Finite Group Databasebeta

← #14544201

Subgroups — #14544201

Order 2176.

286 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 60 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#11763780C211yesmaximal, characteristic, complemented
2#11763751C211yesmaximal, characteristic, complemented
3#11763662C212yesmaximal, complemented
4#11763661C212yesmaximal, complemented
5#11763321C211yesmaximal, characteristic
6C2×#63252C411yescharacteristic, complemented
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×C4⋊#595—21no—
10C2×#63243—21no—
11C17×#117C411yescharacteristic, complemented
12#10587583C411yescharacteristic, complemented
13#10587583C2211yescharacteristic
14#10587581C411yescharacteristic, complemented
15#10587581C2211yescharacteristic
16#10587572—21no—
17#10587568—21no—
18C4×D68—21no—
19C4×D68—21no—
20C22⋊#3624—21no—
21(C2×#595)⋊C4C2212yes—
22#10587543C2212yes—
23C23×D17D411yescharacteristic
24C22×#595—22no—
25C22×C4×C17C2×C411yescharacteristic
26C2×C22⋊D17—21no—
27C22×Dic17D411yescharacteristic
28#63252—42no—
29C2×D68—21no—
30C2×D68—21no—
31C2×D68C2×C411yescharacteristic
32C2×C4×D17—21no—
33C2×C4×D17—42no—
34C2×C4×D17C2×C411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×C4×D17C2311yescharacteristic
38C2×Dic34C2×C411yescharacteristic
39C22⋊#595—42no—
40C2×#3624—22no—
41C4⋊#595—42no—
42#63243—42no—
43C4⋊C4×C17C2×C411yescharacteristic
44C4⋊C4×C17C2×C411yescharacteristic
45C22⋊C4×C17—21no—
46C42×C17—21no—
47#63227—21no—
48#63227—21no—
49#63227—21no—
50#63227—21no—
51C4⋊Dic17—21no—
52C4⋊Dic17—21no—
53C23×C17C22⋊C411yescharacteristic
54C22×D17—21no—
55C22×D17—21no—
56C22×D17—21no—
57C22×D17—41no—
58C22×D17—41no—
59C22×D17C22⋊C411yescharacteristic
60C22×D17C2×D411yescharacteristic
61C2×#595—42no—
62C2×#595—42no—
63C2×#595—42no—
64C2×C4×C17—22no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17C22⋊C411yescharacteristic
68C2×C4×C17C22×C411yescharacteristic
69C22⋊D17—42no—
70C2×Dic17—21no—
71C2×Dic17—41no—
72C2×Dic17C22⋊C411yescharacteristic
73C2×Dic17C2×D411yescharacteristic
74D68—41no—
75D68—42no—
76D68—41no—
77D68—41no—
78C4×D17—22no—
79C4×D17—22no—
80C4×D17—41no—
81C4×D17—41no—
82C4×D17—42no—
83Dic34—41no—
84#3624—42no—
85#1513—171nomaximal, Sylow
86C22×C17—21no—
87C22×C17—21no—
88C22×C17—21no—
89C22×C17C2×C22⋊C411yescharacteristic
90C2×D17—21no—
91C2×D17—21no—
92C2×D17—21no—
93C2×D17—21no—
94C2×D17—21no—
95C2×D17—42no—
96C2×D17—42no—
97C2×D17—41no—
98C2×D17—41no—
99C2×D17—41no—
100C2×D17AGL(1,ℤ/8)11yescharacteristic
101C2×D17#13711yescharacteristic
102#595—82no—
103C4×C17—21no—
104C4×C17—41no—
105C4×C17—41no—
106C4×C17#9912yes—
107Dic17—21no—
108Dic17—21no—
109Dic17—41no—
110#525—171no—
111#407—171no—
112#380—171no—
113#328—172no—
114#327—172no—
115C2×C17—41no—
116C2×C17C2×#9911yescharacteristic
117C2×C17#41711yescharacteristic
118C2×C17#42111yescharacteristic
119D17—21no—
120D17—21no—
121D17—41no—
122D17—81no—
123C2×#41—171no—
124C23×C4—171no—
125C2×M16—341no—
126#123—341no—
127C22⋊Q8—171no—
128C22⋊Q8—171no—
129#121—171no—
130#121—171no—
131#120—341no—
132C4×D4—341no—
133C4×D4—341no—
134#117#59511yescharacteristic, complemented
135C2×C4⋊C4—341no—
136C2×C22⋊C4—341no—
137C22⋊C8—172no—
138C22⋊C8—341no—
139#95—172no—
140C17#151311yescharacteristic, complemented, Sylow
141C24—171no—
142#41—682no—
143C2×Q8—171no—
144C2×D4—171no—
145C2×D4—341no—
146C2×D4—341no—
147C2×D4—341no—
148C22×C4—171no—
149C22×C4—171no—
150C22×C4—171no—
151C22×C4—171no—
152C22×C4—171no—
153C22×C4—341no—
154C22×C4—342no—
155C22×C4—682no—
156C22×C4C2×#59511yescharacteristic
157M16—682no—
158C2×C8—342no—
159C4⋊C4—341no—
160C4⋊C4—341no—
161C4⋊C4—682no—
162C4⋊C4C2×#59511yescharacteristic
163C4⋊C4C2×#59511yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—341no—
166C22⋊C4—341no—
167C22⋊C4—341no—
168C22⋊C4—341no—
169C22⋊C4—682no—
170C42—21no—
171C23—171no—
172C23—171no—
173C23—341no—
174C23—341no—
175C23—341no—
176C23—681no—
177C23—681no—
178C23C22⋊#59511yescharacteristic
179Q8—681no—
180D4—681no—
181D4—681no—
182D4—681no—
183D4—682no—
184D4—682no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—171no—
189C2×C4—171no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—341no—
193C2×C4—681no—
194C2×C4—681no—
195C2×C4—681no—
196C2×C4—682no—
197C2×C4—682no—
198C2×C4—682no—
199C2×C4—682no—
200C2×C4C22⋊#59511yescharacteristic
201C2×C4C22×#59511yescharacteristic
202C8—682no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—171no—
207C22—171no—
208C22—341no—
209C22—341no—
210C22—341no—
211C22—341no—
212C22—341no—
213C22—682no—
214C22—682no—
215C22—681no—
216C22—681no—
217C22—681no—
218C22C2×C22⋊#59511yescharacteristic
219C4—21no—
220C4—41no—
221C4—41no—
222C4—341no—
223C4—341no—
224C4—681no—
225C4—1362no—
226C4#1058753912yes—
227C2—41no—
228C2—341no—
229C2—341no—
230C2—681no—
231C2—1361no—
232C2C2×#1058753911yescharacteristic
233C2#1176380011yescharacteristic
234C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#117C4
socleC22×C17C2×C22⋊C4
prime radical#11763321C2
cosocleC2×C4×D17C23
hypercenter#117#595
hypocenterC17#1513
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #11763780C2semidirect
2. #11763751C2semidirect
3. #11763662C2semidirect
4. #11763661C2semidirect
6. C2×#63252C4semidirect
11. C17×#117C4semidirect
12. #10587583C4semidirect
14. #10587581C4semidirect
134. #117#595semidirect
140. C17#1513semidirect

Maximal subgroups

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

subgroupindex
#117637802
#117637512
#117636622
#117636612
#117633212
#151317