Finite Group Databasebeta

← #14529065

Subgroups — #14529065

Order 2176.

374 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 93 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×#10587505C211yesmaximal, characteristic
2#11762689C211yesmaximal, characteristic
3#11762681C212yesmaximal, complemented
4#11762576C212yesmaximal, complemented
5C17×#518C211yesmaximal, characteristic
6C22×D4×C17C2211yescharacteristic
7C23×Dic17C2211yescharacteristic
8C4×D4×C17—22no—
9C4×D4×C17—22no—
10C17×#117—21no—
11C2×C22⋊C4×C17—21no—
12C2×C22⋊C4×C17C2212yes—
13C2×C22⋊Dic17—21no—
14C2×C22⋊Dic17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×#63225—22no—
17#10587505C414yescomplemented
18#10587502—21no—
19#10587502—22no—
20C24×C17D411yescharacteristic
21C24×C17C2311yescharacteristic
22C2×D4×C17—22no—
23C2×D4×C17—22no—
24C2×D4×C17C2×C412yes—
25C2×D4×C17Q812yes—
26C22×C4×C17—21no—
27C22×C4×C17—21no—
28C22×C4×C17D411yescharacteristic
29C22×C4×C17D412yes—
30C22×C4×C17D412yes—
31C22×Dic17—22no—
32C22×Dic17—21no—
33C22×Dic17—22no—
34C22×Dic17—21no—
35C22×Dic17—42no—
36C4⋊C4×C17—21no—
37C4⋊C4×C17—21no—
38C22⋊C4×C17—22no—
39C22⋊C4×C17—22no—
40C22⋊C4×C17—22no—
41C42×C17—22no—
42C22⋊Dic17—22no—
43C22⋊Dic17—22no—
44C22⋊Dic17—42no—
45C22⋊Dic17C2×C414yescomplemented
46#63225—44no—
47C23×C17—21no—
48C23×C17—21no—
49C23×C17—41no—
50C23×C17—41no—
51C23×C17—41no—
52C23×C17—41no—
53C23×C17C4⋊C412yescomplemented
54C23×C17C22×C411yescharacteristic
55C23×C17C2×D411yescharacteristic
56C23×C17C2×D412yes—
57C23×C17C2×Q811yescharacteristic
58C23×C17#4112yes—
59D4×C17—24no—
60D4×C17—24no—
61C2×C4×C17—21no—
62C2×C4×C17—21no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—42no—
66C2×C4×C17—42no—
67C2×C4×C17—81no—
68C2×C4×C17—81no—
69C2×C4×C17C4⋊C412yescomplemented
70C2×Dic17—22no—
71C2×Dic17—24no—
72C2×Dic17—24no—
73C2×Dic17—42no—
74C2×Dic17—42no—
75C2×Dic17—41no—
76C2×Dic17—42no—
77C2×Dic17—41no—
78C2×Dic17—41no—
79C2×Dic17—41no—
80C2×Dic17—41no—
81#1867—171nomaximal, Sylow
82C22×C17—22no—
83C22×C17—21no—
84C22×C17—22no—
85C22×C17—21no—
86C22×C17—22no—
87C22×C17—41no—
88C22×C17—41no—
89C22×C17—42no—
90C22×C17—42no—
91C22×C17—41no—
92C22×C17—41no—
93C22×C17—81no—
94C22×C17C2×C4⋊C411yescharacteristic
95C22×C17C4×D412yes—
96C22×C17#12011yescharacteristic
97C22×C17C22⋊Q812yes—
98C22×C17#12311yescharacteristic
99C4×C17—22no—
100C4×C17—42no—
101C4×C17—42no—
102Dic17—44no—
103Dic17—42no—
104Dic17—81no—
105#518D1711yescharacteristic
106C2×#99—171no—
107#386—171no—
108#385—172no—
109#342—172no—
110C2×C17—22no—
111C2×C17—21no—
112C2×C17—21no—
113C2×C17—22no—
114C2×C17—42no—
115C2×C17#38511yescharacteristic
116C2×C17B(4,𝔽2)11yescharacteristic
117C2×C17#45811yescharacteristic
118C22×D4C2×D1711yescharacteristic
119C23×C4—171no—
120C4×D4—22no—
121C4×D4—22no—
122#117—21no—
123C2×C4⋊C4—342no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—172no—
126C2×C22⋊C4—171no—
127C2×C22⋊C4—341no—
128C2×C22⋊C4C2×D1712yes—
129#99—174no—
130#95—342no—
131#95—341no—
132C17#186711yescharacteristic, complemented, Sylow
133C24C22⋊D1711yescharacteristic
134C24C22×D1711yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4Dic3412yes—
138C2×D4C4×D1712yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—342no—
142C22×C4—341no—
143C22×C4—342no—
144C22×C4—341no—
145C22×C4—682no—
146C22×C4C22⋊D1711yescharacteristic
147C22×C4C22⋊D1712yes—
148C22×C4C22⋊D1712yes—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—684no—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—174no—
156C22⋊C4—342no—
157C22⋊C4—342no—
158C22⋊C4—682no—
159C42—22no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23#6322512yescomplemented
167C23C2×Dic3411yescharacteristic
168C23C2×C4×D1711yescharacteristic
169C23#6325212yes—
170C23C2×C22⋊D1711yescharacteristic
171C23C2×C22⋊D1712yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—342no—
183C2×C4—344no—
184C2×C4—344no—
185C2×C4—682no—
186C2×C4—682no—
187C2×C4—681no—
188C2×C4—682no—
189C2×C4—681no—
190C2×C4—681no—
191C2×C4—681no—
192C2×C4—681no—
193C2×C4#6322512yescomplemented
194C22—22no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×#6322511yescharacteristic
207C22#1058761012yes—
208C22C4×C22⋊D1712yes—
209C22#1058761511yescharacteristic
210C22#1058763911yescharacteristic
211C4—22no—
212C4—42no—
213C4—42no—
214C4—684no—
215C4—682no—
216C4—1361no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#1176268211yescharacteristic
223C2#1176299611yescharacteristic
224C2#1176299911yescharacteristic
subgroupG/H
centerC22#10587639
commutatorC23×C17C22×C4
FrattiniC24C22×D17
FittingC17×#518C2
socleC22×C17#120
prime radicalC22×D4×C17C22
cosocleC24×C17C23
hypercenter#518D17
hypocenterC17#1867
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
3. #11762681C2semidirect
4. #11762576C2semidirect
17. #10587505C4semidirect
45. C22⋊Dic17C2×C4semidirect
53. C23×C17C4⋊C4semidirect
69. C2×C4×C17C4⋊C4semidirect
132. C17#1867semidirect
166. C23#63225semidirect
193. C2×C4#63225semidirect

Maximal subgroups

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

subgroupindex
C2×#105875052
#117626892
#117626812
#117625762
C17×#5182
#186717