Finite Group Databasebeta

← #16921955

Subgroups — #16921955

Order 4352.

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

normal subgroups only

This group has 198 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#14530129C211yesmaximal, characteristic
2#14526667C211yesmaximal, characteristic
3C17×#1325C211yesmaximal, characteristic
4#11763440C411yescharacteristic
5#11762849—21no—
6C2×C4⋊#3622—21no—
7#11762578C411yescharacteristic
8#11762569—21no—
9C17×#384C2211yescharacteristic
10C17×#324—21no—
11C2×C4⋊C4×C17—41no—
12C2×C4⋊C4×C17C2×C411yescharacteristic
13C2×C4⋊C4×C17D411yescharacteristic
14C2×C42×C17D411yescharacteristic
15#10587612—21no—
16#10587610—41no—
17C2×#63223—21no—
18C22×#3622—41no—
19#10587576—41no—
20#10587574—41no—
21C17×C22⋊C8—41no—
22#10587479—41no—
23#10587478—41no—
24C4⋊#3622—22no—
25C4⋊#3622—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—
34C4⋊C4×C17—81no—
35C4⋊C4×C17—81no—
36C42×C17—21no—
37C42×C17SD1611yescharacteristic
38C42×C17Q1611yescharacteristic
39C22⋊Dic17—41no—
40C4⋊Dic17—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17—41no—
43#63225—42no—
44C4×Dic17—41no—
45#63223—22no—
46#63223—41no—
47C2×#3622—41no—
48C2×#3622—42no—
49C2×#3622—81no—
50C2×#3622—81no—
51#7443—171nomaximal, Sylow
52C23×C17#9911yescharacteristic
53C2×C4×C17—21no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×C4×C17—81no—
64C2×C4×C17—81no—
65C2×C4×C17#10311yescharacteristic
66C2×C4×C17#10411yescharacteristic
67C2×Dic17—41no—
68C2×Dic17—41no—
69Dic34—161no—
70C8×C17—161no—
71#3622—42no—
72#3622—82no—
73#2060—171no—
74#1325D1711yescharacteristic
75#1296—171no—
76C22×C17—21no—
77C22×C17—21no—
78C22×C17#32811yescharacteristic
79C22×C17AΣL(1,𝔽16)11yescharacteristic
80C22×C17#35611yescharacteristic
81C4×C17—21no—
82C4×C17—21no—
83C4×C17—21no—
84C4×C17—21no—
85C4×C17—21no—
86C4×C17—21no—
87C4×C17—81no—
88C4×C17—81no—
89Dic17—81no—
90Dic17—81no—
91#544—171no—
92#428—341no—
93C2×C4⋊C8—341no—
94#384C2×D1711yescharacteristic
95#343—171no—
96#339—341no—
97#324—21no—
98C2×C17—21no—
99C2×C17—21no—
100C2×C17#135411yescharacteristic
101C2×C17#140211yescharacteristic
102C2×C17#140511yescharacteristic
103C2×M16—341no—
104C22×C8—681no—
105C4⋊Q8—681no—
106#125—681no—
107C22⋊Q8—681no—
108#117—341no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4C4×D1711yescharacteristic
111C2×C4⋊C4C22⋊D1711yescharacteristic
112C2×C42D6811yescharacteristic
113#107—681no—
114#106—681no—
115C4⋊C8—342no—
116C4⋊C8—681no—
117C22⋊C8—41no—
118C17#744311yescharacteristic, complemented, Sylow
119C2×Q8—681no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4#6322711yescharacteristic
124M16—342no—
125M16—681no—
126C2×C8—81no—
127C2×C8—681no—
128C2×C8—682no—
129C2×C8—1361no—
130C2×C8—1361no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—81no—
134C4⋊C4—81no—
135C4⋊C4—341no—
136C4⋊C4—341no—
137C4⋊C4—681no—
138C4⋊C4—682no—
139C22⋊C4—681no—
140C42—21no—
141C42—681no—
142C42#6321911yescharacteristic
143C42Dic6811yescharacteristic
144C23#1058747711yescharacteristic
145Q8—2721no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—681no—
159C2×C4—681no—
160C2×C4#1058748711yescharacteristic
161C2×C4#1058749611yescharacteristic
162C8—161no—
163C8—682no—
164C8—1362no—
165C22—21no—
166C22—21no—
167C22#1176254711yescharacteristic
168C22#1176259511yescharacteristic
169C22#1176260711yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—81no—
177C4—81no—
178C4—1361no—
179C4—1361no—
180C2—21no—
181C2—21no—
182C2#1452681511yescharacteristic
183C2#1452692811yescharacteristic
184C2#1452693711yescharacteristic
subgroupG/H
centerC22#11762595
commutatorC2×C4⋊C4×C17C2×C4
Frattini#384C2×D17
FittingC17×#1325C2
socleC22×C17AΣL(1,𝔽16)
prime radicalC23×C17#99
cosocleC17×#384C22
hypercenter#1325D17
hypocenterC17#7443
LCSG ▷ C2×C4⋊C4×C17 ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1325

Complements

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

NH ≅ G/Ntype
118. C17#7443semidirect

Maximal subgroups

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

subgroupindex
#145301292
#145266672
C17×#13252
#744317