Finite Group Databasebeta

← #14534445

Subgroups — #14534445

Order 2176.

444 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 145 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#11762745C212yesmaximal
2#11762721C212yesmaximal
3#11762715C212yesmaximal, complemented
4#11762713C212yesmaximal, complemented
5#11762693C212yesmaximal, complemented
6#11762692C212yesmaximal, complemented
7#11762672C211yesmaximal, characteristic
8C4×C22⋊Dic17C211yesmaximal, characteristic
9C4×C22⋊C4×C17C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C22×Dic34C2212yes—
12C2×C22⋊C4×C17C2212yes—
13C2×C42×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2211yescharacteristic
17C2×C22⋊Dic17C2211yescharacteristic
18C2×C4⋊Dic17C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×C4×Dic17C2211yescharacteristic
22C2×C4×Dic17C2211yescharacteristic
23C17×#95C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C22×C4×C17—41no—
33C22×C4×C17—41no—
34C22×C4×C17Q814yescomplemented
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C2×Dic34—24no—
44C2×Dic34—24no—
45C2×Dic34—42no—
46C22⋊C4×C17—24no—
47C42×C17—41no—
48C42×C17—41no—
49C22⋊Dic17—24no—
50C22⋊Dic17—41no—
51C22⋊Dic17—41no—
52C4⋊Dic17—24no—
53#63225—24no—
54#63225—42no—
55C4×Dic17—41no—
56C4×Dic17—41no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—21no—
60C23×C17—21no—
61C23×C17C2×Q812yes—
62C23×C17C2411yescharacteristic
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—44no—
73C2×C4×C17C2×Q812yes—
74C2×C4×C17C2×Q812yes—
75C2×C4×C17#4114yes—
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—22no—
86C2×Dic17—22no—
87C2×Dic17#4114yes—
88Dic34—48no—
89#2607—171nomaximal, Sylow
90C22×C17—22no—
91C22×C17—22no—
92C22×C17—22no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17C22×Q811yescharacteristic
96C22×C17C2×#4112yes—
97C22×C17C2×#4112yes—
98C22×C17H(5,𝔽2)11yescharacteristic
99C22×C17H(5,𝔽2)11yescharacteristic
100C4×C17—22no—
101C4×C17—22no—
102C4×C17—24no—
103C4×C17—42no—
104Dic17—24no—
105Dic17—42no—
106Dic17—42no—
107Dic17—42no—
108#399—172no—
109#393—172no—
110#391—172no—
111#391—172no—
112#387—172no—
113#387—172no—
114#381—171no—
115C4×C22⋊C4—171no—
116C4×C22⋊C4D1711yescharacteristic
117C2×C17—42no—
118C2×C17#53311yescharacteristic
119C2×C17#53311yescharacteristic
120C2×C17#54311yescharacteristic
121C2×C17#55112yes—
122C2×C17#55912yes—
123C22×Q8—172no—
124C23×C4C2×D1711yescharacteristic
125C2×C4⋊C4—172no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—172no—
128C2×C22⋊C4—172no—
129C2×C22⋊C4—171no—
130C2×C22⋊C4—171no—
131C2×C22⋊C4C2×D1712yes—
132C2×C42—171no—
133C2×C42—171no—
134C2×C42C2×D1711yescharacteristic
135C2×C42C2×D1711yescharacteristic
136#95—172no—
137#95—172no—
138#95—172no—
139#95—172no—
140#95—172no—
141#95—172no—
142#95—172no—
143#95C2×D1712yes—
144C17#260711yescharacteristic, complemented, Sylow
145C24C22×D1711yescharacteristic
146C2×Q8—344no—
147C2×Q8—344no—
148C2×Q8—682no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4—172no—
154C22×C4—172no—
155C22×C4Dic3414yescomplemented
156C22×C4C22×D1712yes—
157C22×C4C22×D1711yescharacteristic
158C22×C4C22×D1711yescharacteristic
159C22×C4C22×D1712yes—
160C4⋊C4—344no—
161C4⋊C4—344no—
162C4⋊C4—682no—
163C22⋊C4—24no—
164C22⋊C4—344no—
165C22⋊C4—681no—
166C22⋊C4—681no—
167C42—41no—
168C42—41no—
169C42—681no—
170C42—681no—
171C23—22no—
172C23—21no—
173C23—21no—
174C23—22no—
175C23C2×Dic3412yes—
176C23C23×D1711yescharacteristic
177Q8—688no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—174no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4C2×Dic3412yes—
201C2×C4C2×Dic3412yes—
202C2×C4#6325214yes—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—42no—
207C22—42no—
208C22C22×Dic3411yescharacteristic
209C22C2×#6325212yes—
210C22#1058769111yescharacteristic
211C22#1058769111yescharacteristic
212C22#41×D1712yes—
213C4—24no—
214C4—22no—
215C4—22no—
216C4—42no—
217C4—344no—
218C4—682no—
219C4—682no—
220C4—682no—
221C2—42no—
222C2#1176336211yescharacteristic
223C2#1176336611yescharacteristic
224C2#1176343511yescharacteristic
225C2#1176348812yes—
226C2#1176354212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC4×C22⋊C4×C17C2
prime radicalC24×C17C23
hypercenterC4×C22⋊C4D17
hypocenterC17#2607
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #11762715C2semidirect
4. #11762713C2semidirect
5. #11762693C2semidirect
6. #11762692C2semidirect
34. C22×C4×C17Q8semidirect
144. C17#2607semidirect
155. C22×C4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627452
#117627212
#117627152
#117627132
#117626932
#117626922
#117626722
C4×C22⋊Dic172
C4×C22⋊C4×C172
#260717