Finite Group Databasebeta

← #14544556

Subgroups — #14544556

Order 2176.

324 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#10587549C212yesmaximal
2#11763768C211yesmaximal, characteristic
3C4×C4⋊#595C211yesmaximal, characteristic
4#11763682C212yesmaximal
5C4×Q8×D17C211yesmaximal, characteristic
6C2×Q8×D17C2211yescharacteristic
7C2×C4⋊#595C2212yes—
8C2×C4×#595—21no—
9C2×#63242C2212yes—
10C4×Q8×C17C411yescharacteristic, complemented
11Q8×Dic17C411yescharacteristic, complemented
12C4⋊C4×D17—21no—
13C4⋊C4×D17C2211yescharacteristic
14#10587573—21no—
15#10587573C411yescharacteristic, complemented
16C42×D17—21no—
17C42×D17C2211yescharacteristic
18C4×Dic34—21no—
19C4×Dic34C411yescharacteristic, complemented
20#10587549C418yescomplemented
21(C2×#595)⋊C4—21no—
22#10587542—21no—
23#10587542—21no—
24C22×#595—22no—
25C2×Q8×C17C2×C411yescharacteristic
26Q8×D17—22no—
27Q8×D17C2×C414yes—
28C2×C4×D17—21no—
29C2×C4×D17—21no—
30C2×C4×D17D411yescharacteristic
31C2×C4×D17D411yescharacteristic
32C2×C4×D17C2311yescharacteristic
33C2×Dic34—21no—
34C2×Dic34C2×C411yescharacteristic
35C2×#3624—22no—
36C4⋊#595—22no—
37C4⋊#595C2×C414yes—
38C4×#595—42no—
39#63242—22no—
40#63242C2×C414yes—
41C4⋊C4×C17—21no—
42C4⋊C4×C17C2×C411yescharacteristic
43C42×C17—21no—
44C42×C17C2×C411yescharacteristic
45C4⋊Dic17—21no—
46C4⋊Dic17C2×C411yescharacteristic
47#63225—21no—
48#63225—21no—
49#63225—21no—
50C4×Dic17—21no—
51C4×Dic17—21no—
52C4×Dic17—21no—
53C4×Dic17C2×C411yescharacteristic
54C4×Dic17D411yescharacteristic
55C4×Dic17D411yescharacteristic
56C22×D17C2×D411yescharacteristic
57C2×#595—24no—
58C2×#595—42no—
59C2×#595—42no—
60Q8×C17—21no—
61Q8×C17C4212yescomplemented
62C2×C4×C17—21no—
63C2×C4×C17—21no—
64C2×C4×C17C22⋊C411yescharacteristic
65C2×C4×C17C22⋊C411yescharacteristic
66C2×C4×C17C22×C411yescharacteristic
67C2×Dic17—21no—
68C2×Dic17—21no—
69C2×Dic17C22⋊C411yescharacteristic
70C2×Dic17C22⋊C411yescharacteristic
71C2×Dic17C2×D411yescharacteristic
72C4×D17—24no—
73C4×D17—21no—
74C4×D17—21no—
75C4×D17—22no—
76C4×D17—41no—
77C4×D17—41no—
78C4×D17C22×C412yes—
79C4×D17#4112yes—
80Dic34—21no—
81Dic34—24no—
82Dic34C4212yescomplemented
83#3624—24no—
84#1763—171nomaximal, Sylow
85C22×C17C2×C22⋊C411yescharacteristic
86C2×D17C4×D412yes—
87C2×D17AGL(1,ℤ/8)11yescharacteristic
88C2×D17#13712yes—
89C2×D17#13711yescharacteristic
90#595—44no—
91C4×C17—22no—
92C4×C17—21no—
93C4×C17—21no—
94C4×C17—41no—
95C4×C17C2×C4211yescharacteristic
96C4×C17#11711yescharacteristic
97Dic17—22no—
98Dic17—21no—
99Dic17—21no—
100Dic17—41no—
101Dic17C4×D412yes—
102C2×C4×Q8—171no—
103C2×#103—172no—
104C2×#97—171no—
105C4×C4⋊C4—171no—
106#340—172no—
107C2×C17C4×C22⋊C411yescharacteristic
108C2×C17#41711yescharacteristic
109C2×C17#41911yescharacteristic
110D17#44012yes—
111D17#44112yes—
112C22×Q8—171no—
113C22×C8—172no—
114C4×Q8—171no—
115C4×Q8—171no—
116C4×Q8—171no—
117C4×Q8—341no—
118C4×Q8—341no—
119C4×Q8#59511yescharacteristic, complemented
120C2×C4⋊C4—172no—
121C2×C4⋊C4—171no—
122C2×C4⋊C4—341no—
123C2×C42—171no—
124C2×C42—341no—
125C2×C42—341no—
126#103—178no—
127#97—341no—
128#97—341no—
129#95—341no—
130C17#176311yescharacteristic, complemented, Sylow
131C2×Q8—174no—
132C2×Q8—171no—
133C2×Q8—341no—
134C2×Q8—342no—
135C2×Q8C2×#59511yescharacteristic
136C22×C4—171no—
137C22×C4—171no—
138C22×C4—171no—
139C22×C4—341no—
140C22×C4—341no—
141C22×C4—342no—
142C2×C8—174no—
143C2×C8—342no—
144C2×C8—342no—
145C4⋊C4—21no—
146C4⋊C4—174no—
147C4⋊C4—171no—
148C4⋊C4—341no—
149C4⋊C4—341no—
150C4⋊C4—341no—
151C4⋊C4—342no—
152C4⋊C4—341no—
153C4⋊C4C2×#59511yescharacteristic
154C42—21no—
155C42—171no—
156C42—171no—
157C42—171no—
158C42—341no—
159C42—341no—
160C42—341no—
161C42—682no—
162C42C2×#59511yescharacteristic
163C23—171no—
164Q8—21no—
165Q8—172no—
166Q8—344no—
167Q8—341no—
168Q8C4×#59512yescomplemented
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—171no—
172C2×C4—172no—
173C2×C4—172no—
174C2×C4—171no—
175C2×C4—171no—
176C2×C4—341no—
177C2×C4—341no—
178C2×C4—342no—
179C2×C4—344no—
180C2×C4—341no—
181C2×C4—344no—
182C2×C4—341no—
183C2×C4—681no—
184C2×C4—681no—
185C2×C4—682no—
186C2×C4—682no—
187C2×C4C22⋊#59511yescharacteristic
188C2×C4C22⋊#59511yescharacteristic
189C2×C4C22×#59511yescharacteristic
190C8—344no—
191C22—172no—
192C22—171no—
193C22—171no—
194C22—172no—
195C22C2×C22⋊#59511yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—172no—
201C4—342no—
202C4—341no—
203C4—341no—
204C4—684no—
205C4—681no—
206C4C2×C4×#59511yescharacteristic
207C4#1058767711yescharacteristic
208C2—172no—
209C2—172no—
210C2C4×C22⋊#59511yescharacteristic
211C2#1176380011yescharacteristic
212C2#1176380211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC4×C17C2×C42
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C2×D4
cosocleC2×C4×D17C23
hypercenterC4×Q8#595
hypocenterC17#1763
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
10. C4×Q8×C17C4semidirect
11. Q8×Dic17C4semidirect
15. #10587573C4semidirect
19. C4×Dic34C4semidirect
20. #10587549C4semidirect
61. Q8×C17C42semidirect
82. Dic34C42semidirect
119. C4×Q8#595semidirect
130. C17#1763semidirect
168. Q8C4×#595semidirect

Maximal subgroups

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

subgroupindex
C2×#105875492
#117637682
C4×C4⋊#5952
#117636822
C4×Q8×D172
#176317