Finite Group Databasebeta

← #14529943

Subgroups — #14529943

Order 2176.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#11763437C211yesmaximal, characteristic
2#11762818C211yesmaximal, characteristic
3#11762716C211yesmaximal, characteristic
4#11762577C211yesmaximal, characteristic
5#11762570C211yesmaximal, characteristic
6#11762568C211yesmaximal, characteristic
7C17×#419C211yesmaximal, characteristic
8C22×Q8×C17C2211yescharacteristic
9C2×M16×C17C2211yescharacteristic
10C17×#117C2211yescharacteristic
11#10587612C2211yescharacteristic
12C2×C4⋊Dic17C2211yescharacteristic
13C2×C4×Dic17C2211yescharacteristic
14C2×#63223C2211yescharacteristic
15#10587576—21no—
16#10587574—21no—
17#10587564—22no—
18#10587556—21no—
19#10587555—21no—
20C17×#103—21no—
21C17×#103—21no—
22#10587506—21no—
23#10587506—21no—
24#10587502—22no—
25C2×Q8×C17—21no—
26C2×Q8×C17—21no—
27C2×Q8×C17—21no—
28C2×Q8×C17—21no—
29C2×Q8×C17—21no—
30C2×Q8×C17Q812yescomplemented
31C22×C4×C17—21no—
32C22×C4×C17C2311yescharacteristic
33C22×Dic17—21no—
34C22×Dic17D412yes—
35C2×Dic34—22no—
36M16×C17—21no—
37M16×C17—21no—
38C2×C8×C17—21no—
39C4⋊C4×C17D411yescharacteristic
40C4⋊C4×C17D411yescharacteristic
41C22⋊C4×C17—21no—
42C42×C17—21no—
43C22⋊Dic17—21no—
44C4⋊Dic17—22no—
45C4⋊Dic17—21no—
46C4⋊Dic17—21no—
47C4⋊Dic17D411yescharacteristic
48C4⋊Dic17D411yescharacteristic
49#63225—22no—
50#63225—22no—
51C4×Dic17—21no—
52C4×Dic17—21no—
53C4×Dic17—21no—
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—21no—
63C2×C4×C17—21no—
64C2×C4×C17—22no—
65C2×C4×C17—41no—
66C2×C4×C17—41no—
67C2×C4×C17C2×D411yescharacteristic
68C2×C4×C17C2×D411yescharacteristic
69C2×C4×C17C2×Q811yescharacteristic
70C2×C4×C17#4111yescharacteristic
71C2×C4×C17#4111yescharacteristic
72C2×C4×C17#4111yescharacteristic
73C2×Dic17—22no—
74C2×Dic17—21no—
75C2×Dic17—21no—
76C2×Dic17—22no—
77C2×Dic17—41no—
78C2×Dic17—41no—
79C2×Dic17—42no—
80Dic34—82no—
81C8×C17—41no—
82#3622—41no—
83#2028—171nomaximal, Sylow
84C22×C17—21no—
85C22×C17—21no—
86C22×C17#12011yescharacteristic
87C22×C17C22⋊Q811yescharacteristic
88C22×C17#12411yescharacteristic
89C4×C17—42no—
90C4×C17—41no—
91C4×C17—41no—
92C4×C17C22⋊Q811yescharacteristic
93C4×C17C22⋊Q811yescharacteristic
94C4×C17#12411yescharacteristic
95C4×C17#12411yescharacteristic
96Dic17—42no—
97Dic17—41no—
98Dic17—42no—
99Dic17—41no—
100#544—171no—
101#419—171no—
102#419D1711yescharacteristic
103#391—171no—
104#343—171no—
105#339—171no—
106#339—171no—
107C2×C17—21no—
108C2×C17—21no—
109C2×C17#39311yescharacteristic
110C2×C17#45511yescharacteristic
111C2×C17#45611yescharacteristic
112C22×Q8C2×D1711yescharacteristic
113C2×M16—171no—
114C2×M16C2×D1711yescharacteristic
115C4⋊Q8—341no—
116C4⋊Q8—341no—
117#125—341no—
118#125—341no—
119C22⋊Q8—342no—
120#117—171no—
121#117C2×D1711yescharacteristic
122C2×C4⋊C4—171no—
123C2×C42—171no—
124#103—21no—
125#103—21no—
126#103—341no—
127#103—341no—
128#95—342no—
129C17#202811yescharacteristic, complemented, Sylow
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—342no—
136C2×Q8Dic3412yescomplemented
137C22×C4—21no—
138C22×C4—172no—
139C22×C4—341no—
140C22×C4C22×D1711yescharacteristic
141M16—21no—
142M16—21no—
143M16—341no—
144M16—341no—
145C2×C8—21no—
146C2×C8—341no—
147C4⋊C4—171no—
148C4⋊C4—171no—
149C4⋊C4—342no—
150C4⋊C4—342no—
151C4⋊C4—342no—
152C4⋊C4—341no—
153C4⋊C4—341no—
154C4⋊C4C22⋊D1711yescharacteristic
155C4⋊C4C22⋊D1711yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—341no—
158C42—21no—
159C42—341no—
160C42—341no—
161C42—341no—
162C23D4×D1711yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—1362no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—342no—
174C2×C4—341no—
175C2×C4—341no—
176C2×C4—342no—
177C2×C4—681no—
178C2×C4—681no—
179C2×C4—682no—
180C2×C4C2×Dic3411yescharacteristic
181C2×C4#6325211yescharacteristic
182C2×C4D4×D1711yescharacteristic
183C2×C4#6325411yescharacteristic
184C2×C4#6325411yescharacteristic
185C2×C4C2×C22⋊D1711yescharacteristic
186C8—41no—
187C8—681no—
188C22—21no—
189C22—21no—
190C22#1058756411yescharacteristic
191C22#1058757111yescharacteristic
192C22#1058762311yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—682no—
197C4—682no—
198C4—681no—
199C4—681no—
200C4#1058756411yescharacteristic
201C4#1058757111yescharacteristic
202C4#1058761011yescharacteristic
203C4#1058762211yescharacteristic
204C2—21no—
205C2—21no—
206C2#1176272111yescharacteristic
207C2#1176298511yescharacteristic
208C2#1176299011yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#419C2
socleC22×C17#120
prime radicalC23×C17C2×D4
hypercenter#419D17
hypocenterC17#2028
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C17Q8semidirect
129. C17#2028semidirect
136. C2×Q8Dic34semidirect

Maximal subgroups

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

subgroupindex
#117634372
#117628182
#117627162
#117625772
#117625702
#117625682
C17×#4192
#202817