Finite Group Databasebeta

← #16953468

Subgroups — #16953468

Order 4352.

572 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 77 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#14528742C213yesmaximal
2#14526911C213yesmaximal, complemented
3C17×#2811C211yesmaximal, characteristic, complemented
4C2×#10587694C2211yescharacteristic
5C2×#10587476—23no—
6#11762808—23no—
7C2×#10587507C2213yes—
8C17×#395—23no—
9C17×#391—23no—
10C17×#391C2213yes—
11C22×Q8×C17—21no—
12C22×Q8×C17—21no—
13C22×Q8×C17C2311yescharacteristic
14#10587694—23no—
15#10587694—42no—
16#10587694Q812yescomplemented
17C2×#63256—41no—
18C2×Q8×D17—41no—
19C2×#63252—23no—
20C2×C4⋊C4×C17—23no—
21C2×C4⋊C4×C17D413yes—
22C2×C42×C17D413yes—
23C2×#63223—23no—
24C17×#95—23no—
25C17×#95—26no—
26#10587507—26no—
27#10587481—43no—
28#10587480—43no—
29#10587476—46no—
30C2×Q8×C17—24no—
31C2×Q8×C17—41no—
32C2×Q8×C17—41no—
33C2×Q8×C17—81no—
34C2×Q8×C17—81no—
35C2×Q8×C17—82no—
36C2×Q8×C17—81no—
37C2×Q8×C17C2×Q811yescharacteristic
38C2×Q8×C17#4113yes—
39C22×C4×C17—23no—
40C22×C4×C17—23no—
41C22×C4×C17C2×D413yes—
42C2×C22⋊D17—41no—
43#63256—42no—
44#63256—46no—
45Q8×D17—42no—
46Q8×D17—46no—
47#63252—26no—
48#63252—26no—
49#63252—43no—
50#63252—43no—
51C2×D68—23no—
52C2×C4×D17—43no—
53C2×Dic34—23no—
54C4⋊C4×C17—43no—
55C4⋊C4×C17—83no—
56C42×C17—43no—
57#63223—43no—
58#63223—43no—
59C2×#3622—43no—
60#13378—171nomaximal, Sylow
61C23×C17#12011yescharacteristic
62C22×D17—41no—
63Q8×C17—41no—
64Q8×C17—43no—
65Q8×C17—84no—
66Q8×C17—84no—
67C2×C4×C17—23no—
68C2×C4×C17—23no—
69C2×C4×C17—26no—
70C2×C4×C17—43no—
71C2×C4×C17—43no—
72C2×C4×C17—46no—
73C2×C4×C17—43no—
74C2×C4×C17C22⋊Q813yes—
75C2×C4×C17#12413yes—
76C22⋊D17—42no—
77C22⋊D17—42no—
78C2×Dic17—41no—
79D68—26no—
80D68—43no—
81C4×D17—46no—
82C4×D17—46no—
83Dic34—26no—
84Dic34—43no—
85#3622—83no—
86#2811D1711yescharacteristic, complemented
87#1793—173no—
88#1394—173no—
89C22×C17—21no—
90C22×C17—21no—
91C22×C17#39311yescharacteristic
92C22×C17B(4,𝔽2)11yescharacteristic
93C22×C17B(4,𝔽2)11yescharacteristic
94C2×D17—42no—
95C2×D17—81no—
96C2×D17—81no—
97C4×C17—23no—
98C4×C17—23no—
99C4×C17—46no—
100C4×C17—83no—
101Dic17—42no—
102C2×2−1+4—171no—
103C2×#104—343no—
104#417—343no—
105C2×#101—173no—
106#395—23no—
107#391—23no—
108#391C2×D1713yes—
109C2×C17—21no—
110C2×C17—21no—
111C2×C17#202911yescharacteristic
112C2×C17#220211yescharacteristic
113C2×C17#220511yescharacteristic
114D17—82no—
1152−1+4—172no—
1162−1+4—343no—
1172−1+4—682no—
118C2×#41—343no—
119C2×#41—681no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—681no—
123C22×Q8C22×D1711yescharacteristic
124C2×M16—343no—
125C2×C4⋊C4—23no—
126C2×C4⋊C4C22⋊D1713yes—
127C2×C42C22⋊D1713yes—
128#104—686no—
129#103—683no—
130#102—683no—
131#101—346no—
132#95—23no—
133#95—26no—
134C17#1337811yescharacteristic, complemented, Sylow
135#41—346no—
136#41—346no—
137#41—683no—
138#41—682no—
139#41—686no—
140#41—683no—
141C2×Q8—24no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—81no—
145C2×Q8—81no—
146C2×Q8—82no—
147C2×Q8—81no—
148C2×Q8—343no—
149C2×Q8—682no—
150C2×Q8—686no—
151C2×Q8Q8×D1711yescharacteristic
152C2×Q8#6325613yes—
153C2×D4—343no—
154C2×D4—681no—
155C22×C4—23no—
156C22×C4—23no—
157C22×C4—683no—
158C22×C4C2×C22⋊D1713yes—
159M16—683no—
160M16—683no—
161C2×C8—683no—
162C4⋊C4—43no—
163C4⋊C4—83no—
164C42—43no—
165C23—681no—
166C23#1058763911yescharacteristic
167Q8—41no—
168Q8—43no—
169Q8—84no—
170Q8—84no—
171Q8—346no—
172Q8—683no—
173D4—346no—
174D4—683no—
175D4—682no—
176D4—682no—
177C2×C4—23no—
178C2×C4—23no—
179C2×C4—26no—
180C2×C4—43no—
181C2×C4—43no—
182C2×C4—46no—
183C2×C4—43no—
184C2×C4—686no—
185C2×C4—686no—
186C2×C4—681no—
187C2×C4#1058763213yes—
188C2×C4#1058763313yes—
189C8—1363no—
190C22—21no—
191C22—21no—
192C22—682no—
193C22—1361no—
194C22—1361no—
195C22#1176272011yescharacteristic
196C22#1176299611yescharacteristic
197C22#1176299611yescharacteristic
198C4—23no—
199C4—23no—
200C4—46no—
201C4—83no—
202C4—682no—
203C2—21no—
204C2—21no—
205C2—1362no—
206C2#1452994711yescharacteristic
207C2#1453080111yescharacteristic
208C2#1453081411yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC22×Q8×C17C23
FrattiniC22×Q8C22×D17
FittingC17×#2811C2
socleC22×C17B(4,𝔽2)
prime radicalC2×#10587694C22
hypercenter#2811D17
hypocenterC17#13378
LCSG ▷ C22×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2811

Complements

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

NH ≅ G/Ntype
2. #14526911C2semidirect
3. C17×#2811C2semidirect
16. #10587694Q8semidirect
86. #2811D17semidirect
134. C17#13378semidirect

Maximal subgroups

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

subgroupindex
#145287422
#145269112
C17×#28112
#1337817