Finite Group Databasebeta

← #14544276

Subgroups — #14544276

Order 2176.

194 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 56 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#11763832C211yesmaximal, characteristic
2#11763832C211yesmaximal, characteristic
3#11763666C214yesmaximal, complemented
4#11763364C211yesmaximal, characteristic
5C17×C4⋊Q8C411yescharacteristic
6#10587632—21no—
7Q8×Dic17—21no—
8#10587583—21no—
9#10587578C412yes—
10#10587576C2212yes—
11#10587573—21no—
12C42×D17C2211yescharacteristic
13#10587555C411yescharacteristic
14#10587544—21no—
15#10587544—21no—
16#10587543—21no—
17#10587543—21no—
18#10587542C2212yes—
19#10587542C2212yes—
20C2×Q8×C17—21no—
21C2×C4×D17C2×C411yescharacteristic
22C2×C4×D17D411yescharacteristic
23C2×C4×D17D411yescharacteristic
24C2×Dic34—21no—
25C2×#3624—22no—
26C2×#3624—22no—
27C4⋊C4×C17—21no—
28C4⋊C4×C17C2×C412yes—
29C42×C17C2×C411yescharacteristic
30#63227—22no—
31C4⋊Dic17—21no—
32C4⋊Dic17—22no—
33C4⋊Dic17C2×C412yes—
34#63225—22no—
35C4×Dic17—22no—
36C4×Dic17D411yescharacteristic
37C4×Dic17D411yescharacteristic
38C4×Dic17C2311yescharacteristic
39C22×D17C22⋊C411yescharacteristic
40Q8×C17—42no—
41C2×C4×C17—22no—
42C2×C4×C17C22⋊C411yescharacteristic
43C2×C4×C17C22⋊C411yescharacteristic
44C2×C4×C17C22×C411yescharacteristic
45C2×Dic17—22no—
46C2×Dic17C22⋊C411yescharacteristic
47C2×Dic17C2×D411yescharacteristic
48C2×Dic17C2×D411yescharacteristic
49C4×D17—22no—
50C4×D17—41no—
51C4×D17—41no—
52Dic34—42no—
53#3624—42no—
54#3624—42no—
55#1537—171nomaximal, Sylow
56C22×C17C2×C22⋊C411yescharacteristic
57C2×D17—21no—
58C2×D17—21no—
59C2×D17—21no—
60C4×C17—21no—
61C4×C17—21no—
62C4×C17—21no—
63C4×C17—42no—
64Dic17—21no—
65Dic17—21no—
66Dic17—42no—
67Dic17#10112yes—
68#533—171no—
69#432—171no—
70#432—171no—
71#330—174no—
72C2×C17C2×#10111yescharacteristic
73C2×C17#42111yescharacteristic
74C2×C17#42111yescharacteristic
75D17—41no—
76C4⋊Q8—171no—
77C4⋊Q8#59511yescharacteristic
78#125—172no—
79C22⋊Q8—341no—
80C22⋊Q8—341no—
81C4×Q8—341no—
82C4×Q8—341no—
83#117—172no—
84C2×C42—171no—
85C4⋊C8—341no—
86C4⋊C8—341no—
87C22⋊C8—341no—
88C22⋊C8—341no—
89#97—172no—
90#97—172no—
91C17#153711yescharacteristic, complemented, Sylow
92C2×Q8—21no—
93C2×Q8—341no—
94C22×C4—171no—
95C22×C4—171no—
96C22×C4—171no—
97C2×C8—342no—
98C2×C8—342no—
99C4⋊C4—21no—
100C4⋊C4—172no—
101C4⋊C4—342no—
102C4⋊C4—342no—
103C4⋊C4—341no—
104C4⋊C4C2×#59512yes—
105C22⋊C4—342no—
106C42—171no—
107C42—171no—
108C42—171no—
109C42—342no—
110C42C2×#59511yescharacteristic
111C23—171no—
112Q8—42no—
113Q8—682no—
114C2×C4—22no—
115C2×C4—171no—
116C2×C4—171no—
117C2×C4—171no—
118C2×C4—342no—
119C2×C4—342no—
120C2×C4—681no—
121C2×C4—681no—
122C2×C4C22⋊#59511yescharacteristic
123C2×C4C22⋊#59511yescharacteristic
124C2×C4C22×#59511yescharacteristic
125C8—682no—
126C8—682no—
127C22—341no—
128C22—341no—
129C22—341no—
130C22C2×C22⋊#59511yescharacteristic
131C4—21no—
132C4—21no—
133C4—21no—
134C4—42no—
135C4—172no—
136C4—341no—
137C4—341no—
138C4—682no—
139C2—681no—
140C2D17⋊#10111yescharacteristic
141C2#1176380711yescharacteristic
142C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C4⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C22⋊C4
cosocleC4×Dic17C23
hypercenterC4⋊Q8#595
hypocenterC17#1537
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #11763666C2semidirect
91. C17#1537semidirect

Maximal subgroups

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

subgroupindex
#117638322
#117636662
#117633642
#153717