Finite Group Databasebeta

← #14544053

Subgroups — #14544053

Order 2176.

256 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 60 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#11763829C211yesmaximal, characteristic, complemented
2#11763825C211yesmaximal, characteristic, complemented
3#11763653C212yesmaximal, complemented
4#11763652C212yesmaximal, complemented
5C2×#10587571C211yesmaximal, characteristic
6C22×C22⋊D17C411yescharacteristic
7C22×Dic34C411yescharacteristic
8C2×C22⋊C4×C17C411yescharacteristic
9C2×C22⋊Dic17C2211yescharacteristic
10C2×#63227C411yescharacteristic
11C2×#63227C2211yescharacteristic
12C2×C4×Dic17C2211yescharacteristic
13#10587571—21no—
14#10587571—21no—
15#10587571—21no—
16#10587571—21no—
17C22⋊#3624C2212yes—
18C22⋊#3624C2212yes—
19#10587544—21no—
20#10587542—21no—
21C4⋊#3624—21no—
22C4×#3624—21no—
23C24×C17C2×C411yescharacteristic
24C23×D17C2×C411yescharacteristic
25C22×C4×C17C2×C411yescharacteristic
26C22×C4×C17C2×C411yescharacteristic
27C2×C22⋊D17—21no—
28C2×C22⋊D17—21no—
29C2×C22⋊D17—42no—
30C22×Dic17C2×C411yescharacteristic
31C22×Dic17C2×C411yescharacteristic
32C22×Dic17C2311yescharacteristic
33C2×Dic34—21no—
34C2×Dic34—21no—
35C2×Dic34—42no—
36C2×#3624—22no—
37C2×#3624—22no—
38C22⋊C4×C17—21no—
39C22⋊C4×C17—21no—
40C22⋊Dic17—21no—
41C22⋊Dic17—21no—
42#63227—21no—
43#63227—21no—
44#63227—21no—
45#63227—21no—
46C4×Dic17D411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17D411yescharacteristic
50C23×C17—21no—
51C23×C17—22no—
52C23×C17—41no—
53C23×C17—41no—
54C23×C17C22×C411yescharacteristic
55C22×D17—21no—
56C22×D17—22no—
57C22×D17—41no—
58C22×D17—41no—
59C2×C4×C17—22no—
60C2×C4×C17—21no—
61C2×C4×C17—41no—
62C2×C4×C17C22⋊C411yescharacteristic
63C2×C4×C17C22⋊C411yescharacteristic
64C22⋊D17—42no—
65C22⋊D17—42no—
66C2×Dic17—22no—
67C2×Dic17—21no—
68C2×Dic17—41no—
69C2×Dic17—41no—
70C2×Dic17C22⋊C411yescharacteristic
71C2×Dic17C22⋊C411yescharacteristic
72C2×Dic17C2×D411yescharacteristic
73C2×Dic17C2×D411yescharacteristic
74Dic34—42no—
75Dic34—42no—
76#3624—42no—
77#3624—42no—
78#1469—171nomaximal, Sylow
79C22×C17—21no—
80C22×C17—21no—
81C22×C17—41no—
82C22×C17—41no—
83C22×C17—41no—
84C22×C17—82no—
85C22×C17C2×C22⋊C411yescharacteristic
86C22×C17#13112yes—
87C2×D17—41no—
88C2×D17—41no—
89C2×D17—41no—
90C2×D17—82no—
91C4×C17—21no—
92C4×C17—21no—
93C4×C17—41no—
94Dic17—21no—
95Dic17—21no—
96Dic17—21no—
97Dic17—41no—
98Dic17#10012yes—
99C2×#124—171no—
100#432—171no—
101#431—171no—
102#323—172no—
103#323—172no—
104C2×C17—22no—
105C2×C17—81no—
106C2×C17#40811yescharacteristic
107C2×C17#41011yescharacteristic
108C2×C17C2×#10011yescharacteristic
109D17—81no—
110C22×Q8—171no—
111C22×D4—171no—
112#124—341no—
113#124—341no—
114#124—341no—
115#124—341no—
116C2×C22⋊C4—171no—
117C2×C22⋊C4—171no—
118C2×C22⋊C4—171no—
119C2×C22⋊C4#59511yescharacteristic
120C2×C42—171no—
121C4⋊C8—341no—
122C4⋊C8—341no—
123C22⋊C8—172no—
124C22⋊C8—172no—
125#97—341no—
126C4×C8—341no—
127C17#146911yescharacteristic, complemented, Sylow
128C24—171no—
129C24C2×#59511yescharacteristic
130C2×Q8—341no—
131C2×Q8—341no—
132C2×Q8—682no—
133C2×D4—341no—
134C2×D4—341no—
135C2×D4—682no—
136C22×C4—171no—
137C22×C4—171no—
138C22×C4—171no—
139C22×C4C2×#59511yescharacteristic
140C22×C4C2×#59511yescharacteristic
141C2×C8—342no—
142C2×C8—342no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C22⋊C4—341no—
146C22⋊C4—341no—
147C22⋊C4—341no—
148C22⋊C4—341no—
149C22⋊C4—341no—
150C22⋊C4—341no—
151C42—171no—
152C42—171no—
153C42—171no—
154C42—171no—
155C23—21no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—341no—
160C23—342no—
161C23—681no—
162C23—681no—
163C23C22×#59511yescharacteristic
164Q8—682no—
165Q8—682no—
166D4—682no—
167D4—682no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—171no—
172C2×C4—171no—
173C2×C4—171no—
174C2×C4—171no—
175C2×C4—342no—
176C2×C4—341no—
177C2×C4—681no—
178C2×C4—681no—
179C2×C4C22⋊#59511yescharacteristic
180C2×C4C22⋊#59511yescharacteristic
181C8—682no—
182C8—682no—
183C22—21no—
184C22—21no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—82no—
189C22—681no—
190C22—681no—
191C22—681no—
192C22—1362no—
193C22#1058767812yes—
194C22C2×C22⋊#59511yescharacteristic
195C4—21no—
196C4—21no—
197C4—41no—
198C4—172no—
199C4—341no—
200C4—341no—
201C4—341no—
202C4—681no—
203C2—22no—
204C2—81no—
205C2—1361no—
206C2#1176378211yescharacteristic
207C2#1176378811yescharacteristic
208C2D17⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C22⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×C22⋊D17C4
cosocleC22×Dic17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#1469
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11763829C2semidirect
2. #11763825C2semidirect
3. #11763653C2semidirect
4. #11763652C2semidirect
127. C17#1469semidirect

Maximal subgroups

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

subgroupindex
#117638292
#117638252
#117636532
#117636522
C2×#105875712
#146917