Finite Group Databasebeta

← #14530804

Subgroups — #14530804

Order 2176.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#11763559C211yesmaximal, characteristic, complemented
2#11762992C212yesmaximal, complemented
3#11762607C212yesmaximal, complemented
4#11762606C211yesmaximal, characteristic, complemented
5C17×#454C211yesmaximal, characteristic
6C17×2−1+4C2211yescharacteristic
7C17×AGL(1,ℤ/8)—21no—
8C17×#124C2211yescharacteristic
9#10587637—22no—
10#10587630C2212yes—
11#10587621—21no—
12#10587565—22no—
13#10587564—22no—
14#10587556—21no—
15C17×#104—21no—
16C17×#101C2211yescharacteristic
17#10587508—22no—
18#10587507C2212yes—
19#41×C17—21no—
20#41×C17—22no—
21#41×C17—41no—
22C2×Q8×C17—41no—
23C2×Q8×C17C2311yescharacteristic
24C2×D4×C17D411yescharacteristic
25C22×Dic17—21no—
26C2×Dic34D412yes—
27SD16×C17—41no—
28D8×C17—41no—
29M16×C17—21no—
30C22⋊C4×C17—22no—
31C42×C17D411yescharacteristic
32C22⋊Dic17—22no—
33C22⋊Dic17—21no—
34#63231—42no—
35#63229—42no—
36C4⋊Dic17—21no—
37C4⋊Dic17—21no—
38#63225—22no—
39#63225—22no—
40#63225—22no—
41C4×Dic17D412yes—
42#63223—22no—
43C23×C17—21no—
44Q8×C17—21no—
45Q8×C17—22no—
46Q8×C17—41no—
47D4×C17—21no—
48D4×C17—22no—
49D4×C17—41no—
50D4×C17—41no—
51C2×C4×C17—21no—
52C2×C4×C17—42no—
53C2×C4×C17—41no—
54C2×C4×C17C2×D412yes—
55C2×C4×C17C2×D411yescharacteristic
56C2×Dic17—22no—
57C2×Dic17—22no—
58C2×Dic17—81no—
59Dic34—42no—
60C8×C17—41no—
61#3622—42no—
62#2203—171nomaximal, Sylow
63C22×C17—42no—
64C22×C17—41no—
65C22×C17—41no—
66C22×C17#12011yescharacteristic
67C4×C17—22no—
68C4×C17—21no—
69C4×C17—41no—
70C4×C17—41no—
71Dic17—42no—
72Dic17—42no—
73#563—171no—
74#456—172no—
75#454D1711yescharacteristic
76#356—172no—
77#355—171no—
78C2×C17—21no—
79C2×C17—81no—
80C2×C17—81no—
81C2×C17B(4,𝔽2)11yescharacteristic
822−1+4C2×D1711yescharacteristic
83#137—342no—
84AGL(1,ℤ/8)—21no—
85C4⋊Q8—172no—
86#126—342no—
87#125—341no—
88#124C2×D1711yescharacteristic
89#123—341no—
90C22⋊Q8—342no—
91#104—21no—
92#104—342no—
93#101—172no—
94#101C2×D1711yescharacteristic
95C17#220311yescharacteristic, complemented, Sylow
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—172no—
101C2×Q8C22×D1711yescharacteristic
102C2×D4C22⋊D1711yescharacteristic
103C22×C4—341no—
104Q16—682no—
105SD16—41no—
106SD16—682no—
107D8—41no—
108M16—21no—
109M16—342no—
110C4⋊C4—341no—
111C4⋊C4—342no—
112C4⋊C4—341no—
113C4⋊C4—342no—
114C4⋊C4—342no—
115C22⋊C4—22no—
116C22⋊C4—342no—
117C22⋊C4—341no—
118C42—172no—
119C42C22⋊D1711yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—682no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—342no—
133C2×C4—342no—
134C2×C4—1361no—
135C2×C4D4×D1712yes—
136C2×C4C2×C22⋊D1711yescharacteristic
137C8—41no—
138C8—682no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#1058762311yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—682no—
148C4—682no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#1176299511yescharacteristic
subgroupG/H
centerC2#11762995
commutator, cosocleC2×Q8×C17C23
FrattiniC2×Q8C22×D17
Fitting, prime radicalC17×#454C2
socleC2×C17B(4,𝔽2)
hypercenter#454D17
hypocenterC17#2203
LCSG ▷ C2×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #11763559C2semidirect
2. #11762992C2semidirect
3. #11762607C2semidirect
4. #11762606C2semidirect
95. C17#2203semidirect

Maximal subgroups

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

subgroupindex
#117635592
#117629922
#117626072
#117626062
C17×#4542
#220317