Finite Group Databasebeta

← #14544049

Subgroups — #14544049

Order 2176.

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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#11763779C211yesmaximal, characteristic, complemented
2C2×C22⋊#3624C211yesmaximal, characteristic, complemented
3#11763655C212yesmaximal, complemented
4#11763652C212yesmaximal, complemented
5C2×#10587621C211yesmaximal, characteristic
6C22×D4×C17C411yescharacteristic
7C23×Dic17C2211yescharacteristic
8C2×#63247—21no—
9C22×#3624—21no—
10C2×C22⋊Dic17C411yescharacteristic
11C2×C22⋊Dic17C411yescharacteristic
12C2×C22⋊Dic17C2211yescharacteristic
13#10587621—21no—
14#10587621—21no—
15#10587621—21no—
16#10587621—21no—
17C2×#63225C411yescharacteristic
18C2×#63225C2211yescharacteristic
19C22⋊#3624—21no—
20C22⋊#3624—21no—
21C22⋊#3624C2212yes—
22C22⋊#3624C2212yes—
23C24×C17C2×C411yescharacteristic
24C24×C17C2×C411yescharacteristic
25C2×D4×C17—21no—
26C2×D4×C17—21no—
27C2×D4×C17—42no—
28C22×C4×C17C2×C411yescharacteristic
29C22×Dic17—42no—
30C22×Dic17C2×C411yescharacteristic
31C22×Dic17C2×C411yescharacteristic
32C22×Dic17C2×C411yescharacteristic
33C22×Dic17D411yescharacteristic
34C22×Dic17D411yescharacteristic
35C22×Dic17D411yescharacteristic
36C22×Dic17D411yescharacteristic
37C22×Dic17C2311yescharacteristic
38#63247—42no—
39C2×#3624—22no—
40C2×#3624—22no—
41C2×#3624—42no—
42C22⋊Dic17—21no—
43C22⋊Dic17—21no—
44C22⋊Dic17—21no—
45C22⋊Dic17—21no—
46C22⋊Dic17—21no—
47C22⋊Dic17—21no—
48#63225—21no—
49#63225—21no—
50#63225—21no—
51#63225—21no—
52C23×C17—22no—
53C23×C17—22no—
54C23×C17—21no—
55C23×C17—22no—
56C23×C17—41no—
57C23×C17—41no—
58C23×C17—41no—
59C23×C17C22⋊C411yescharacteristic
60C23×C17C22⋊C411yescharacteristic
61C23×C17C22×C411yescharacteristic
62D4×C17—42no—
63D4×C17—42no—
64C2×C4×C17—21no—
65C2×C4×C17—41no—
66C2×Dic17—22no—
67C2×Dic17—22no—
68C2×Dic17—21no—
69C2×Dic17—21no—
70C2×Dic17—22no—
71C2×Dic17—41no—
72C2×Dic17—41no—
73C2×Dic17—41no—
74C2×Dic17—41no—
75C2×Dic17—41no—
76C2×Dic17C22⋊C411yescharacteristic
77C2×Dic17C22⋊C411yescharacteristic
78C2×Dic17C2×D411yescharacteristic
79C2×Dic17C2×D411yescharacteristic
80#3624—42no—
81#3624—42no—
82#1468—171nomaximal, Sylow
83C22×C17—21no—
84C22×C17—21no—
85C22×C17—21no—
86C22×C17—21no—
87C22×C17—21no—
88C22×C17—21no—
89C22×C17—21no—
90C22×C17—44no—
91C22×C17—41no—
92C22×C17—41no—
93C22×C17—41no—
94C22×C17—82no—
95C22×C17#9912yes—
96C22×C17C2×C22⋊C411yescharacteristic
97C22×C17#13112yes—
98C4×C17—41no—
99Dic17—21no—
100Dic17—21no—
101Dic17—21no—
102Dic17—21no—
103Dic17—41no—
104Dic17—41no—
105C2×#123—171no—
106#407—171no—
107C2×C22⋊C8—171no—
108#324—172no—
109#323—172no—
110C2×C17—22no—
111C2×C17—22no—
112C2×C17—41no—
113C2×C17—81no—
114C2×C17#40811yescharacteristic
115C2×C17C2×#9911yescharacteristic
116C2×C17#41311yescharacteristic
117C22×D4#59511yescharacteristic
118C23×C4—171no—
119C2×M16—341no—
120C22×C8—341no—
121#123—341no—
122#123—341no—
123#123—341no—
124#123—341no—
125C2×C4⋊C4—171no—
126C2×C4⋊C4—171no—
127C2×C22⋊C4—171no—
128C2×C22⋊C4—171no—
129C2×C22⋊C4—171no—
130C22⋊C8—172no—
131C22⋊C8—172no—
132C22⋊C8—341no—
133C22⋊C8—341no—
134C17#146811yescharacteristic, complemented, Sylow
135C24C2×#59511yescharacteristic
136C24C2×#59511yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C22×C4—171no—
141C22×C4—171no—
142C22×C4—171no—
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4—682no—
149C22×C4C2×#59511yescharacteristic
150M16—682no—
151C2×C8—342no—
152C2×C8—342no—
153C2×C8—682no—
154C4⋊C4—341no—
155C4⋊C4—341no—
156C4⋊C4—341no—
157C4⋊C4—341no—
158C22⋊C4—341no—
159C22⋊C4—341no—
160C22⋊C4—341no—
161C22⋊C4—341no—
162C22⋊C4—341no—
163C22⋊C4—341no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22⋊#59511yescharacteristic
172C23C22⋊#59511yescharacteristic
173C23C22×#59511yescharacteristic
174D4—42no—
175D4—42no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—171no—
179C2×C4—171no—
180C2×C4—171no—
181C2×C4—171no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—341no—
186C2×C4—341no—
187C2×C4—681no—
188C2×C4—681no—
189C2×C4—681no—
190C2×C4—681no—
191C2×C4—681no—
192C8—682no—
193C8—682no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—82no—
206C22#1058755112yes—
207C22#1058767812yes—
208C22C2×C22⋊#59511yescharacteristic
209C4—41no—
210C4—341no—
211C4—341no—
212C4—341no—
213C4—341no—
214C4—681no—
215C4—681no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2#1176378211yescharacteristic
221C2C2×#1058755111yescharacteristic
222C2#1176379611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
Fitting, prime radicalC22×D4×C17C4
socleC22×C17C2×C22⋊C4
cosocleC22×Dic17C23
hypercenterC22×D4#595
hypocenterC17#1468
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #11763779C2semidirect
2. C2×C22⋊#3624C2semidirect
3. #11763655C2semidirect
4. #11763652C2semidirect
134. C17#1468semidirect

Maximal subgroups

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

subgroupindex
#117637792
C2×C22⋊#36242
#117636552
#117636522
C2×#105876212
#146817