Finite Group Databasebeta

← #14539986

Subgroups — #14539986

Order 2176.

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

normal subgroups only

This group has 324 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
1C2×#10587556C211yesmaximal, characteristic
2#11763215C212yesmaximal, complemented
3#11763177C212yesmaximal, complemented
4#11763142C212yesmaximal
5#11763137C212yesmaximal
6#11763116C214yesmaximal, complemented
7#11762866C211yesmaximal, characteristic
8C17×#533C211yesmaximal, characteristic
9C17×C4⋊Q8C2211yescharacteristic
10C17×C4⋊Q8C2211yescharacteristic
11C17×#125—21no—
12C17×C22⋊Q8—21no—
13C17×C22⋊Q8C2212yes—
14C4×Q8×C17—21no—
15C4×Q8×C17C2212yes—
16C17×#117—21no—
17C2×C42×C17C2211yescharacteristic
18C2×C4⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2211yescharacteristic
20C2×#63225—21no—
21C2×#63225—21no—
22#10587556—21no—
23#10587556—21no—
24#10587556C2212yes—
25#10587556C2212yes—
26#10587506C2214yes—
27#10587506C2214yes—
28C22⋊#3622C2212yes—
29#10587479C2214yes—
30#10587478C2214yes—
31C4⋊#3622C2212yes—
32#10587474C2212yes—
33C2×Q8×C17—21no—
34C2×Q8×C17C2312yes—
35C22×C4×C17D411yescharacteristic
36C22×C4×C17D411yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×Dic17—21no—
39C22×Dic17—21no—
40C4⋊C4×C17—22no—
41C4⋊C4×C17—21no—
42C4⋊C4×C17—22no—
43C4⋊C4×C17—21no—
44C4⋊C4×C17—21no—
45C4⋊C4×C17C2312yes—
46C22⋊C4×C17—22no—
47C42×C17—22no—
48C42×C17D411yescharacteristic
49C42×C17D411yescharacteristic
50C42×C17C2311yescharacteristic
51C42×C17C2311yescharacteristic
52C4⋊Dic17—21no—
53C4⋊Dic17—21no—
54C4⋊Dic17C2312yes—
55C4⋊Dic17C2312yes—
56#63225—22no—
57#63225—22no—
58#63225—22no—
59#63225—22no—
60C2×#3622C2314yes—
61C23×C17C2×D411yescharacteristic
62Q8×C17—24no—
63Q8×C17—42no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—41no—
69C2×C4×C17C2×D411yescharacteristic
70C2×C4×C17C2×D411yescharacteristic
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2×D411yescharacteristic
74C2×C4×C17C2411yescharacteristic
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—42no—
78C2×Dic17—42no—
79#3622—24no—
80#3236—171nomaximal, Sylow
81C22×C17—21no—
82C22×C17—21no—
83C22×C17#13712yes—
84C22×C17C22×D411yescharacteristic
85C4×C17—21no—
86C4×C17—21no—
87C4×C17—21no—
88C4×C17—21no—
89C4×C17—21no—
90C4×C17—42no—
91C4×C17—42no—
92C4×C172−1+412yes—
93Dic17—42no—
94Dic17—42no—
95#533D1711yescharacteristic
96C2×#125—171no—
97#501—172no—
98#491—172no—
99#484—172no—
100#483—172no—
101#479—174no—
102#432—171no—
103C2×C17—22no—
104C2×C17#53611yescharacteristic
105C2×C17C2×#13711yescharacteristic
106C2×C17#57511yescharacteristic
107C4⋊Q8C2×D1711yescharacteristic
108C4⋊Q8C2×D1711yescharacteristic
109#125—21no—
110#125—172no—
111#125—172no—
112#125—341no—
113#125—341no—
114C22⋊Q8—21no—
115C22⋊Q8C2×D1712yes—
116C4×Q8—21no—
117C4×Q8C2×D1712yes—
118#117—21no—
119C2×C4⋊C4—171no—
120C2×C4⋊C4—171no—
121C2×C4⋊C4—341no—
122C2×C4⋊C4—341no—
123C2×C42C2×D1711yescharacteristic
124#107—174no—
125#106—174no—
126C4⋊C8—172no—
127#103—174no—
128#103—174no—
129C22⋊C8—172no—
130#97—172no—
131C17#323611yescharacteristic, complemented, Sylow
132C2×Q8—21no—
133C2×Q8C22×D1712yes—
134C22×C4—341no—
135C22×C4—341no—
136C22×C4C22⋊D1711yescharacteristic
137C22×C4C22⋊D1711yescharacteristic
138C22×C4C22×D1711yescharacteristic
139C2×C8—174no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—172no—
146C4⋊C4—172no—
147C4⋊C4—341no—
148C4⋊C4—341no—
149C4⋊C4—342no—
150C4⋊C4—342no—
151C4⋊C4—342no—
152C4⋊C4—342no—
153C4⋊C4C22×D1712yes—
154C22⋊C4—22no—
155C42—22no—
156C42C22⋊D1711yescharacteristic
157C42C22⋊D1711yescharacteristic
158C42C22×D1711yescharacteristic
159C42C22×D1711yescharacteristic
160C23C2×C22⋊D1711yescharacteristic
161Q8—24no—
162Q8—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—682no—
171C2×C4—682no—
172C2×C4C2×C22⋊D1711yescharacteristic
173C2×C4C2×C22⋊D1711yescharacteristic
174C2×C4C2×C22⋊D1711yescharacteristic
175C2×C4C2×C22⋊D1711yescharacteristic
176C2×C4C2×C22⋊D1711yescharacteristic
177C2×C4C23×D1711yescharacteristic
178C8—344no—
179C22—21no—
180C22—21no—
181C22#1058762812yes—
182C22C22×C22⋊D1711yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—42no—
189C4—42no—
190C4—682no—
191C4—682no—
192C4#1058769712yes—
193C2—22no—
194C2#1176338011yescharacteristic
195C2C2×#1058762811yescharacteristic
196C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#533C2
socleC22×C17C22×D4
prime radicalC23×C17C2×D4
hypercenter#533D17
hypocenterC17#3236
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #11763215C2semidirect
3. #11763177C2semidirect
6. #11763116C2semidirect
131. C17#3236semidirect

Maximal subgroups

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

subgroupindex
C2×#105875562
#117632152
#117631772
#117631422
#117631372
#117631162
#117628662
C17×#5332
#323617