Finite Group Databasebeta

← #16921952

Subgroups — #16921952

Order 4352.

194 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 41 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#14530134C211yesmaximal, characteristic
2#14526665C211yesmaximal, characteristic
3C17×#1324C211yesmaximal, characteristic
4#11763440C411yescharacteristic
5#11762849—21no—
6C2×#10587474—21no—
7#11762578C411yescharacteristic
8#11762569—21no—
9C17×#382C2211yescharacteristic
10C17×#324—21no—
11C2×C4⋊C4×C17C2×C411yescharacteristic
12C2×C4⋊C4×C17D411yescharacteristic
13C2×C42×C17D411yescharacteristic
14#10587612—21no—
15#10587610—41no—
16C2×#63223—21no—
17C22×#3622—41no—
18#10587576—41no—
19#10587574—41no—
20C17×C22⋊C8—41no—
21C17×#95—41no—
22#10587479—41no—
23#10587478—41no—
24#10587474—22no—
25#10587474—41no—
26C22×C4×C17—21no—
27C22×C4×C17—21no—
28C22×C4×C17—21no—
29C22×C4×C17C22⋊C411yescharacteristic
30C2×Dic34—41no—
31C2×C8×C17—81no—
32C4⋊C4×C17—41no—
33C4⋊C4×C17—41no—
34C42×C17—21no—
35C42×C17SD1611yescharacteristic
36C42×C17Q1611yescharacteristic
37C22⋊Dic17—41no—
38C4⋊Dic17—21no—
39C4⋊Dic17—21no—
40C4⋊Dic17—41no—
41#63225—42no—
42C4×Dic17—41no—
43#63223—22no—
44#63223—41no—
45C2×#3622—42no—
46C2×#3622—41no—
47C2×#3622—42no—
48C2×#3622—42no—
49#7442—171nomaximal, Sylow
50C23×C17#9911yescharacteristic
51C2×C4×C17—21no—
52C2×C4×C17—21no—
53C2×C4×C17—21no—
54C2×C4×C17—21no—
55C2×C4×C17—41no—
56C2×C4×C17—41no—
57C2×C4×C17—41no—
58C2×C4×C17—41no—
59C2×C4×C17—81no—
60C2×C4×C17—81no—
61C2×C4×C17#10311yescharacteristic
62C2×C4×C17#10411yescharacteristic
63C2×Dic17—41no—
64C2×Dic17—41no—
65Dic34—161no—
66C8×C17—161no—
67#3622—42no—
68#3622—44no—
69#2061—171no—
70#1324D1711yescharacteristic
71#1295—171no—
72C22×C17—21no—
73C22×C17—21no—
74C22×C17#32811yescharacteristic
75C22×C17AΣL(1,𝔽16)11yescharacteristic
76C22×C17#35611yescharacteristic
77C4×C17—21no—
78C4×C17—21no—
79C4×C17—41no—
80C4×C17—41no—
81C4×C17—81no—
82C4×C17—81no—
83Dic17—81no—
84Dic17—81no—
85#544—171no—
86#428—341no—
87C2×#97—341no—
88#382C2×D1711yescharacteristic
89#343—171no—
90#339—341no—
91#324—21no—
92C2×C17—21no—
93C2×C17—21no—
94C2×C17#135411yescharacteristic
95C2×C17#140311yescharacteristic
96C2×C17#140411yescharacteristic
97C2×M16—341no—
98C22×C8—681no—
99C4⋊Q8—681no—
100#125—681no—
101C22⋊Q8—681no—
102#117—341no—
103C2×C4⋊C4C4×D1711yescharacteristic
104C2×C4⋊C4C22⋊D1711yescharacteristic
105C2×C42D6811yescharacteristic
106#107—681no—
107#106—681no—
108C22⋊C8—41no—
109#97—342no—
110#97—681no—
111#95—41no—
112C17#744211yescharacteristic, complemented, Sylow
113C2×Q8—681no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4#6322711yescharacteristic
118M16—342no—
119M16—681no—
120C2×C8—81no—
121C2×C8—682no—
122C2×C8—681no—
123C2×C8—682no—
124C2×C8—682no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4—681no—
130C4⋊C4—682no—
131C22⋊C4—681no—
132C42—21no—
133C42—681no—
134C42#6321911yescharacteristic
135C42Dic6811yescharacteristic
136C23#1058747711yescharacteristic
137Q8—2721no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—681no—
149C2×C4—681no—
150C2×C4#1058748711yescharacteristic
151C2×C4#1058749611yescharacteristic
152C8—161no—
153C8—682no—
154C8—684no—
155C22—21no—
156C22—21no—
157C22#1176254711yescharacteristic
158C22#1176259511yescharacteristic
159C22#1176260711yescharacteristic
160C4—21no—
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—81no—
166C4—1361no—
167C4—1361no—
168C2—21no—
169C2—21no—
170C2#1452681511yescharacteristic
171C2#1452693111yescharacteristic
172C2#1452693411yescharacteristic
subgroupG/H
centerC22#11762595
commutatorC2×C4⋊C4×C17C2×C4
Frattini#382C2×D17
FittingC17×#1324C2
socleC22×C17AΣL(1,𝔽16)
prime radicalC23×C17#99
cosocleC17×#382C22
hypercenter#1324D17
hypocenterC17#7442
LCSG ▷ C2×C4⋊C4×C17 ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ #1324

Complements

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

NH ≅ G/Ntype
112. C17#7442semidirect

Maximal subgroups

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

subgroupindex
#145301342
#145266652
C17×#13242
#744217