Finite Group Databasebeta

← #14544762

Subgroups — #14544762

Order 2176.

408 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C22×#63243C211yesmaximal, characteristic, complemented
2C2×#10587543C211yesmaximal, characteristic, complemented
3C2×C22⋊#3624C212yesmaximal, complemented
4#11763675C212yesmaximal, complemented
5C2×#10587567C211yesmaximal, characteristic
6C22×C22⋊D17C411yescharacteristic
7C23×Dic17C2211yescharacteristic
8C22×C4×D17C2211yescharacteristic
9C22×#3624C2212yes—
10C22×#3624C2212yes—
11C2×#63243—22no—
12C2×#63243—24no—
13C2×C22⋊C4×C17C411yescharacteristic
14C2×#63227C411yescharacteristic
15C2×#63225C411yescharacteristic
16C2×C4×Dic17C2211yescharacteristic
17#10587567—22no—
18#10587567—22no—
19C22⋊#3624—24no—
20#10587543—22no—
21C24×C17C2×C411yescharacteristic
22C23×D17C2×C411yescharacteristic
23C22×C4×C17C2×C411yescharacteristic
24C22×C4×C17C2×C411yescharacteristic
25C2×C22⋊D17—22no—
26C2×C22⋊D17—22no—
27C2×C22⋊D17—22no—
28C22×Dic17—22no—
29C22×Dic17—21no—
30C22×Dic17—21no—
31C22×Dic17—22no—
32C22×Dic17C2×C411yescharacteristic
33C22×Dic17C2×C411yescharacteristic
34C22×Dic17C2311yescharacteristic
35C2×C4×D17—22no—
36C2×C4×D17—22no—
37C2×C4×D17D412yes—
38C2×C4×D17D412yes—
39C2×#3624—22no—
40C2×#3624—22no—
41C2×#3624—22no—
42C2×#3624—22no—
43C2×#3624—22no—
44C2×#3624D414yescomplemented
45#63243—28no—
46C22⋊C4×C17—22no—
47#63227—22no—
48#63225—22no—
49C4×Dic17—21no—
50C4×Dic17—21no—
51C23×C17—21no—
52C23×C17—22no—
53C23×C17—21no—
54C23×C17—22no—
55C23×C17M1612yescomplemented
56C23×C17C22×C411yescharacteristic
57C22×D17—22no—
58C22×D17—21no—
59C22×D17—21no—
60C22×D17—22no—
61C22×D17C22⋊C412yes—
62C2×C4×C17—22no—
63C2×C4×C17—21no—
64C2×C4×C17—22no—
65C2×C4×C17C22⋊C412yes—
66C22⋊D17—28no—
67C2×Dic17—21no—
68C2×Dic17—21no—
69C2×Dic17—21no—
70C2×Dic17—22no—
71C2×Dic17—41no—
72C2×Dic17—42no—
73C2×Dic17—41no—
74C2×Dic17M1612yescomplemented
75C2×Dic17C2×D412yes—
76C2×Dic17C2×D411yescharacteristic
77C2×Dic17C2×D411yescharacteristic
78C2×Dic17#4112yes—
79C4×D17—22no—
80C4×D17—24no—
81C4×D17—22no—
82#3624—24no—
83#3624—42no—
84#1870—171nomaximal, Sylow
85C22×C17—22no—
86C22×C17—24no—
87C22×C17—41no—
88C22×C17—41no—
89C22×C17—42no—
90C22×C17C2×C22⋊C411yescharacteristic
91C22×C17C4×D412yes—
92C22×C17C2×M1611yescharacteristic
93C22×C17C2×M1611yescharacteristic
94C22×C17#13112yes—
95C2×D17—22no—
96C2×D17—22no—
97C2×D17—22no—
98C2×D17—42no—
99C2×D17—42no—
100C4×C17—22no—
101C4×C17—41no—
102Dic17—22no—
103Dic17—41no—
104Dic17#12011yescharacteristic
105Dic17#12112yes—
106Dic17#12311yescharacteristic
107C22×M16—171no—
108C2×C4×D4—171no—
109C2×C22⋊C8—172no—
110C2×C22⋊C8—171no—
111#336—172no—
112C2×C17—42no—
113C2×C17#38611yescharacteristic
114C2×C17#40711yescharacteristic
115C2×C17#40811yescharacteristic
116C2×C17#43512yes—
117C2×C17#43612yes—
118D17—42no—
119C22×D4—171no—
120C23×C4—171no—
121C23×C4—171no—
122C2×M16—344no—
123C2×M16—342no—
124C22×C8—172no—
125C22×C8—172no—
126C4×D4—342no—
127C4×D4—342no—
128C2×C4⋊C4—171no—
129C2×C22⋊C4—171no—
130C2×C22⋊C4#59511yescharacteristic
131C2×C42—171no—
132C22⋊C8—344no—
133C22⋊C8—342no—
134C17#187011yescharacteristic, complemented, Sylow
135C24—171no—
136C24C2×#59511yescharacteristic
137C2×D4—342no—
138C2×D4—342no—
139C2×D4—342no—
140C22×C4—171no—
141C22×C4—171no—
142C22×C4—172no—
143C22×C4—172no—
144C22×C4—171no—
145C22×C4—342no—
146C22×C4—342no—
147C22×C4—342no—
148C22×C4—341no—
149C22×C4—341no—
150C22×C4—342no—
151C22×C4C2×#59511yescharacteristic
152C22×C4C2×#59511yescharacteristic
153M16—348no—
154C2×C8—174no—
155C2×C8—342no—
156C2×C8—342no—
157C2×C8—342no—
158C2×C8—342no—
159C2×C8—342no—
160C4⋊C4—342no—
161C22⋊C4—22no—
162C22⋊C4—342no—
163C42—341no—
164C42—341no—
165C23—21no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—172no—
170C23—341no—
171C23—341no—
172C23—342no—
173C23—342no—
174C23#6324312yescomplemented
175C23C22×#59511yescharacteristic
176D4—348no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—172no—
181C2×C4—172no—
182C2×C4—171no—
183C2×C4—171no—
184C2×C4—172no—
185C2×C4—342no—
186C2×C4—344no—
187C2×C4—342no—
188C2×C4—341no—
189C2×C4—341no—
190C2×C4—342no—
191C2×C4—341no—
192C2×C4—681no—
193C2×C4—682no—
194C2×C4—681no—
195C2×C4C22⋊#59512yes—
196C8—344no—
197C8—682no—
198C22—22no—
199C22—24no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—342no—
204C22—342no—
205C22—342no—
206C22—682no—
207C22—682no—
208C22C2×#6324311yescharacteristic
209C22D17⋊M1611yescharacteristic
210C22#1058767812yes—
211C22D4×#59512yes—
212C22C2×C22⋊#59511yescharacteristic
213C4—22no—
214C4—41no—
215C4—171no—
216C4—172no—
217C4—171no—
218C4—342no—
219C4—681no—
220C2—42no—
221C2—682no—
222C2#1176375711yescharacteristic
223C2#1176378011yescharacteristic
224C2#1176378311yescharacteristic
225C2#1176383812yes—
226C2#1176384212yes—
subgroupG/H
center, FrattiniC23C22×#595
commutator, socleC23×C17C22×C4
FittingC2×C22⋊C4×C17C4
prime radicalC22×C22⋊D17C4
cosocleC22×Dic17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#1870
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#63243C2semidirect
2. C2×#10587543C2semidirect
3. C2×C22⋊#3624C2semidirect
4. #11763675C2semidirect
44. C2×#3624D4semidirect
55. C23×C17M16semidirect
74. C2×Dic17M16semidirect
134. C17#1870semidirect
174. C23#63243semidirect

Maximal subgroups

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

subgroupindex
C22×#632432
C2×#105875432
C2×C22⋊#36242
#117636752
C2×#105875672
#187017