Finite Group Databasebeta

← #14535997

Subgroups — #14535997

Order 2176.

464 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 137 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#11762745C212yesmaximal, complemented
2#11762742C212yesmaximal, complemented
3#11762732C212yesmaximal, complemented
4#11762693C212yesmaximal, complemented
5#11762689C211yesmaximal, characteristic
6#11762689C211yesmaximal, characteristic
7#11762681C212yesmaximal, complemented
8#11762667C212yesmaximal, complemented
9C17×#390C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×Dic17C2211yescharacteristic
12C23×Dic17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2212yes—
16C2×C42×C17C2211yescharacteristic
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C4⋊Dic17C2212yes—
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C24×C17C2311yescharacteristic
33C2×D4×C17—41no—
34C2×D4×C17—41no—
35C2×D4×C17Q814yes—
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×Dic17—24no—
40C22×Dic17—41no—
41C22×Dic17—42no—
42C22×Dic17—41no—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C4⋊C4×C17—41no—
48C22⋊C4×C17—42no—
49C22⋊C4×C17—42no—
50C42×C17—41no—
51C22⋊Dic17—24no—
52C22⋊Dic17—42no—
53C22⋊Dic17—41no—
54C22⋊Dic17—41no—
55C4⋊Dic17—42no—
56#63225—24no—
57#63225—42no—
58#63225—42no—
59#63225—42no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—21no—
64C23×C17—21no—
65C23×C17—22no—
66C23×C17—21no—
67C23×C17—21no—
68C23×C17C2×Q812yes—
69C23×C17C2×Q812yes—
70C23×C17C2411yescharacteristic
71D4×C17—44no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17C2×Q812yes—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—44no—
93C2×Dic17—44no—
94C2×Dic17#4114yes—
95#2748—171nomaximal, Sylow
96C22×C17—22no—
97C22×C17—22no—
98C22×C17—24no—
99C22×C17—24no—
100C22×C17—42no—
101C22×C17—42no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17—41no—
105C22×C17—41no—
106C22×C17C22×Q811yescharacteristic
107C22×C17C2×#4112yes—
108C22×C17H(5,𝔽2)11yescharacteristic
109C22×C17H(5,𝔽2)12yes—
110C22×C17H(5,𝔽2)11yescharacteristic
111C4×C17—22no—
112C4×C17—42no—
113C4×C17—42no—
114Dic17—24no—
115Dic17—42no—
116Dic17—42no—
117Dic17—42no—
118#399—172no—
119#398—172no—
120#396—172no—
121#390D1711yescharacteristic
122#387—172no—
123#386—171no—
124#386—171no—
125#385—172no—
126#380—172no—
127C2×C17—42no—
128C2×C17—42no—
129C2×C17#54012yes—
130C2×C17#54311yescharacteristic
131C2×C17#55111yescharacteristic
132C2×C17#55412yes—
133C2×C17#56011yescharacteristic
134C22×D4C2×D1711yescharacteristic
135C23×C4—171no—
136C23×C4—171no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4C2×D1711yescharacteristic
143C2×C22⋊C4—172no—
144C2×C22⋊C4—172no—
145C2×C22⋊C4—171no—
146C2×C22⋊C4—171no—
147C2×C22⋊C4C2×D1712yes—
148C2×C22⋊C4C2×D1712yes—
149C2×C42C2×D1711yescharacteristic
150#95—172no—
151#95—172no—
152#95—172no—
153#95—172no—
154#95—172no—
155C17#274811yescharacteristic, complemented, Sylow
156C24C22×D1711yescharacteristic
157C24C22×D1711yescharacteristic
158C2×D4—41no—
159C2×D4—41no—
160C2×D4Dic3414yes—
161C22×C4—172no—
162C22×C4—172no—
163C22×C4—172no—
164C22×C4—172no—
165C22×C4—344no—
166C22×C4—681no—
167C22×C4—682no—
168C22×C4—681no—
169C22×C4C22×D1712yes—
170C22×C4C22×D1712yes—
171C22×C4C22×D1711yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—344no—
174C4⋊C4—682no—
175C4⋊C4—682no—
176C4⋊C4—682no—
177C4⋊C4—682no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—344no—
181C22⋊C4—681no—
182C22⋊C4—681no—
183C22⋊C4—682no—
184C42—41no—
185C23—22no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23C2×Dic3412yes—
194C23C2×Dic3412yes—
195C23C23×D1711yescharacteristic
196D4—44no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—174no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—684no—
218C2×C4—684no—
219C2×C4C2×Dic3412yes—
220C22—24no—
221C22—24no—
222C22—22no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22—42no—
230C22C22×Dic3411yescharacteristic
231C22#1058769111yescharacteristic
232C22#1058769112yes—
233C22#1058769111yescharacteristic
234C22#41×D1712yes—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—344no—
239C4—682no—
240C4—682no—
241C4—682no—
242C2—42no—
243C2—42no—
244C2#1176340912yes—
245C2#1176343511yescharacteristic
246C2#1176348911yescharacteristic
247C2#1176351812yes—
248C2#1176354611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC22×D4×C17C22
hypercenter#390D17
hypocenterC17#2748
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #11762745C2semidirect
2. #11762742C2semidirect
3. #11762732C2semidirect
4. #11762693C2semidirect
7. #11762681C2semidirect
8. #11762667C2semidirect
155. C17#2748semidirect

Maximal subgroups

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

subgroupindex
#117627452
#117627422
#117627322
#117626932
#117626892
#117626812
#117626672
C17×#3902
#274817