Finite Group Databasebeta

← #14536748

Subgroups — #14536748

Order 2176.

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

normal subgroups only

This group has 392 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#11762742C212yesmaximal
2#11762741C212yesmaximal
3#11762730C212yesmaximal
4#11762716C212yesmaximal
5#11762715C211yesmaximal, characteristic
6#11762707C211yesmaximal, characteristic
7#11762706C212yesmaximal
8#11762692C212yesmaximal
9C17×#391C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14C2×C4⋊Dic17C2211yescharacteristic
15C2×C4⋊Dic17C2211yescharacteristic
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×C4×Dic17C2212yes—
23C17×#95C2212yes—
24C17×#95C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C2×Q8×C17—41no—
32C2×Q8×C17—41no—
33C2×Q8×C17Q814yescomplemented
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C2×Dic34—22no—
44C2×Dic34—41no—
45C2×Dic34—41no—
46C4⋊C4×C17—41no—
47C42×C17—41no—
48C4⋊Dic17—42no—
49C4⋊Dic17—42no—
50C4⋊Dic17—41no—
51C4⋊Dic17—41no—
52C4⋊Dic17Q814yescomplemented
53#63225—24no—
54#63225—42no—
55#63225—42no—
56C4×Dic17—42no—
57C23×C17C2411yescharacteristic
58Q8×C17—44no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—21no—
63C2×C4×C17—21no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17C2×Q812yes—
71C2×C4×C17C2×Q814yes—
72C2×C4×C17C2×Q812yes—
73C2×C4×C17#4112yes—
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—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17C2×Q814yes—
86Dic34—44no—
87#2806—171nomaximal, Sylow
88C22×C17C22×Q811yescharacteristic
89C22×C17C22×Q811yescharacteristic
90C22×C17C2×#4111yescharacteristic
91C22×C17H(5,𝔽2)11yescharacteristic
92C22×C17H(5,𝔽2)11yescharacteristic
93C22×C17H(5,𝔽2)11yescharacteristic
94C22×C172−1+411yescharacteristic
95C4×C17—22no—
96C4×C17—22no—
97C4×C17—24no—
98C4×C17—22no—
99C4×C17—42no—
100Dic17—24no—
101Dic17—42no—
102Dic17—42no—
103Dic17—42no—
104#398—172no—
105#398—172no—
106#395—172no—
107#391—172no—
108#391—171no—
109#391D1711yescharacteristic
110#389—171no—
111#389—172no—
112#387—172no—
113C2×C17#53811yescharacteristic
114C2×C17#54311yescharacteristic
115C2×C17#54411yescharacteristic
116C2×C17#55411yescharacteristic
117C2×C17#55711yescharacteristic
118C2×C17Q8211yescharacteristic
119C2×C17#56211yescharacteristic
120C22×Q8—171no—
121C22×Q8C2×D1711yescharacteristic
122C2×C4⋊C4—171no—
123C2×C4⋊C4—172no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—171no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4C2×D1711yescharacteristic
131C2×C42—172no—
132C2×C42C2×D1711yescharacteristic
133#95—172no—
134#95—172no—
135#95—172no—
136#95—172no—
137#95—172no—
138#95—172no—
139#95C2×D1712yes—
140#95C2×D1712yes—
141C17#280611yescharacteristic, complemented, Sylow
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—342no—
145C2×Q8—681no—
146C2×Q8—681no—
147C2×Q8Dic3414yescomplemented
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4C22×D1712yes—
153C22×C4C22×D1711yescharacteristic
154C22×C4C22×D1712yes—
155C22×C4C22×D1711yescharacteristic
156C22×C4C22×D1711yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—174no—
159C4⋊C4—344no—
160C4⋊C4—682no—
161C4⋊C4—682no—
162C4⋊C4—681no—
163C4⋊C4—682no—
164C4⋊C4—681no—
165C4⋊C4—682no—
166C42—41no—
167C42—682no—
168C23C23×D1711yescharacteristic
169Q8—44no—
170Q8—684no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—174no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4C2×Dic3412yes—
195C2×C4C2×Dic3414yes—
196C2×C4Q8×D1712yes—
197C2×C4#6325612yes—
198C22C22×Dic3411yescharacteristic
199C22#1058769111yescharacteristic
200C22#1058769111yescharacteristic
201C22C2×Q8×D1711yescharacteristic
202C22C2×#6325611yescharacteristic
203C22#1058769611yescharacteristic
204C22#1058769711yescharacteristic
205C4—22no—
206C4—22no—
207C4—24no—
208C4—22no—
209C4—42no—
210C4—344no—
211C4—682no—
212C4—682no—
213C4—682no—
214C2#1176339911yescharacteristic
215C2#1176343511yescharacteristic
216C2#1176343811yescharacteristic
217C2#1176352011yescharacteristic
218C2#1176353811yescharacteristic
219C2Q8×Dic3411yescharacteristic
220C2#1176355311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#391C2
hypercenter#391D17
hypocenterC17#2806
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
33. C2×Q8×C17Q8semidirect
52. C4⋊Dic17Q8semidirect
141. C17#2806semidirect
147. C2×Q8Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627422
#117627412
#117627302
#117627162
#117627152
#117627072
#117627062
#117626922
C17×#3912
#280617