Finite Group Databasebeta

← #14534980

Subgroups — #14534980

Order 2176.

408 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 137 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#11762743C212yesmaximal, complemented
3#11762741C212yesmaximal, complemented
4#11762732C212yesmaximal
5#11762706C212yesmaximal, complemented
6#11762696C212yesmaximal, complemented
7#11762669C211yesmaximal, characteristic
8C4×C22⋊Dic17C211yesmaximal, characteristic
9C17×#381C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C2×C22⋊C4×C17C2212yes—
12C2×C22⋊Dic17C2212yes—
13C2×C22⋊Dic17C2211yescharacteristic
14C2×C22⋊Dic17C2211yescharacteristic
15C2×C4⋊Dic17C2212yes—
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×C4×Dic17C2212yes—
23C17×#95C2211yescharacteristic
24C17×#95C2211yescharacteristic
25C17×#95C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2211yescharacteristic
31C24×C17C2311yescharacteristic
32C22×C4×C17—42no—
33C22×C4×C17Q814yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22⋊C4×C17—42no—
43C22⋊Dic17—24no—
44C22⋊Dic17—41no—
45C22⋊Dic17—41no—
46C4⋊Dic17—42no—
47C4⋊Dic17—42no—
48C4⋊Dic17—42no—
49#63225—24no—
50#63225—24no—
51#63225—42no—
52#63225—42no—
53C4×Dic17—42no—
54C23×C17—22no—
55C23×C17—21no—
56C23×C17—22no—
57C23×C17—21no—
58C23×C17C2×Q812yes—
59C23×C17C2411yescharacteristic
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—44no—
69C2×C4×C17C2×Q812yes—
70C2×C4×C17C2×Q812yes—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17#4114yes—
83#2667—171nomaximal, Sylow
84C22×C17—22no—
85C22×C17—22no—
86C22×C17—22no—
87C22×C17—42no—
88C22×C17—42no—
89C22×C17C22×Q811yescharacteristic
90C22×C17C2×#4112yes—
91C22×C17H(5,𝔽2)12yes—
92C22×C17H(5,𝔽2)11yescharacteristic
93C22×C172−1+411yescharacteristic
94C4×C17—22no—
95C4×C17—22no—
96C4×C17—42no—
97C4×C17—42no—
98Dic17—24no—
99Dic17—42no—
100Dic17—42no—
101Dic17—42no—
102#399—172no—
103#398—172no—
104#398—172no—
105#396—172no—
106#389—172no—
107#387—172no—
108#381D1711yescharacteristic
109#380—171no—
110C4×C22⋊C4—171no—
111C2×C17—42no—
112C2×C17#53311yescharacteristic
113C2×C17#53912yes—
114C2×C17#54012yes—
115C2×C17#54311yescharacteristic
116C2×C17#54411yescharacteristic
117C23×C4C2×D1711yescharacteristic
118C2×C4⋊C4—172no—
119C2×C4⋊C4—172no—
120C2×C4⋊C4—172no—
121C2×C4⋊C4—172no—
122C2×C4⋊C4—172no—
123C2×C4⋊C4—172no—
124C2×C4⋊C4—172no—
125C2×C22⋊C4—172no—
126C2×C22⋊C4—171no—
127C2×C22⋊C4—171no—
128C2×C22⋊C4C2×D1712yes—
129C2×C42—172no—
130#95—171no—
131#95—171no—
132#95—172no—
133#95—172no—
134#95—172no—
135#95C2×D1711yescharacteristic
136#95C2×D1711yescharacteristic
137#95C2×D1712yes—
138C17#266711yescharacteristic, complemented, Sylow
139C24C22×D1711yescharacteristic
140C22×C4—42no—
141C22×C4—172no—
142C22×C4—172no—
143C22×C4—172no—
144C22×C4—172no—
145C22×C4Dic3414yes—
146C22×C4C22×D1712yes—
147C22×C4C22×D1712yes—
148C22×C4C22×D1711yescharacteristic
149C22×C4C22×D1711yescharacteristic
150C4⋊C4—344no—
151C4⋊C4—344no—
152C4⋊C4—682no—
153C4⋊C4—682no—
154C4⋊C4—682no—
155C4⋊C4—682no—
156C4⋊C4—682no—
157C22⋊C4—42no—
158C22⋊C4—344no—
159C22⋊C4—681no—
160C22⋊C4—681no—
161C42—682no—
162C23—22no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23C2×Dic3412yes—
167C23C23×D1711yescharacteristic
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—44no—
177C2×C4—174no—
178C2×C4—342no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4C2×Dic3412yes—
190C2×C4C2×Dic3412yes—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22C22×Dic3411yescharacteristic
197C22#1058769112yes—
198C22#41×D1712yes—
199C22#1058769611yescharacteristic
200C22#1058769711yescharacteristic
201C4—22no—
202C4—22no—
203C4—42no—
204C4—42no—
205C4—344no—
206C4—682no—
207C4—682no—
208C4—682no—
209C2—42no—
210C2#1176336211yescharacteristic
211C2#1176340212yes—
212C2#1176341112yes—
213C2#1176343511yescharacteristic
214C2#1176343711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radicalC24×C17C23
hypercenter#381D17
hypocenterC17#2667
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #11762743C2semidirect
3. #11762741C2semidirect
5. #11762706C2semidirect
6. #11762696C2semidirect
138. C17#2667semidirect

Maximal subgroups

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

subgroupindex
#117627452
#117627432
#117627412
#117627322
#117627062
#117626962
#117626692
C4×C22⋊Dic172
C17×#3812
#266717