Finite Group Databasebeta

← #14544268

Subgroups — #14544268

Order 2176.

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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, complemented
2#11763831C211yesmaximal, characteristic, complemented
3#11763666C212yesmaximal, complemented
4#11763664C212yesmaximal, complemented
5#11763341C211yesmaximal, characteristic
6C4×D4×C17C411yescharacteristic
7#10587624C411yescharacteristic
8#10587622C2211yescharacteristic
9C4×C22⋊D17—21no—
10#10587612—21no—
11#10587583C411yescharacteristic
12#10587576C2211yescharacteristic
13#10587569—21no—
14#10587565—21no—
15C42×D17C2211yescharacteristic
16C4×Dic34C411yescharacteristic
17#10587544—21no—
18#10587543—21no—
19#10587543—21no—
20#10587542C2212yes—
21#10587542C2212yes—
22C4⋊#3624—21no—
23C2×D4×C17C2×C411yescharacteristic
24C22×C4×C17—21no—
25C2×C22⋊D17—21no—
26C2×C4×D17C2×C411yescharacteristic
27C2×C4×D17D411yescharacteristic
28C2×C4×D17D411yescharacteristic
29C2×Dic34C2×C411yescharacteristic
30C2×#3624—22no—
31C2×#3624—22no—
32C4⋊C4×C17C2×C411yescharacteristic
33C22⋊C4×C17—21no—
34C42×C17C2×C411yescharacteristic
35C22⋊Dic17—21no—
36C22⋊Dic17—21no—
37C22⋊Dic17—21no—
38#63227—21no—
39C4⋊Dic17—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17C2×C411yescharacteristic
42#63225—21no—
43#63225—21no—
44C4×Dic17—21no—
45C4×Dic17D411yescharacteristic
46C4×Dic17D411yescharacteristic
47C4×Dic17C2311yescharacteristic
48C23×C17—21no—
49C22×D17C22⋊C411yescharacteristic
50D4×C17—41no—
51C2×C4×C17—21no—
52C2×C4×C17—42no—
53C2×C4×C17C22⋊C411yescharacteristic
54C2×C4×C17C22⋊C411yescharacteristic
55C2×C4×C17C22×C411yescharacteristic
56C22⋊D17—42no—
57C2×Dic17—21no—
58C2×Dic17—21no—
59C2×Dic17C22⋊C411yescharacteristic
60C2×Dic17C2×D411yescharacteristic
61C2×Dic17C2×D411yescharacteristic
62C4×D17—41no—
63C4×D17—41no—
64C4×D17—41no—
65Dic34—41no—
66#3624—42no—
67#3624—42no—
68#1535—171nomaximal, Sylow
69C22×C17—41no—
70C22×C17—41no—
71C22×C17—41no—
72C22×C17C2×C22⋊C411yescharacteristic
73C2×D17—21no—
74C2×D17—21no—
75C2×D17—21no—
76C4×C17—21no—
77C4×C17—21no—
78C4×C17—21no—
79C4×C17—41no—
80Dic17—21no—
81Dic17—21no—
82Dic17—21no—
83Dic17—41no—
84Dic17—41no—
85#529—171no—
86#432—171no—
87#432—171no—
88#330—172no—
89#329—172no—
90C2×C17—81no—
91C2×C17#41311yescharacteristic
92C2×C17#42111yescharacteristic
93C2×C17#42111yescharacteristic
94D17—41no—
95#126—341no—
96#125—171no—
97#124—171no—
98#123—341no—
99C22⋊Q8—171no—
100#121—171no—
101C4×Q8—171no—
102C4×D4—341no—
103C4×D4#59511yescharacteristic
104#117—341no—
105C2×C42—171no—
106C4⋊C8—341no—
107C4⋊C8—341no—
108C22⋊C8—341no—
109C22⋊C8—341no—
110#97—172no—
111#97—172no—
112C17#153511yescharacteristic, complemented, Sylow
113C2×Q8—171no—
114C2×D4—341no—
115C2×D4C2×#59511yescharacteristic
116C22×C4—21no—
117C22×C4—171no—
118C22×C4—171no—
119C22×C4—171no—
120C2×C8—342no—
121C2×C8—342no—
122C4⋊C4—171no—
123C4⋊C4—341no—
124C4⋊C4—341no—
125C4⋊C4—341no—
126C4⋊C4—341no—
127C4⋊C4C2×#59511yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—341no—
130C22⋊C4—341no—
131C22⋊C4—341no—
132C22⋊C4—341no—
133C42—171no—
134C42—171no—
135C42—171no—
136C42—341no—
137C42C2×#59511yescharacteristic
138C23—21no—
139C23—171no—
140Q8—681no—
141D4—41no—
142D4—682no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—171no—
146C2×C4—171no—
147C2×C4—171no—
148C2×C4—341no—
149C2×C4—341no—
150C2×C4—681no—
151C2×C4—681no—
152C2×C4—681no—
153C2×C4C22⋊#59511yescharacteristic
154C2×C4C22⋊#59511yescharacteristic
155C2×C4C22×#59511yescharacteristic
156C8—682no—
157C8—682no—
158C22—41no—
159C22—41no—
160C22—41no—
161C22—341no—
162C22—341no—
163C22—341no—
164C22C2×C22⋊#59511yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—41no—
169C4—341no—
170C4—341no—
171C4—341no—
172C4—681no—
173C4—681no—
174C2—81no—
175C2—681no—
176C2#1176379511yescharacteristic
177C2#1176380611yescharacteristic
178C2#1176380711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radical#10587624C4
cosocleC4×Dic17C23
hypercenterC4×D4#595
hypocenterC17#1535
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763832C2semidirect
2. #11763831C2semidirect
3. #11763666C2semidirect
4. #11763664C2semidirect
112. C17#1535semidirect

Maximal subgroups

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

subgroupindex
#117638322
#117638312
#117636662
#117636642
#117633412
#153517