Finite Group Databasebeta

← #14533809

Subgroups — #14533809

Order 2176.

556 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 181 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×#10587574C212yesmaximal
2C2×#10587630C212yesmaximal
3#11762730C212yesmaximal
4#11762729C211yesmaximal, characteristic
5#11762706C212yesmaximal
6#11762704C212yesmaximal
7#11762678C211yesmaximal, characteristic
8#11762660C212yesmaximal
9C17×#395C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C22×Dic34C2212yes—
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15#10587630—28no—
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2211yescharacteristic
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×C4×Dic17C2212yes—
25C2×C4×Dic17C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27#10587574—28no—
28C17×#95C2211yescharacteristic
29C17×#95C2212yes—
30#10587502C2211yescharacteristic
31#10587502C2212yes—
32#10587502C2212yes—
33C2×Q8×C17—24no—
34C2×Q8×C17—41no—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C2×Dic34—24no—
45C2×Dic34—24no—
46C2×Dic34—24no—
47C2×Dic34—41no—
48C2×Dic34—42no—
49C4⋊C4×C17—24no—
50C4⋊C4×C17—41no—
51C4⋊Dic17—24no—
52C4⋊Dic17—41no—
53#63225—24no—
54#63225—24no—
55#63225—24no—
56#63225—24no—
57#63225D418yes—
58#63225Q818yescomplemented
59C4×Dic17—24no—
60C4×Dic17—24no—
61C4×Dic17Q814yescomplemented
62C23×C17C2411yescharacteristic
63Q8×C17—44no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17C2×D414yes—
75C2×C4×C17C2×Q812yes—
76C2×C4×C17C2×Q814yes—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17C2×D414yes—
85C2×Dic17C2×D414yes—
86C2×Dic17C2×Q814yes—
87C2×Dic17C2×Q814yes—
88C2×Dic17C2×Q814yes—
89Dic34—44no—
90Dic34—48no—
91#2551—171nomaximal, Sylow
92C22×C17C22×D412yes—
93C22×C17C22×Q811yescharacteristic
94C22×C17C22×Q812yes—
95C22×C17H(5,𝔽2)11yescharacteristic
96C22×C172−1+411yescharacteristic
97C4×C17—22no—
98C4×C17—24no—
99C4×C17—24no—
100C4×C17—41no—
101C4×C17—41no—
102Dic17—24no—
103Dic17—24no—
104Dic17—42no—
105Dic17C4⋊Q818yescomplemented
106C2×C4⋊Q8—172no—
107C2×C4⋊Q8—172no—
108#395—171no—
109#395—172no—
110#395D1711yescharacteristic
111#389—172no—
112#389—172no—
113#384—171no—
114C4×C4⋊C4—172no—
115C2×C17C2×C4⋊Q812yes—
116C2×C17#54411yescharacteristic
117C2×C17D4211yescharacteristic
118C2×C17D4×Q812yes—
119C2×C17Q8211yescharacteristic
120C22×Q8—171no—
121C22×Q8—172no—
122C22×Q8C2×D1711yescharacteristic
123C4⋊Q8—348no—
124C4⋊Q8—348no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—172no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—171no—
133C2×C4⋊C4C2×D1712yes—
134C2×C4⋊C4C2×D1711yescharacteristic
135C2×C42—171no—
136C2×C42—172no—
137C2×C42—172no—
138#95—172no—
139#95—171no—
140#95—172no—
141#95C2×D1712yes—
142#95C2×D1711yescharacteristic
143C17#255111yescharacteristic, complemented, Sylow
144C2×Q8—24no—
145C2×Q8—41no—
146C2×Q8—344no—
147C2×Q8—344no—
148C2×Q8—344no—
149C2×Q8—682no—
150C2×Q8—681no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4—172no—
154C22×C4—172no—
155C22×C4C22×D1711yescharacteristic
156C22×C4C22×D1711yescharacteristic
157C22×C4C22×D1712yes—
158C22×C4C22×D1711yescharacteristic
159C22×C4C22×D1712yes—
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—178no—
163C4⋊C4—178no—
164C4⋊C4—344no—
165C4⋊C4—344no—
166C4⋊C4—344no—
167C4⋊C4—344no—
168C4⋊C4—344no—
169C4⋊C4—681no—
170C42—174no—
171C42—344no—
172C42—344no—
173C23C23×D1711yescharacteristic
174Q8—44no—
175Q8—688no—
176Q8—684no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—174no—
188C2×C4—174no—
189C2×C4—174no—
190C2×C4—174no—
191C2×C4—174no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4D4×D1714yes—
200C2×C4Q8×D1712yes—
201C2×C4Q8×D1714yes—
202C22C2×D4×D1712yes—
203C22#1058769111yescharacteristic
204C22C2×Q8×D1712yes—
205C22C2×Q8×D1711yescharacteristic
206C22#1058769711yescharacteristic
207C4—24no—
208C4—22no—
209C4—24no—
210C4—41no—
211C4—41no—
212C4—178no—
213C4—344no—
214C4—344no—
215C4—682no—
216C2C4⋊Q8×D1712yes—
217C2#1176343811yescharacteristic
218C2#1176344111yescharacteristic
219C2#1176347212yes—
220C2#1176353911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#395C2
hypercenter#395D17
hypocenterC17#2551
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
58. #63225Q8semidirect
61. C4×Dic17Q8semidirect
105. Dic17C4⋊Q8semidirect
143. C17#2551semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105876302
#117627302
#117627292
#117627062
#117627042
#117626782
#117626602
C17×#3952
#255117