Finite Group Databasebeta

← #14544613

Subgroups — #14544613

Order 2176.

612 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 86 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×#10587539C212yesmaximal, complemented
2C2×#10587551C212yesmaximal, complemented
3#11763752C212yesmaximal, complemented
4C22×#63252C211yesmaximal, characteristic
5C23×C4×C17C411yescharacteristic, complemented
6C22×C22⋊D17C411yescharacteristic, complemented
7C22×C22⋊D17C2211yescharacteristic
8C2×#63252—24no—
9C2×#63252—24no—
10C2×#63252—24no—
11C22×D68C2211yescharacteristic
12C22×C4×D17C411yescharacteristic, complemented
13C22×C4×D17C2211yescharacteristic
14C22×Dic34C411yescharacteristic, complemented
15C2×C22⋊#595C2214yes—
16#10587551—24no—
17(C2×#595)⋊C4—24no—
18#10587539—24no—
19C24×C17C2×C411yescharacteristic
20C23×D17C2×C411yescharacteristic
21C23×D17C2311yescharacteristic
22C22×#595—24no—
23C22×C4×C17—22no—
24C22×C4×C17—22no—
25C22×C4×C17—22no—
26C22×C4×C17C2×C411yescharacteristic
27C22×C4×C17C2×C411yescharacteristic
28C2×C22⋊D17—22no—
29C2×C22⋊D17—22no—
30C2×C22⋊D17—22no—
31C2×C22⋊D17—22no—
32C2×C22⋊D17—22no—
33C2×C22⋊D17D414yes—
34C22×Dic17C2×C411yescharacteristic
35C22×Dic17C2×C411yescharacteristic
36#63252—232no—
37C2×D68—22no—
38C2×D68—22no—
39C2×D68D414yes—
40C2×C4×D17—22no—
41C2×C4×D17—22no—
42C2×C4×D17—22no—
43C2×C4×D17—22no—
44C2×C4×D17—22no—
45C2×C4×D17D414yes—
46C2×Dic34—22no—
47C2×Dic34—22no—
48C2×Dic34—22no—
49C22⋊#595—24no—
50C22⋊#595—24no—
51C23×C17—21no—
52C23×C17—21no—
53C23×C17—42no—
54C23×C17C22⋊C412yes—
55C23×C17C22×C411yescharacteristic
56C22×D17—21no—
57C22×D17—21no—
58C22×D17—42no—
59C22×D17—42no—
60C22×D17C22⋊C412yes—
61C22×D17C2×D412yes—
62C22×D17C2×D412yes—
63C22×D17C2×D412yes—
64C2×#595—48no—
65C2×#595—44no—
66C2×C4×C17—28no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17C22⋊C412yes—
72C2×C4×C17C22⋊C412yes—
73C22⋊D17—28no—
74C22⋊D17—28no—
75C2×Dic17—21no—
76C2×Dic17—21no—
77C2×Dic17—21no—
78C2×Dic17—21no—
79C2×Dic17C22⋊C412yes—
80C2×Dic17C22⋊C412yes—
81D68—28no—
82C4×D17—28no—
83C4×D17—28no—
84Dic34—28no—
85#1782—171nomaximal, Sylow
86C22×C17—22no—
87C22×C17—22no—
88C22×C17—42no—
89C22×C17—42no—
90C22×C17—42no—
91C22×C17C2×C22⋊C411yescharacteristic
92C22×C17C2×C22⋊C411yescharacteristic
93C22×C17C2×C22⋊C411yescharacteristic
94C2×D17—24no—
95C2×D17—22no—
96C2×D17—42no—
97C2×D17—42no—
98C2×D17—42no—
99C2×D17—42no—
100C2×D17—42no—
101C2×D17#12014yes—
102#595—84no—
103C4×C17—22no—
104C4×C17—22no—
105Dic17—22no—
106Dic17—22no—
107C22×#41—171no—
108C2×#99—172no—
109C2×#99—172no—
110#381—172no—
111C2×C17—21no—
112C2×C17—21no—
113C2×C17—42no—
114C2×C17#37911yescharacteristic
115C2×C17#41012yes—
116D17—24no—
117D17—42no—
118C2×#41—344no—
119C2×#41—344no—
120C2×#41—344no—
121C22×Q8—171no—
122C22×D4—171no—
123C22×D4—171no—
124C22×D4—171no—
125C23×C4—171no—
126C23×C4—171no—
127C23×C4#59511yescharacteristic, complemented
128C2×C22⋊C4—174no—
129#99—344no—
130#99—344no—
131#95—344no—
132C17#178211yescharacteristic, complemented, Sylow
133C24—171no—
134C24—171no—
135C24C2×#59511yescharacteristic
136#41—3432no—
137C2×Q8—342no—
138C2×Q8—342no—
139C2×Q8—342no—
140C2×D4—174no—
141C2×D4—174no—
142C2×D4—342no—
143C2×D4—342no—
144C2×D4—342no—
145C2×D4—342no—
146C2×D4—342no—
147C2×D4—342no—
148C2×D4—342no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—171no—
153C22×C4—171no—
154C22×C4—174no—
155C22×C4—342no—
156C22×C4—342no—
157C22×C4—342no—
158C22×C4—342no—
159C22×C4—342no—
160C22×C4—344no—
161C22×C4C2×#59511yescharacteristic
162C22×C4C2×#59511yescharacteristic
163C22⋊C4—344no—
164C22⋊C4—344no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23—172no—
169C23—172no—
170C23—172no—
171C23—172no—
172C23—341no—
173C23—341no—
174C23—682no—
175C23—682no—
176C23C22⋊#59512yes—
177C23C22×#59511yescharacteristic
178Q8—348no—
179D4—348no—
180D4—348no—
181D4—348no—
182C2×C4—28no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—172no—
188C2×C4—172no—
189C2×C4—348no—
190C2×C4—348no—
191C2×C4—341no—
192C2×C4—341no—
193C2×C4—341no—
194C2×C4—341no—
195C2×C4—684no—
196C2×C4—688no—
197C2×C4C22⋊#59512yes—
198C2×C4C22⋊#59512yes—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—174no—
205C22—344no—
206C22—342no—
207C22—682no—
208C22—682no—
209C22—682no—
210C22—682no—
211C22—682no—
212C22C2×C22⋊#59511yescharacteristic
213C22C2×C22⋊#59511yescharacteristic
214C22C2×C22⋊#59511yescharacteristic
215C4—22no—
216C4—22no—
217C4—342no—
218C4—342no—
219C4—1364no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—344no—
224C2—682no—
225C2#1176375011yescharacteristic
226C2#1176378812yes—
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC23×C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×#63252C2
cosocleC23×D17C23
hypercenterC23×C4#595
hypocenterC17#1782
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#10587539C2semidirect
2. C2×#10587551C2semidirect
3. #11763752C2semidirect
5. C23×C4×C17C4semidirect
6. C22×C22⋊D17C4semidirect
12. C22×C4×D17C4semidirect
14. C22×Dic34C4semidirect
127. C23×C4#595semidirect
132. C17#1782semidirect

Maximal subgroups

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

subgroupindex
C2×#105875392
C2×#105875512
#117637522
C22×#632522
#178217