Finite Group Databasebeta

← #14544928

Subgroups — #14544928

Order 2176.

312 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 70 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#11763812C211yesmaximal, characteristic, complemented
2C2×#10587545C212yesmaximal, complemented
3C2×#10587553C212yesmaximal
4C4×#63247C211yesmaximal, characteristic, complemented
5C2×#10587571C211yesmaximal, characteristic
6C22×C22⋊D17C2211yescharacteristic
7C22×Dic34C2211yescharacteristic
8C2×#63247C2212yes—
9C2×#63247C2212yes—
10C2×C22⋊C4×C17C411yescharacteristic
11C2×C22⋊Dic17C411yescharacteristic
12C2×#63227C411yescharacteristic
13C2×#63227C411yescharacteristic
14C2×C4×Dic17C2211yescharacteristic
15#10587571—21no—
16#10587571—21no—
17#10587571—21no—
18#10587571—21no—
19#10587553—24no—
20#10587545—24no—
21#10587544—21no—
22#10587542—21no—
23C4⋊#3624—21no—
24C4×#3624—21no—
25C24×C17C2×C411yescharacteristic
26C23×D17C2×C411yescharacteristic
27C22×C4×C17C2×C411yescharacteristic
28C22×C4×C17C2×C411yescharacteristic
29C2×C22⋊D17—22no—
30C2×C22⋊D17—21no—
31C2×C22⋊D17—21no—
32C2×C22⋊D17—22no—
33C22×Dic17C2×C411yescharacteristic
34C22×Dic17C2×C411yescharacteristic
35C22×Dic17C2311yescharacteristic
36C2×Dic34—22no—
37C2×Dic34—21no—
38C2×Dic34—21no—
39C2×Dic34—22no—
40#63247—22no—
41#63247—22no—
42#63247—22no—
43#63247D414yes—
44C2×#3624—22no—
45C2×#3624—22no—
46C22⋊C4×C17—21no—
47C22⋊C4×C17—21no—
48C22⋊Dic17—21no—
49C22⋊Dic17—21no—
50#63227—21no—
51#63227—21no—
52#63227—21no—
53#63227—21no—
54C4×Dic17D411yescharacteristic
55C4×Dic17D411yescharacteristic
56C4×Dic17D411yescharacteristic
57C4×Dic17D411yescharacteristic
58C23×C17—21no—
59C23×C17—22no—
60C23×C17—41no—
61C23×C17—41no—
62C23×C17C22×C411yescharacteristic
63C22×D17—21no—
64C22×D17—22no—
65C22×D17—41no—
66C22×D17—41no—
67C2×C4×C17—21no—
68C2×C4×C17—22no—
69C2×C4×C17—41no—
70C2×C4×C17C22⋊C411yescharacteristic
71C2×C4×C17C22⋊C411yescharacteristic
72C22⋊D17—42no—
73C22⋊D17—41no—
74C22⋊D17—41no—
75C2×Dic17—21no—
76C2×Dic17—22no—
77C2×Dic17—41no—
78C2×Dic17C22⋊C411yescharacteristic
79C2×Dic17C22⋊C411yescharacteristic
80C2×Dic17C2×D411yescharacteristic
81C2×Dic17C2×D412yes—
82C2×Dic17C2×D411yescharacteristic
83C2×Dic17#4112yes—
84Dic34—42no—
85Dic34—41no—
86Dic34—41no—
87#3624—24no—
88#3624—42no—
89#1967—171nomaximal, Sylow
90C22×C17—21no—
91C22×C17—21no—
92C22×C17—41no—
93C22×C17—41no—
94C22×C17—41no—
95C22×C17—82no—
96C22×C17C2×C22⋊C411yescharacteristic
97C22×C17C4×D412yes—
98C2×D17—41no—
99C2×D17—41no—
100C2×D17—41no—
101C2×D17—82no—
102C4×C17—21no—
103C4×C17—21no—
104C4×C17—41no—
105Dic17—21no—
106Dic17—21no—
107Dic17—41no—
108Dic17#12112yes—
109Dic17#12411yescharacteristic
110Dic17#12711yescharacteristic
111C2×#124—171no—
112#423—171no—
113C2×#101—172no—
114C2×#100—172no—
115C4×M16—171no—
116C2×C17—22no—
117C2×C17—81no—
118C2×C17#39011yescharacteristic
119C2×C17#41311yescharacteristic
120C2×C17#41311yescharacteristic
121D17—81no—
122C22×Q8—171no—
123C22×D4—171no—
124C2×M16—172no—
125C2×M16—172no—
126#124—341no—
127#124—341no—
128#124—341no—
129#124—341no—
130C2×C22⋊C4—171no—
131C2×C22⋊C4—171no—
132C2×C22⋊C4—171no—
133C2×C22⋊C4#59511yescharacteristic
134C2×C42—171no—
135C4⋊C8—341no—
136C4⋊C8—341no—
137#101—344no—
138#100—344no—
139#97—341no—
140C4×C8—341no—
141C17#196711yescharacteristic, complemented, Sylow
142C24—171no—
143C24C2×#59511yescharacteristic
144C2×Q8—342no—
145C2×Q8—341no—
146C2×Q8—341no—
147C2×Q8—342no—
148C2×D4—342no—
149C2×D4—341no—
150C2×D4—341no—
151C2×D4—342no—
152C22×C4—171no—
153C22×C4—171no—
154C22×C4—171no—
155C22×C4C2×#59511yescharacteristic
156C22×C4C2×#59511yescharacteristic
157M16—174no—
158M16—342no—
159M16—342no—
160M16—342no—
161C2×C8—342no—
162C2×C8—342no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—341no—
166C22⋊C4—341no—
167C22⋊C4—341no—
168C22⋊C4—341no—
169C22⋊C4—341no—
170C22⋊C4—341no—
171C42—171no—
172C42—171no—
173C42—171no—
174C42—171no—
175C23—21no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—341no—
180C23—342no—
181C23—681no—
182C23—681no—
183C23C22×#59511yescharacteristic
184Q8—682no—
185Q8—681no—
186Q8—681no—
187D4—682no—
188D4—681no—
189D4—681no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—172no—
194C2×C4—172no—
195C2×C4—171no—
196C2×C4—171no—
197C2×C4—171no—
198C2×C4—171no—
199C2×C4—342no—
200C2×C4—341no—
201C2×C4—681no—
202C2×C4C22⋊#59511yescharacteristic
203C2×C4C22⋊#59511yescharacteristic
204C8—344no—
205C8—682no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—82no—
212C22—681no—
213C22—681no—
214C22—681no—
215C22—1362no—
216C22D4×#59512yes—
217C22C2×C22⋊#59511yescharacteristic
218C4—21no—
219C4—21no—
220C4—41no—
221C4—172no—
222C4—171no—
223C4—171no—
224C4—341no—
225C4—341no—
226C4—681no—
227C2—22no—
228C2—81no—
229C2—1361no—
230C2#1176376111yescharacteristic
231C2#1176379511yescharacteristic
232C2#1176379611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C22⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×C22⋊D17C22
cosocleC22×Dic17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#1967
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. #11763812C2semidirect
2. C2×#10587545C2semidirect
4. C4×#63247C2semidirect
141. C17#1967semidirect

Maximal subgroups

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

subgroupindex
#117638122
C2×#105875452
C2×#105875532
C4×#632472
C2×#105875712
#196717