Finite Group Databasebeta

← #14539916

Subgroups — #14539916

Order 2176.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#11763366C211yesmaximal, characteristic, complemented
2#11763212C211yesmaximal, characteristic, complemented
3#11763205C211yesmaximal, characteristic, complemented
4#11763155C211yesmaximal, characteristic, complemented
5#11763155C211yesmaximal, characteristic, complemented
6#11763130C214yesmaximal, complemented
7#11763109C214yesmaximal, complemented
8#11762862C211yesmaximal, characteristic, complemented
9C17×#532C211yesmaximal, characteristic
10C2×#41×C17—21no—
11C17×C4⋊Q8C2211yescharacteristic
12C17×#127C2211yescharacteristic
13C17×#124—21no—
14C17×#121—21no—
15C17×#121C2212yes—
16C4×D4×C17—21no—
17C4×D4×C17C2212yes—
18C2×C42×C17C2211yescharacteristic
19#10587612C2212yes—
20#10587610—21no—
21#10587610—21no—
22#10587556C2212yes—
23#10587555C2211yescharacteristic
24#10587555C2211yescharacteristic
25C4×Dic34—21no—
26C4×Dic34—21no—
27#10587503C2214yes—
28#10587503C2214yes—
29C22⋊#3622C2212yes—
30#10587479C2214yes—
31#10587478C2214yes—
32C4⋊#3622C2212yes—
33C4×#3622C2211yescharacteristic
34C4×#3622C2211yescharacteristic
35#41×C17—44no—
36C2×Q8×C17—21no—
37C2×D4×C17—22no—
38C2×D4×C17—21no—
39C2×D4×C17—21no—
40C2×D4×C17C2312yes—
41C22×C4×C17—22no—
42C22×C4×C17D411yescharacteristic
43C22×C4×C17D411yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×Dic34—21no—
46C2×Dic34—21no—
47C4⋊C4×C17—21no—
48C4⋊C4×C17C2312yes—
49C22⋊C4×C17—22no—
50C22⋊C4×C17—22no—
51C42×C17D411yescharacteristic
52C42×C17D411yescharacteristic
53C42×C17C2311yescharacteristic
54C42×C17C2311yescharacteristic
55C22⋊Dic17—22no—
56C4⋊Dic17—21no—
57C4⋊Dic17—21no—
58C4⋊Dic17C2312yes—
59C4⋊Dic17C2312yes—
60#63225—22no—
61#63225—22no—
62C4×Dic17—22no—
63C2×#3622C2312yes—
64C2×#3622C2312yes—
65C23×C17—22no—
66C23×C17C2×D411yescharacteristic
67Q8×C17—42no—
68D4×C17—24no—
69D4×C17—44no—
70D4×C17—42no—
71D4×C17—42no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—44no—
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2×D411yescharacteristic
82C2×C4×C17C2411yescharacteristic
83C2×Dic17—22no—
84C2×Dic17—22no—
85Dic34—42no—
86Dic34—42no—
87#3622—22no—
88#3622—22no—
89#3228—171nomaximal, Sylow
90C22×C17—21no—
91C22×C17—21no—
92C22×C17—21no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17—42no—
96C22×C17C22×D411yescharacteristic
97C4×C17—21no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—42no—
101C4×C17#13512yes—
102C4×C17AGL(1,ℤ/8)12yes—
103C4×C172−1+412yes—
104Dic17—42no—
105Dic17—42no—
106#533—171no—
107#532D1711yescharacteristic
108#500—171no—
109#498—171no—
110#486—171no—
111#486—171no—
112#482—174no—
113#478—174no—
114#431—171no—
115C2×C17—41no—
116C2×C17—82no—
117C2×C17#53611yescharacteristic
118C2×C17C2×#13511yescharacteristic
119C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
120C2×#41—21no—
121C4⋊Q8—171no—
122C4⋊Q8—171no—
123C4⋊Q8C2×D1711yescharacteristic
124#127C2×D1711yescharacteristic
125#125—172no—
126#124—21no—
127C22⋊Q8—341no—
128C22⋊Q8—341no—
129#121—21no—
130#121C2×D1712yes—
131C4×Q8—341no—
132C4×Q8—341no—
133C4×D4—21no—
134C4×D4C2×D1712yes—
135#117—172no—
136C2×C42C2×D1711yescharacteristic
137#107—174no—
138#106—174no—
139C4⋊C8—172no—
140#102—174no—
141#102—174no—
142C22⋊C8—172no—
143C4×C8—171no—
144C4×C8—171no—
145C17#322811yescharacteristic, complemented, Sylow
146#41—44no—
147C2×Q8—21no—
148C2×Q8—341no—
149C2×Q8—341no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D1712yes—
154C22×C4—22no—
155C22×C4C22⋊D1711yescharacteristic
156C22×C4C22⋊D1711yescharacteristic
157C22×C4C22×D1711yescharacteristic
158C2×C8—172no—
159C2×C8—172no—
160C4⋊C4—21no—
161C4⋊C4—172no—
162C4⋊C4—172no—
163C4⋊C4—342no—
164C4⋊C4—342no—
165C4⋊C4—341no—
166C4⋊C4—341no—
167C4⋊C4C22×D1712yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C22⋊C4—342no—
171C42—342no—
172C42C22⋊D1711yescharacteristic
173C42C22⋊D1711yescharacteristic
174C42C22×D1711yescharacteristic
175C42C22×D1711yescharacteristic
176C23—22no—
177C23C2×C22⋊D1711yescharacteristic
178Q8—42no—
179Q8—682no—
180Q8—682no—
181D4—24no—
182D4—44no—
183D4—42no—
184D4—42no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4C2×C22⋊D1711yescharacteristic
193C2×C4C2×C22⋊D1711yescharacteristic
194C2×C4C2×C22⋊D1711yescharacteristic
195C2×C4C2×C22⋊D1711yescharacteristic
196C2×C4C2×C22⋊D1711yescharacteristic
197C2×C4C23×D1711yescharacteristic
198C8—342no—
199C8—342no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22C22×C22⋊D1711yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—682no—
212C4—682no—
213C4#1058761812yes—
214C4#1058763612yes—
215C4#1058769712yes—
216C2—41no—
217C2—82no—
218C2#1176338011yescharacteristic
219C2C2×#1058763611yescharacteristic
220C2C2×#1058761811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#532C2
socleC22×C17C22×D4
hypercenter#532D17
hypocenterC17#3228
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #11763366C2semidirect
2. #11763212C2semidirect
3. #11763205C2semidirect
4. #11763155C2semidirect
5. #11763155C2semidirect
6. #11763130C2semidirect
7. #11763109C2semidirect
8. #11762862C2semidirect
145. C17#3228semidirect

Maximal subgroups

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

subgroupindex
#117633662
#117632122
#117632052
#117631552
#117631302
#117631092
#117628622
C17×#5322
#322817