Finite Group Databasebeta

← #14527809

Subgroups — #14527809

Order 2176.

236 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 57 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#11763142C212yesmaximal, complemented
2#11762745C211yesmaximal, characteristic
3#11762546C212yesmaximal, complemented
4#11762533C211yesmaximal, characteristic
5C17×#543C211yesmaximal, characteristic
6C17×C22⋊Q8—22no—
7C17×C22⋊Q8—21no—
8C17×C22⋊Q8—22no—
9C17×C22⋊Q8C2212yes—
10C2×C22⋊C4×C17—21no—
11C2×C22⋊C4×C17C2211yescharacteristic
12C2×C22⋊Dic17—21no—
13C2×C4⋊Dic17C2211yescharacteristic
14C2×#63225—21no—
15#10587506—22no—
16#10587502C2211yescharacteristic
17C22⋊#3622C2212yes—
18#10587478—22no—
19C24×C17D411yescharacteristic
20C2×Q8×C17—21no—
21C2×Q8×C17D412yes—
22C22×C4×C17—21no—
23C22×C4×C17—21no—
24C22×C4×C17D411yescharacteristic
25C22×C4×C17C2311yescharacteristic
26C22×Dic17—21no—
27C22×Dic17—21no—
28C4⋊C4×C17—21no—
29C4⋊C4×C17—22no—
30C4⋊C4×C17—22no—
31C4⋊C4×C17D412yes—
32C22⋊C4×C17—22no—
33C22⋊C4×C17—22no—
34C22⋊C4×C17—22no—
35C22⋊Dic17—42no—
36C4⋊Dic17—22no—
37#63225—42no—
38C2×#3622—22no—
39C23×C17—21no—
40C23×C17—21no—
41C23×C17—41no—
42C23×C17—41no—
43C23×C17Q1612yes—
44C23×C17C2×D411yescharacteristic
45Q8×C17—42no—
46Q8×C17—81no—
47C2×C4×C17—22no—
48C2×C4×C17—21no—
49C2×C4×C17—22no—
50C2×C4×C17—41no—
51C2×C4×C17—41no—
52C2×C4×C17—81no—
53C2×C4×C17—81no—
54C2×C4×C17C2×D412yes—
55C2×Dic17—22no—
56C2×Dic17—41no—
57C2×Dic17—41no—
58C2×Dic17—41no—
59C2×Dic17—41no—
60C2×Dic17—41no—
61#3622—42no—
62#1614—171nomaximal, Sylow
63C22×C17—21no—
64C22×C17—21no—
65C22×C17—22no—
66C22×C17—42no—
67C22×C17—41no—
68C22×C17—41no—
69C22×C17—81no—
70C22×C17#12011yescharacteristic
71C22×C17C2×Q1611yescharacteristic
72C22×C17#13711yescharacteristic
73C4×C17—22no—
74C4×C17—41no—
75C4×C17—42no—
76C4×C17—42no—
77Dic17—42no—
78Dic17—81no—
79#543D1711yescharacteristic
80#484—172no—
81#399—171no—
82#328—172no—
83#323—171no—
84C2×C17—21no—
85C2×C17—21no—
86C2×C17—42no—
87C2×C17#45111yescharacteristic
88C2×C17#45311yescharacteristic
89C2×C17#45811yescharacteristic
90C22⋊Q8—22no—
91C22⋊Q8—21no—
92C22⋊Q8—22no—
93C22⋊Q8C2×D1712yes—
94C2×C4⋊C4—171no—
95C2×C4⋊C4—341no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—341no—
98C2×C22⋊C4C2×D1711yescharacteristic
99#107—342no—
100#103—342no—
101C22⋊C8—172no—
102#95—171no—
103C17#161411yescharacteristic, complemented, Sylow
104C24C22⋊D1711yescharacteristic
105C2×Q8—21no—
106C2×Q8C22⋊D1712yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—341no—
110C22×C4—341no—
111C22×C4C22⋊D1711yescharacteristic
112C22×C4C22×D1711yescharacteristic
113C2×C8—342no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—342no—
118C4⋊C4—682no—
119C4⋊C4C22⋊D1712yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—682no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23#6323112yes—
129C23C2×C22⋊D1711yescharacteristic
130Q8—42no—
131Q8—81no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—342no—
140C2×C4—681no—
141C2×C4—681no—
142C2×C4—681no—
143C2×C4—681no—
144C2×C4—681no—
145C2×C4C2×C22⋊D1712yes—
146C8—682no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1058762811yescharacteristic
155C22C2×#6323111yescharacteristic
156C22#1058763911yescharacteristic
157C4—22no—
158C4—41no—
159C4—42no—
160C4—42no—
161C4—682no—
162C4—1361no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#1176296511yescharacteristic
167C2#1176297711yescharacteristic
168C2#1176299911yescharacteristic
subgroupG/H
centerC22#10587639
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#543C2
socleC22×C17#120
prime radicalC24×C17D4
hypercenter#543D17
hypocenterC17#1614
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #11763142C2semidirect
3. #11762546C2semidirect
103. C17#1614semidirect

Maximal subgroups

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

subgroupindex
#117631422
#117627452
#117625462
#117625332
C17×#5432
#161417