Finite Group Databasebeta

← #14533796

Subgroups — #14533796

Order 2176.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 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×#10587564C212yesmaximal
2#11762742C212yesmaximal, complemented
3#11762729C212yesmaximal, complemented
4#11762713C212yesmaximal, complemented
5#11762706C212yesmaximal, complemented
6#11762681C212yesmaximal
7C4×C22⋊Dic17C211yesmaximal, characteristic
8C2×#10587502C211yesmaximal, characteristic
9C17×#380C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C23×Dic17C2212yes—
12C22×Dic34C2212yes—
13C2×C4⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊Dic17C2211yescharacteristic
16C2×C22⋊Dic17C2211yescharacteristic
17C2×C4⋊Dic17C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×C4×Dic17C2212yes—
23#10587564—28no—
24C17×#95C2211yescharacteristic
25C17×#95C2211yescharacteristic
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32C24×C17C2311yescharacteristic
33C22×C4×C17—42no—
34C22×C4×C17Q814yescomplemented
35C22×C4×C17C2312yes—
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×Dic17—24no—
40C22×Dic17—24no—
41C22×Dic17—42no—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C2×Dic34—24no—
47C2×Dic34—24no—
48C2×Dic34—42no—
49C4⋊C4×C17—42no—
50C22⋊C4×C17—24no—
51C22⋊Dic17D414yes—
52C22⋊Dic17Q814yescomplemented
53C4⋊Dic17—24no—
54#63225—24no—
55#63225—24no—
56#63225—24no—
57#63225—42no—
58C4×Dic17—42no—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17C2×D412yes—
62C23×C17C2×Q812yes—
63C23×C17C2×Q812yes—
64C23×C17C2411yescharacteristic
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—44no—
74C2×C4×C17C2×Q812yes—
75C2×C4×C17C2×Q812yes—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—48no—
86C2×Dic17C2×D414yes—
87C2×Dic17C2×Q814yes—
88C2×Dic17#4114yes—
89Dic34—48no—
90#2549—171nomaximal, Sylow
91C22×C17—22no—
92C22×C17—22no—
93C22×C17—24no—
94C22×C17—24no—
95C22×C17C4⋊Q814yes—
96C22×C17C22×D411yescharacteristic
97C22×C17C22×Q811yescharacteristic
98C22×C17C22×Q811yescharacteristic
99C22×C17C2×#4112yes—
100C22×C17H(5,𝔽2)12yes—
101C4×C17—22no—
102C4×C17—22no—
103C4×C17—42no—
104C4×C17—42no—
105Dic17—24no—
106Dic17—24no—
107Dic17—24no—
108Dic17—42no—
109C2×C22⋊Q8—172no—
110#398—172no—
111#395—172no—
112#391—172no—
113#389—172no—
114#385—172no—
115#380D1711yescharacteristic
116C4×C22⋊C4—171no—
117C2×#95—171no—
118C2×C17—24no—
119C2×C17C2×C4⋊Q811yescharacteristic
120C2×C17#53311yescharacteristic
121C2×C17#54311yescharacteristic
122C2×C17#54612yes—
123C2×C17#55412yes—
124C22×Q8—172no—
125C23×C4—172no—
126C23×C4C2×D1711yescharacteristic
127C22⋊Q8—348no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—172no—
133C2×C4⋊C4C2×D1712yes—
134C2×C22⋊C4—171no—
135C2×C22⋊C4—171no—
136C2×C22⋊C4C2×D1712yes—
137C2×C42—172no—
138#95—172no—
139#95—171no—
140#95—171no—
141#95—172no—
142#95—172no—
143#95—172no—
144#95C2×D1711yescharacteristic
145#95C2×D1711yescharacteristic
146C17#254911yescharacteristic, complemented, Sylow
147C24C22×D1711yescharacteristic
148C2×Q8—344no—
149C2×Q8—344no—
150C2×Q8—682no—
151C22×C4—42no—
152C22×C4—172no—
153C22×C4—172no—
154C22×C4—172no—
155C22×C4—172no—
156C22×C4—344no—
157C22×C4—344no—
158C22×C4—682no—
159C22×C4Dic3414yescomplemented
160C22×C4C22×D1712yes—
161C22×C4C22×D1712yes—
162C22×C4C22×D1711yescharacteristic
163C22×C4C22×D1711yescharacteristic
164C4⋊C4—42no—
165C4⋊C4—344no—
166C4⋊C4—344no—
167C4⋊C4—344no—
168C4⋊C4—344no—
169C4⋊C4—682no—
170C22⋊C4—24no—
171C22⋊C4—174no—
172C22⋊C4—174no—
173C42—682no—
174C23—22no—
175C23—22no—
176C23C2×Dic3412yes—
177C23D4×D1712yes—
178C23Q8×D1712yes—
179C23C23×D1711yescharacteristic
180Q8—688no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—174no—
191C2×C4—174no—
192C2×C4—174no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—688no—
203C2×C4C2×Dic3412yes—
204C2×C4C2×Dic3412yes—
205C22—24no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22#1058757414yes—
210C22C22×Dic3411yescharacteristic
211C22C2×D4×D1711yescharacteristic
212C22#1058769112yes—
213C22C2×Q8×D1711yescharacteristic
214C22#41×D1712yes—
215C4—22no—
216C4—22no—
217C4—42no—
218C4—42no—
219C4—344no—
220C4—344no—
221C4—344no—
222C4—682no—
223C2—24no—
224C2C2×#1058757411yescharacteristic
225C2#1176336211yescharacteristic
226C2#1176343511yescharacteristic
227C2#1176344812yes—
228C2#1176351312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#380C2
prime radicalC24×C17C23
hypercenter#380D17
hypocenterC17#2549
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #11762742C2semidirect
3. #11762729C2semidirect
4. #11762713C2semidirect
5. #11762706C2semidirect
34. C22×C4×C17Q8semidirect
52. C22⋊Dic17Q8semidirect
146. C17#2549semidirect
159. C22×C4Dic34semidirect

Maximal subgroups

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

subgroupindex
C2×#105875642
#117627422
#117627292
#117627132
#117627062
#117626812
C4×C22⋊Dic172
C2×#105875022
C17×#3802
#254917