Finite Group Databasebeta

← #14528754

Subgroups — #14528754

Order 2176.

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

normal subgroups only

This group has 588 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#11762826C212yesmaximal, complemented
2#11762814C212yesmaximal, complemented
3#11762793C211yesmaximal, characteristic, complemented
4#11762783C211yesmaximal, characteristic, complemented
5C2×C17×H(5,𝔽2)C211yesmaximal, characteristic
6C17×H(5,𝔽2)—24no—
7C17×H(5,𝔽2)—24no—
8C2×#41×C17—21no—
9C2×#41×C17C412yescomplemented
10C2×#41×C17C2211yescharacteristic
11C2×#41×C17C2211yescharacteristic
12C22×D4×C17—22no—
13C22×D4×C17—21no—
14C22×D4×C17C412yescomplemented
15C22×D4×C17C2211yescharacteristic
16#10587612C2212yescomplemented
17C2×#63223C2212yescomplemented
18#10587508—24no—
19#10587507—21no—
20#10587505—22no—
21#10587504—21no—
22#10587503—24no—
23C24×C17—22no—
24C24×C17—21no—
25#41×C17—28no—
26#41×C17—24no—
27#41×C17—24no—
28#41×C17—22no—
29#41×C17—22no—
30#41×C17—22no—
31#41×C17D414yes—
32C2×Q8×C17C2×C411yescharacteristic
33C2×Q8×C17D411yescharacteristic
34C2×D4×C17—216no—
35C2×D4×C17—28no—
36C2×D4×C17—22no—
37C2×D4×C17—21no—
38C2×D4×C17—24no—
39C2×D4×C17—24no—
40C2×D4×C17—22no—
41C2×D4×C17—21no—
42C2×D4×C17—22no—
43C2×D4×C17C2×C411yescharacteristic
44C2×D4×C17C2×C412yes—
45C2×D4×C17D412yes—
46C2×D4×C17D411yescharacteristic
47C2×D4×C17D414yes—
48C22×C4×C17—22no—
49C22×C4×C17—21no—
50C22×C4×C17C2×C412yes—
51C22×C4×C17C2311yescharacteristic
52C22⋊Dic17—22no—
53C4⋊Dic17—22no—
54C4×Dic17—22no—
55#63223—22no—
56#63223—22no—
57C2×#3622—22no—
58C23×C17—28no—
59C23×C17—22no—
60C23×C17—21no—
61C23×C17—24no—
62C23×C17—82no—
63C23×C17—84no—
64C23×C17C22⋊C412yes—
65C23×C17C2×D411yescharacteristic
66Q8×C17—21no—
67Q8×C17—22no—
68Q8×C17C22⋊C412yes—
69D4×C17—28no—
70D4×C17—28no—
71D4×C17—24no—
72D4×C17—24no—
73D4×C17—22no—
74D4×C17—21no—
75D4×C17—22no—
76D4×C17—24no—
77D4×C17C22⋊C412yes—
78D4×C17C22⋊C414yes—
79C2×C4×C17—28no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—24no—
84C2×C4×C17C22⋊C412yes—
85C2×C4×C17C22×C411yescharacteristic
86C2×C4×C17C2×D412yes—
87C2×C4×C17C2×D412yes—
88C2×C4×C17C2×D411yescharacteristic
89C2×Dic17—42no—
90#3622—42no—
91#1795—171nomaximal, Sylow
92C22×C17—22no—
93C22×C17—24no—
94C22×C17—22no—
95C22×C17—41no—
96C22×C17—42no—
97C22×C17—44no—
98C22×C17—42no—
99C22×C17—84no—
100C22×C17—88no—
101C22×C17C2×C22⋊C411yescharacteristic
102C22×C17#12012yes—
103C4×C17—22no—
104C4×C17—22no—
105C4×C17C2×C22⋊C411yescharacteristic
106C4×C17C2×C22⋊C411yescharacteristic
107C4×C17#12012yes—
108Dic17—82no—
109C2×H(5,𝔽2)D1711yescharacteristic
110#421—172no—
111#418—172no—
112#413—171no—
113#410—171no—
114C2×C17—22no—
115C2×C17—21no—
116C2×C17—44no—
117C2×C17—42no—
118C2×C17—42no—
119C2×C17#37911yescharacteristic
120H(5,𝔽2)—24no—
121H(5,𝔽2)—24no—
122C2×#41—21no—
123C2×#41Dic1712yescomplemented
124C2×#41C2×D1711yescharacteristic
125C2×#41C2×D1711yescharacteristic
126C22×D4—22no—
127C22×D4—21no—
128C22×D4Dic1712yescomplemented
129C22×D4C2×D1711yescharacteristic
130C2×M16—172no—
131#117—172no—
132#104—344no—
133#102—344no—
134#101—341no—
135#100—341no—
136#99—342no—
137C17#179511yescharacteristic, complemented, Sylow
138C24—22no—
139C24—21no—
140#41—28no—
141#41—24no—
142#41—24no—
143#41—22no—
144#41—22no—
145#41—22no—
146#41C22⋊D1714yes—
147C2×Q8C2×Dic1711yescharacteristic
148C2×Q8C22⋊D1711yescharacteristic
149C2×D4—216no—
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4C2×Dic1711yescharacteristic
159C2×D4C2×Dic1712yes—
160C2×D4C22⋊D1712yes—
161C2×D4C22⋊D1711yescharacteristic
162C2×D4C22⋊D1714yes—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4C2×Dic1712yes—
166C22×C4C22×D1711yescharacteristic
167M16—342no—
168M16—342no—
169C2×C8—342no—
170C4⋊C4—342no—
171C22⋊C4—342no—
172C42—342no—
173C23—28no—
174C23—22no—
175C23—21no—
176C23—24no—
177C23—82no—
178C23—84no—
179C23C22⋊Dic1712yes—
180C23C2×C22⋊D1711yescharacteristic
181Q8—21no—
182Q8—22no—
183Q8C22⋊Dic1712yes—
184D4—28no—
185D4—28no—
186D4—24no—
187D4—24no—
188D4—22no—
189D4—21no—
190D4—22no—
191D4—24no—
192D4C22⋊Dic1712yes—
193D4C22⋊Dic1714yes—
194C2×C4—28no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—682no—
200C2×C4C22⋊Dic1712yes—
201C2×C4C22×Dic1711yescharacteristic
202C2×C4C2×C22⋊D1712yes—
203C2×C4C2×C22⋊D1712yes—
204C2×C4C2×C22⋊D1711yescharacteristic
205C8—682no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—41no—
210C22—42no—
211C22—44no—
212C22—42no—
213C22—84no—
214C22—88no—
215C22C2×C22⋊Dic1711yescharacteristic
216C22#1058763912yes—
217C4—22no—
218C4—22no—
219C4—1362no—
220C4C2×C22⋊Dic1711yescharacteristic
221C4C2×C22⋊Dic1711yescharacteristic
222C4#1058763912yes—
223C2—22no—
224C2—21no—
225C2—44no—
226C2—42no—
227C2—42no—
228C2#1176266511yescharacteristic
subgroupG/H
centerC2#11762665
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
Fitting, prime radicalC2×C17×H(5,𝔽2)C2
socleC2×C17#379
cosocleC22×C4×C17C23
hypercenterC2×H(5,𝔽2)D17
hypocenterC17#1795
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #11762826C2semidirect
2. #11762814C2semidirect
3. #11762793C2semidirect
4. #11762783C2semidirect
9. C2×#41×C17C4semidirect
14. C22×D4×C17C4semidirect
16. #10587612C22semidirect
17. C2×#63223C22semidirect
123. C2×#41Dic17semidirect
128. C22×D4Dic17semidirect
137. C17#1795semidirect

Maximal subgroups

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

subgroupindex
#117628262
#117628142
#117627932
#117627832
C2×C17×H(5,𝔽2)2
#179517