Finite Group Databasebeta

← #14536608

Subgroups — #14536608

Order 2176.

344 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 121 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#11762750C212yesmaximal
2#11762750C212yesmaximal
3#11762748C211yesmaximal, characteristic
4#11762696C212yesmaximal
5#11762695C212yesmaximal
6#11762690C212yesmaximal
7#11762690C212yesmaximal
8#11762678C211yesmaximal, characteristic
9C17×#398C211yesmaximal, characteristic
10C2×C4⋊C4×C17C2212yes—
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊Dic17C2211yescharacteristic
14C2×C4⋊Dic17C2211yescharacteristic
15C2×#63225C2212yes—
16C2×#63225C2212yes—
17C2×#63225C2212yes—
18C2×C4×Dic17C2211yescharacteristic
19C2×C4×Dic17C2212yes—
20C17×#95C2212yes—
21C17×#95C2211yescharacteristic
22#10587502C2212yes—
23#10587502C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31C22×C4×C17C2312yes—
32C22×C4×C17C2312yes—
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×Dic17C2312yes—
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C4⋊C4×C17—24no—
41C4⋊C4×C17—41no—
42C4⋊C4×C17—41no—
43C4⋊Dic17—41no—
44C4⋊Dic17—41no—
45#63225—24no—
46#63225—24no—
47#63225—24no—
48C4×Dic17—22no—
49C4×Dic17—42no—
50C23×C17C2411yescharacteristic
51C2×C4×C17—21no—
52C2×C4×C17—21no—
53C2×C4×C17—22no—
54C2×C4×C17—22no—
55C2×C4×C17—22no—
56C2×C4×C17—22no—
57C2×C4×C17—22no—
58C2×C4×C17—21no—
59C2×C4×C17—22no—
60C2×C4×C17—21no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17#4112yes—
64C2×Dic17—22no—
65C2×Dic17—22no—
66C2×Dic17—22no—
67C2×Dic17—22no—
68C2×Dic17—22no—
69C2×Dic17—22no—
70C2×Dic17—22no—
71C2×Dic17—22no—
72C2×Dic17—24no—
73C2×Dic17—22no—
74C2×Dic17#4114yes—
75#2794—171nomaximal, Sylow
76C22×C17C2×#4111yescharacteristic
77C22×C17C2×#4112yes—
78C22×C17H(5,𝔽2)11yescharacteristic
79C22×C172−1+412yes—
80C22×C172−1+411yescharacteristic
81C4×C17—22no—
82C4×C17—42no—
83C4×C17—42no—
84C4×C17—41no—
85C4×C17—41no—
86Dic17—24no—
87Dic17—42no—
88Dic17—42no—
89Dic17—42no—
90#400—171no—
91#400—172no—
92#400—172no—
93#398D1711yescharacteristic
94#387—172no—
95#387—172no—
96#387—172no—
97#387—172no—
98#384—171no—
99C2×C17#53911yescharacteristic
100C2×C17#54112yes—
101C2×C17#55212yes—
102C2×C17#55911yescharacteristic
103C2×C17#56311yescharacteristic
104C2×C4⋊C4—172no—
105C2×C4⋊C4—172no—
106C2×C4⋊C4—172no—
107C2×C4⋊C4—171no—
108C2×C4⋊C4—171no—
109C2×C4⋊C4C2×D1712yes—
110C2×C4⋊C4C2×D1711yescharacteristic
111C2×C4⋊C4C2×D1711yescharacteristic
112C2×C42—171no—
113C2×C42—172no—
114#95—172no—
115#95—172no—
116#95—172no—
117#95—172no—
118#95—172no—
119#95—172no—
120#95—172no—
121#95—172no—
122#95—171no—
123#95C2×D1712yes—
124#95C2×D1711yescharacteristic
125C17#279411yescharacteristic, complemented, Sylow
126C22×C4—172no—
127C22×C4—172no—
128C22×C4—172no—
129C22×C4—172no—
130C22×C4C22×D1712yes—
131C22×C4C22×D1712yes—
132C22×C4C22×D1711yescharacteristic
133C22×C4C22×D1711yescharacteristic
134C22×C4C22×D1711yescharacteristic
135C4⋊C4—24no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—344no—
139C4⋊C4—344no—
140C4⋊C4—344no—
141C4⋊C4—681no—
142C4⋊C4—681no—
143C42—342no—
144C42—682no—
145C23C23×D1711yescharacteristic
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—174no—
159C2×C4—342no—
160C2×C4—342no—
161C2×C4—342no—
162C2×C4—342no—
163C2×C4—344no—
164C2×C4—342no—
165C2×C4—342no—
166C2×C4—342no—
167C2×C4—342no—
168C2×C4—342no—
169C2×C4#6325412yes—
170C22C2×#6325411yescharacteristic
171C22#1058769111yescharacteristic
172C22#1058769412yes—
173C22#41×D1712yes—
174C22#1058769711yescharacteristic
175C4—22no—
176C4—42no—
177C4—41no—
178C4—41no—
179C4—42no—
180C4—344no—
181C4—682no—
182C4—682no—
183C4—682no—
184C2#1176340411yescharacteristic
185C2#1176341812yes—
186C2#1176349312yes—
187C2#1176354311yescharacteristic
188C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#398C2
hypercenter#398D17
hypocenterC17#2794
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
125. C17#2794semidirect

Maximal subgroups

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

subgroupindex
#117627502
#117627482
#117626962
#117626952
#117626902
#117626782
C17×#3982
#279417