Finite Group Databasebeta

← #14539946

Subgroups — #14539946

Order 2176.

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

normal subgroups only

This group has 326 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#11763364C211yesmaximal, characteristic
2#11763364C211yesmaximal, characteristic
3#11763212C211yesmaximal, characteristic, complemented
4#11763161C211yesmaximal, characteristic, complemented
5#11763160C211yesmaximal, characteristic, complemented
6#11763153C214yesmaximal
7#11763119C214yesmaximal, complemented
8#11762860C211yesmaximal, characteristic
9C17×#498C211yesmaximal, characteristic, complemented
10C17×C4⋊Q8C2211yescharacteristic
11C17×C4⋊Q8C2211yescharacteristic
12#10587632—21no—
13#10587632—21no—
14Q8×Dic17—21no—
15Q8×Dic17—21no—
16#10587583—21no—
17#10587583C2212yes—
18#10587578—21no—
19#10587578C2212yes—
20#10587576—21no—
21#10587576C2212yes—
22#10587573—21no—
23#10587573C2212yes—
24C42×D17C2211yescharacteristic
25#10587555C2211yescharacteristic
26#10587555C2211yescharacteristic
27C17×#106C2214yes—
28C4×C8×C17C2211yescharacteristic
29#10587491C2212yes—
30#10587487C2214yes—
31#10587485C2212yes—
32#10587481C2214yes—
33#10587478C2214yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17—21no—
37C2×C4×D17D411yescharacteristic
38C2×C4×D17D411yescharacteristic
39C2×C4×D17C2311yescharacteristic
40C2×Dic34—21no—
41C2×Dic34C2312yes—
42C2×C8×C17C2312yes—
43C4⋊C4×C17—21no—
44C4⋊C4×C17—21no—
45C4⋊C4×C17C2312yes—
46C4⋊C4×C17C2312yes—
47C42×C17C2311yescharacteristic
48#63227—22no—
49#63227—22no—
50C4⋊Dic17—21no—
51C4⋊Dic17—21no—
52C4⋊Dic17—21no—
53C4⋊Dic17—22no—
54C4⋊Dic17—22no—
55C4⋊Dic17C2312yes—
56#63225—22no—
57#63225—22no—
58C4×Dic17—22no—
59C4×Dic17—22no—
60C4×Dic17D411yescharacteristic
61C4×Dic17D411yescharacteristic
62C4×Dic17C2311yescharacteristic
63C2×#3622C2312yes—
64C22×D17C2×D411yescharacteristic
65Q8×C17—42no—
66Q8×C17—42no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17C2×D411yescharacteristic
70C2×C4×C17C2×D411yescharacteristic
71C2×C4×C17C2411yescharacteristic
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17C2×D411yescharacteristic
75C2×Dic17C2×D411yescharacteristic
76C2×Dic17C2×D411yescharacteristic
77C4×D17—22no—
78C4×D17—22no—
79C4×D17—22no—
80Dic34—24no—
81Dic34—42no—
82C8×C17—22no—
83#3622—22no—
84#3231—171nomaximal, Sylow
85C22×C17C22×D411yescharacteristic
86C2×D17—21no—
87C2×D17—21no—
88C2×D17—21no—
89C4×C17—42no—
90C4×C17—42no—
91C4×C17#13512yes—
92C4×C17#13712yes—
93C4×C172−1+412yes—
94Dic17—21no—
95Dic17—21no—
96Dic17—21no—
97Dic17—42no—
98Dic17—42no—
99#533—171no—
100#533—171no—
101#500—171no—
102#498D1711yescharacteristic, complemented
103#487—171no—
104#487—171no—
105#485—174no—
106#479—174no—
107#431—171no—
108C2×C17#53611yescharacteristic
109C2×C17C2×#13511yescharacteristic
110C2×C17C2×#13711yescharacteristic
111D17—41no—
112C4⋊Q8—171no—
113C4⋊Q8—171no—
114C4⋊Q8C2×D1711yescharacteristic
115C4⋊Q8C2×D1711yescharacteristic
116#125—172no—
117#125—341no—
118C22⋊Q8—172no—
119C22⋊Q8—341no—
120C22⋊Q8—341no—
121C22⋊Q8—341no—
122C4×Q8—172no—
123C4×Q8—341no—
124C4×Q8—341no—
125C4×Q8—341no—
126#117—172no—
127#117—341no—
128C2×C42—171no—
129#107—174no—
130#106C2×D1714yes—
131C4⋊C8—172no—
132#103—174no—
133#103—174no—
134C22⋊C8—172no—
135C4×C8—171no—
136C4×C8C2×D1711yescharacteristic
137C17#323111yescharacteristic, complemented, Sylow
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—172no—
141C2×Q8—341no—
142C22×C4—171no—
143C22×C4—171no—
144C22×C4—171no—
145C2×C8—172no—
146C2×C8C22×D1712yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—172no—
150C4⋊C4—342no—
151C4⋊C4—342no—
152C4⋊C4—342no—
153C4⋊C4—342no—
154C4⋊C4—341no—
155C4⋊C4—341no—
156C4⋊C4—341no—
157C4⋊C4C22×D1712yes—
158C4⋊C4C22×D1712yes—
159C22⋊C4—342no—
160C22⋊C4—342no—
161C42—171no—
162C42—171no—
163C42—171no—
164C42—342no—
165C42—342no—
166C42C22×D1711yescharacteristic
167C23—171no—
168Q8—42no—
169Q8—42no—
170Q8—344no—
171Q8—682no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—171no—
175C2×C4—171no—
176C2×C4—171no—
177C2×C4—342no—
178C2×C4—342no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4D4×D1711yescharacteristic
183C2×C4D4×D1711yescharacteristic
184C2×C4C23×D1711yescharacteristic
185C8—22no—
186C8—342no—
187C22—341no—
188C22—341no—
189C22—341no—
190C22C2×D4×D1711yescharacteristic
191C4—42no—
192C4—42no—
193C4—341no—
194C4—341no—
195C4—341no—
196C4—682no—
197C4—682no—
198C4#1058760112yes—
199C4#1058760212yes—
200C4#1058769412yes—
201C2—681no—
202C2#1176338411yescharacteristic
203C2C2×#1058760211yescharacteristic
204C2C2×#1058760111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#498C2
socleC22×C17C22×D4
prime radicalC22×D17C2×D4
hypercenter#498D17
hypocenterC17#3231
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
3. #11763212C2semidirect
4. #11763161C2semidirect
5. #11763160C2semidirect
7. #11763119C2semidirect
9. C17×#498C2semidirect
102. #498D17semidirect
137. C17#3231semidirect

Maximal subgroups

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

subgroupindex
#117633642
#117632122
#117631612
#117631602
#117631532
#117631192
#117628602
C17×#4982
#323117