Finite Group Databasebeta

← #14544166

Subgroups — #14544166

Order 2176.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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#11763827C211yesmaximal, characteristic
2C4×C4⋊#595C211yesmaximal, characteristic
3#11763663C212yesmaximal
4#11763662C212yesmaximal
5#11763363C211yesmaximal, characteristic
6C2×C4⋊#595—21no—
7C2×C4×#595—21no—
8C4×Q8×C17C411yescharacteristic, complemented
9#10587632C2211yescharacteristic
10#10587630C411yescharacteristic, complemented
11#10587583C2211yescharacteristic
12#10587582—21no—
13#10587575—21no—
14#10587574C411yescharacteristic, complemented
15#10587558—21no—
16C42×D17C2211yescharacteristic
17C4×Dic34—21no—
18C4×Dic34C411yescharacteristic, complemented
19(C2×#595)⋊C4C2212yes—
20#10587544—21no—
21#10587543C2212yes—
22C4×#3624—21no—
23C22×#595—22no—
24C2×Q8×C17C2×C411yescharacteristic
25C2×C4×D17D411yescharacteristic
26C2×C4×D17D411yescharacteristic
27C2×C4×D17C2311yescharacteristic
28C2×Dic34—21no—
29C2×Dic34C2×C411yescharacteristic
30C2×#3624—22no—
31C4⋊#595—24no—
32C4×#595—42no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17C2×C411yescharacteristic
35C42×C17—21no—
36C42×C17C2×C411yescharacteristic
37#63227—21no—
38#63227—21no—
39C4⋊Dic17—21no—
40C4⋊Dic17C2×C411yescharacteristic
41#63225—21no—
42#63225—21no—
43#63225—21no—
44#63225—21no—
45#63225—21no—
46C4×Dic17—21no—
47C4×Dic17C2×C411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17D411yescharacteristic
50C22×D17C2×D411yescharacteristic
51C2×#595—24no—
52C2×#595—42no—
53C2×#595—42no—
54Q8×C17—41no—
55C2×C4×C17—21no—
56C2×C4×C17—21no—
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22⋊C411yescharacteristic
59C2×C4×C17C22×C411yescharacteristic
60C2×Dic17—21no—
61C2×Dic17—21no—
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C22⋊C411yescharacteristic
64C2×Dic17C2×D411yescharacteristic
65C4×D17—22no—
66C4×D17—41no—
67C4×D17SD1612yes—
68C4×D17Q1612yes—
69Dic34—42no—
70Dic34—41no—
71#3624—42no—
72#1503—171nomaximal, Sylow
73C22×C17C2×C22⋊C411yescharacteristic
74C2×D17—21no—
75C2×D17—21no—
76C2×D17C2×SD1611yescharacteristic
77C2×D17C2×Q1611yescharacteristic
78#595—44no—
79C4×C17—21no—
80C4×C17—21no—
81C4×C17—41no—
82C4×C17—41no—
83C4×C17#10312yes—
84Dic17—21no—
85Dic17—41no—
86Dic17—41no—
87Dic17#10312yes—
88Dic17#10412yes—
89#533—171no—
90#431—171no—
91C4×C4⋊C4—171no—
92#328—172no—
93#328—172no—
94C2×C17#41011yescharacteristic
95C2×C17C2×#10311yescharacteristic
96C2×C17C2×#10411yescharacteristic
97D17—21no—
98D17—21no—
99C4⋊Q8—171no—
100C4⋊Q8—171no—
101#125—341no—
102C22⋊Q8—171no—
103C22⋊Q8—171no—
104C22⋊Q8—341no—
105C4×Q8—171no—
106C4×Q8—341no—
107C4×Q8#59511yescharacteristic, complemented
108#117—341no—
109C2×C4⋊C4—341no—
110C2×C42—171no—
111C2×C42—341no—
112C4⋊C8—341no—
113C22⋊C8—172no—
114C4×C8—341no—
115#95—172no—
116C17#150311yescharacteristic, complemented, Sylow
117C2×Q8—171no—
118C2×Q8—341no—
119C2×Q8C2×#59511yescharacteristic
120C22×C4—171no—
121C22×C4—171no—
122C22×C4—171no—
123C22×C4—342no—
124C2×C8—342no—
125C4⋊C4—21no—
126C4⋊C4—171no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4—341no—
130C4⋊C4—341no—
131C4⋊C4—341no—
132C4⋊C4—344no—
133C4⋊C4—341no—
134C4⋊C4C2×#59511yescharacteristic
135C22⋊C4—341no—
136C22⋊C4—341no—
137C42—21no—
138C42—171no—
139C42—171no—
140C42—171no—
141C42—341no—
142C42—682no—
143C42C2×#59511yescharacteristic
144C23—171no—
145Q8—41no—
146Q8—681no—
147Q8—682no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—171no—
151C2×C4—171no—
152C2×C4—172no—
153C2×C4—172no—
154C2×C4—171no—
155C2×C4—342no—
156C2×C4—344no—
157C2×C4—341no—
158C2×C4—341no—
159C2×C4—681no—
160C2×C4—682no—
161C2×C4—682no—
162C2×C4C22⋊#59511yescharacteristic
163C2×C4C22⋊#59511yescharacteristic
164C2×C4C22×#59511yescharacteristic
165C8—682no—
166C22—171no—
167C22—171no—
168C22—341no—
169C22—341no—
170C22C2×C22⋊#59511yescharacteristic
171C4—21no—
172C4—21no—
173C4—41no—
174C4—41no—
175C4—172no—
176C4—172no—
177C4—341no—
178C4—684no—
179C4—681no—
180C4—681no—
181C4#1058754912yes—
182C2—341no—
183C2—341no—
184C2#1176378811yescharacteristic
185C2C2×#1058754911yescharacteristic
186C2D17⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C2×D4
cosocleC2×C4×D17C23
hypercenterC4×Q8#595
hypocenterC17#1503
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
8. C4×Q8×C17C4semidirect
10. #10587630C4semidirect
14. #10587574C4semidirect
18. C4×Dic34C4semidirect
107. C4×Q8#595semidirect
116. C17#1503semidirect

Maximal subgroups

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

subgroupindex
#117638272
C4×C4⋊#5952
#117636632
#117636622
#117633632
#150317