Finite Group Databasebeta

← #14539498

Subgroups — #14539498

Order 2176.

402 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 129 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#11763366C211yesmaximal, characteristic, complemented
2#11763212C211yesmaximal, characteristic, complemented
3#11763181C211yesmaximal, characteristic, complemented
4#11763161C211yesmaximal, characteristic, complemented
5#11763155C211yesmaximal, characteristic, complemented
6#11763053C214yesmaximal, complemented
7#11762909C214yesmaximal, complemented
8C4×#63223C211yesmaximal, characteristic, complemented
9C17×#532C211yesmaximal, characteristic
10C2×#41×C17—21no—
11C17×C4⋊Q8C2211yescharacteristic
12C17×#127C2211yescharacteristic
13C17×#124—21no—
14C17×#121—21no—
15C17×#121C2212yes—
16C4×D4×C17—21no—
17C4×D4×C17C2212yes—
18C2×C42×C17C2211yescharacteristic
19C2×#63229C2214yes—
20#10587612—21no—
21#10587610—21no—
22#10587610C2212yes—
23C2×#63223C2212yes—
24#10587556—21no—
25#10587555C2211yescharacteristic
26#10587555C2211yescharacteristic
27C4×Dic34—21no—
28C4×Dic34C2212yes—
29#10587503C2214yes—
30#10587481C2214yes—
31#10587478C2214yes—
32#10587474C2212yes—
33C4×#3622C2211yescharacteristic
34C4×#3622C2211yescharacteristic
35#41×C17—44no—
36C2×Q8×C17—21no—
37C2×D4×C17—22no—
38C2×D4×C17—21no—
39C2×D4×C17—21no—
40C2×D4×C17C2312yes—
41C22×C4×C17—22no—
42C22×C4×C17D411yescharacteristic
43C22×C4×C17D411yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×Dic34—21no—
46C2×Dic34C2312yes—
47C4⋊C4×C17—21no—
48C4⋊C4×C17C2312yes—
49C22⋊C4×C17—22no—
50C22⋊C4×C17—22no—
51C42×C17D411yescharacteristic
52C42×C17D411yescharacteristic
53C42×C17C2311yescharacteristic
54C42×C17C2311yescharacteristic
55C22⋊Dic17—22no—
56#63229—28no—
57C4⋊Dic17—21no—
58C4⋊Dic17—21no—
59C4⋊Dic17—21no—
60C4⋊Dic17C2312yes—
61#63225—22no—
62#63225—22no—
63C4×Dic17—22no—
64#63223—24no—
65C2×#3622C2312yes—
66C2×#3622C2312yes—
67C23×C17—22no—
68C23×C17C2×D411yescharacteristic
69Q8×C17—42no—
70D4×C17—24no—
71D4×C17—44no—
72D4×C17—42no—
73D4×C17—42no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—44no—
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2×D411yescharacteristic
82C2×C4×C17C2×D411yescharacteristic
83C2×C4×C17C2×D411yescharacteristic
84C2×C4×C17C2411yescharacteristic
85C2×Dic17—22no—
86C2×Dic17—22no—
87Dic34—24no—
88Dic34—42no—
89#3622—22no—
90#3622#4114yescomplemented
91#3182—171nomaximal, Sylow
92C22×C17—21no—
93C22×C17—21no—
94C22×C17—21no—
95C22×C17—42no—
96C22×C17—42no—
97C22×C17—42no—
98C22×C17C22×D411yescharacteristic
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17—42no—
103C4×C17AGL(1,ℤ/8)12yes—
104C4×C17#13712yes—
105C4×C17C2×#4112yes—
106Dic17—42no—
107Dic17—42no—
108#533—171no—
109#532D1711yescharacteristic
110#500—171no—
111#492—171no—
112#487—171no—
113#486—171no—
114#468—174no—
115#440—174no—
116C4×M16—171no—
117C2×C17—41no—
118C2×C17—82no—
119C2×C17#53211yescharacteristic
120C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C17C2×#13711yescharacteristic
122C2×#41—21no—
123C2×SD16—174no—
124C2×M16—172no—
125C4⋊Q8—171no—
126C4⋊Q8—171no—
127C4⋊Q8C2×D1711yescharacteristic
128#127C2×D1711yescharacteristic
129#125—341no—
130#124—21no—
131C22⋊Q8—172no—
132C22⋊Q8—341no—
133#121—21no—
134#121C2×D1712yes—
135C4×Q8—172no—
136C4×Q8—341no—
137C4×D4—21no—
138C4×D4C2×D1712yes—
139#117—341no—
140C2×C42C2×D1711yescharacteristic
141#107—174no—
142#103—174no—
143#102—174no—
144#97—172no—
145C4×C8—171no—
146C4×C8—171no—
147C17#318211yescharacteristic, complemented, Sylow
148#41—44no—
149C2×Q8—21no—
150C2×Q8—172no—
151C2×Q8—341no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4C22×D1712yes—
156C22×C4—22no—
157C22×C4C22⋊D1711yescharacteristic
158C22×C4C22⋊D1711yescharacteristic
159C22×C4C22×D1711yescharacteristic
160SD16—348no—
161M16—344no—
162C2×C8—172no—
163C2×C8—172no—
164C4⋊C4—21no—
165C4⋊C4—172no—
166C4⋊C4—341no—
167C4⋊C4—341no—
168C4⋊C4—342no—
169C4⋊C4—342no—
170C4⋊C4—341no—
171C4⋊C4C22×D1712yes—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—342no—
175C42—342no—
176C42C22⋊D1711yescharacteristic
177C42C22⋊D1711yescharacteristic
178C42C22×D1711yescharacteristic
179C42C22×D1711yescharacteristic
180C23—22no—
181C23C2×C22⋊D1711yescharacteristic
182Q8—42no—
183Q8—344no—
184Q8—682no—
185D4—24no—
186D4—44no—
187D4—42no—
188D4—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—44no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4C2×C22⋊D1711yescharacteristic
197C2×C4C2×C22⋊D1711yescharacteristic
198C2×C4C2×C22⋊D1711yescharacteristic
199C2×C4C2×C22⋊D1711yescharacteristic
200C2×C4C2×C22⋊D1711yescharacteristic
201C2×C4C23×D1711yescharacteristic
202C8—174no—
203C8—342no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C22×C22⋊D1711yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—682no—
216C4—682no—
217C4#1058761812yes—
218C4#1058763712yes—
219C4#41×D1712yes—
220C2—41no—
221C2—82no—
222C2#1176335511yescharacteristic
223C2C2×#1058761811yescharacteristic
224C2C2×#1058763711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#532C2
socleC22×C17C22×D4
hypercenter#532D17
hypocenterC17#3182
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #11763366C2semidirect
2. #11763212C2semidirect
3. #11763181C2semidirect
4. #11763161C2semidirect
5. #11763155C2semidirect
6. #11763053C2semidirect
7. #11762909C2semidirect
8. C4×#63223C2semidirect
90. #3622#41semidirect
147. C17#3182semidirect

Maximal subgroups

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

subgroupindex
#117633662
#117632122
#117631812
#117631612
#117631552
#117630532
#117629092
C4×#632232
C17×#5322
#318217