Finite Group Databasebeta

← #14530813

Subgroups — #14530813

Order 2176.

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

normal subgroups only

This group has 224 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#11763539C211yesmaximal, characteristic, complemented
2#11762992C212yesmaximal
3#11762608C211yesmaximal, characteristic, complemented
4#11762607C212yesmaximal, complemented
5C17×#456C211yesmaximal, characteristic
6C17×2−1+4C2211yescharacteristic
7C17×#137—21no—
8C17×C4⋊Q8C2211yescharacteristic
9#10587637—22no—
10#10587630—21no—
11#10587630C2212yes—
12#10587574—22no—
13#10587573—22no—
14C4×Dic34—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×C17D411yescharacteristic
24C2×Q8×C17C2311yescharacteristic
25C2×Dic34—21no—
26C2×Dic34—21no—
27C2×Dic34D412yes—
28Q16×C17—41no—
29SD16×C17—41no—
30M16×C17—21no—
31C4⋊C4×C17—22no—
32C42×C17D411yescharacteristic
33#63231—42no—
34#63229—42no—
35C4⋊Dic17—21no—
36#63225—22no—
37#63225—22no—
38#63225—22no—
39C4×Dic17—22no—
40C4×Dic17—21no—
41C4×Dic17D412yes—
42#63223—22no—
43Q8×C17—21no—
44Q8×C17—22no—
45Q8×C17—41no—
46Q8×C17—41no—
47D4×C17—21no—
48D4×C17—22no—
49D4×C17—41no—
50C2×C4×C17—21no—
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—
58Dic34—24no—
59Dic34—42no—
60Dic34—81no—
61C8×C17—41no—
62#3622—42no—
63#2205—171nomaximal, Sylow
64C22×C17—41no—
65C22×C17#12011yescharacteristic
66C4×C17—22no—
67C4×C17—22no—
68C4×C17—21no—
69C4×C17—41no—
70C4×C17—41no—
71Dic17—24no—
72Dic17—42no—
73Q82—171no—
74#456—172no—
75#456D1711yescharacteristic
76#356—171no—
77#356—172no—
78C2×C17—21no—
79C2×C17—81no—
80C2×C17B(4,𝔽2)11yescharacteristic
812−1+4C2×D1711yescharacteristic
82#137—21no—
83#137—342no—
84C4⋊Q8—172no—
85C4⋊Q8—342no—
86C4⋊Q8—341no—
87C4⋊Q8C2×D1711yescharacteristic
88C4×Q8—342no—
89C4×Q8—341no—
90#104—21no—
91#104—342no—
92#101—172no—
93#101C2×D1711yescharacteristic
94C17#220511yescharacteristic, complemented, Sylow
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—172no—
100C2×Q8—341no—
101C2×Q8—341no—
102C2×Q8C22⋊D1711yescharacteristic
103C2×Q8C22×D1711yescharacteristic
104Q16—41no—
105Q16—682no—
106SD16—41no—
107SD16—682no—
108M16—21no—
109M16—342no—
110C4⋊C4—22no—
111C4⋊C4—342no—
112C4⋊C4—341no—
113C4⋊C4—342no—
114C4⋊C4—342no—
115C42—172no—
116C42—342no—
117C42—341no—
118C42C22⋊D1711yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—344no—
124Q8—682no—
125Q8—1361no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—342no—
134C2×C4—342no—
135C2×C4D4×D1712yes—
136C2×C4C2×C22⋊D1711yescharacteristic
137C8—41no—
138C8—682no—
139C22—41no—
140C22#1058762311yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—344no—
147C4—682no—
148C2—21no—
149C2—81no—
150C2#1176299511yescharacteristic
subgroupG/H
centerC2#11762995
commutator, cosocleC2×Q8×C17C23
FrattiniC2×Q8C22×D17
FittingC17×#456C2
socleC2×C17B(4,𝔽2)
prime radicalC17×2−1+4C22
hypercenter#456D17
hypocenterC17#2205
LCSG ▷ C2×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #11763539C2semidirect
3. #11762608C2semidirect
4. #11762607C2semidirect
94. C17#2205semidirect

Maximal subgroups

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

subgroupindex
#117635392
#117629922
#117626082
#117626072
C17×#4562
#220517