Finite Group Databasebeta

← #14531585

Subgroups — #14531585

Order 2176.

428 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 177 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×#10587556C211yesmaximal, characteristic
2#11762706C212yesmaximal
3#11762696C212yesmaximal
4#11762694C212yesmaximal
5#11762692C212yesmaximal
6#11762691C212yesmaximal
7C4×C4⋊Dic17C211yesmaximal, characteristic
8#11762660C212yesmaximal
9C17×#383C211yesmaximal, characteristic
10C2×C42×C17C2211yescharacteristic
11C2×C4⋊Dic17C2212yes—
12C2×C4⋊Dic17C2211yescharacteristic
13C2×C4⋊Dic17C2211yescharacteristic
14C2×#63225C2212yes—
15C2×#63225C2212yes—
16C2×#63225C2212yes—
17C2×#63225C2212yes—
18C2×C4×Dic17C2212yes—
19C2×C4×Dic17C2211yescharacteristic
20C2×C4×Dic17C2212yes—
21C2×C4×Dic17C2211yescharacteristic
22#10587556C418yescomplemented
23C17×#95C2212yes—
24C17×#95C2212yes—
25C17×#95C2211yescharacteristic
26C17×#95C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32C22×C4×C17C2312yes—
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C42×C17C2×C414yes—
42C4⋊Dic17—24no—
43C4⋊Dic17C2×C414yes—
44C4⋊Dic17C2×C414yes—
45#63225—24no—
46#63225—42no—
47#63225C2×C418yes—
48#63225C2×C418yes—
49C4×Dic17—24no—
50C4×Dic17—41no—
51C4×Dic17—42no—
52C4×Dic17—41no—
53C23×C17C2411yescharacteristic
54C2×C4×C17—22no—
55C2×C4×C17—21no—
56C2×C4×C17—22no—
57C2×C4×C17—21no—
58C2×C4×C17—22no—
59C2×C4×C17—22no—
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17C22×C412yes—
65C2×C4×C17C22×C412yes—
66C2×C4×C17C22×C412yes—
67C2×Dic17—22no—
68C2×Dic17—22no—
69C2×Dic17—22no—
70C2×Dic17—22no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17C22×C414yes—
77C2×Dic17C22×C414yes—
78C2×Dic17#4114yes—
79#2336—171nomaximal, Sylow
80C22×C17C23×C411yescharacteristic
81C22×C17C2×#4112yes—
82C22×C17H(5,𝔽2)11yescharacteristic
83C22×C172−1+412yes—
84C22×C172−1+411yescharacteristic
85C4×C17—22no—
86C4×C17—22no—
87C4×C17—22no—
88C4×C17—42no—
89C4×C17—42no—
90Dic17—24no—
91Dic17—24no—
92Dic17—24no—
93Dic17—42no—
94C2×#125—171no—
95#389—172no—
96#387—172no—
97#387—172no—
98#387—172no—
99#387—172no—
100#383D1711yescharacteristic
101C4×C4⋊C4—171no—
102C4×C4⋊C4—172no—
103C2×C17C4×#4111yescharacteristic
104C2×C17#51911yescharacteristic
105C2×C17#52011yescharacteristic
106C2×C17#53912yes—
107C2×C17#54112yes—
108#125—178no—
109C2×C4⋊C4—172no—
110C2×C4⋊C4—171no—
111C2×C4⋊C4—171no—
112C2×C4⋊C4—172no—
113C2×C4⋊C4—172no—
114C2×C4⋊C4—172no—
115C2×C4⋊C4—172no—
116C2×C42—172no—
117C2×C42—171no—
118C2×C42—171no—
119C2×C42—172no—
120C2×C42C2×D1711yescharacteristic
121#95—172no—
122#95—172no—
123#95—172no—
124#95—172no—
125#95—172no—
126#95C2×D1712yes—
127#95C2×D1711yescharacteristic
128#95C2×D1712yes—
129#95C2×D1711yescharacteristic
130C17#233611yescharacteristic, complemented, Sylow
131C22×C4—172no—
132C22×C4—172no—
133C22×C4—172no—
134C22×C4—172no—
135C22×C4C22×D1712yes—
136C22×C4C22×D1712yes—
137C22×C4C22×D1711yescharacteristic
138C22×C4C22×D1711yescharacteristic
139C22×C4C22×D1711yescharacteristic
140C4⋊C4—178no—
141C4⋊C4—178no—
142C4⋊C4—174no—
143C4⋊C4—174no—
144C4⋊C4—344no—
145C4⋊C4—344no—
146C4⋊C4—682no—
147C42—344no—
148C42—681no—
149C42—681no—
150C42—682no—
151C42C4×D1714yes—
152C23C23×D1711yescharacteristic
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—174no—
164C2×C4—174no—
165C2×C4—174no—
166C2×C4—342no—
167C2×C4—342no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—342no—
173C2×C4—342no—
174C2×C4—342no—
175C2×C4C2×C4×D1712yes—
176C2×C4C2×C4×D1712yes—
177C2×C4C2×C4×D1712yes—
178C22C22×C4×D1711yescharacteristic
179C22#41×D1712yes—
180C22#1058769611yescharacteristic
181C22#1058769712yes—
182C22#1058769711yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—42no—
188C4—344no—
189C4—344no—
190C4—344no—
191C4—682no—
192C2#1176327711yescharacteristic
193C2#1176328611yescharacteristic
194C2#1176328911yescharacteristic
195C2#1176340012yes—
196C2#1176341612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutatorC22×C17C23×C4
FittingC17×#383C2
socle, prime radical, cosocleC23×C17C24
hypercenter#383D17
hypocenterC17#2336
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
22. #10587556C4semidirect
130. C17#2336semidirect

Maximal subgroups

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

subgroupindex
C2×#105875562
#117627062
#117626962
#117626942
#117626922
#117626912
C4×C4⋊Dic172
#117626602
C17×#3832
#233617