Finite Group Databasebeta

← #14539528

Subgroups — #14539528

Order 2176.

358 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 121 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#11763364C211yesmaximal, characteristic
2#11763364C211yesmaximal, characteristic
3#11763205C211yesmaximal, characteristic, complemented
4#11763188C211yesmaximal, characteristic, complemented
5#11763160C211yesmaximal, characteristic, complemented
6#11763066C214yesmaximal
7#11762913C214yesmaximal, complemented
8C4×#63218C211yesmaximal, characteristic
9C17×#487C211yesmaximal, characteristic, complemented
10C17×C4⋊Q8C2211yescharacteristic
11C17×C4⋊Q8C2211yescharacteristic
12#10587632—21no—
13#10587632C2212yes—
14Q8×Dic17—21no—
15Q8×Dic17C2212yes—
16C2×#63231C2214yes—
17C2×#63218C2212yes—
18#10587583—21no—
19#10587583C2212yes—
20#10587578—21no—
21#10587578—21no—
22#10587576—21no—
23#10587576—21no—
24#10587573—21no—
25#10587573C2212yes—
26C42×D17C2211yescharacteristic
27#10587555C2211yescharacteristic
28#10587555C2211yescharacteristic
29C17×#103C2214yes—
30C4×C8×C17C2211yescharacteristic
31#10587487C2214yes—
32#10587486C2212yes—
33#10587479C2214yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17C2312yes—
37C2×C4×D17D411yescharacteristic
38C2×C4×D17D411yescharacteristic
39C2×C4×D17C2311yescharacteristic
40C2×Dic34—21no—
41C2×Dic34C2312yes—
42C2×C8×C17C2312yes—
43C4⋊C4×C17—21no—
44C4⋊C4×C17—21no—
45C4⋊C4×C17—21no—
46C4⋊C4×C17C2312yes—
47C42×C17C2311yescharacteristic
48#63231—28no—
49#63227—22no—
50#63227—22no—
51C4⋊Dic17—21no—
52C4⋊Dic17—21no—
53C4⋊Dic17—21no—
54C4⋊Dic17—22no—
55C4⋊Dic17—22no—
56C4⋊Dic17C2312yes—
57#63225—22no—
58#63225—22no—
59C4×Dic17—22no—
60C4×Dic17—22no—
61C4×Dic17D411yescharacteristic
62C4×Dic17D411yescharacteristic
63C4×Dic17C2311yescharacteristic
64C2×#3622C2312yes—
65#63218—24no—
66C22×D17C2×D411yescharacteristic
67Q8×C17—24no—
68Q8×C17—42no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2411yescharacteristic
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17C2×D411yescharacteristic
77C2×Dic17C2×D411yescharacteristic
78C2×Dic17C2×D411yescharacteristic
79C4×D17—22no—
80C4×D17—22no—
81C4×D17—22no—
82Dic34—24no—
83Dic34—42no—
84C8×C17—22no—
85#3622#4114yes—
86#3185—171nomaximal, Sylow
87C22×C17C22×D411yescharacteristic
88C2×D17—21no—
89C2×D17—21no—
90C2×D17—21no—
91C4×C17—42no—
92C4×C17—42no—
93C4×C17#13712yes—
94C4×C17#13712yes—
95C4×C17C2×#4112yes—
96Dic17—21no—
97Dic17—21no—
98Dic17—21no—
99Dic17—42no—
100Dic17—42no—
101#533—171no—
102#533—171no—
103#498—171no—
104#494—171no—
105#487—171no—
106#487D1711yescharacteristic, complemented
107#470—174no—
108#441—174no—
109C4×M16—171no—
110C2×C17#53211yescharacteristic
111C2×C17C2×#13711yescharacteristic
112C2×C17C2×#13711yescharacteristic
113D17—41no—
114C2×Q16—174no—
115C2×M16—172no—
116C4⋊Q8—171no—
117C4⋊Q8—171no—
118C4⋊Q8C2×D1711yescharacteristic
119C4⋊Q8C2×D1711yescharacteristic
120#125—341no—
121#125—341no—
122C22⋊Q8—172no—
123C22⋊Q8—172no—
124C22⋊Q8—341no—
125C22⋊Q8—341no—
126C4×Q8—172no—
127C4×Q8—172no—
128C4×Q8—341no—
129C4×Q8—341no—
130#117—341no—
131#117—341no—
132C2×C42—171no—
133#106—174no—
134#103—174no—
135#103C2×D1714yes—
136#97—172no—
137C4×C8—171no—
138C4×C8C2×D1711yescharacteristic
139C17#318511yescharacteristic, complemented, Sylow
140C2×Q8—21no—
141C2×Q8—172no—
142C2×Q8—341no—
143C2×Q8C22×D1712yes—
144C22×C4—171no—
145C22×C4—171no—
146C22×C4—171no—
147Q16—348no—
148M16—344no—
149C2×C8—172no—
150C2×C8C22×D1712yes—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—172no—
155C4⋊C4—342no—
156C4⋊C4—342no—
157C4⋊C4—342no—
158C4⋊C4—342no—
159C4⋊C4—341no—
160C4⋊C4—341no—
161C4⋊C4—341no—
162C4⋊C4C22×D1712yes—
163C22⋊C4—342no—
164C22⋊C4—342no—
165C42—171no—
166C42—171no—
167C42—171no—
168C42—342no—
169C42—342no—
170C42C22×D1711yescharacteristic
171C23—171no—
172Q8—24no—
173Q8—42no—
174Q8—344no—
175Q8—682no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—171no—
179C2×C4—171no—
180C2×C4—171no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4D4×D1711yescharacteristic
187C2×C4D4×D1711yescharacteristic
188C2×C4C23×D1711yescharacteristic
189C8—22no—
190C8—174no—
191C22—341no—
192C22—341no—
193C22—341no—
194C22C2×D4×D1711yescharacteristic
195C4—42no—
196C4—42no—
197C4—341no—
198C4—341no—
199C4—341no—
200C4—682no—
201C4—682no—
202C4#1058760112yes—
203C4#1058760412yes—
204C4#41×D1712yes—
205C2—681no—
206C2#1176335811yescharacteristic
207C2C2×#1058760111yescharacteristic
208C2C2×#1058760411yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#487C2
socleC22×C17C22×D4
prime radicalC22×D17C2×D4
hypercenter#487D17
hypocenterC17#3185
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #11763205C2semidirect
4. #11763188C2semidirect
5. #11763160C2semidirect
7. #11762913C2semidirect
9. C17×#487C2semidirect
106. #487D17semidirect
139. C17#3185semidirect

Maximal subgroups

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

subgroupindex
#117633642
#117632052
#117631882
#117631602
#117630662
#117629132
C4×#632182
C17×#4872
#318517