D17C2×C411yescharacteristic
32C2×C4×D17D411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×Dic34—21no—
35C2×Dic34C2×C411yescharacteristic
36C2×#3624—22no—
37C2×#3624—22no—
38#63243—24no—
39C22⋊C4×C17—21no—
40C22⋊C4×C17—21no—
41C42×C17C2×C411yescharacteristic
42C22⋊Dic17—21no—
43C22⋊Dic17—21no—
44#63227—21no—
45#63227—21no—
46#63227—21no—
47#63227—21no—
48C4×Dic17D411yescharacteristic
49C4×Dic17D411yescharacteristic
50C4×Dic17C2311yescharacteristic
51C23×C17—21no—
52C22×D17—21no—
53C22×D17—42no—
54C22×D17—42no—
55C22×D17—42no—
56C22×D17C22⋊C411yescharacteristic
57Q8×C17—41no—
58D4×C17—41no—
59C2×C4×C17—21no—
60C2×C4×C17C22⋊C411yescharacteristic
61C2×C4×C17C22⋊C411yescharacteristic
62C2×C4×C17C22×C411yescharacteristic
63C22⋊D17—42no—
64C2×Dic17—21no—
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C2×D411yescharacteristic
67C2×Dic17C2×D411yescharacteristic
68D68—41no—
69C4×D17—22no—
70C4×D17—22no—
71C4×D17—41no—
72C4×D17—42no—
73Dic34—42no—
74Dic34—41no—
75#3624—24no—
76#3624—42no—
77#1527—171nomaximal, Sylow
78C22×C17—41no—
79C22×C17—41no—
80C22×C17—41no—
81C22×C17C2×C22⋊C411yescharacteristic
82C2×D17—21no—
83C2×D17—21no—
84C2×D17—41no—
85C2×D17—41no—
86C2×D17—41no—
87C2×D17—84no—
88C2×D17#10412yes—
89C4×C17—21no—
90C4×C17—21no—
91C4×C17—21no—
92C4×C17—41no—
93Dic17—21no—
94Dic17—21no—
95Dic17—41no—
96Dic17#10012yes—
97C2×#124—171no—
98#432—171no—
99C4×M16—171no—
100#329—172no—
101#329—172no—
102C2×C17—81no—
103C2×C17C2×#10011yescharacteristic
104C2×C17C2×#10411yescharacteristic
105C2×C17#42111yescharacteristic
106D17—22no—
107D17—81no—
108C22×Q8—171no—
109C22×D4—171no—
110C2×M16—341no—
111#124—171no—
112#124—171no—
113#124—171no—
114#124—341no—
115#124—341no—
116#124#59511yescharacteristic
117C2×C22⋊C4—341no—
118C2×C22⋊C4—341no—
119C2×C42—171no—
120C4⋊C8—341no—
121C22⋊C8—341no—
122#97—172no—
123#97—172no—
124C4×C8—341no—
125C17#152711yescharacteristic, complemented, Sylow
126C24—341no—
127C2×Q8—171no—
128C2×Q8—341no—
129C2×Q8—682no—
130C2×Q8C2×#59511yescharacteristic
131C2×D4—171no—
132C2×D4—341no—
133C2×D4—682no—
134C2×D4C2×#59511yescharacteristic
135C22×C4—171no—
136C22×C4—171no—
137C22×C4—171no—
138C22×C4—341no—
139M16—344no—
140C2×C8—342no—
141C2×C8—342no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—341no—
145C22⋊C4—341no—
146C22⋊C4—341no—
147C22⋊C4—341no—
148C22⋊C4—341no—
149C22⋊C4—341no—
150C42—171no—
151C42—171no—
152C42—171no—
153C42C2×#59511yescharacteristic
154C23—21no—
155C23—171no—
156C23—341no—
157C23—682no—
158C23—682no—
159C23—682no—
160Q8—41no—
161Q8—681no—
162Q8—682no—
163D4—41no—
164D4—681no—
165D4—682no—
166C2×C4—21no—
167C2×C4—171no—
168C2×C4—171no—
169C2×C4—171no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—341no—
173C2×C4—681no—
174C2×C4—682no—
175C2×C4C22⋊#59511yescharacteristic
176C2×C4C22⋊#59511yescharacteristic
177C2×C4C22×#59511yescharacteristic
178C8—344no—
179C8—682no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—172no—
184C22—341no—
185C22—341no—
186C22—681no—
187C22—681no—
188C22—681no—
189C22—1364no—
190C22C2×C22⋊#59511yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—41no—
195C4—172no—
196C4—341no—
197C4—341no—
198C4—681no—
199C2—81no—
200C2—342no—
201C2—1361no—
202C2D17⋊#10011yescharacteristic
203C2D17⋊#10411yescharacteristic
204C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC22×C17C2×C22⋊C4
prime radicalC2×D4×D17C4
cosocleC4×Dic17C23
hypercenter#124#595
hypocenterC17#1527
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #11763832C2semidirect
2. C4×#63243C2semidirect
3. #11763665C2semidirect
4. #11763664C2semidirect
20. #10587542C22semidirect
21. #10587542C22semidirect
125. C17#1527semidirect

Maximal subgroups

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

subgroupindex
#117638322
C4×#632432
#117636652
#117636642
#124×D172
#152717