Finite Group Databasebeta

← #14530133

Subgroups — #14530133

Order 2176.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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#11763437C211yesmaximal, characteristic
2#11762852C211yesmaximal, characteristic
3#11762851C211yesmaximal, characteristic
4#11762675C211yesmaximal, characteristic
5#11762577C211yesmaximal, characteristic
6#11762569C211yesmaximal, characteristic
7C17×#339C211yesmaximal, characteristic
8C2×M16×C17C2211yescharacteristic
9C17×#117C2211yescharacteristic
10C2×C42×C17C2211yescharacteristic
11#10587612C2211yescharacteristic
12C2×C4⋊Dic17C2211yescharacteristic
13C2×C4⋊Dic17C2211yescharacteristic
14C2×#63223C2211yescharacteristic
15#10587576—21no—
16#10587574—21no—
17#10587564—22no—
18#10587556—21no—
19#10587555—21no—
20#10587502—22no—
21#10587489—21no—
22#10587488—21no—
23#10587479—21no—
24#10587478—21no—
25C22×C4×C17—21no—
26C22×C4×C17C2311yescharacteristic
27C22×Dic17—21no—
28C22×Dic17D412yes—
29C2×Dic34—22no—
30M16×C17—21no—
31M16×C17Q812yescomplemented
32C2×C8×C17—21no—
33C4⋊C4×C17D411yescharacteristic
34C4⋊C4×C17D411yescharacteristic
35C22⋊C4×C17—21no—
36C42×C17—21no—
37C42×C17—21no—
38C42×C17Q812yescomplemented
39C22⋊Dic17—21no—
40C4⋊Dic17—22no—
41C4⋊Dic17—21no—
42C4⋊Dic17—21no—
43C4⋊Dic17—21no—
44C4⋊Dic17—21no—
45C4⋊Dic17D411yescharacteristic
46C4⋊Dic17D411yescharacteristic
47#63225—22no—
48#63225—22no—
49C4×Dic17—21no—
50#63223—21no—
51#63223Q812yescomplemented
52C2×#3622—21no—
53C23×C17C2×D411yescharacteristic
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17—22no—
57C2×C4×C17—41no—
58C2×C4×C17—41no—
59C2×C4×C17C2×D411yescharacteristic
60C2×C4×C17C2×D411yescharacteristic
61C2×C4×C17C2×Q811yescharacteristic
62C2×C4×C17C2×Q811yescharacteristic
63C2×C4×C17C2×Q811yescharacteristic
64C2×C4×C17#4111yescharacteristic
65C2×Dic17—21no—
66C2×Dic17—21no—
67C2×Dic17—22no—
68C2×Dic17—42no—
69C2×Dic17—42no—
70C2×Dic17—41no—
71Dic34—82no—
72C8×C17—22no—
73#3622—22no—
74#2061—171nomaximal, Sylow
75C22×C17—21no—
76C22×C17—21no—
77C22×C17#12011yescharacteristic
78C22×C17C22⋊Q811yescharacteristic
79C22×C17C4⋊Q811yescharacteristic
80C4×C17—42no—
81C4×C17—41no—
82C4×C17—41no—
83C4×C17C22⋊Q811yescharacteristic
84C4×C17C22⋊Q811yescharacteristic
85C4×C17C4⋊Q811yescharacteristic
86C4×C17C4⋊Q811yescharacteristic
87Dic17—41no—
88Dic17—41no—
89Dic17—42no—
90Dic17—81no—
91#544—171no—
92#428—171no—
93#428—171no—
94#382—171no—
95#343—171no—
96#339—171no—
97#339D1711yescharacteristic
98C2×C17—21no—
99C2×C17—21no—
100C2×C17#39511yescharacteristic
101C2×C17#45411yescharacteristic
102C2×C17#45511yescharacteristic
103C2×M16—171no—
104C2×M16C2×D1711yescharacteristic
105C4⋊Q8—341no—
106C4⋊Q8—341no—
107#125—341no—
108#125—341no—
109C22⋊Q8—342no—
110#117—171no—
111#117C2×D1711yescharacteristic
112C2×C4⋊C4—171no—
113C2×C4⋊C4—171no—
114C2×C42C2×D1711yescharacteristic
115#107—341no—
116#107—341no—
117#106—341no—
118#106—341no—
119#95—342no—
120C17#206111yescharacteristic, complemented, Sylow
121C2×Q8—342no—
122C22×C4—21no—
123C22×C4—172no—
124C22×C4—341no—
125C22×C4C22×D1711yescharacteristic
126M16—21no—
127M16—172no—
128M16—341no—
129M16Dic3412yescomplemented
130C2×C8—21no—
131C2×C8—341no—
132C4⋊C4—171no—
133C4⋊C4—171no—
134C4⋊C4—342no—
135C4⋊C4—342no—
136C4⋊C4—342no—
137C4⋊C4—341no—
138C4⋊C4—341no—
139C4⋊C4—341no—
140C4⋊C4—341no—
141C4⋊C4D6811yescharacteristic
142C4⋊C4D6811yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—341no—
145C42—21no—
146C42—21no—
147C42—341no—
148C42Dic3412yescomplemented
149C23D4×D1711yescharacteristic
150Q8—1362no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—341no—
157C2×C4—341no—
158C2×C4—342no—
159C2×C4—682no—
160C2×C4—682no—
161C2×C4—681no—
162C2×C4C2×Dic3411yescharacteristic
163C2×C4C2×Dic3411yescharacteristic
164C2×C4C2×D6811yescharacteristic
165C2×C4D4×D1711yescharacteristic
166C2×C4#6325411yescharacteristic
167C2×C4Q8×D1711yescharacteristic
168C8—22no—
169C8—342no—
170C22—21no—
171C22—21no—
172C22#1058756411yescharacteristic
173C22#1058756811yescharacteristic
174C22#1058757411yescharacteristic
175C4—42no—
176C4—41no—
177C4—41no—
178C4—681no—
179C4—682no—
180C4—681no—
181C4—1361no—
182C4#1058755511yescharacteristic
183C4#1058756411yescharacteristic
184C4#1058757411yescharacteristic
185C4#1058758311yescharacteristic
186C2—21no—
187C2—21no—
188C2#1176273011yescharacteristic
189C2#1176298311yescharacteristic
190C2#1176298811yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#339C2
socleC22×C17#120
prime radicalC23×C17C2×D4
hypercenter#339D17
hypocenterC17#2061
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
31. M16×C17Q8semidirect
38. C42×C17Q8semidirect
51. #63223Q8semidirect
120. C17#2061semidirect
129. M16Dic34semidirect
148. C42Dic34semidirect

Maximal subgroups

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

subgroupindex
#117634372
#117628522
#117628512
#117626752
#117625772
#117625692
C17×#3392
#206117