Finite Group Databasebeta

← #16931928

Subgroups — #16931928

Order 4352.

400 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 107 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#14539166C211yesmaximal, characteristic
2#14527522C211yesmaximal, characteristic
3#14527468C212yesmaximal
4#14526675C212yesmaximal
5C17×#2986C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7#11763179C414yes—
8#11763161—22no—
9#11762833C2212yes—
10#11762817—22no—
11C4×#63223C2211yescharacteristic
12#11762556—22no—
13#11762556—22no—
14#11762555—22no—
15C17×#533C2211yescharacteristic
16C17×#446—22no—
17C8×Q8×C17—22no—
18C17×#433—22no—
19C17×#431C2212yes—
20C22×Dic34—21no—
21C17×C4⋊Q8D412yes—
22C17×#125C2×C412yes—
23C17×C22⋊Q8—22no—
24C4×Q8×C17—22no—
25C17×#117D412yes—
26C2×C42×C17C2311yescharacteristic
27C2×C4⋊Dic17—22no—
28C2×#63223—21no—
29C2×#63223—22no—
30#10587555—21no—
31#10587555—22no—
32#10587555C2×C412yes—
33C17×C4⋊C8—24no—
34C17×C4⋊C8D412yes—
35C17×C4⋊C8D412yes—
36C17×C22⋊C8—22no—
37C17×#97—22no—
38C4×C8×C17—22no—
39C4×C8×C17—22no—
40#10587481—28no—
41C4⋊#3622—22no—
42C4⋊#3622—22no—
43#10587474C2×C412yes—
44C4×#3622—21no—
45C2×Q8×C17—22no—
46C22×C4×C17C2×D411yescharacteristic
47C22×C4×C17#4112yes—
48C22×Dic17—41no—
49C2×Dic34—22no—
50C2×Dic34—44no—
51C2×Dic34—41no—
52C2×C8×C17—22no—
53C2×C8×C17—22no—
54C4⋊C4×C17—24no—
55C4⋊C4×C17—22no—
56C4⋊C4×C17C22⋊C414yes—
57C22⋊C4×C17—22no—
58C42×C17—22no—
59C42×C17C22×C411yescharacteristic
60C42×C17C2×D411yescharacteristic
61C42×C17C2×D412yes—
62C4⋊Dic17—24no—
63C4⋊Dic17—42no—
64#63223—42no—
65#63223—82no—
66C2×#3622—22no—
67C2×#3622—42no—
68#9436—171nomaximal, Sylow
69C23×C17#12311yescharacteristic
70Q8×C17—42no—
71Q8×C17—42no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—24no—
76C2×C4×C17C2×C22⋊C411yescharacteristic
77C2×C4×C17C4×D412yes—
78C2×C4×C17#12011yescharacteristic
79C2×C4×C17#12112yes—
80C2×Dic17—42no—
81C2×Dic17—82no—
82Dic34—44no—
83Dic34—82no—
84C8×C17—42no—
85C8×C17—42no—
86#3622—42no—
87#3622—82no—
88#3139—171no—
89#2986D1711yescharacteristic
90#1561—171no—
91#1549—172no—
92#1299—172no—
93C22×C17—21no—
94C22×C17—22no—
95C22×C17#38611yescharacteristic
96C4×C17—21no—
97C4×C17—22no—
98C4×C17—44no—
99C4×C17#45412yes—
100C4×C17#47112yes—
101C4×C17#47312yes—
102Dic17—82no—
103#533C2×D1711yescharacteristic
104C2×C4⋊Q8—171no—
105#491—174no—
106#487—342no—
107#446—22no—
108C8×Q8—22no—
109#433—22no—
110#431C2×D1712yes—
111#423—172no—
112#419—342no—
113C4×M16—171no—
114#332—342no—
115#332—342no—
116#332—342no—
117C2×C17—41no—
118C2×C17#188811yescharacteristic
119C2×C17#190812yes—
120C22×Q8—341no—
121C2×M16—341no—
122C2×M16—342no—
123C4⋊Q8—172no—
124C4⋊Q8—341no—
125C4⋊Q8—342no—
126C4⋊Q8D6812yes—
127#125C4×D1712yes—
128C22⋊Q8—22no—
129C4×Q8—22no—
130#117C22⋊D1712yes—
131C2×C4⋊C4—342no—
132C2×C42C22×D1711yescharacteristic
133C4⋊C8—24no—
134C4⋊C8—342no—
135C4⋊C8—342no—
136C4⋊C8C22⋊D1712yes—
137C4⋊C8C22⋊D1712yes—
138#103—348no—
139C22⋊C8—22no—
140#97—22no—
141#97—172no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—341no—
145C17#943611yescharacteristic, complemented, Sylow
146C2×Q8—22no—
147C2×Q8—342no—
148C2×Q8—684no—
149C2×Q8—681no—
150C22×C4—681no—
151C22×C4#6325212yes—
152C22×C4C2×C22⋊D1711yescharacteristic
153M16—682no—
154M16—1362no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—342no—
158C2×C8—682no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—344no—
162C4⋊C4—682no—
163C4⋊C4#6322714yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D1711yescharacteristic
167C42C2×D6811yescharacteristic
168C42C2×C22⋊D1712yes—
169C23#1058761511yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—684no—
173Q8—1362no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—682no—
179C2×C4—1362no—
180C2×C4C2×#6322711yescharacteristic
181C2×C4C4×C22⋊D1712yes—
182C2×C4#1058761612yes—
183C2×C4#1058763911yescharacteristic
184C8—42no—
185C8—42no—
186C8—682no—
187C8—1362no—
188C22—22no—
189C22—21no—
190C22#1176268811yescharacteristic
191C4—22no—
192C4—21no—
193C4—44no—
194C4—1362no—
195C4#1176298212yes—
196C4#1176307212yes—
197C4#1176308412yes—
198C2—41no—
199C2#1452918711yescharacteristic
200C2#1452929912yes—
subgroupG/H
centerC22#11762688
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2986C2
socleC22×C17#386
prime radicalC23×C17#123
cosocleC2×C42×C17C23
hypercenter#2986D17
hypocenterC17#9436
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
145. C17#9436semidirect

Maximal subgroups

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

subgroupindex
#145391662
#145275222
#145274682
#145266752
C17×#29862
#943617