Finite Group Databasebeta

← #14541819

Subgroups — #14541819

Order 2176.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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#11763529C211yesmaximal, characteristic
3#11763200C212yesmaximal
4#11763188C211yesmaximal, characteristic, complemented
5#11763029C212yesmaximal
6#11763020C211yesmaximal, characteristic, complemented
7#11763018C212yesmaximal, complemented
8#11762936C211yesmaximal, characteristic, complemented
9#11762913C212yesmaximal, complemented
10C4×#63229C211yesmaximal, characteristic
11C17×#461C211yesmaximal, characteristic
12C2×SD16×C17C2212yes—
13C17×C4⋊Q8C2211yescharacteristic
14C4×D4×C17C2211yescharacteristic
15Q8×Dic17C2212yes—
16#10587630C2212yes—
17C2×#63231C2212yes—
18C2×#63231C2212yes—
19#10587622C2212yes—
20C2×#63229C2211yescharacteristic
21#10587612—21no—
22#10587610—21no—
23C2×Dic68C2212yes—
24#10587576—21no—
25#10587575—22no—
26#10587573—21no—
27#10587573C2212yes—
28#10587565—22no—
29#10587555C2211yescharacteristic
30C4×Dic34C2211yescharacteristic
31C4×Dic34C2211yescharacteristic
32C17×C4⋊C8C2211yescharacteristic
33C17×#103C2212yes—
34#10587503C2211yescharacteristic
35#10587487C2212yes—
36#10587486C2212yes—
37#10587485C2212yes—
38#10587481C2211yescharacteristic
39#10587479C2211yescharacteristic
40C4×#3622C2211yescharacteristic
41C2×Q8×C17C2312yes—
42C2×D4×C17C2311yescharacteristic
43C22×C4×C17—21no—
44C2×Dic34D411yescharacteristic
45C2×Dic34C2311yescharacteristic
46C2×Dic34C2312yes—
47SD16×C17—42no—
48C2×C8×C17C2312yes—
49C4⋊C4×C17—21no—
50C4⋊C4×C17—21no—
51C4⋊C4×C17C2311yescharacteristic
52C22⋊C4×C17—21no—
53C42×C17C2311yescharacteristic
54C22⋊Dic17—22no—
55C22⋊Dic17—22no—
56#63231—24no—
57#63231—42no—
58#63229—22no—
59C4⋊Dic17—21no—
60C4⋊Dic17—21no—
61C4⋊Dic17—22no—
62C4⋊Dic17D411yescharacteristic
63C4⋊Dic17C2311yescharacteristic
64#63225—21no—
65#63225—22no—
66#63225—22no—
67#63225—22no—
68#63225D412yes—
69C4×Dic17—21no—
70C4×Dic17—22no—
71C4×Dic17—22no—
72C4×Dic17C2312yes—
73C2×#3622C2311yescharacteristic
74C2×#3622C2311yescharacteristic
75Dic68—42no—
76C23×C17—21no—
77Q8×C17—22no—
78Q8×C17—22no—
79D4×C17—21no—
80D4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—42no—
84C2×C4×C17C2×D411yescharacteristic
85C2×C4×C17C2×D411yescharacteristic
86C2×C4×C17C2411yescharacteristic
87C2×Dic17—22no—
88C2×Dic17—21no—
89C2×Dic17—21no—
90C2×Dic17C2×D412yes—
91C2×Dic17C2×D412yes—
92Dic34—22no—
93Dic34—21no—
94Dic34—22no—
95Dic34—21no—
96Dic34—22no—
97C8×C17—22no—
98#3622—21no—
99#3622#4112yes—
100#3398—171nomaximal, Sylow
101C22×C17—41no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17C22×D411yescharacteristic
105C4×C17—21no—
106C4×C17—42no—
107C4×C17—41no—
108C4×C17#13712yes—
109C4×C17C2×#4111yescharacteristic
110C4×C17H(5,𝔽2)11yescharacteristic
111Dic17—22no—
112Dic17—22no—
113Dic17—41no—
114Dic17—42no—
115Dic17—41no—
116#557—171no—
117#556—171no—
118#497—172no—
119#494—171no—
120#464—172no—
121#462—171no—
122#462—172no—
123#461D1711yescharacteristic
124#446—171no—
125#441—172no—
126C4×SD16—171no—
127C2×C17—81no—
128C2×C17#55011yescharacteristic
129C2×C17C2×#13711yescharacteristic
130C2×C17#57811yescharacteristic
131C2×Q16—172no—
132C2×Q16—172no—
133C2×Q16—172no—
134C2×SD16—171no—
135C2×SD16C2×D1712yes—
136C4⋊Q8—172no—
137C4⋊Q8—171no—
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×D4C2×D1711yescharacteristic
150#117—341no—
151#106—171no—
152C4⋊C8—172no—
153C4⋊C8C2×D1711yescharacteristic
154#103—172no—
155#103—171no—
156#103C2×D1712yes—
157#102—171no—
158#97—172no—
159C4×C8—171no—
160C17#339811yescharacteristic, complemented, Sylow
161C2×Q8—171no—
162C2×Q8—172no—
163C2×Q8—171no—
164C2×Q8C22×D1712yes—
165C2×D4C22×D1711yescharacteristic
166C22×C4—21no—
167Q16—344no—
168Q16—682no—
169Q16—682no—
170SD16—42no—
171SD16—342no—
172C2×C8—171no—
173C2×C8—171no—
174C2×C8C22×D1712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—171no—
178C4⋊C4—172no—
179C4⋊C4—171no—
180C4⋊C4—342no—
181C4⋊C4—342no—
182C4⋊C4—342no—
183C4⋊C4—342no—
184C4⋊C4—341no—
185C4⋊C4—341no—
186C4⋊C4—341no—
187C4⋊C4C22×D1711yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—342no—
190C22⋊C4—342no—
191C42—172no—
192C42—342no—
193C42—342no—
194C42—341no—
195C42C22×D1711yescharacteristic
196C23—21no—
197Q8—22no—
198Q8—22no—
199Q8—342no—
200Q8—341no—
201Q8—342no—
202Q8—341no—
203Q8—342no—
204D4—21no—
205D4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—172no—
210C2×C4—172no—
211C2×C4—341no—
212C2×C4—342no—
213C2×C4—341no—
214C2×C4D4×D1711yescharacteristic
215C2×C4D4×D1711yescharacteristic
216C2×C4C23×D1711yescharacteristic
217C8—22no—
218C8—172no—
219C8—341no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22C2×D4×D1711yescharacteristic
224C4—21no—
225C4—42no—
226C4—41no—
227C4—342no—
228C4—342no—
229C4—682no—
230C4—681no—
231C4—681no—
232C4#1058760112yes—
233C4#1058769111yescharacteristic
234C4#41×D1711yescharacteristic
235C2—81no—
236C2#1176347711yescharacteristic
237C2C2×#1058760111yescharacteristic
238C2#1176362511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#461C2
socleC22×C17C22×D4
prime radicalC2×D4×C17C23
hypercenter#461D17
hypocenterC17#3398
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #11763534C2semidirect
4. #11763188C2semidirect
6. #11763020C2semidirect
7. #11763018C2semidirect
8. #11762936C2semidirect
9. #11762913C2semidirect
160. C17#3398semidirect

Maximal subgroups

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

subgroupindex
#117635342
#117635292
#117632002
#117631882
#117630292
#117630202
#117630182
#117629362
#117629132
C4×#632292
C17×#4612
#339817