Finite Group Databasebeta

← #14529923

Subgroups — #14529923

Order 2176.

290 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 89 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×#10587574C211yesmaximal, characteristic
2C2×#10587506C211yesmaximal, characteristic
3#11762716C211yesmaximal, characteristic
4#11762572C211yesmaximal, characteristic
5#11762571C211yesmaximal, characteristic
6#11762563C211yesmaximal, characteristic
7C2×C17×#103C211yesmaximal, characteristic
8C22×Q8×C17C2211yescharacteristic
9C22×Dic34—21no—
10C22×C8×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C4⋊Dic17C2211yescharacteristic
13C2×C4⋊Dic17C2211yescharacteristic
14C2×#63225—21no—
15C2×C4×Dic17C2211yescharacteristic
16C22×#3622C2211yescharacteristic
17#10587574—22no—
18#10587574—22no—
19C17×#103—22no—
20#10587506—22no—
21#10587502—21no—
22#10587502—21no—
23C2×Q8×C17—21no—
24C2×Q8×C17—21no—
25C2×Q8×C17—21no—
26C2×Q8×C17—21no—
27C2×Q8×C17—21no—
28C2×Q8×C17Q812yescomplemented
29C22×C4×C17—21no—
30C22×C4×C17—21no—
31C22×C4×C17C2311yescharacteristic
32C22×Dic17—21no—
33C22×Dic17—21no—
34C22×Dic17D411yescharacteristic
35C22×Dic17D411yescharacteristic
36C2×Dic34—24no—
37C2×Dic34—42no—
38C2×Dic34—42no—
39C2×C8×C17—21no—
40C2×C8×C17—21no—
41C2×C8×C17—21no—
42C4⋊C4×C17—21no—
43C4⋊C4×C17D412yes—
44C4⋊Dic17—21no—
45C4⋊Dic17—21no—
46C4⋊Dic17—21no—
47C4⋊Dic17D412yes—
48#63225—24no—
49C4×Dic17—21no—
50C4×Dic17—21no—
51C2×#3622—21no—
52C2×#3622—21no—
53C2×#3622—21no—
54C23×C17C2×D411yescharacteristic
55Q8×C17—22no—
56Q8×C17—22no—
57Q8×C17—41no—
58Q8×C17—41no—
59C2×C4×C17—22no—
60C2×C4×C17—22no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×C4×C17—41no—
64C2×C4×C17—41no—
65C2×C4×C17C2×D411yescharacteristic
66C2×C4×C17C2×D411yescharacteristic
67C2×C4×C17C2×Q811yescharacteristic
68C2×C4×C17#4111yescharacteristic
69C2×C4×C17#4111yescharacteristic
70C2×C4×C17#4111yescharacteristic
71C2×Dic17—21no—
72C2×Dic17—21no—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—22no—
76C2×Dic17—41no—
77C2×Dic17—41no—
78C2×Dic17—41no—
79C2×Dic17—41no—
80C2×Dic17—41no—
81C2×Dic17SD1612yes—
82C2×Dic17Q1612yes—
83Dic34—48no—
84C8×C17—41no—
85#3622—41no—
86#2024—171nomaximal, Sylow
87C22×C17#12011yescharacteristic
88C22×C17C22⋊Q811yescharacteristic
89C22×C17#12411yescharacteristic
90C22×C17C2×SD1611yescharacteristic
91C22×C17C2×Q1611yescharacteristic
92C22×C17AGL(1,ℤ/8)11yescharacteristic
93C22×C17#13711yescharacteristic
94C4×C17—42no—
95C4×C17—42no—
96C4×C17C22⋊Q811yescharacteristic
97C4×C17C22⋊Q811yescharacteristic
98C4×C17#12411yescharacteristic
99C4×C17#12411yescharacteristic
100Dic17—22no—
101Dic17—22no—
102Dic17—42no—
103Dic17—81no—
104C2×C4⋊Q8—171no—
105C2×#103—171no—
106C2×#103D1711yescharacteristic
107#391—171no—
108#340—171no—
109#340—171no—
110#336—171no—
111C2×C17#39311yescharacteristic
112C2×C17#45011yescharacteristic
113C2×C17#45111yescharacteristic
114C2×C17#47511yescharacteristic
115C2×C17#47711yescharacteristic
116C2×C17#48711yescharacteristic
117C2×C17#48911yescharacteristic
118C22×Q8—341no—
119C22×Q8C2×D1711yescharacteristic
120C22×C8—171no—
121C22×C8C2×D1711yescharacteristic
122C4⋊Q8—342no—
123C4⋊Q8—342no—
124C2×C4⋊C4—171no—
125C2×C4⋊C4—171no—
126C2×C4⋊C4—341no—
127C2×C4⋊C4C2×D1711yescharacteristic
128C2×C42—171no—
129#103—22no—
130#103—342no—
131#95—341no—
132#95—341no—
133C17#202411yescharacteristic, complemented, Sylow
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—344no—
140C2×Q8—682no—
141C2×Q8—682no—
142C2×Q8Dic3412yescomplemented
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—341no—
148C22×C4—341no—
149C22×C4C22×D1711yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—341no—
154C2×C8—341no—
155C2×C8—341no—
156C4⋊C4—21no—
157C4⋊C4—172no—
158C4⋊C4—344no—
159C4⋊C4—341no—
160C4⋊C4—341no—
161C4⋊C4—341no—
162C4⋊C4C22⋊D1712yes—
163C42—341no—
164C42—341no—
165C23D4×D1711yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—688no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—172no—
178C2×C4—172no—
179C2×C4—341no—
180C2×C4—341no—
181C2×C4—341no—
182C2×C4—341no—
183C2×C4—342no—
184C2×C4—681no—
185C2×C4—681no—
186C2×C4—681no—
187C2×C4—681no—
188C2×C4—681no—
189C2×C4C2×Dic3411yescharacteristic
190C2×C4#6325211yescharacteristic
191C2×C4D4×D1711yescharacteristic
192C2×C4#6325411yescharacteristic
193C2×C4#6325411yescharacteristic
194C2×C4C2×C22⋊D1711yescharacteristic
195C8—41no—
196C8—681no—
197C22#1058756411yescharacteristic
198C22#1058757111yescharacteristic
199C22SD16×D1711yescharacteristic
200C22#1058760011yescharacteristic
201C22Q16×D1711yescharacteristic
202C22#1058760411yescharacteristic
203C22#1058762311yescharacteristic
204C4—42no—
205C4—42no—
206C4—342no—
207C4—342no—
208C4—682no—
209C4—1361no—
210C4#1058756411yescharacteristic
211C4#1058757111yescharacteristic
212C4#1058761011yescharacteristic
213C4#1058762211yescharacteristic
214C2#1176272111yescharacteristic
215C2#1176295811yescharacteristic
216C2#1176296311yescharacteristic
217C2#1176309211yescharacteristic
218C2#1176310211yescharacteristic
219C2#1176315811yescharacteristic
220C2#1176317011yescharacteristic
subgroupG/H
centerC23D4×D17
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC2×C17×#103C2
socle, prime radicalC23×C17C2×D4
hypercenterC2×#103D17
hypocenterC17#2024
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
28. C2×Q8×C17Q8semidirect
133. C17#2024semidirect
142. C2×Q8Dic34semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875062
#117627162
#117625722
#117625712
#117625632
C2×C17×#1032
#202417