Finite Group Databasebeta

← #16931846

Subgroups — #16931846

Order 4352.

384 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 99 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#14527502C212yesmaximal
3#14527422C211yesmaximal, characteristic
4#14526655C212yesmaximal
5C17×#2987C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7#11763179—22no—
8#11763161C414yes—
9#11762866C2212yes—
10#11762817—22no—
11C4×#63223C2211yescharacteristic
12#11762556—24no—
13#11762551—22no—
14C17×#533C2211yescharacteristic
15C17×#446—24no—
16C17×#433—22no—
17C17×#432C2212yes—
18C22×Dic34—21no—
19C17×C4⋊Q8C2×C412yes—
20C17×#125D412yes—
21C17×C22⋊Q8—22no—
22C4×Q8×C17—22no—
23C17×#117D412yes—
24C2×C42×C17C2311yescharacteristic
25C2×C4⋊Dic17—22no—
26C2×#63223—21no—
27#10587555—21no—
28#10587555—22no—
29#10587555C2×C412yes—
30C17×C4⋊C8—24no—
31C17×C4⋊C8D414yes—
32C17×C22⋊C8—22no—
33C17×#97—22no—
34C17×#97—22no—
35C4×C8×C17—22no—
36C22⋊#3622—22no—
37#10587481—28no—
38C4⋊#3622—22no—
39#10587474—21no—
40#10587474—22no—
41C4×#3622C2×C412yes—
42C2×Q8×C17—22no—
43C22×C4×C17C2×D411yescharacteristic
44C22×C4×C17#4112yes—
45C22×Dic17—41no—
46C2×Dic34—22no—
47C2×Dic34—44no—
48C2×Dic34—41no—
49C2×C8×C17—24no—
50C4⋊C4×C17—22no—
51C4⋊C4×C17—24no—
52C4⋊C4×C17C22⋊C414yes—
53C22⋊C4×C17—22no—
54C42×C17—22no—
55C42×C17C22×C411yescharacteristic
56C42×C17C2×D411yescharacteristic
57C42×C17C2×D412yes—
58C4⋊Dic17—24no—
59C4⋊Dic17—42no—
60#63223—42no—
61C2×#3622—22no—
62C2×#3622—42no—
63#9422—171nomaximal, Sylow
64C23×C17#12311yescharacteristic
65Q8×C17—82no—
66C2×C4×C17—22no—
67C2×C4×C17—24no—
68C2×C4×C17—42no—
69C2×C4×C17C2×C22⋊C411yescharacteristic
70C2×C4×C17C4×D412yes—
71C2×C4×C17#12011yescharacteristic
72C2×C4×C17#12112yes—
73C2×Dic17—42no—
74C2×Dic17—82no—
75Dic34—44no—
76Dic34—82no—
77C8×C17—44no—
78#3622—42no—
79#3622—82no—
80#3139—171no—
81#2987D1711yescharacteristic
82#1556—172no—
83#1537—171no—
84#1291—172no—
85C22×C17—21no—
86C22×C17—22no—
87C22×C17#38611yescharacteristic
88C4×C17—21no—
89C4×C17—22no—
90C4×C17—22no—
91C4×C17—44no—
92C4×C17#45412yes—
93Dic17—82no—
94#533C2×D1711yescharacteristic
95C2×C4⋊Q8—171no—
96#491—342no—
97#487—174no—
98#446—24no—
99#433—22no—
100#432—172no—
101#432C2×D1712yes—
102#419—342no—
103C4×M16—171no—
104#332—344no—
105#330—342no—
106C2×C17—41no—
107C2×C17#188811yescharacteristic
108C2×C17#190212yes—
109C22×Q8—341no—
110C2×M16—341no—
111C4⋊Q8—172no—
112C4⋊Q8—341no—
113C4⋊Q8—342no—
114C4⋊Q8C4×D1712yes—
115#125D6812yes—
116C22⋊Q8—22no—
117C4×Q8—22no—
118#117C22⋊D1712yes—
119C2×C4⋊C4—342no—
120C2×C42C22×D1711yescharacteristic
121C4⋊C8—24no—
122C4⋊C8—342no—
123C4⋊C8C22⋊D1714yes—
124#103—348no—
125C22⋊C8—22no—
126C22⋊C8—342no—
127#97—22no—
128#97—22no—
129#97—341no—
130#97—342no—
131C4×C8—22no—
132C4×C8—172no—
133C17#942211yescharacteristic, complemented, Sylow
134C2×Q8—22no—
135C2×Q8—342no—
136C2×Q8—684no—
137C2×Q8—681no—
138C22×C4—681no—
139C22×C4#6325212yes—
140C22×C4C2×C22⋊D1711yescharacteristic
141M16—682no—
142C2×C8—24no—
143C2×C8—342no—
144C2×C8—682no—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—344no—
148C4⋊C4—682no—
149C4⋊C4#6322714yes—
150C22⋊C4—22no—
151C42—22no—
152C42C2×C4×D1711yescharacteristic
153C42C2×D6811yescharacteristic
154C42C2×C22⋊D1712yes—
155C23#1058761511yescharacteristic
156Q8—82no—
157Q8—684no—
158Q8—1362no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—42no—
162C2×C4—682no—
163C2×C4—1362no—
164C2×C4C2×#6322711yescharacteristic
165C2×C4C4×C22⋊D1712yes—
166C2×C4#1058761612yes—
167C2×C4#1058763911yescharacteristic
168C8—44no—
169C8—682no—
170C8—1362no—
171C22—21no—
172C22—22no—
173C22#1176268811yescharacteristic
174C4—21no—
175C4—22no—
176C4—22no—
177C4—44no—
178C4—1362no—
179C4#1176298212yes—
180C2—41no—
181C2#1452918711yescharacteristic
182C2#1452925712yes—
subgroupG/H
centerC22#11762688
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2987C2
socleC22×C17#386
prime radicalC23×C17#123
cosocleC2×C42×C17C23
hypercenter#2987D17
hypocenterC17#9422
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
133. C17#9422semidirect

Maximal subgroups

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

subgroupindex
#145391662
#145275022
#145274222
#145266552
C17×#29872
#942217