Finite Group Databasebeta

← #14534614

Subgroups — #14534614

Order 2176.

472 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 165 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, complemented
2#11762742C214yesmaximal, complemented
3#11762732C214yesmaximal
4#11762707C212yesmaximal, complemented
5#11762679C211yesmaximal, characteristic, complemented
6#11762669C212yesmaximal
7C2×C17×#117C211yesmaximal, characteristic
8C23×C4×C17C2211yescharacteristic
9C17×#117—24no—
10C2×C4⋊C4×C17C2212yes—
11C2×C22⋊C4×C17C2212yes—
12C2×C42×C17C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14C2×C22⋊Dic17C2214yes—
15C2×C4⋊Dic17C2212yes—
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2214yes—
18C2×#63225C2214yes—
19C2×#63225C2214yes—
20#10587556—24no—
21#10587502C2214yes—
22#10587502C2214yes—
23C24×C17C2311yescharacteristic
24C22×C4×C17—22no—
25C22×C4×C17—22no—
26C22×C4×C17Q814yes—
27C22×C4×C17C2312yes—
28C22×C4×C17C2312yes—
29C22×C4×C17C2311yescharacteristic
30C22×C4×C17C2311yescharacteristic
31C22×Dic17C2314yes—
32C22×Dic17C2314yes—
33C4⋊C4×C17—24no—
34C22⋊C4×C17—24no—
35C42×C17—22no—
36C42×C17D414yes—
37C22⋊Dic17—44no—
38C4⋊Dic17—24no—
39C4⋊Dic17—24no—
40C4⋊Dic17—44no—
41#63225—28no—
42#63225—44no—
43C23×C17—21no—
44C23×C17—21no—
45C23×C17—22no—
46C23×C17—22no—
47C23×C17C2×Q812yes—
48C23×C17C2411yescharacteristic
49C2×C4×C17—21no—
50C2×C4×C17—21no—
51C2×C4×C17—22no—
52C2×C4×C17—22no—
53C2×C4×C17—28no—
54C2×C4×C17—22no—
55C2×C4×C17—22no—
56C2×C4×C17—22no—
57C2×C4×C17C2×D414yes—
58C2×C4×C17C2×D412yes—
59C2×C4×C17C2×Q812yes—
60C2×C4×C17C2×Q812yes—
61C2×C4×C17#4112yes—
62C2×Dic17—24no—
63C2×Dic17—24no—
64C2×Dic17—24no—
65C2×Dic17—24no—
66C2×Dic17—24no—
67C2×Dic17—24no—
68#2629—171nomaximal, Sylow
69C22×C17—22no—
70C22×C17—24no—
71C22×C17—42no—
72C22×C17—41no—
73C22×C17—41no—
74C22×C17C22×D411yescharacteristic
75C22×C17C22×Q811yescharacteristic
76C22×C17C2×#4111yescharacteristic
77C22×C17H(5,𝔽2)12yes—
78C22×C172−1+412yes—
79C4×C17—22no—
80C4×C17—24no—
81C4×C17—42no—
82C4×C17C22⋊Q814yes—
83Dic17—44no—
84Dic17—44no—
85C2×#125—171no—
86C2×#117D1711yescharacteristic
87#398—174no—
88#396—174no—
89#389—172no—
90#384—171no—
91#380—172no—
92C2×C17—42no—
93C2×C17C2×C22⋊Q811yescharacteristic
94C2×C17#53712yes—
95C2×C17#54011yescharacteristic
96C2×C17#54111yescharacteristic
97C2×C17#54412yes—
98C23×C4C2×D1711yescharacteristic
99#125—344no—
100#117—24no—
101C2×C4⋊C4—172no—
102C2×C4⋊C4—174no—
103C2×C4⋊C4—172no—
104C2×C4⋊C4—174no—
105C2×C4⋊C4—174no—
106C2×C4⋊C4C2×D1712yes—
107C2×C22⋊C4—174no—
108C2×C22⋊C4C2×D1712yes—
109C2×C42C2×D1711yescharacteristic
110C2×C42C2×D1711yescharacteristic
111#95—174no—
112#95—174no—
113C17#262911yescharacteristic, complemented, Sylow
114C24C22×D1711yescharacteristic
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—174no—
118C22×C4—174no—
119C22×C4Dic3414yes—
120C22×C4C22×D1712yes—
121C22×C4C22×D1711yescharacteristic
122C22×C4C22×D1712yes—
123C22×C4C22×D1711yescharacteristic
124C4⋊C4—24no—
125C4⋊C4—344no—
126C4⋊C4—348no—
127C4⋊C4—344no—
128C4⋊C4—684no—
129C4⋊C4—684no—
130C22⋊C4—24no—
131C22⋊C4—684no—
132C42—22no—
133C42C22⋊D1714yes—
134C23—21no—
135C23—22no—
136C23—22no—
137C23—21no—
138C23C2×Dic3412yes—
139C23C23×D1711yescharacteristic
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—28no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—344no—
149C2×C4—344no—
150C2×C4—344no—
151C2×C4—344no—
152C2×C4—344no—
153C2×C4—344no—
154C2×C4C2×Dic3412yes—
155C2×C4C2×Dic3412yes—
156C2×C4#6325212yes—
157C2×C4C2×C22⋊D1714yes—
158C2×C4C2×C22⋊D1712yes—
159C22—22no—
160C22—24no—
161C22—42no—
162C22—41no—
163C22—41no—
164C22C22×Dic3411yescharacteristic
165C22C2×#6325211yescharacteristic
166C22#1058769612yes—
167C22#1058769712yes—
168C22C22×C22⋊D1711yescharacteristic
169C4—22no—
170C4—24no—
171C4—42no—
172C4—684no—
173C4—684no—
174C4#1058761014yes—
175C2—42no—
176C2C2×#1058761011yescharacteristic
177C2#1176338712yes—
178C2#1176341411yescharacteristic
179C2#1176342111yescharacteristic
180C2#1176343712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#117C2
prime radicalC24×C17C23
hypercenterC2×#117D17
hypocenterC17#2629
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#10587556C2semidirect
2. #11762742C2semidirect
4. #11762707C2semidirect
5. #11762679C2semidirect
113. C17#2629semidirect

Maximal subgroups

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

subgroupindex
C2×#105875562
#117627422
#117627322
#117627072
#117626792
#117626692
C2×C17×#1172
#262917