Finite Group Databasebeta

← #14529949

Subgroups — #14529949

Order 2176.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#11762664C211yesmaximal, characteristic
3#11762664C212yesmaximal
4#11762575C212yesmaximal, complemented
5C17×#342C211yesmaximal, characteristic, complemented
6C24×D17C2211yescharacteristic
7C2×C22⋊C4×C17C2212yes—
8C2×C22⋊C4×C17C2211yescharacteristic
9C2×C22⋊Dic17C2211yescharacteristic
10C2×C22⋊Dic17C2212yes—
11C2×#63227—22no—
12C2×#63227—24no—
13#10587610—22no—
14#10587564—24no—
15C24×C17C2311yescharacteristic
16C23×D17—21no—
17C23×D17—22no—
18C23×D17—22no—
19C23×D17—21no—
20C23×D17—44no—
21C23×D17Q812yescomplemented
22C22×C4×C17—22no—
23C22×C4×C17D412yes—
24C22×Dic17—21no—
25C22×Dic17D414yes—
26C2×Dic34—22no—
27C22⋊C4×C17—22no—
28C22⋊C4×C17—22no—
29C22⋊C4×C17—22no—
30C22⋊Dic17—22no—
31C22⋊Dic17—22no—
32C22⋊Dic17—22no—
33#63227—48no—
34#63227—44no—
35C4⋊Dic17—22no—
36#63225—24no—
37C23×C17—42no—
38C23×C17C2×D411yescharacteristic
39C23×C17C2×D412yes—
40C23×C17C2×Q811yescharacteristic
41C23×C17#4112yes—
42C23×C17#4111yescharacteristic
43C22×D17—24no—
44C22×D17—28no—
45C22×D17—24no—
46C22×D17—22no—
47C22×D17—44no—
48C22×D17—48no—
49C22×D17—44no—
50C22×D17—42no—
51C22×D17—44no—
52C22×D17—41no—
53C22×D17—41no—
54C22×D17—42no—
55C2×C4×C17—22no—
56C2×C4×C17—42no—
57C2×C4×C17—44no—
58C2×C4×C17—42no—
59C2×Dic17—24no—
60C2×Dic17—44no—
61C2×Dic17—42no—
62C2×Dic17—41no—
63Dic34—82no—
64#2029—171nomaximal, Sylow
65C22×C17—22no—
66C22×C17—24no—
67C22×C17—41no—
68C22×C17—42no—
69C22×C17—41no—
70C22×C17#12011yescharacteristic
71C22×C17C22⋊Q812yes—
72C22×C17C22⋊Q811yescharacteristic
73C22×C17#12411yescharacteristic
74C22×C17#12412yes—
75C2×D17—28no—
76C2×D17—24no—
77C2×D17—44no—
78C2×D17—48no—
79C2×D17—44no—
80C2×D17—42no—
81C2×D17—42no—
82C2×D17—44no—
83C4×C17—42no—
84C4×C17—82no—
85Dic17—44no—
86Dic17—81no—
87#543—171no—
88#379—172no—
89#379—171no—
90#342—172no—
91#342D1711yescharacteristic, complemented
92C2×C17—21no—
93C2×C17—22no—
94C2×C17—22no—
95C2×C17—21no—
96C2×C17#39311yescharacteristic
97C2×C17B(4,𝔽2)12yes—
98D17—44no—
99C25—171no—
100C22⋊Q8—344no—
101C22⋊Q8—342no—
102C2×C22⋊C4—172no—
103C2×C22⋊C4—171no—
104C2×C22⋊C4—344no—
105C2×C22⋊C4—342no—
106C2×C22⋊C4C2×D1712yes—
107C2×C22⋊C4C2×D1711yescharacteristic
108C17#202911yescharacteristic, complemented, Sylow
109C24—172no—
110C24—341no—
111C24—342no—
112C24—342no—
113C24—341no—
114C24—684no—
115C24C22×D1711yescharacteristic
116C2×Q8—342no—
117C22×C4—22no—
118C22×C4—174no—
119C22×C4—341no—
120C22×C4D6812yes—
121C4⋊C4—344no—
122C4⋊C4—342no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—342no—
127C22⋊C4—342no—
128C22⋊C4—342no—
129C22⋊C4—684no—
130C22⋊C4—688no—
131C23—42no—
132C23—344no—
133C23—348no—
134C23—344no—
135C23—342no—
136C23—682no—
137C23—684no—
138C23—684no—
139C23—688no—
140C23—684no—
141C23—681no—
142C23—681no—
143C23—682no—
144C23C2×D6811yescharacteristic
145C23#6325212yes—
146C23D4×D1712yes—
147C23#6325411yescharacteristic
148C23Q8×D1711yescharacteristic
149Q8—1362no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—344no—
155C2×C4—684no—
156C2×C4—682no—
157C2×C4—681no—
158C22—22no—
159C22—24no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22—348no—
164C22—344no—
165C22—688no—
166C22—684no—
167C22—684no—
168C22—682no—
169C22—682no—
170C22—684no—
171C22#1058756111yescharacteristic
172C22#1058756811yescharacteristic
173C22#1058757112yes—
174C22#1058758212yes—
175C22#1058758311yescharacteristic
176C4—42no—
177C4—82no—
178C4—684no—
179C4—1361no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—684no—
185C2#1176272211yescharacteristic
186C2#1176299712yes—
subgroupG/H
centerC22#10587568
commutator, cosocleC24×C17C23
FrattiniC24C22×D17
FittingC17×#342C2
socleC22×C17#120
prime radicalC24×D17C22
hypercenter#342D17
hypocenterC17#2029
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #11763435C2semidirect
4. #11762575C2semidirect
5. C17×#342C2semidirect
21. C23×D17Q8semidirect
91. #342D17semidirect
108. C17#2029semidirect

Maximal subgroups

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

subgroupindex
#117634352
#117626642
#117625752
C17×#3422
#202917