Finite Group Databasebeta

← #14539439

Subgroups — #14539439

Order 2176.

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

normal subgroups only

This group has 338 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#11763417C212yesmaximal
2#11763210C211yesmaximal, characteristic, complemented
3#11763187C211yesmaximal, characteristic, complemented
4#11763176C212yesmaximal, complemented
5#11763066C212yesmaximal
6#11763051C211yesmaximal, characteristic
7#11763047C211yesmaximal, characteristic
8#11762914C212yesmaximal, complemented
9C4×Dic68C211yesmaximal, characteristic, complemented
10#11762763C211yesmaximal, characteristic
11C4×Q16×C17C211yesmaximal, characteristic, complemented
12C2×Q16×C17C2211yescharacteristic
13C4×Q8×C17C2212yes—
14#10587632C2212yes—
15#10587630C2212yes—
16C2×#63231C2212yes—
17C2×Dic68C2211yescharacteristic
18C2×#63218C2211yescharacteristic
19C2×C8×D17C2211yescharacteristic
20#10587584—22no—
21#10587583C2212yes—
22#10587576C2212yes—
23#10587575—22no—
24#10587562—22no—
25#10587558C2211yescharacteristic
26#10587556—22no—
27C4×Dic34C2212yes—
28C17×#107C2211yescharacteristic
29C17×#103C2212yes—
30C4×C8×C17C2211yescharacteristic
31#10587506C2212yes—
32#10587489C2211yescharacteristic
33#10587487C2212yes—
34#10587486C2211yescharacteristic
35C8×Dic17C2211yescharacteristic
36#10587481C2212yes—
37#10587479C2212yes—
38#10587474C2211yescharacteristic
39C2×Q8×C17C2312yes—
40C2×C4×D17C2311yescharacteristic
41C2×Dic34C2312yes—
42Q16×C17—22no—
43C2×C8×C17C2311yescharacteristic
44C2×C8×C17C2311yescharacteristic
45C4⋊C4×C17—22no—
46C4⋊C4×C17C2312yes—
47C42×C17—22no—
48C42×C17C2311yescharacteristic
49#63231—24no—
50#63227—22no—
51#63227—22no—
52#63227D411yescharacteristic
53#63227D411yescharacteristic
54C4⋊Dic17—22no—
55C4⋊Dic17—22no—
56C4⋊Dic17C2312yes—
57#63225—22no—
58#63225—22no—
59#63225—22no—
60#63225—22no—
61#63225—22no—
62#63225D411yescharacteristic
63#63225D411yescharacteristic
64C4×Dic17—22no—
65C4×Dic17C2311yescharacteristic
66C2×#3622C2311yescharacteristic
67C2×#3622C2311yescharacteristic
68Dic68—22no—
69#63218—22no—
70C8×D17—22no—
71C22×D17C2×D411yescharacteristic
72Q8×C17—22no—
73Q8×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17C2×D411yescharacteristic
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2411yescharacteristic
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17C2×D411yescharacteristic
82C2×Dic17C2×D411yescharacteristic
83C2×Dic17C2×D411yescharacteristic
84C4×D17—21no—
85C4×D17—21no—
86Dic34—22no—
87Dic34—22no—
88C8×C17—21no—
89C8×C17#4112yes—
90#3622—21no—
91#3622#4112yes—
92#3177—171nomaximal, Sylow
93C22×C17C22×D411yescharacteristic
94C2×D17—21no—
95C2×D17—21no—
96C2×D17—21no—
97C4×C17—21no—
98C4×C17—21no—
99C4×C17—42no—
100C4×C17—42no—
101C4×C17C2×#4111yescharacteristic
102C4×C17C2×#4111yescharacteristic
103Dic17—21no—
104Dic17—21no—
105Dic17—21no—
106Dic17—42no—
107Dic17—42no—
108#541—172no—
109#499—171no—
110#494—171no—
111#491—172no—
112#470—172no—
113#467—171no—
114#467—171no—
115#441—172no—
116C4×Q16—171no—
117C4×Q16D1711yescharacteristic, complemented
118#405—171no—
119C2×C17#53211yescharacteristic
120C2×C17#57811yescharacteristic
121C2×C17#57811yescharacteristic
122D17—41no—
123C2×Q16—172no—
124C2×Q16—171no—
125C2×Q16C2×D1711yescharacteristic
126C2×M16—171no—
127C22×C8—171no—
128C4⋊Q8—172no—
129#126—342no—
130#126—342no—
131#125—172no—
132#125—342no—
133#125—342no—
134C22⋊Q8—172no—
135C22⋊Q8—172no—
136C4×Q8—172no—
137C4×Q8C2×D1712yes—
138#117—171no—
139#107—171no—
140#107C2×D1711yescharacteristic
141#106—172no—
142#103—172no—
143#103—172no—
144#103—172no—
145#103C2×D1712yes—
146#97—171no—
147#97—171no—
148C4×C8—171no—
149C4×C8C2×D1711yescharacteristic
150C17#317711yescharacteristic, complemented, Sylow
151C2×Q8—172no—
152C2×Q8C22×D1712yes—
153C22×C4—171no—
154Q16—22no—
155Q16—344no—
156Q16—342no—
157M16—342no—
158C2×C8—171no—
159C2×C8—171no—
160C2×C8—342no—
161C2×C8C22×D1711yescharacteristic
162C2×C8C22×D1711yescharacteristic
163C4⋊C4—22no—
164C4⋊C4—171no—
165C4⋊C4—171no—
166C4⋊C4—172no—
167C4⋊C4—342no—
168C4⋊C4—342no—
169C4⋊C4—342no—
170C4⋊C4—342no—
171C4⋊C4—342no—
172C4⋊C4—342no—
173C4⋊C4—342no—
174C4⋊C4C22×D1712yes—
175C22⋊C4—171no—
176C22⋊C4—171no—
177C22⋊C4—342no—
178C22⋊C4—342no—
179C42—22no—
180C42—171no—
181C42—342no—
182C42C22×D1711yescharacteristic
183C23—171no—
184Q8—22no—
185Q8—22no—
186Q8—342no—
187Q8—342no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—171no—
191C2×C4—171no—
192C2×C4—171no—
193C2×C4—341no—
194C2×C4—341no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4D4×D1711yescharacteristic
198C2×C4D4×D1711yescharacteristic
199C2×C4C23×D1711yescharacteristic
200C8—21no—
201C8—172no—
202C8—341no—
203C8#6325212yes—
204C22—341no—
205C22—341no—
206C22—341no—
207C22C2×D4×D1711yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—341no—
213C4—341no—
214C4—341no—
215C4—682no—
216C4—682no—
217C4C2×#6325211yescharacteristic
218C4#41×D1711yescharacteristic
219C2—681no—
220C2#1176335611yescharacteristic
221C2#1176362211yescharacteristic
222C2#1176362311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC4×Q16×C17C2
socleC22×C17C22×D4
prime radicalC22×D17C2×D4
hypercenterC4×Q16D17
hypocenterC17#3177
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #11763210C2semidirect
3. #11763187C2semidirect
4. #11763176C2semidirect
8. #11762914C2semidirect
9. C4×Dic68C2semidirect
11. C4×Q16×C17C2semidirect
117. C4×Q16D17semidirect
150. C17#3177semidirect

Maximal subgroups

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

subgroupindex
#117634172
#117632102
#117631872
#117631762
#117630662
#117630512
#117630472
#117629142
C4×Dic682
#117627632
C4×Q16×C172
#317717