Finite Group Databasebeta

← #14544070

Subgroups — #14544070

Order 2176.

344 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 94 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#11763810C211yesmaximal, characteristic, complemented
2C4×#63242C211yesmaximal, characteristic, complemented
3#11763657C212yesmaximal, complemented
4D4⋊#3624C212yesmaximal, complemented
5C4×D4×D17C211yesmaximal, characteristic
6C2×D4×D17C411yescharacteristic
7C22×C4×D17—21no—
8C2×#63242—21no—
9C2×#63242—21no—
10C4×D4×C17C411yescharacteristic
11D4×Dic17C2211yescharacteristic
12C4×C22⋊D17—21no—
13#10587579C2211yescharacteristic
14C4⋊C4×D17C411yescharacteristic
15#10587567—21no—
16C22⋊C4×D17—21no—
17C4×D68C411yescharacteristic
18C42×D17C2211yescharacteristic
19#10587544—21no—
20C4⋊#3624C2212yescomplemented
21C4×#3624—21no—
22C4×#3624C2212yescomplemented
23C23×D17—21no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17—21no—
26C2×C22⋊D17—21no—
27C22×Dic17—21no—
28D4×D17—22no—
29D4×D17C814yescomplemented
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#63242—24no—
42#63242—42no—
43C4⋊C4×C17C2×C411yescharacteristic
44C22⋊C4×C17—21no—
45C42×C17C2×C411yescharacteristic
46C22⋊Dic17—21no—
47#63227—21no—
48#63227—21no—
49C4⋊Dic17C2×C411yescharacteristic
50#63225—21no—
51C4×Dic17D411yescharacteristic
52C4×Dic17D411yescharacteristic
53C4×Dic17C2311yescharacteristic
54C23×C17—21no—
55C22×D17—21no—
56C22×D17—24no—
57C22×D17—42no—
58C22×D17—42no—
59C22×D17C22⋊C411yescharacteristic
60D4×C17—21no—
61D4×C17C2×C812yescomplemented
62C2×C4×C17—21no—
63C2×C4×C17—42no—
64C2×C4×C17C22⋊C411yescharacteristic
65C2×C4×C17C22⋊C411yescharacteristic
66C2×C4×C17C22×C411yescharacteristic
67C22⋊D17—24no—
68C2×Dic17—21no—
69C2×Dic17—41no—
70C2×Dic17—41no—
71C2×Dic17C22⋊C411yescharacteristic
72C2×Dic17C2×D411yescharacteristic
73C2×Dic17C2×D411yescharacteristic
74D68—21no—
75D68C2×C812yescomplemented
76C4×D17—22no—
77C4×D17—42no—
78C4×D17—42no—
79C4×D17—41no—
80C4×D17—41no—
81C4×D17C2×C812yes—
82C4×D17M1612yes—
83C4×D17D812yes—
84C4×D17SD1612yes—
85#3624—24no—
86#3624—42no—
87C2×D4⋊C8—171nomaximal, Sylow
88C22×C17—22no—
89C22×C17—41no—
90C22×C17—41no—
91C22×C17C2×C22⋊C411yescharacteristic
92C2×D17—22no—
93C2×D17—41no—
94C2×D17—41no—
95C2×D17—48no—
96C2×D17C22⋊C812yes—
97C2×D17#10212yes—
98C2×D17#10412yes—
99C4×C17—21no—
100C4×C17—41no—
101C4×C17#10212yes—
102C4×C17C22×C811yescharacteristic
103C4×C17C2×M1611yescharacteristic
104Dic17—21no—
105Dic17—41no—
106Dic17C22⋊C812yes—
107Dic17C2×D811yescharacteristic
108Dic17C2×SD1611yescharacteristic
109C2×C4×D4—171no—
110C2×C4⋊C8—171no—
111C2×C4×C8—171no—
112D4⋊C8—172no—
113D4⋊C8—172no—
114C2×C17—42no—
115C2×C17C2×C22⋊C811yescharacteristic
116C2×C17C2×#10211yescharacteristic
117C2×C17C2×#10411yescharacteristic
118D17—42no—
119D17D4⋊C814yescomplemented
120C22×D4—171no—
121C23×C4—341no—
122C22×C8—341no—
123C22×C8—341no—
124C4×D4—171no—
125C4×D4—171no—
126C4×D4—171no—
127C4×D4—341no—
128C4×D4—341no—
129C4×D4#59511yescharacteristic
130C2×C4⋊C4—171no—
131C2×C22⋊C4—341no—
132C2×C42—171no—
133C4⋊C8—172no—
134C4⋊C8—341no—
135C4×C8—172no—
136C4×C8—341no—
137C17C2×D4⋊C811yescharacteristic, complemented, Sylow
138C24—341no—
139C2×D4—174no—
140C2×D4—171no—
141C2×D4—342no—
142C2×D4—341no—
143C2×D4C2×#59511yescharacteristic
144C22×C4—21no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4—341no—
149C22×C4—341no—
150C22×C4—341no—
151C22×C4—341no—
152C22×C4—682no—
153C22×C4—682no—
154C2×C8—342no—
155C2×C8—344no—
156C2×C8—342no—
157C2×C8—682no—
158C4⋊C4—171no—
159C4⋊C4—341no—
160C4⋊C4C2×#59511yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—341no—
163C22⋊C4—341no—
164C22⋊C4—341no—
165C42—171no—
166C42—171no—
167C42—171no—
168C42C2×#59511yescharacteristic
169C23—21no—
170C23—171no—
171C23—344no—
172C23—341no—
173C23—682no—
174C23—682no—
175D4—21no—
176D4—172no—
177D4—344no—
178D4—341no—
179D4#6324212yescomplemented
180C2×C4—21no—
181C2×C4—42no—
182C2×C4—171no—
183C2×C4—171no—
184C2×C4—172no—
185C2×C4—172no—
186C2×C4—172no—
187C2×C4—172no—
188C2×C4—171no—
189C2×C4—342no—
190C2×C4—341no—
191C2×C4—681no—
192C2×C4—681no—
193C2×C4—682no—
194C2×C4—681no—
195C2×C4—681no—
196C2×C4—682no—
197C2×C4C22⋊#59511yescharacteristic
198C2×C4C22⋊#59511yescharacteristic
199C2×C4C22×#59511yescharacteristic
200C8—344no—
201C8—682no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22—172no—
206C22—172no—
207C22—172no—
208C22—342no—
209C22—688no—
210C22—681no—
211C22—681no—
212C22C2×C22⋊#59511yescharacteristic
213C4—21no—
214C4—41no—
215C4—172no—
216C4—171no—
217C4—171no—
218C4—341no—
219C4—681no—
220C4#1058754712yes—
221C4C2×#6324211yescharacteristic
222C4D17⋊M1611yescharacteristic
223C2—42no—
224C2—174no—
225C2—682no—
226C2#1176377711yescharacteristic
227C2C2×#1058754711yescharacteristic
228C2D17⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#595
commutatorC4×C17C22×C8
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×D4×D17C4
cosocleC4×Dic17C23
hypercenterC4×D4#595
hypocenterC17C2×D4⋊C8
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763810C2semidirect
2. C4×#63242C2semidirect
3. #11763657C2semidirect
4. D4⋊#3624C2semidirect
20. C4⋊#3624C22semidirect
22. C4×#3624C22semidirect
29. D4×D17C8semidirect
61. D4×C17C2×C8semidirect
75. D68C2×C8semidirect
119. D17D4⋊C8semidirect
137. C17C2×D4⋊C8semidirect
179. D4#63242semidirect

Maximal subgroups

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

subgroupindex
#117638102
C4×#632422
#117636572
D4⋊#36242
C4×D4×D172
C2×D4⋊C817