Finite Group Databasebeta

← #14545045

Subgroups — #14545045

Order 2176.

266 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 54 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#11763795C211yesmaximal, characteristic, complemented
2#11763788C211yesmaximal, characteristic, complemented
3#11763710C214yesmaximal, complemented
4#11763378C211yesmaximal, characteristic
5C2×#63252C2211yescharacteristic
6C22×D68C411yescharacteristic, complemented
7#10587677—21no—
8D17⋊M16—21no—
9C17×#117C411yescharacteristic, complemented
10#10587581—21no—
11#10587581—21no—
12#10587568C412yescomplemented
13#10587568C2212yes—
14#10587561—21no—
15#10587560—21no—
16#10587553C2212yes—
17#10587551C2212yes—
18#10587545—21no—
19#10587539—21no—
20C23×D17C2×C412yes—
21C22×C4×C17C2×C411yescharacteristic
22C2×C22⋊D17C2×C411yescharacteristic
23C2×C22⋊D17C2311yescharacteristic
24#63252—41no—
25#63252—41no—
26C2×D68—21no—
27C2×D68—21no—
28C2×D68—22no—
29C2×D68—22no—
30C2×D68—41no—
31C2×D68—41no—
32C2×D68D411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×Dic34D411yescharacteristic
36C22⋊#595—22no—
37#63247—22no—
38C4⋊#595—41no—
39C4×#595—41no—
40#63243—41no—
41#63242—41no—
42C4⋊C4×C17—21no—
43C22⋊C4×C17C2×C412yes—
44C42×C17—21no—
45#63227—22no—
46#63227—22no—
47C23×C17C22×C411yescharacteristic
48C22×D17—22no—
49C22×D17—42no—
50C22×D17—82no—
51C22×D17C22⋊C411yescharacteristic
52C22×D17C2×D411yescharacteristic
53C2×#595—42no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17—22no—
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22⋊C411yescharacteristic
59C22⋊D17—41no—
60C22⋊D17—41no—
61C2×Dic17C22⋊C411yescharacteristic
62C2×Dic17C2×D411yescharacteristic
63D68—41no—
64D68—42no—
65D68—41no—
66D68—41no—
67D68—82no—
68D68—82no—
69C4×D17—21no—
70C4×D17—21no—
71C4×D17—21no—
72C4×D17—21no—
73Dic34—41no—
74#3624—42no—
75#2122—171nomaximal, Sylow
76C22×C17—21no—
77C22×C17—41no—
78C22×C17C2×C22⋊C411yescharacteristic
79C2×D17—21no—
80C2×D17—21no—
81C2×D17—41no—
82C2×D17—41no—
83C2×D17—42no—
84C2×D17—42no—
85C2×D17—42no—
86C2×D17—84no—
87#595—82no—
88C4×C17—21no—
89C4×C17—42no—
90C4×C17#9911yescharacteristic
91C4×C17#9911yescharacteristic
92Dic17—21no—
93Dic17—21no—
94#535—171no—
95#413—171no—
96#410—171no—
97AΣL(1,𝔽16)—174no—
98C2×C17—21no—
99C2×C17—41no—
100C2×C17C2×#9911yescharacteristic
101D17—41no—
102D17—41no—
103D17—82no—
104C2×#41—171no—
105C22×D4—171no—
106C2×M16—341no—
107#127—341no—
108#124—341no—
109#121—341no—
110#121—341no—
111#120—172no—
112#120—172no—
113#117—341no—
114#117#59511yescharacteristic, complemented
115#101—341no—
116#100—172no—
117#99—172no—
118#99—341no—
119C17#212211yescharacteristic, complemented, Sylow
120C24—172no—
121#41—681no—
122#41—681no—
123C2×Q8—171no—
124C2×D4—171no—
125C2×D4—171no—
126C2×D4—171no—
127C2×D4—342no—
128C2×D4—342no—
129C2×D4—341no—
130C2×D4—341no—
131C2×D4—681no—
132C2×D4—681no—
133C22×C4—171no—
134C22×C4—171no—
135C22×C4C2×#59511yescharacteristic
136M16—342no—
137M16—681no—
138C2×C8—681no—
139C4⋊C4—21no—
140C4⋊C4—681no—
141C22⋊C4—342no—
142C22⋊C4—342no—
143C22⋊C4—342no—
144C22⋊C4C2×#59512yes—
145C42—21no—
146C42—681no—
147C23—171no—
148C23—171no—
149C23—342no—
150C23—682no—
151C23—1362no—
152C23C22×#59511yescharacteristic
153Q8—681no—
154D4—681no—
155D4—682no—
156D4—681no—
157D4—681no—
158D4—681no—
159D4—681no—
160D4—1362no—
161D4—1362no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—171no—
166C2×C4—171no—
167C2×C4—341no—
168C2×C4—341no—
169C2×C4—341no—
170C2×C4—341no—
171C2×C4—682no—
172C2×C4C22⋊#59511yescharacteristic
173C2×C4C22⋊#59511yescharacteristic
174C8—682no—
175C22—21no—
176C22—41no—
177C22—341no—
178C22—341no—
179C22—681no—
180C22—681no—
181C22—682no—
182C22—682no—
183C22—682no—
184C22—1364no—
185C22C2×C22⋊#59511yescharacteristic
186C4—21no—
187C4—42no—
188C4—341no—
189C4—341no—
190C4—1362no—
191C4#1058753911yescharacteristic
192C4#1058753911yescharacteristic
193C2—21no—
194C2—41no—
195C2—681no—
196C2—681no—
197C2—1362no—
198C2C2×#1058753911yescharacteristic
subgroupG/H
centerC2C2×#10587539
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC17×#117C4
socleC2×C17C2×#99
prime radical#11763378C2
cosocleC2×C22⋊D17C23
hypercenter#117#595
hypocenterC17#2122
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #11763795C2semidirect
2. #11763788C2semidirect
3. #11763710C2semidirect
6. C22×D68C4semidirect
9. C17×#117C4semidirect
12. #10587568C4semidirect
114. #117#595semidirect
119. C17#2122semidirect

Maximal subgroups

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

subgroupindex
#117637952
#117637882
#117637102
#117633782
#212217