Finite Group Databasebeta

← #14544110

Subgroups — #14544110

Order 2176.

312 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 76 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#11763828C211yesmaximal, characteristic, complemented
2#11763813C211yesmaximal, characteristic, complemented
3#11763657C212yesmaximal, complemented
4D4⋊#3624C212yesmaximal, complemented
5C4×D4×D17C211yesmaximal, characteristic
6C2×D4×D17C411yescharacteristic
7C22×C4×D17—21no—
8C2×#63243—21no—
9C4×D4×C17C411yescharacteristic
10D4×Dic17C2211yescharacteristic
11C4×C22⋊D17—21no—
12#10587579C2211yescharacteristic
13C4⋊C4×D17C411yescharacteristic
14#10587567—21no—
15C22⋊C4×D17—21no—
16C4×D68C411yescharacteristic
17C42×D17C2211yescharacteristic
18#10587543—21no—
19C4⋊#3624—21no—
20C4⋊#3624—21no—
21C4⋊#3624C2212yescomplemented
22C4×#3624C2212yescomplemented
23C23×D17—21no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17—21no—
26C2×C22⋊D17—21no—
27C22×Dic17—21no—
28D4×D17—22no—
29D4×D17—22no—
30C2×D68C2×C411yescharacteristic
31C2×C4×D17—21no—
32C2×C4×D17—21no—
33C2×C4×D17—21no—
34C2×C4×D17—42no—
35C2×C4×D17—42no—
36C2×C4×D17C2×C411yescharacteristic
37C2×C4×D17D411yescharacteristic
38C2×C4×D17D411yescharacteristic
39C2×#3624—22no—
40C2×#3624—22no—
41#63243—42no—
42C4⋊C4×C17C2×C411yescharacteristic
43C22⋊C4×C17—21no—
44C42×C17C2×C411yescharacteristic
45C22⋊Dic17—21no—
46#63227—21no—
47#63227—21no—
48C4⋊Dic17C2×C411yescharacteristic
49#63225—21no—
50C4×Dic17D411yescharacteristic
51C4×Dic17D411yescharacteristic
52C4×Dic17C2311yescharacteristic
53C23×C17—21no—
54C22×D17—24no—
55C22×D17—21no—
56C22×D17—42no—
57C22×D17—42no—
58C22×D17C22⋊C411yescharacteristic
59D4×C17—21no—
60D4×C17M1612yescomplemented
61C2×C4×C17—21no—
62C2×C4×C17—42no—
63C2×C4×C17C22⋊C411yescharacteristic
64C2×C4×C17C22⋊C411yescharacteristic
65C2×C4×C17C22×C411yescharacteristic
66C22⋊D17—24no—
67C2×Dic17—21no—
68C2×Dic17—41no—
69C2×Dic17—41no—
70C2×Dic17C22⋊C411yescharacteristic
71C2×Dic17C2×D411yescharacteristic
72C2×Dic17C2×D411yescharacteristic
73D68—21no—
74D68M1612yescomplemented
75C4×D17—21no—
76C4×D17—21no—
77C4×D17—41no—
78C4×D17—42no—
79C4×D17—42no—
80C4×D17—41no—
81C4×D17—41no—
82C4×D17D812yes—
83C4×D17SD1612yes—
84#3624—42no—
85#3624—42no—
86#1486—171nomaximal, Sylow
87C22×C17—22no—
88C22×C17—41no—
89C22×C17—41no—
90C22×C17C2×C22⋊C411yescharacteristic
91C2×D17—21no—
92C2×D17—21no—
93C2×D17—22no—
94C2×D17—48no—
95C2×D17—41no—
96C2×D17—41no—
97C2×D17#10212yes—
98C4×C17—21no—
99C4×C17—41no—
100C4×C17#10212yes—
101C4×C17C2×M1611yescharacteristic
102C4×C17C2×M1611yescharacteristic
103Dic17—21no—
104Dic17—21no—
105Dic17—41no—
106Dic17C2×D811yescharacteristic
107Dic17C2×SD1611yescharacteristic
108C2×C4×D4—171no—
109#431—171no—
110#423—171no—
111D4⋊C8—172no—
112D4⋊C8—172no—
113C2×C17—42no—
114C2×C17#40711yescharacteristic
115C2×C17C2×#10211yescharacteristic
116C2×C17#42111yescharacteristic
117D17—22no—
118D17—42no—
119C22×D4—171no—
120C23×C4—341no—
121C2×M16—341no—
122C4×D4—171no—
123C4×D4—171no—
124C4×D4—171no—
125C4×D4—341no—
126C4×D4—341no—
127C4×D4#59511yescharacteristic
128C2×C4⋊C4—171no—
129C2×C22⋊C4—341no—
130C2×C42—171no—
131C4⋊C8—172no—
132C4⋊C8—341no—
133C4⋊C8—341no—
134C22⋊C8—341no—
135C4×C8—172no—
136C17#148611yescharacteristic, complemented, Sylow
137C24—341no—
138C2×D4—171no—
139C2×D4—341no—
140C2×D4—342no—
141C2×D4—342no—
142C2×D4C2×#59511yescharacteristic
143C22×C4—21no—
144C22×C4—171no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—341no—
148C22×C4—341no—
149C22×C4—341no—
150C22×C4—341no—
151C22×C4—682no—
152C22×C4—682no—
153M16—682no—
154C2×C8—342no—
155C2×C8—342no—
156C4⋊C4—171no—
157C4⋊C4—341no—
158C4⋊C4C2×#59511yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—341no—
161C22⋊C4—341no—
162C22⋊C4—341no—
163C42—171no—
164C42—171no—
165C42—171no—
166C42C2×#59511yescharacteristic
167C23—21no—
168C23—171no—
169C23—344no—
170C23—341no—
171C23—682no—
172C23—682no—
173D4—21no—
174D4—172no—
175D4—344no—
176D4—341no—
177D4#6324312yescomplemented
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—171no—
181C2×C4—172no—
182C2×C4—172no—
183C2×C4—171no—
184C2×C4—171no—
185C2×C4—341no—
186C2×C4—341no—
187C2×C4—341no—
188C2×C4—681no—
189C2×C4—681no—
190C2×C4—681no—
191C2×C4—682no—
192C2×C4—681no—
193C2×C4—681no—
194C2×C4—682no—
195C2×C4C22⋊#59511yescharacteristic
196C2×C4C22⋊#59511yescharacteristic
197C2×C4C22×#59511yescharacteristic
198C8—682no—
199C8—682no—
200C22—22no—
201C22—41no—
202C22—41no—
203C22—172no—
204C22—341no—
205C22—341no—
206C22—342no—
207C22—688no—
208C22—681no—
209C22—681no—
210C22C2×C22⋊#59511yescharacteristic
211C4—21no—
212C4—41no—
213C4—171no—
214C4—171no—
215C4—341no—
216C4—341no—
217C4—681no—
218C4#1058754712yes—
219C4C2×#6324311yescharacteristic
220C4D17⋊M1611yescharacteristic
221C2—42no—
222C2—342no—
223C2—682no—
224C2#1176378011yescharacteristic
225C2C2×#1058754711yescharacteristic
226C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×D4×D17C4
cosocleC4×Dic17C23
hypercenterC4×D4#595
hypocenterC17#1486
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763828C2semidirect
2. #11763813C2semidirect
3. #11763657C2semidirect
4. D4⋊#3624C2semidirect
21. C4⋊#3624C22semidirect
22. C4×#3624C22semidirect
60. D4×C17M16semidirect
74. D68M16semidirect
136. C17#1486semidirect
177. D4#63243semidirect

Maximal subgroups

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

subgroupindex
#117638282
#117638132
#117636572
D4⋊#36242
C4×D4×D172
#148617