Finite Group Databasebeta

← #14539869

Subgroups — #14539869

Order 2176.

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

normal subgroups only

This group has 364 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#11763421C211yesmaximal, characteristic, complemented
2#11763385C211yesmaximal, characteristic
3#11763203C211yesmaximal, characteristic, complemented
4#11763189C211yesmaximal, characteristic, complemented
5#11763178C211yesmaximal, characteristic, complemented
6#11763164C211yesmaximal, characteristic, complemented
7#11763032C212yesmaximal, complemented
8#11763020C212yesmaximal, complemented
9#11762973C212yesmaximal
10#11762967C212yesmaximal, complemented
11C17×#433C211yesmaximal, characteristic, complemented
12C2×#63252C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C17×#117C2211yescharacteristic
15#10587610C2212yes—
16C2×Dic68C2212yes—
17C2×Dic68C2212yes—
18C2×Dic68C2212yes—
19C2×#63219C2212yes—
20#10587583—22no—
21#10587575—22no—
22#10587572—22no—
23#10587565—22no—
24#10587561C2211yescharacteristic
25#10587561C2211yescharacteristic
26#10587556C2211yescharacteristic
27#10587555C2211yescharacteristic
28#10587555C2211yescharacteristic
29#10587555C2211yescharacteristic
30C4×Dic34C2212yes—
31C17×C4⋊C8C2212yes—
32C17×C22⋊C8C2212yes—
33C17×#97C2211yescharacteristic
34C4×C8×C17C2211yescharacteristic
35#10587492C2212yes—
36#10587487C2212yes—
37#10587487C2212yes—
38#10587487C2212yes—
39C22×C4×C17C2311yescharacteristic
40C2×C22⋊D17—21no—
41C22×Dic17—21no—
42#63252—42no—
43C2×D68C2311yescharacteristic
44C2×C4×D17—21no—
45C2×Dic34—21no—
46C2×Dic34—42no—
47C2×Dic34C2311yescharacteristic
48C2×Dic34C2311yescharacteristic
49C2×Dic34C2311yescharacteristic
50C2×Dic34C2312yes—
51C2×C8×C17C2312yes—
52C2×C8×C17C2312yes—
53C4⋊C4×C17D412yes—
54C22⋊C4×C17D412yes—
55C42×C17C2311yescharacteristic
56C42×C17C2311yescharacteristic
57C22⋊Dic17—22no—
58#63227—22no—
59#63227—22no—
60C4⋊Dic17—22no—
61C4⋊Dic17—22no—
62C4⋊Dic17—22no—
63C4⋊Dic17C2312yes—
64#63225—22no—
65#63225—22no—
66#63225—22no—
67C4×Dic17—22no—
68Dic68—42no—
69Dic68—42no—
70Dic68—42no—
71#63219—42no—
72C23×C17C2×D411yescharacteristic
73C22×D17—21no—
74C2×C4×C17—22no—
75C2×C4×C17C2×D412yes—
76C2×C4×C17C2×D412yes—
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2411yescharacteristic
79C22⋊D17—42no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—21no—
83C2×Dic17—21no—
84C2×Dic17—21no—
85C2×Dic17—42no—
86D68—22no—
87C4×D17—42no—
88Dic34—22no—
89Dic34—22no—
90Dic34—22no—
91Dic34—22no—
92Dic34—22no—
93Dic34—42no—
94C8×C17—22no—
95C8×C17—22no—
96#3223—171nomaximal, Sylow
97C22×C17—21no—
98C22×C17—22no—
99C22×C17C22×D411yescharacteristic
100C2×D17—41no—
101C2×D17—42no—
102C4×C17—21no—
103C4×C17—22no—
104C4×C17—22no—
105C4×C17H(5,𝔽2)12yes—
106Dic17—41no—
107Dic17—41no—
108Dic17—41no—
109Dic17—42no—
110Dic17—42no—
111#541—171no—
112#536—171no—
113#497—171no—
114#494—171no—
115#491—171no—
116#488—171no—
117#464—172no—
118#462—172no—
119#452—172no—
120#451—172no—
121#433D1711yescharacteristic, complemented
122C2×C17—41no—
123C2×C17#53511yescharacteristic
124C2×C17#57812yes—
125D17—81no—
126C2×#41—171no—
127C22×Q8—171no—
128C2×Q16—172no—
129C2×Q16—172no—
130C2×Q16—172no—
131C2×SD16—172no—
132C4⋊Q8—171no—
133C4⋊Q8—171no—
134C4⋊Q8—171no—
135#126—342no—
136#125—171no—
137#125—342no—
138#124—171no—
139#124—171no—
140#123—342no—
141C22⋊Q8—172no—
142C22⋊Q8—342no—
143C4×Q8—172no—
144#117C2×D1711yescharacteristic
145C4⋊C8C2×D1712yes—
146#103—172no—
147#103—172no—
148#103—172no—
149#102—172no—
150C22⋊C8C2×D1712yes—
151#97C2×D1711yescharacteristic
152C4×C8C2×D1711yescharacteristic
153C17#322311yescharacteristic, complemented, Sylow
154#41—682no—
155C2×Q8—171no—
156C2×Q8—171no—
157C2×Q8—171no—
158C2×Q8—172no—
159C2×Q8—341no—
160C2×Q8—682no—
161C2×D4—171no—
162C2×D4—341no—
163C22×C4—341no—
164C22×C4—341no—
165C22×C4C22×D1711yescharacteristic
166Q16—682no—
167Q16—682no—
168Q16—682no—
169SD16—682no—
170C2×C8C22×D1712yes—
171C2×C8C22×D1712yes—
172C4⋊C4—172no—
173C4⋊C4—342no—
174C4⋊C4—342no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4D6812yes—
180C22⋊C4—342no—
181C22⋊C4—342no—
182C22⋊C4—342no—
183C22⋊C4D6812yes—
184C42—342no—
185C42C22×D1711yescharacteristic
186C42C22×D1711yescharacteristic
187C23—341no—
188C23C2×D6811yescharacteristic
189Q8—342no—
190Q8—342no—
191Q8—342no—
192Q8—342no—
193Q8—342no—
194Q8—682no—
195D4—342no—
196D4—682no—
197C2×C4—22no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—341no—
201C2×C4—341no—
202C2×C4—341no—
203C2×C4—682no—
204C2×C4—682no—
205C2×C4C2×D6812yes—
206C2×C4C2×D6812yes—
207C2×C4C2×D6811yescharacteristic
208C2×C4C23×D1711yescharacteristic
209C8—22no—
210C8—22no—
211C22—22no—
212C22—21no—
213C22—682no—
214C22—681no—
215C22C22×D6811yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—682no—
220C4—681no—
221C4—681no—
222C4—681no—
223C4—682no—
224C4#1058769612yes—
225C2—41no—
226C2—1361no—
227C2#1176337811yescharacteristic
228C2#1176362612yes—
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#433C2
socleC22×C17C22×D4
prime radicalC2×#63252C22
hypercenter#433D17
hypocenterC17#3223
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #11763421C2semidirect
3. #11763203C2semidirect
4. #11763189C2semidirect
5. #11763178C2semidirect
6. #11763164C2semidirect
7. #11763032C2semidirect
8. #11763020C2semidirect
10. #11762967C2semidirect
11. C17×#433C2semidirect
121. #433D17semidirect
153. C17#3223semidirect

Maximal subgroups

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

subgroupindex
#117634212
#117633852
#117632032
#117631892
#117631782
#117631642
#117630322
#117630202
#117629732
#117629672
C17×#4332
#322317