Finite Group Databasebeta

← #14529174

Subgroups — #14529174

Order 2176.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#10587630C211yesmaximal, characteristic
2#11763286C211yesmaximal, characteristic
3#11762818C211yesmaximal, characteristic
4C2×#10587495C211yesmaximal, characteristic
5#11762570C211yesmaximal, characteristic
6#11762568C211yesmaximal, characteristic
7C17×#419C211yesmaximal, characteristic
8C22×Q8×C17C2211yescharacteristic
9C22×Dic34C2211yescharacteristic
10C2×M16×C17C2211yescharacteristic
11C17×#117C2211yescharacteristic
12#10587630—22no—
13#10587630C414yescomplemented
14#10587612C2211yescharacteristic
15C2×#63225—22no—
16C2×C4×Dic17C2211yescharacteristic
17C2×#63223C2211yescharacteristic
18#10587573—22no—
19#10587563—22no—
20C4×Dic34—22no—
21C17×#103—22no—
22#10587506—22no—
23#10587495—22no—
24C2×Q8×C17—21no—
25C2×Q8×C17—21no—
26C2×Q8×C17—21no—
27C2×Q8×C17—21no—
28C2×Q8×C17—21no—
29C2×Q8×C17C2×C412yes—
30C22×C4×C17—21no—
31C22×C4×C17C2311yescharacteristic
32C22×Dic17—21no—
33C22×Dic17D412yes—
34C2×Dic34—21no—
35C2×Dic34—21no—
36C2×Dic34—21no—
37C2×Dic34—21no—
38C2×Dic34C2×C412yes—
39C2×Dic34D412yes—
40M16×C17—21no—
41M16×C17—21no—
42C2×C8×C17—21no—
43C4⋊C4×C17D412yes—
44C22⋊C4×C17—21no—
45C42×C17—21no—
46C22⋊Dic17—21no—
47C4⋊Dic17D412yes—
48#63225—24no—
49#63225—28no—
50C4×Dic17—24no—
51C4×Dic17—21no—
52C4×Dic17—21no—
53C4×Dic17C2×C412yes—
54#63223—21no—
55#63223—21no—
56C2×#3622—21no—
57C23×C17C2×D411yescharacteristic
58Q8×C17—22no—
59Q8×C17—22no—
60Q8×C17—41no—
61Q8×C17—41no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—41no—
65C2×C4×C17—41no—
66C2×C4×C17C22×C411yescharacteristic
67C2×C4×C17C2×D411yescharacteristic
68C2×C4×C17C2×D411yescharacteristic
69C2×C4×C17C2×D411yescharacteristic
70C2×C4×C17#4111yescharacteristic
71C2×C4×C17#4111yescharacteristic
72C2×Dic17—24no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17C22⋊C414yes—
78Dic34—22no—
79Dic34—22no—
80Dic34—22no—
81Dic34—22no—
82C8×C17—41no—
83#3622—41no—
84#1886—171nomaximal, Sylow
85C22×C17—22no—
86C22×C17C2×C22⋊C411yescharacteristic
87C22×C17#12011yescharacteristic
88C22×C17#12311yescharacteristic
89C4×C17—42no—
90C4×C17—42no—
91C4×C17C4×D411yescharacteristic
92C4×C17C4×D411yescharacteristic
93C4×C17#12111yescharacteristic
94C4×C17#12111yescharacteristic
95Dic17—24no—
96Dic17—24no—
97Dic17—42no—
98C2×C4⋊Q8—171no—
99#519—171no—
100#419—171no—
101#419D1711yescharacteristic
102C2×#101—171no—
103#339—171no—
104#339—171no—
105C2×C17—21no—
106C2×C17—21no—
107C2×C17#38611yescharacteristic
108C2×C17#45612yes—
109C22×Q8—171no—
110C22×Q8C2×D1711yescharacteristic
111C2×M16—171no—
112C2×M16C2×D1711yescharacteristic
113C4⋊Q8—174no—
114C4⋊Q8—342no—
115C4×Q8—342no—
116C4×Q8—342no—
117#117—171no—
118#117—342no—
119#117C2×D1711yescharacteristic
120C2×C4⋊C4—342no—
121C2×C42—171no—
122#103—22no—
123#103—342no—
124#101—342no—
125C17#188611yescharacteristic, complemented, Sylow
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—172no—
132C2×Q8—172no—
133C2×Q8—341no—
134C2×Q8—341no—
135C2×Q8—341no—
136C2×Q8—341no—
137C2×Q8C4×D1712yes—
138C22×C4—21no—
139C22×C4—172no—
140C22×C4—341no—
141C22×C4C22×D1711yescharacteristic
142M16—21no—
143M16—21no—
144M16—341no—
145M16—341no—
146C2×C8—21no—
147C2×C8—341no—
148C4⋊C4—172no—
149C4⋊C4—348no—
150C4⋊C4—344no—
151C4⋊C4C22⋊D1712yes—
152C22⋊C4—21no—
153C22⋊C4—341no—
154C42—21no—
155C42—172no—
156C42—344no—
157C42—341no—
158C42—341no—
159C23D4×D1711yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—342no—
165Q8—342no—
166Q8—342no—
167Q8—342no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—174no—
173C2×C4—344no—
174C2×C4—342no—
175C2×C4—342no—
176C2×C4—342no—
177C2×C4—342no—
178C2×C4C2×C4×D1711yescharacteristic
179C2×C4#6325211yescharacteristic
180C2×C4D4×D1711yescharacteristic
181C2×C4D4×D1711yescharacteristic
182C2×C4#6325411yescharacteristic
183C2×C4C2×C22⋊D1711yescharacteristic
184C8—41no—
185C8—681no—
186C22—22no—
187C22C22⋊C4×D1711yescharacteristic
188C22#1058756911yescharacteristic
189C22#1058762311yescharacteristic
190C4—42no—
191C4—42no—
192C4—344no—
193C4—344no—
194C4—682no—
195C4#1058756711yescharacteristic
196C4#1058757011yescharacteristic
197C4C4×C22⋊D1711yescharacteristic
198C4#1058762411yescharacteristic
199C2—21no—
200C2—21no—
201C2#1176268511yescharacteristic
202C2#1176299012yes—
subgroupG/H
centerC22#10587623
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#419C2
socleC22×C17#120
prime radicalC23×C17C2×D4
cosocleC22×C4×C17C23
hypercenter#419D17
hypocenterC17#1886
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
13. #10587630C4semidirect
125. C17#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
#117632862
#117628182
C2×#105874952
#117625702
#117625682
C17×#4192
#188617