Finite Group Databasebeta

← #14527706

Subgroups — #14527706

Order 2176.

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

normal subgroups only

This group has 316 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#11763435C211yesmaximal, characteristic, complemented
2#11762960C212yesmaximal, complemented
3#11762545C212yesmaximal, complemented
4#11762533C211yesmaximal, characteristic, complemented
5C17×#392C211yesmaximal, characteristic
6C22×D4×C17—21no—
7C22×D4×C17C2211yescharacteristic
8C2×C22⋊C4×C17—21no—
9C2×C22⋊C4×C17C2211yescharacteristic
10C2×C22⋊Dic17—21no—
11C2×#63229—22no—
12#10587610—21no—
13#10587610C2212yes—
14#10587564—22no—
15#10587564—22no—
16C17×#95C2211yescharacteristic
17#10587503—22no—
18C22⋊#3622C2212yes—
19C24×C17—21no—
20C24×C17D411yescharacteristic
21C2×D4×C17—22no—
22C2×D4×C17—42no—
23C2×D4×C17—42no—
24C2×D4×C17—42no—
25C2×D4×C17—42no—
26C22×C4×C17—21no—
27C22×C4×C17D411yescharacteristic
28C22×C4×C17C2311yescharacteristic
29C22×Dic17—21no—
30C22×Dic17—21no—
31C2×Dic34—21no—
32C2×Dic34D412yescomplemented
33C22⋊C4×C17—22no—
34C22⋊C4×C17—42no—
35C22⋊Dic17—22no—
36C22⋊Dic17—22no—
37#63229—44no—
38C4⋊Dic17—21no—
39C4⋊Dic17D412yescomplemented
40#63225—22no—
41#63225—22no—
42C2×#3622—22no—
43C23×C17—21no—
44C23×C17—21no—
45C23×C17—22no—
46C23×C17—41no—
47C23×C17—41no—
48C23×C17—41no—
49C23×C17—41no—
50C23×C17—41no—
51C23×C17—41no—
52C23×C17—41no—
53C23×C17—41no—
54C23×C17SD1612yescomplemented
55C23×C17C2×D411yescharacteristic
56D4×C17—24no—
57D4×C17—42no—
58D4×C17—48no—
59C2×C4×C17—22no—
60C2×C4×C17—21no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×C4×C17—41no—
64C2×C4×C17—41no—
65C2×C4×C17C2×D412yes—
66C2×Dic17—22no—
67C2×Dic17—22no—
68C2×Dic17—81no—
69C2×Dic17—81no—
70Dic34—42no—
71Dic34—81no—
72#3622—42no—
73#1596—171nomaximal, Sylow
74C22×C17—21no—
75C22×C17—21no—
76C22×C17—22no—
77C22×C17—42no—
78C22×C17—42no—
79C22×C17—42no—
80C22×C17—41no—
81C22×C17—41no—
82C22×C17—42no—
83C22×C17—81no—
84C22×C17—81no—
85C22×C17—81no—
86C22×C17—81no—
87C22×C17—81no—
88C22×C17#12011yescharacteristic
89C22×C17C2×SD1611yescharacteristic
90C22×C17#13711yescharacteristic
91C4×C17—22no—
92C4×C17—42no—
93C4×C17—41no—
94Dic17—42no—
95Dic17—42no—
96#543—171no—
97#450—172no—
98#392D1711yescharacteristic
99#328—172no—
100#323—171no—
101C2×C17—21no—
102C2×C17—21no—
103C2×C17—42no—
104C2×C17—82no—
105C2×C17#44811yescharacteristic
106C2×C17#45311yescharacteristic
107C2×C17B(4,𝔽2)11yescharacteristic
108C22×D4—21no—
109C22×D4C2×D1711yescharacteristic
110C2×SD16—342no—
111C22⋊Q8—172no—
112C22⋊Q8—342no—
113C22⋊Q8—342no—
114C22⋊Q8—341no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—341no—
117C2×C22⋊C4C2×D1711yescharacteristic
118#102—342no—
119C22⋊C8—172no—
120#95C2×D1711yescharacteristic
121C17#159611yescharacteristic, complemented, Sylow
122C24—21no—
123C24C22⋊D1711yescharacteristic
124C2×Q8—172no—
125C2×Q8—341no—
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C22×C4—21no—
132C22×C4—341no—
133C22×C4—341no—
134C22×C4C22⋊D1711yescharacteristic
135C22×C4C22×D1711yescharacteristic
136SD16—684no—
137C2×C8—342no—
138C4⋊C4—172no—
139C4⋊C4—342no—
140C4⋊C4—341no—
141C4⋊C4—342no—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4—342no—
145C22⋊C4—342no—
146C23—21no—
147C23—21no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23#6322912yescomplemented
158C23C2×C22⋊D1711yescharacteristic
159Q8—682no—
160Q8—1361no—
161D4—24no—
162D4—42no—
163D4—48no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—1361no—
173C2×C4—1361no—
174C2×C4D4×D1712yes—
175C8—682no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—42no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22C2×#6322911yescharacteristic
191C22#1058762311yescharacteristic
192C22#1058763711yescharacteristic
193C4—22no—
194C4—42no—
195C4—41no—
196C4—682no—
197C4—682no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—82no—
202C2#1176294711yescharacteristic
203C2#1176297611yescharacteristic
204C2#1176299511yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
Fitting, prime radicalC17×#392C2
socleC22×C17#120
hypercenter#392D17
hypocenterC17#1596
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #11763435C2semidirect
2. #11762960C2semidirect
3. #11762545C2semidirect
4. #11762533C2semidirect
32. C2×Dic34D4semidirect
39. C4⋊Dic17D4semidirect
54. C23×C17SD16semidirect
121. C17#1596semidirect
157. C23#63229semidirect

Maximal subgroups

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

subgroupindex
#117634352
#117629602
#117625452
#117625332
C17×#3922
#159617