Finite Group Databasebeta

← #14533735

Subgroups — #14533735

Order 2176.

676 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 185 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×#10587574C211yesmaximal, characteristic, complemented
2C2×#10587564C214yesmaximal, complemented
3C2×D4×Dic17C211yesmaximal, characteristic
4#11762745C214yesmaximal, complemented
5#11762712C211yesmaximal, characteristic
6#11762679C211yesmaximal, characteristic, complemented
7#11762678C211yesmaximal, characteristic, complemented
8C4×#63225C211yesmaximal, characteristic, complemented
9C17×#390C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×Dic17C2212yes—
12C22×Dic34C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2214yes—
15C2×C42×C17C2211yescharacteristic
16C2×C22⋊Dic17C2214yes—
17C2×C22⋊Dic17C2212yes—
18D4×Dic17—24no—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2214yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×#63225C2212yes—
23C2×#63225C2214yes—
24C2×#63225C2212yes—
25C2×C4×Dic17C2211yescharacteristic
26C2×C4×Dic17C2211yescharacteristic
27#10587574—24no—
28#10587564—216no—
29#10587502C2212yes—
30C24×C17C2312yes—
31C2×D4×C17—22no—
32C2×D4×C17—22no—
33C2×D4×C17Q814yescomplemented
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×Dic17—28no—
38C22×Dic17—24no—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2311yescharacteristic
42C22×Dic17C2312yes—
43C22×Dic17C2311yescharacteristic
44C2×Dic34—28no—
45C2×Dic34—24no—
46C4⋊C4×C17—22no—
47C22⋊C4×C17—28no—
48C42×C17—22no—
49C22⋊Dic17—28no—
50C22⋊Dic17—24no—
51C4⋊Dic17—28no—
52C4⋊Dic17—22no—
53C4⋊Dic17—22no—
54#63225—28no—
55#63225—24no—
56#63225D418yescomplemented
57C4×Dic17—22no—
58C4×Dic17D414yes—
59C23×C17—24no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—24no—
63C23×C17C2×Q814yes—
64C23×C17C2411yescharacteristic
65D4×C17—28no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—24no—
69C2×C4×C17—22no—
70C2×C4×C17C2×D414yes—
71C2×C4×C17C2×D412yes—
72C2×C4×C17C2×Q812yes—
73C2×C4×C17#4112yes—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—24no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—24no—
81C2×Dic17—48no—
82C2×Dic17C2×D412yes—
83C2×Dic17C2×D414yes—
84C2×Dic17C2×D412yes—
85C2×Dic17C2×D414yes—
86Dic34—48no—
87#2542—171nomaximal, Sylow
88C22×C17—24no—
89C22×C17—28no—
90C22×C17—42no—
91C22×C17—42no—
92C22×C17—44no—
93C22×C17C22×D411yescharacteristic
94C22×C17C22×D412yes—
95C22×C17C22×Q811yescharacteristic
96C22×C17C2×#4111yescharacteristic
97C22×C172−1+412yes—
98C4×C17—24no—
99C4×C17—42no—
100C4×C17C22⋊Q814yescomplemented
101Dic17—22no—
102Dic17—24no—
103Dic17—42no—
104Dic17—42no—
105Dic17#12714yes—
106C2×C4⋊Q8—171no—
107C2×C22⋊Q8—174no—
108C2×C4×D4—171no—
109#399—174no—
110#390—171no—
111#390D1711yescharacteristic
112#384—171no—
113#384—171no—
114C4×C4⋊C4—171no—
115C2×C17—44no—
116C2×C17C2×C22⋊Q811yescharacteristic
117C2×C17C2×#12711yescharacteristic
118C2×C17#53611yescharacteristic
119C2×C17#54712yes—
120C2×C17D4×Q812yes—
121C22×Q8—172no—
122C22×D4C2×D1711yescharacteristic
123C23×C4—172no—
124C4⋊Q8—344no—
125C22⋊Q8—3416no—
126C4×D4—344no—
127C2×C4⋊C4—172no—
128C2×C4⋊C4—171no—
129C2×C4⋊C4—174no—
130C2×C4⋊C4—174no—
131C2×C4⋊C4—171no—
132C2×C4⋊C4—172no—
133C2×C4⋊C4C2×D1711yescharacteristic
134C2×C22⋊C4—172no—
135C2×C22⋊C4—174no—
136C2×C22⋊C4C2×D1714yes—
137C2×C42—171no—
138C2×C42—171no—
139C2×C42C2×D1711yescharacteristic
140#95—172no—
141C17#254211yescharacteristic, complemented, Sylow
142C24C22×D1712yes—
143C2×Q8—348no—
144C2×Q8—344no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4Dic3414yescomplemented
148C22×C4—171no—
149C22×C4—171no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4—344no—
154C22×C4—348no—
155C22×C4C22×D1712yes—
156C22×C4C22×D1712yes—
157C22×C4C22×D1711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—178no—
160C4⋊C4—348no—
161C4⋊C4—344no—
162C4⋊C4—342no—
163C4⋊C4—348no—
164C4⋊C4—342no—
165C22⋊C4—28no—
166C22⋊C4—344no—
167C22⋊C4—348no—
168C42—22no—
169C42—174no—
170C42—342no—
171C23—24no—
172C23—22no—
173C23—22no—
174C23—24no—
175C23C2×Dic3414yes—
176C23C23×D1711yescharacteristic
177Q8—688no—
178D4—28no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—172no—
184C2×C4—174no—
185C2×C4—172no—
186C2×C4—174no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—344no—
192C2×C4—344no—
193C2×C4—342no—
194C2×C4—688no—
195C2×C4C2×Dic3412yes—
196C2×C4D4×D1714yes—
197C2×C4D4×D1712yes—
198C2×C4#6325412yes—
199C22—28no—
200C22—24no—
201C22—44no—
202C22—42no—
203C22—42no—
204C22C22×Dic3411yescharacteristic
205C22C2×D4×D1712yes—
206C22C2×D4×D1711yescharacteristic
207C22C2×#6325411yescharacteristic
208C22#1058769712yes—
209C4—24no—
210C4—42no—
211C4—174no—
212C4—344no—
213C4—342no—
214C4—682no—
215C4—682no—
216C4#1058756414yescomplemented
217C2—44no—
218C2C2×#1058756411yescharacteristic
219C2#127×D1711yescharacteristic
220C2#1176337911yescharacteristic
221C2#1176345912yes—
222C2D4×Dic3412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC22×D4×C17C22
hypercenter#390D17
hypocenterC17#2542
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#10587574C2semidirect
2. C2×#10587564C2semidirect
4. #11762745C2semidirect
6. #11762679C2semidirect
7. #11762678C2semidirect
8. C4×#63225C2semidirect
33. C2×D4×C17Q8semidirect
56. #63225D4semidirect
100. C4×C17C22⋊Q8semidirect
141. C17#2542semidirect
147. C2×D4Dic34semidirect
216. C4#10587564semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875642
C2×D4×Dic172
#117627452
#117627122
#117626792
#117626782
C4×#632252
C17×#3902
#254217