Finite Group Databasebeta

← #14541820

Subgroups — #14541820

Order 2176.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#11763534C211yesmaximal, characteristic, complemented
2#11763533C211yesmaximal, characteristic
3#11763202C212yesmaximal
4#11763033C212yesmaximal
5#11763017C212yesmaximal, complemented
6#11763016C211yesmaximal, characteristic, complemented
7#11763011C211yesmaximal, characteristic
8#11762939C211yesmaximal, characteristic, complemented
9#11762916C212yesmaximal, complemented
10C4×#63219C211yesmaximal, characteristic
11C17×#494C211yesmaximal, characteristic, complemented
12C2×Q16×C17C2212yes—
13C2×Q16×C17C2212yes—
14C17×C4⋊Q8C2211yescharacteristic
15C17×C4⋊Q8C2211yescharacteristic
16#10587633C2212yes—
17Q8×Dic17C2212yes—
18Q8×Dic17C2212yes—
19#10587630C2212yes—
20C2×#63231C2212yes—
21C2×#63230C2212yes—
22C2×#63219C2211yescharacteristic
23#10587584—22no—
24#10587583—21no—
25#10587578—21no—
26#10587576—21no—
27#10587575—22no—
28#10587573—21no—
29C4×D68C2211yescharacteristic
30C4×Dic34C2211yescharacteristic
31C4×Dic34C2211yescharacteristic
32C4×C8×C17C2211yescharacteristic
33#10587506C2212yes—
34#10587506C2212yes—
35#10587492C2211yescharacteristic
36#10587488C2211yescharacteristic
37#10587487C2211yescharacteristic
38#10587486C2212yes—
39#10587485C2212yes—
40C4⋊#3622C2211yescharacteristic
41C2×Q8×C17C2312yes—
42C2×Q8×C17C2312yes—
43C2×D68C2311yescharacteristic
44C2×C4×D17—21no—
45C2×Dic34D411yescharacteristic
46C2×Dic34C2311yescharacteristic
47Q16×C17—24no—
48Q16×C17—42no—
49C2×C8×C17C2311yescharacteristic
50C2×C8×C17C2311yescharacteristic
51C4⋊C4×C17—21no—
52C4⋊C4×C17—21no—
53C4⋊C4×C17—21no—
54C4⋊C4×C17—21no—
55C42×C17C2311yescharacteristic
56#63231—42no—
57#63230—42no—
58#63227—21no—
59#63227—22no—
60#63227—22no—
61C4⋊Dic17—22no—
62C4⋊Dic17—22no—
63C4⋊Dic17D411yescharacteristic
64C4⋊Dic17C2311yescharacteristic
65C4⋊Dic17C2311yescharacteristic
66#63225—21no—
67#63225—22no—
68#63225—22no—
69#63225D412yes—
70C4×Dic17—21no—
71C4×Dic17—22no—
72C4×Dic17—22no—
73C4×Dic17C2312yes—
74C2×#3622C2312yes—
75#63219—22no—
76C22×D17—21no—
77Q8×C17—22no—
78Q8×C17—22no—
79Q8×C17—22no—
80Q8×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17C2×D411yescharacteristic
84C2×C4×C17C2×D411yescharacteristic
85C2×C4×C17C2411yescharacteristic
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17C2×D412yes—
90C2×Dic17C2×D412yes—
91D68—21no—
92D68—21no—
93C4×D17—42no—
94Dic34—22no—
95Dic34—21no—
96Dic34—21no—
97C8×C17—21no—
98C8×C17#4112yes—
99#3622—22no—
100#3398—171nomaximal, Sylow
101C22×C17C22×D411yescharacteristic
102C2×D17—41no—
103C2×D17—41no—
104C2×D17—41no—
105C4×C17—21no—
106C4×C17—42no—
107C4×C17—42no—
108C4×C17#13712yes—
109C4×C17C2×#4111yescharacteristic
110C4×C17H(5,𝔽2)11yescharacteristic
111Dic17—22no—
112Dic17—22no—
113Dic17—41no—
114Dic17—41no—
115Dic17—41no—
116#557—171no—
117#556—171no—
118#497—172no—
119#494D1711yescharacteristic, complemented
120#464—172no—
121#462—172no—
122#462—171no—
123#461—171no—
124#446—171no—
125#441—172no—
126C4×SD16—171no—
127C2×C17#55011yescharacteristic
128C2×C17C2×#13711yescharacteristic
129C2×C17#57811yescharacteristic
130D17—81no—
131C2×Q16—172no—
132C2×Q16C2×D1712yes—
133C2×Q16C2×D1712yes—
134C2×SD16—172no—
135C2×SD16—171no—
136C4⋊Q8—172no—
137C4⋊Q8C2×D1711yescharacteristic
138C4⋊Q8C2×D1711yescharacteristic
139#126—342no—
140#125—342no—
141#125—341no—
142#124—172no—
143C22⋊Q8—341no—
144C4×Q8—171no—
145C4×Q8—172no—
146C4×Q8—172no—
147C4×Q8—171no—
148C4×Q8—341no—
149C4×D4—171no—
150#117—341no—
151#106—171no—
152C4⋊C8—172no—
153C4⋊C8—171no—
154#103—172no—
155#103—172no—
156#103—171no—
157#102—171no—
158#97—172no—
159C4×C8C2×D1711yescharacteristic
160C17#339811yescharacteristic, complemented, Sylow
161C2×Q8—171no—
162C2×Q8—171no—
163C2×Q8C22×D1712yes—
164C2×Q8C22×D1712yes—
165C2×D4—171no—
166C22×C4—341no—
167Q16—24no—
168Q16—42no—
169Q16—682no—
170SD16—342no—
171SD16—682no—
172C2×C8—172no—
173C2×C8C22×D1711yescharacteristic
174C2×C8C22×D1711yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—171no—
180C4⋊C4—172no—
181C4⋊C4—171no—
182C4⋊C4—171no—
183C4⋊C4—342no—
184C4⋊C4—342no—
185C4⋊C4—342no—
186C4⋊C4—342no—
187C4⋊C4—341no—
188C22⋊C4—342no—
189C22⋊C4—342no—
190C22⋊C4—341no—
191C42—172no—
192C42—342no—
193C42—342no—
194C42—341no—
195C42C22×D1711yescharacteristic
196C23—341no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—22no—
201Q8—342no—
202Q8—341no—
203Q8—341no—
204D4—341no—
205D4—341no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—172no—
209C2×C4—172no—
210C2×C4—341no—
211C2×C4—341no—
212C2×C4—341no—
213C2×C4—682no—
214C2×C4D4×D1711yescharacteristic
215C2×C4D4×D1711yescharacteristic
216C2×C4C23×D1711yescharacteristic
217C8—21no—
218C8—342no—
219C8#6325412yes—
220C22—681no—
221C22—681no—
222C22—681no—
223C22C2×D4×D1711yescharacteristic
224C4—21no—
225C4—42no—
226C4—42no—
227C4—342no—
228C4—342no—
229C4—681no—
230C4—681no—
231C4—681no—
232C4#1058760412yes—
233C4C2×#6325411yescharacteristic
234C4#1058769111yescharacteristic
235C2—1361no—
236C2#1176347911yescharacteristic
237C2C2×#1058760411yescharacteristic
238C2#1176362211yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#494C2
socleC22×C17C22×D4
prime radicalC2×D68C23
hypercenter#494D17
hypocenterC17#3398
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #11763534C2semidirect
5. #11763017C2semidirect
6. #11763016C2semidirect
8. #11762939C2semidirect
9. #11762916C2semidirect
11. C17×#494C2semidirect
119. #494D17semidirect
160. C17#3398semidirect

Maximal subgroups

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

subgroupindex
#117635342
#117635332
#117632022
#117630332
#117630172
#117630162
#117630112
#117629392
#117629162
C4×#632192
C17×#4942
#339817