Finite Group Databasebeta

← #14534196

Subgroups — #14534196

Order 2176.

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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#11762739C211yesmaximal, characteristic
2#11762730C212yesmaximal
3#11762716C212yesmaximal
4#11762714C211yesmaximal, characteristic
5#11762708C212yesmaximal
6#11762696C212yesmaximal
7#11762690C212yesmaximal
8#11762660C212yesmaximal
9C17×#395C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊Dic17C2212yes—
15C2×C4⋊Dic17C2211yescharacteristic
16C2×C4⋊Dic17C2212yes—
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×C4×Dic17C2212yes—
20C2×C4×Dic17C2212yes—
21C2×C4×Dic17C2211yescharacteristic
22C17×#95C2211yescharacteristic
23C17×#95C2212yes—
24#10587502C2211yescharacteristic
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C2×Q8×C17—24no—
32C2×Q8×C17—41no—
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C2×Dic34—22no—
43C2×Dic34—41no—
44C2×Dic34—41no—
45C4⋊C4×C17—24no—
46C4⋊C4×C17—41no—
47C4⋊Dic17—42no—
48C4⋊Dic17—42no—
49C4⋊Dic17—41no—
50#63225—24no—
51#63225Q818yescomplemented
52C4×Dic17—24no—
53C4×Dic17—41no—
54C4×Dic17—42no—
55C23×C17C2411yescharacteristic
56Q8×C17—44no—
57C2×C4×C17—22no—
58C2×C4×C17—22no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17C2×Q814yes—
68C2×C4×C17#4114yes—
69C2×C4×C17#4112yes—
70C2×Dic17—22no—
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×Dic17C2×Q814yes—
81C2×Dic17C2×Q814yes—
82Dic34—44no—
83#2586—171nomaximal, Sylow
84C22×C17C22×Q812yes—
85C22×C17C2×#4112yes—
86C22×C17C2×#4111yescharacteristic
87C22×C17H(5,𝔽2)11yescharacteristic
88C22×C17H(5,𝔽2)11yescharacteristic
89C4×C17—22no—
90C4×C17—24no—
91C4×C17—24no—
92C4×C17—41no—
93C4×C17—41no—
94Dic17—24no—
95Dic17—24no—
96Dic17—42no—
97Dic17—42no—
98#398—171no—
99#395—172no—
100#395D1711yescharacteristic
101#391—172no—
102#391—171no—
103#389—172no—
104#387—172no—
105#387—172no—
106C4×C4⋊C4—172no—
107C2×C17#53312yes—
108C2×C17#53811yescharacteristic
109C2×C17#55111yescharacteristic
110C2×C17#55412yes—
111C2×C17Q8211yescharacteristic
112C22×Q8—171no—
113C22×Q8C2×D1711yescharacteristic
114C2×C4⋊C4—172no—
115C2×C4⋊C4—172no—
116C2×C4⋊C4—172no—
117C2×C4⋊C4—171no—
118C2×C4⋊C4—172no—
119C2×C4⋊C4C2×D1712yes—
120C2×C4⋊C4C2×D1711yescharacteristic
121C2×C42—172no—
122C2×C42—171no—
123C2×C42—172no—
124#95—172no—
125#95—172no—
126#95—171no—
127#95—172no—
128#95—172no—
129#95—172no—
130#95—172no—
131#95C2×D1712yes—
132#95C2×D1711yescharacteristic
133C17#258611yescharacteristic, complemented, Sylow
134C2×Q8—24no—
135C2×Q8—41no—
136C2×Q8—342no—
137C2×Q8—681no—
138C2×Q8—681no—
139C22×C4—172no—
140C22×C4—172no—
141C22×C4—172no—
142C22×C4—172no—
143C22×C4C22×D1711yescharacteristic
144C22×C4C22×D1711yescharacteristic
145C22×C4C22×D1712yes—
146C22×C4C22×D1712yes—
147C22×C4C22×D1711yescharacteristic
148C4⋊C4—24no—
149C4⋊C4—41no—
150C4⋊C4—178no—
151C4⋊C4—344no—
152C4⋊C4—682no—
153C4⋊C4—682no—
154C4⋊C4—681no—
155C42—344no—
156C42—682no—
157C42—681no—
158C23C23×D1711yescharacteristic
159Q8—44no—
160Q8—684no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—174no—
172C2×C4—174no—
173C2×C4—342no—
174C2×C4—342no—
175C2×C4—342no—
176C2×C4—342no—
177C2×C4—342no—
178C2×C4—342no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4#6325414yes—
184C2×C4Q8×D1714yes—
185C2×C4#6325612yes—
186C22C2×#6325412yes—
187C22#1058769111yescharacteristic
188C22C2×Q8×D1712yes—
189C22C2×#6325611yescharacteristic
190C22#1058769611yescharacteristic
191C4—24no—
192C4—24no—
193C4—22no—
194C4—41no—
195C4—41no—
196C4—344no—
197C4—344no—
198C4—682no—
199C4—682no—
200C2#1176336412yes—
201C2#1176339911yescharacteristic
202C2#1176349011yescharacteristic
203C2#1176351712yes—
204C2#1176353911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#395C2
hypercenter#395D17
hypocenterC17#2586
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
51. #63225Q8semidirect
133. C17#2586semidirect

Maximal subgroups

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

subgroupindex
#117627392
#117627302
#117627162
#117627142
#117627082
#117626962
#117626902
#117626602
C17×#3952
#258617