Finite Group Databasebeta

← #14527718

Subgroups — #14527718

Order 2176.

310 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 53 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#11763390C211yesmaximal, characteristic, complemented
2#11762943C212yesmaximal, complemented
3#11762542C212yesmaximal, complemented
4#11762535C211yesmaximal, characteristic, complemented
5C17×#394C211yesmaximal, characteristic, complemented
6C22×D4×C17C2211yescharacteristic
7C2×#63252—21no—
8C2×C4⋊C4×C17C2211yescharacteristic
9C2×C22⋊C4×C17—21no—
10C2×C22⋊C4×C17—21no—
11C2×#63228—22no—
12#10587616—21no—
13#10587616C2212yes—
14#10587583—22no—
15#10587581—22no—
16C17×#95C2211yescharacteristic
17#10587503—22no—
18C22⋊#3622C2212yes—
19C24×C17—21no—
20C2×D4×C17—22no—
21C2×D4×C17—42no—
22C22×C4×C17—21no—
23C22×C4×C17D411yescharacteristic
24C22×C4×C17D411yescharacteristic
25C22×C4×C17C2311yescharacteristic
26C2×C22⋊D17—22no—
27#63252—42no—
28#63252—42no—
29C2×D68—21no—
30C2×D68—21no—
31C2×D68D412yes—
32C2×C4×D17—22no—
33C2×Dic34—21no—
34C4⋊C4×C17—22no—
35C22⋊C4×C17—42no—
36C22⋊C4×C17—42no—
37#63228—44no—
38#63227—22no—
39#63227—22no—
40C4⋊Dic17—21no—
41C4⋊Dic17D412yes—
42C2×#3622—22no—
43C23×C17—22no—
44C23×C17—41no—
45C23×C17—41no—
46C23×C17—41no—
47C23×C17—41no—
48C23×C17—41no—
49C23×C17—41no—
50C23×C17C2×D411yescharacteristic
51C22×D17—22no—
52D4×C17—24no—
53D4×C17—42no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17—21no—
57C2×C4×C17—41no—
58C2×C4×C17—41no—
59C2×C4×C17—41no—
60C2×C4×C17—41no—
61C2×C4×C17—41no—
62C2×C4×C17D812yescomplemented
63C2×C4×C17C2×D412yes—
64C22⋊D17—42no—
65C22⋊D17—42no—
66C2×Dic17—22no—
67D68—42no—
68D68—42no—
69D68—81no—
70C4×D17—44no—
71Dic34—42no—
72#3622—42no—
73#1598—171nomaximal, Sylow
74C22×C17—21no—
75C22×C17—21no—
76C22×C17—42no—
77C22×C17—42no—
78C22×C17—42no—
79C22×C17—81no—
80C22×C17—81no—
81C22×C17—81no—
82C22×C17—81no—
83C22×C17#12011yescharacteristic
84C22×C17C2×D811yescharacteristic
85C22×C17AGL(1,ℤ/8)11yescharacteristic
86C2×D17—42no—
87C2×D17—42no—
88C2×D17—42no—
89C4×C17—22no—
90C4×C17—22no—
91C4×C17—41no—
92C4×C17—81no—
93Dic17—42no—
94#537—171no—
95#447—172no—
96#394D1711yescharacteristic, complemented
97#327—172no—
98#324—171no—
99C2×C17—21no—
100C2×C17—21no—
101C2×C17—82no—
102C2×C17#44711yescharacteristic
103C2×C17#45411yescharacteristic
104C2×C17B(4,𝔽2)11yescharacteristic
105D17—82no—
106C2×#41—341no—
107C22×D4C2×D1711yescharacteristic
108C2×D8—342no—
109C22⋊Q8—342no—
110#121—172no—
111#121—342no—
112#121—341no—
113C2×C4⋊C4C2×D1711yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116#102—342no—
117C22⋊C8—172no—
118#95C2×D1711yescharacteristic
119C17#159811yescharacteristic, complemented, Sylow
120C24—21no—
121#41—682no—
122#41—682no—
123C2×Q8—341no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—172no—
127C2×D4—341no—
128C2×D4—341no—
129C2×D4—342no—
130C22×C4—21no—
131C22×C4—342no—
132C22×C4C22⋊D1711yescharacteristic
133C22×C4C22⋊D1711yescharacteristic
134C22×C4C22×D1711yescharacteristic
135D8—684no—
136C2×C8—342no—
137C4⋊C4—22no—
138C4⋊C4—172no—
139C4⋊C4—341no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—342no—
143C22⋊C4—342no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—342no—
152C23C2×C22⋊D1711yescharacteristic
153Q8—682no—
154D4—24no—
155D4—42no—
156D4—682no—
157D4—682no—
158D4—682no—
159D4—682no—
160D4—1361no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—342no—
170C2×C4—684no—
171C2×C4#6322812yescomplemented
172C2×C4D4×D1712yes—
173C8—682no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—682no—
184C22—682no—
185C22—682no—
186C22C2×#6322811yescharacteristic
187C22#1058762311yescharacteristic
188C22#1058763511yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—682no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—1362no—
198C2#1176294211yescharacteristic
199C2#1176298111yescharacteristic
200C2#1176299511yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#394C2
socleC22×C17#120
hypercenter#394D17
hypocenterC17#1598
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #11763390C2semidirect
2. #11762943C2semidirect
3. #11762542C2semidirect
4. #11762535C2semidirect
5. C17×#394C2semidirect
62. C2×C4×C17D8semidirect
96. #394D17semidirect
119. C17#1598semidirect
171. C2×C4#63228semidirect

Maximal subgroups

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

subgroupindex
#117633902
#117629432
#117625422
#117625352
C17×#3942
#159817