Finite Group Databasebeta

← #14527707

Subgroups — #14527707

Order 2176.

316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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#11762959C212yesmaximal, complemented
2#11762718C211yesmaximal, characteristic
3#11762546C212yesmaximal, complemented
4#11762533C211yesmaximal, characteristic, complemented
5C17×#543C211yesmaximal, characteristic, complemented
6C22×C22⋊D17—21no—
7C22×D68C2211yescharacteristic
8C17×C22⋊Q8—22no—
9C17×C22⋊Q8—21no—
10C17×C22⋊Q8—22no—
11C17×C22⋊Q8C2212yes—
12C2×C22⋊C4×C17—21no—
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×#63230—22no—
15C2×#63227—21no—
16#10587502C2211yescharacteristic
17C22⋊#3622C2212yes—
18#10587480—22no—
19C24×C17D411yescharacteristic
20C23×D17—21no—
21C2×Q8×C17—21no—
22C2×Q8×C17D412yescomplemented
23C22×C4×C17—21no—
24C22×C4×C17—21no—
25C22×C4×C17D411yescharacteristic
26C22×C4×C17C2311yescharacteristic
27C2×C22⋊D17—42no—
28C2×C22⋊D17—42no—
29C2×C22⋊D17—42no—
30C22×Dic17—21no—
31C2×D68—22no—
32C2×D68—42no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17—22no—
35C4⋊C4×C17—22no—
36C4⋊C4×C17D412yescomplemented
37C22⋊C4×C17—22no—
38C22⋊C4×C17—22no—
39C22⋊C4×C17—22no—
40#63230—44no—
41#63227—42no—
42C2×#3622—22no—
43C23×C17—21no—
44C23×C17—21no—
45C23×C17—41no—
46C23×C17—41no—
47C23×C17SD1612yescomplemented
48C23×C17C2×D411yescharacteristic
49C22×D17—22no—
50C22×D17—41no—
51C22×D17—41no—
52C22×D17—41no—
53C22×D17—41no—
54C22×D17—41no—
55C22×D17—41no—
56Q8×C17—42no—
57Q8×C17—81no—
58C2×C4×C17—22no—
59C2×C4×C17—21no—
60C2×C4×C17—22no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×C4×C17—81no—
64C2×C4×C17—81no—
65C2×C4×C17C2×D412yes—
66C22⋊D17—48no—
67C2×Dic17—22no—
68C2×Dic17—41no—
69C2×Dic17—41no—
70D68—24no—
71D68—42no—
72#3622—42no—
73#1596—171nomaximal, Sylow
74C22×C17—21no—
75C22×C17—21no—
76C22×C17—22no—
77C22×C17—42no—
78C22×C17—41no—
79C22×C17—41no—
80C22×C17—81no—
81C22×C17#12011yescharacteristic
82C22×C17C2×SD1611yescharacteristic
83C22×C17#13711yescharacteristic
84C2×D17—42no—
85C2×D17—42no—
86C2×D17—42no—
87C2×D17—81no—
88C2×D17—81no—
89C2×D17—81no—
90C2×D17—81no—
91C4×C17—22no—
92C4×C17—41no—
93C4×C17—42no—
94C4×C17—42no—
95Dic17—42no—
96#543D1711yescharacteristic, complemented
97#450—172no—
98#392—171no—
99#328—172no—
100#323—171no—
101C2×C17—21no—
102C2×C17—21no—
103C2×C17—42no—
104C2×C17#44811yescharacteristic
105C2×C17#45311yescharacteristic
106C2×C17B(4,𝔽2)11yescharacteristic
107D17—82no—
108C22×D4—171no—
109C22×D4—341no—
110C2×SD16—342no—
111C22⋊Q8—22no—
112C22⋊Q8—21no—
113C22⋊Q8—22no—
114C22⋊Q8C2×D1712yes—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—341no—
117C2×C22⋊C4C2×D1711yescharacteristic
118#102—342no—
119C22⋊C8—172no—
120#95—171no—
121C17#159611yescharacteristic, complemented, Sylow
122C24—341no—
123C24C22⋊D1711yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D1712yescomplemented
126C2×D4—342no—
127C2×D4—682no—
128C2×D4—682no—
129C2×D4—682no—
130C2×D4—682no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—341no—
134C22×C4C22⋊D1711yescharacteristic
135C22×C4C22×D1711yescharacteristic
136SD16—684no—
137C2×C8—342no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D1712yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—682no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—342no—
151C23—681no—
152C23—681no—
153C23—681no—
154C23—681no—
155C23—681no—
156C23—681no—
157C23#6323012yescomplemented
158C23C2×C22⋊D1711yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—344no—
162D4—682no—
163D4—688no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—342no—
172C2×C4—681no—
173C2×C4—681no—
174C2×C4C2×C22⋊D1712yes—
175C8—682no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—682no—
184C22—682no—
185C22—682no—
186C22—1361no—
187C22—1361no—
188C22—1361no—
189C22—1361no—
190C22C2×#6323011yescharacteristic
191C22#1058762811yescharacteristic
192C22#1058763911yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—682no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—1362no—
202C2#1176294811yescharacteristic
203C2#1176297711yescharacteristic
204C2#1176299611yescharacteristic
subgroupG/H
centerC22#10587639
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#543C2
socleC22×C17#120
prime radical#11762718C2
hypercenter#543D17
hypocenterC17#1596
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #11762959C2semidirect
3. #11762546C2semidirect
4. #11762533C2semidirect
5. C17×#543C2semidirect
22. C2×Q8×C17D4semidirect
36. C4⋊C4×C17D4semidirect
47. C23×C17SD16semidirect
96. #543D17semidirect
121. C17#1596semidirect
125. C2×Q8C22⋊D17semidirect
141. C4⋊C4C22⋊D17semidirect
157. C23#63230semidirect

Maximal subgroups

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

subgroupindex
#117629592
#117627182
#117625462
#117625332
C17×#5432
#159617