Finite Group Databasebeta

← #14541663

Subgroups — #14541663

Order 2176.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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#11763516C211yesmaximal, characteristic
3#11763210C212yesmaximal, complemented
4#11763205C211yesmaximal, characteristic, complemented
5#11763115C212yesmaximal, complemented
6#11763107C212yesmaximal
7#11763105C211yesmaximal, characteristic, complemented
8#11762933C211yesmaximal, characteristic, complemented
9#11762900C212yesmaximal
10C4×#63229C211yesmaximal, characteristic
11C17×#474C211yesmaximal, characteristic
12C17×C4⋊Q8C2211yescharacteristic
13C4×D4×C17C2211yescharacteristic
14Q8×Dic17—21no—
15#10587630—21no—
16D4×Dic17C2212yes—
17C2×#63229C2211yescharacteristic
18C2×C4⋊Dic17—21no—
19#10587610—21no—
20#10587576C2212yes—
21#10587576C2212yes—
22#10587575—22no—
23#10587573C2212yes—
24#10587564—22no—
25#10587555C2211yescharacteristic
26C4×Dic34C2211yescharacteristic
27C4×Dic34C2211yescharacteristic
28C17×#107C2212yes—
29C17×C4⋊C8C2211yescharacteristic
30C17×#102C2212yes—
31#10587503C2211yescharacteristic
32#10587489C2212yes—
33#10587487C2212yes—
34#10587485C2212yes—
35C8×Dic17C2212yes—
36#10587481C2211yescharacteristic
37#10587479C2212yes—
38#10587479C2212yes—
39#10587479C2211yescharacteristic
40C4×#3622C2211yescharacteristic
41C2×Q8×C17—21no—
42C2×D4×C17C2311yescharacteristic
43C22×C4×C17—21no—
44C22×Dic17—22no—
45C2×Dic34—21no—
46C2×Dic34D411yescharacteristic
47C2×Dic34C2311yescharacteristic
48C2×C8×C17C2312yes—
49C4⋊C4×C17—21no—
50C4⋊C4×C17C2312yes—
51C4⋊C4×C17C2311yescharacteristic
52C22⋊C4×C17—21no—
53C42×C17C2311yescharacteristic
54C22⋊Dic17—22no—
55#63229Q814yescomplemented
56C4⋊Dic17—21no—
57C4⋊Dic17—22no—
58C4⋊Dic17—22no—
59C4⋊Dic17D411yescharacteristic
60C4⋊Dic17C2312yes—
61C4⋊Dic17C2311yescharacteristic
62#63225—21no—
63#63225—22no—
64#63225—22no—
65#63225—22no—
66#63225D412yes—
67C4×Dic17—21no—
68C4×Dic17—22no—
69C4×Dic17C2312yes—
70C2×#3622C2311yescharacteristic
71C2×#3622C2311yescharacteristic
72C23×C17—21no—
73Q8×C17—42no—
74D4×C17—21no—
75D4×C17C2×Q812yes—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—42no—
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2411yescharacteristic
82C2×Dic17—21no—
83C2×Dic17—22no—
84C2×Dic17—21no—
85C2×Dic17—82no—
86C2×Dic17C2×D412yes—
87C2×Dic17C2×D412yes—
88Dic34—22no—
89Dic34—21no—
90Dic34—81no—
91Dic34C2×Q812yes—
92C8×C17—22no—
93#3622—21no—
94#3622C2×Q812yes—
95#3381—171nomaximal, Sylow
96C22×C17—22no—
97C22×C17—41no—
98C22×C17—41no—
99C22×C17C22×D411yescharacteristic
100C4×C17—21no—
101C4×C17—41no—
102C4×C17—42no—
103C4×C17#13512yes—
104C4×C17C22×Q811yescharacteristic
105C4×C172−1+411yescharacteristic
106Dic17—22no—
107Dic17—22no—
108Dic17—22no—
109Dic17—42no—
110Dic17—41no—
111#557—171no—
112#554—171no—
113#499—172no—
114#498—171no—
115#479—172no—
116#478—172no—
117#477—171no—
118#474D1711yescharacteristic
119C8×Q8—171no—
120C4×SD16—171no—
121C4×SD16—172no—
122C2×C17—42no—
123C2×C17D4×Q811yescharacteristic
124C2×C17C2×#13511yescharacteristic
125C2×C17#57711yescharacteristic
126C2×SD16—171no—
127C4⋊Q8—171no—
128C4⋊Q8—341no—
129C4⋊Q8C2×D1711yescharacteristic
130#125—172no—
131#125—172no—
132#125—342no—
133C22⋊Q8—342no—
134C22⋊Q8—341no—
135C4×Q8—171no—
136C4×Q8—172no—
137C4×Q8—171no—
138C4×Q8—341no—
139C4×D4—172no—
140C4×D4C2×D1711yescharacteristic
141C2×C4⋊C4—341no—
142#107—172no—
143#107C2×D1712yes—
144#106—172no—
145#106—171no—
146#106—172no—
147C4⋊C8—172no—
148C4⋊C8C2×D1711yescharacteristic
149#103—172no—
150#103—171no—
151#102—171no—
152#102C2×D1712yes—
153C4×C8—172no—
154C4×C8—171no—
155C17#338111yescharacteristic, complemented, Sylow
156C2×Q8—21no—
157C2×Q8—171no—
158C2×Q8—171no—
159C2×Q8—341no—
160C2×D4C22×D1711yescharacteristic
161C22×C4—21no—
162C22×C4—342no—
163SD16—174no—
164C2×C8—171no—
165C2×C8—171no—
166C2×C8C22×D1712yes—
167C4⋊C4—21no—
168C4⋊C4—171no—
169C4⋊C4—172no—
170C4⋊C4—171no—
171C4⋊C4—172no—
172C4⋊C4—342no—
173C4⋊C4—342no—
174C4⋊C4—342no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—341no—
178C4⋊C4—341no—
179C4⋊C4C22×D1712yes—
180C4⋊C4C22×D1711yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—342no—
183C42—172no—
184C42—342no—
185C42—341no—
186C42C22×D1711yescharacteristic
187C23—21no—
188Q8—42no—
189Q8—172no—
190Q8—341no—
191Q8—342no—
192Q8—1361no—
193D4—21no—
194D4Q8×D1712yes—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—172no—
199C2×C4—172no—
200C2×C4—341no—
201C2×C4—341no—
202C2×C4—342no—
203C2×C4—1362no—
204C2×C4D4×D1711yescharacteristic
205C2×C4D4×D1711yescharacteristic
206C2×C4C23×D1711yescharacteristic
207C8—22no—
208C8—172no—
209C8—341no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22C2×D4×D1711yescharacteristic
214C4—21no—
215C4—41no—
216C4—42no—
217C4—342no—
218C4—342no—
219C4—342no—
220C4—682no—
221C4—681no—
222C4#1058759812yes—
223C4C2×Q8×D1711yescharacteristic
224C4#1058769711yescharacteristic
225C2—42no—
226C2#1176347211yescharacteristic
227C2C2×#1058759811yescharacteristic
228C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#474C2
socleC22×C17C22×D4
prime radicalC2×D4×C17C23
hypercenter#474D17
hypocenterC17#3381
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #11763534C2semidirect
3. #11763210C2semidirect
4. #11763205C2semidirect
5. #11763115C2semidirect
7. #11763105C2semidirect
8. #11762933C2semidirect
55. #63229Q8semidirect
155. C17#3381semidirect

Maximal subgroups

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

subgroupindex
#117635342
#117635162
#117632102
#117632052
#117631152
#117631072
#117631052
#117629332
#117629002
C4×#632292
C17×#4742
#338117