Finite Group Databasebeta

← #14544621

Subgroups — #14544621

Order 2176.

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

normal subgroups only

This group has 584 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×#10587545C212yesmaximal, complemented
2C2×#10587553C212yesmaximal, complemented
3#11763779C212yesmaximal, complemented
4C22×#63254C211yesmaximal, characteristic
5C22×D4×C17C411yescharacteristic
6C22×C22⋊D17C411yescharacteristic
7C22×C22⋊D17C2211yescharacteristic
8C23×Dic17C411yescharacteristic
9C23×Dic17C2211yescharacteristic
10C2×#63254—24no—
11C2×#63254—24no—
12C2×#63254—24no—
13C22×C4×D17C411yescharacteristic
14C22×Dic34C2211yescharacteristic
15C2×#63247C2214yes—
16#10587553—24no—
17C22⋊#3624—24no—
18#10587545—24no—
19C24×C17C2×C411yescharacteristic
20C24×C17C2×C411yescharacteristic
21C23×D17C2×C411yescharacteristic
22C2×D4×C17—22no—
23C2×D4×C17—22no—
24C2×D4×C17—22no—
25C22×C4×C17C2×C411yescharacteristic
26C2×C22⋊D17—22no—
27C2×C22⋊D17—22no—
28C2×C22⋊D17—22no—
29C2×C22⋊D17—22no—
30C2×C22⋊D17—22no—
31C2×C22⋊D17D414yes—
32C22×Dic17—22no—
33C22×Dic17—22no—
34C22×Dic17—22no—
35C22×Dic17—22no—
36C22×Dic17—22no—
37C22×Dic17C2×C411yescharacteristic
38C22×Dic17C2×C411yescharacteristic
39C22×Dic17D414yes—
40C22×Dic17C2311yescharacteristic
41#63254—232no—
42C2×C4×D17—22no—
43C2×C4×D17—22no—
44C2×C4×D17—22no—
45C2×Dic34—22no—
46C2×Dic34—22no—
47C2×Dic34D414yes—
48#63247—24no—
49#63247—24no—
50C2×#3624—24no—
51C23×C17—21no—
52C23×C17—21no—
53C23×C17—21no—
54C23×C17—21no—
55C23×C17—42no—
56C23×C17—42no—
57C23×C17C22⋊C412yes—
58C23×C17C22⋊C412yes—
59C23×C17C22×C411yescharacteristic
60C22×D17—21no—
61C22×D17—21no—
62C22×D17—42no—
63C22×D17C22⋊C412yes—
64D4×C17—28no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17C22⋊C412yes—
68C22⋊D17—28no—
69C22⋊D17—28no—
70C2×Dic17—28no—
71C2×Dic17—21no—
72C2×Dic17—21no—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—28no—
76C2×Dic17C22⋊C412yes—
77C2×Dic17C22⋊C412yes—
78C2×Dic17C2×D412yes—
79C2×Dic17C2×D412yes—
80C2×Dic17C2×D412yes—
81C4×D17—28no—
82Dic34—28no—
83#3624—44no—
84#1785—171nomaximal, Sylow
85C22×C17—22no—
86C22×C17—22no—
87C22×C17—22no—
88C22×C17—24no—
89C22×C17—42no—
90C22×C17—42no—
91C22×C17—42no—
92C22×C17—42no—
93C22×C17—42no—
94C22×C17C2×C22⋊C411yescharacteristic
95C22×C17C2×C22⋊C411yescharacteristic
96C22×C17C2×C22⋊C411yescharacteristic
97C2×D17—22no—
98C2×D17—42no—
99C2×D17—42no—
100C2×D17—42no—
101C4×C17—22no—
102Dic17—22no—
103Dic17—22no—
104Dic17#12014yes—
105C22×#41—171no—
106C2×#101—172no—
107C2×#100—172no—
108#407—172no—
109C2×C17—21no—
110C2×C17—21no—
111C2×C17—42no—
112C2×C17—42no—
113C2×C17#37911yescharacteristic
114C2×C17#41312yes—
115D17—42no—
116C2×#41—344no—
117C2×#41—344no—
118C2×#41—344no—
119C22×Q8—171no—
120C22×D4—171no—
121C22×D4—171no—
122C22×D4#59511yescharacteristic
123C23×C4—171no—
124C23×C4—171no—
125C23×C4—171no—
126C2×M16—174no—
127#101—344no—
128#100—344no—
129C22⋊C8—344no—
130C17#178511yescharacteristic, complemented, Sylow
131C24—171no—
132C24C2×#59511yescharacteristic
133C24C2×#59511yescharacteristic
134#41—3432no—
135C2×Q8—174no—
136C2×Q8—342no—
137C2×Q8—342no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—174no—
142C2×D4—342no—
143C2×D4—342no—
144C2×D4—342no—
145C2×D4—342no—
146C2×D4—342no—
147C22×C4—171no—
148C22×C4—174no—
149C22×C4—171no—
150C22×C4—171no—
151C22×C4—342no—
152C22×C4—342no—
153C22×C4—342no—
154C22×C4—342no—
155C22×C4—342no—
156C22×C4—342no—
157C22×C4—342no—
158C22×C4—342no—
159C22×C4C2×#59511yescharacteristic
160M16—344no—
161M16—344no—
162C2×C8—344no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23—42no—
169C23—172no—
170C23—341no—
171C23—341no—
172C23—682no—
173C23C22⋊#59512yes—
174C23C22⋊#59512yes—
175C23C22×#59511yescharacteristic
176Q8—348no—
177D4—28no—
178D4—348no—
179D4—348no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—172no—
183C2×C4—172no—
184C2×C4—172no—
185C2×C4—172no—
186C2×C4—172no—
187C2×C4—348no—
188C2×C4—348no—
189C2×C4—348no—
190C2×C4—341no—
191C2×C4—341no—
192C2×C4—341no—
193C2×C4—341no—
194C2×C4C22⋊#59512yes—
195C8—684no—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—24no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—342no—
206C22—682no—
207C22—682no—
208C22—682no—
209C22C2×C22⋊#59511yescharacteristic
210C22C2×C22⋊#59511yescharacteristic
211C22C2×C22⋊#59511yescharacteristic
212C4—22no—
213C4—174no—
214C4—342no—
215C4—342no—
216C2—21no—
217C2—21no—
218C2—42no—
219C2—42no—
220C2—682no—
221C2#1176375011yescharacteristic
222C2#1176379612yes—
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC22×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×#63254C2
cosocleC22×Dic17C23
hypercenterC22×D4#595
hypocenterC17#1785
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. C2×#10587545C2semidirect
2. C2×#10587553C2semidirect
3. #11763779C2semidirect
130. C17#1785semidirect

Maximal subgroups

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

subgroupindex
C2×#105875452
C2×#105875532
#117637792
C22×#632542
#178517