Finite Group Databasebeta

← #14544033

Subgroups — #14544033

Order 2176.

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

normal subgroups only

This group has 312 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#11763780C212yesmaximal
2#11763655C212yesmaximal, complemented
3#11763654C212yesmaximal, complemented
4C2×C4⋊C4×D17C211yesmaximal, characteristic
5C22×C4×D17C411yescharacteristic
6C22×C4×D17C411yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×#63243—22no—
9C2×C4⋊C4×C17C411yescharacteristic
10C2×C4⋊Dic17C2211yescharacteristic
11C2×#63225C411yescharacteristic
12C2×#63225C2211yescharacteristic
13C4⋊C4×D17—22no—
14C4⋊C4×D17—22no—
15C22⋊#3624C2214yes—
16#10587543—22no—
17C23×D17C2×C411yescharacteristic
18C22×C4×C17C2×C411yescharacteristic
19C22×C4×C17C2×C411yescharacteristic
20C22×C4×C17C2×C411yescharacteristic
21C22×Dic17C2×C411yescharacteristic
22C22×Dic17C2×C411yescharacteristic
23C22×Dic17C2311yescharacteristic
24C2×C4×D17—21no—
25C2×C4×D17—21no—
26C2×C4×D17—22no—
27C2×C4×D17—22no—
28C2×C4×D17—21no—
29C2×C4×D17—21no—
30C2×C4×D17—42no—
31C2×C4×D17—42no—
32C2×C4×D17D412yes—
33C2×C4×D17D412yes—
34C2×#3624—24no—
35#63243—44no—
36C4⋊C4×C17—22no—
37C4⋊Dic17—22no—
38#63225—22no—
39#63225—22no—
40C23×C17C22×C411yescharacteristic
41C22×D17—21no—
42C22×D17—21no—
43C22×D17—22no—
44C22×D17—22no—
45C22×D17C22⋊C411yescharacteristic
46C22×D17C22⋊C411yescharacteristic
47C2×C4×C17—21no—
48C2×C4×C17—21no—
49C2×C4×C17—21no—
50C2×C4×C17—41no—
51C2×C4×C17—41no—
52C2×C4×C17C22⋊C412yes—
53C2×C4×C17M1612yescomplemented
54C2×Dic17—21no—
55C2×Dic17—21no—
56C2×Dic17—21no—
57C2×Dic17—41no—
58C2×Dic17—41no—
59C2×Dic17M1612yescomplemented
60C2×Dic17C2×D412yes—
61C4×D17—24no—
62C4×D17—24no—
63C4×D17—24no—
64C4×D17—42no—
65C4×D17—42no—
66C4×D17—42no—
67#3624—44no—
68#1463—171nomaximal, Sylow
69C22×C17—21no—
70C22×C17—21no—
71C22×C17C2×C22⋊C411yescharacteristic
72C22×C17C2×M1611yescharacteristic
73C22×C17C2×M1611yescharacteristic
74C2×D17—24no—
75C2×D17—24no—
76C2×D17—21no—
77C2×D17—21no—
78C2×D17—21no—
79C2×D17—21no—
80C2×D17#9912yes—
81C2×D17#10112yes—
82C4×C17—22no—
83C4×C17—22no—
84C4×C17—41no—
85Dic17—22no—
86Dic17—22no—
87Dic17—41no—
88C22×C4⋊C4—171no—
89#407—172no—
90#324—172no—
91#324—172no—
92C2×C17—21no—
93C2×C17—21no—
94C2×C17#40711yescharacteristic
95C2×C17C2×#9911yescharacteristic
96C2×C17C2×#10111yescharacteristic
97D17—22no—
98D17—22no—
99C23×C4—171no—
100C23×C4—171no—
101C23×C4—171no—
102C2×M16—342no—
103C2×C4⋊C4—171no—
104C2×C4⋊C4—171no—
105C2×C4⋊C4—171no—
106C2×C4⋊C4—342no—
107C2×C4⋊C4—342no—
108C2×C4⋊C4#59511yescharacteristic
109C22⋊C8—174no—
110C22⋊C8—342no—
111C17#146311yescharacteristic, complemented, Sylow
112C24—171no—
113C22×C4—171no—
114C22×C4—171no—
115C22×C4—172no—
116C22×C4—172no—
117C22×C4—171no—
118C22×C4—342no—
119C22×C4—341no—
120C22×C4—341no—
121C22×C4—341no—
122C22×C4—341no—
123C22×C4—342no—
124C22×C4—682no—
125C22×C4—682no—
126C22×C4C2×#59511yescharacteristic
127C22×C4C2×#59511yescharacteristic
128C22×C4C2×#59511yescharacteristic
129M16—684no—
130C2×C8—344no—
131C4⋊C4—22no—
132C4⋊C4—342no—
133C4⋊C4—342no—
134C4⋊C4—342no—
135C23—171no—
136C23—171no—
137C23—342no—
138C23—342no—
139C23—341no—
140C23—341no—
141C23C22×#59511yescharacteristic
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—172no—
148C2×C4—172no—
149C2×C4—344no—
150C2×C4—344no—
151C2×C4—344no—
152C2×C4—341no—
153C2×C4—341no—
154C2×C4—341no—
155C2×C4—682no—
156C2×C4—682no—
157C2×C4—682no—
158C2×C4—681no—
159C2×C4—681no—
160C2×C4#6324312yescomplemented
161C2×C4C22⋊#59512yes—
162C8—684no—
163C22—21no—
164C22—21no—
165C22—172no—
166C22—172no—
167C22—344no—
168C22—344no—
169C22—341no—
170C22—341no—
171C22—341no—
172C22—341no—
173C22C2×#6324311yescharacteristic
174C22D17⋊M1611yescharacteristic
175C22C2×C22⋊#59511yescharacteristic
176C4—22no—
177C4—22no—
178C4—41no—
179C4—342no—
180C4—342no—
181C4—681no—
182C2—21no—
183C2—21no—
184C2—342no—
185C2—342no—
186C2#1176378011yescharacteristic
187C2D17⋊#9911yescharacteristic
188C2D17⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17C2×C4
cosocleC22×Dic17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1463
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11763655C2semidirect
3. #11763654C2semidirect
53. C2×C4×C17M16semidirect
59. C2×Dic17M16semidirect
111. C17#1463semidirect
160. C2×C4#63243semidirect

Maximal subgroups

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

subgroupindex
#117637802
#117636552
#117636542
C2×C4⋊C4×D172
#146317