Finite Group Databasebeta

← #14539314

Subgroups — #14539314

Order 2176.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#11763440C211yesmaximal, characteristic
2#11763437C211yesmaximal, characteristic
3#11763217C212yesmaximal, complemented
4#11763216C212yesmaximal, complemented
5#11763210C212yesmaximal, complemented
6#11763206C212yesmaximal, complemented
7#11762851C211yesmaximal, characteristic
8#11762845C211yesmaximal, characteristic
9C2×#10587479C211yesmaximal, characteristic
10#11762767C211yesmaximal, characteristic
11C17×#428C211yesmaximal, characteristic
12C2×M16×C17C2211yescharacteristic
13C17×#117C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15#10587612C2211yescharacteristic
16#10587612C2211yescharacteristic
17C2×C4⋊Dic17C2211yescharacteristic
18#10587610—22no—
19C22×#3622C2211yescharacteristic
20#10587576—21no—
21#10587576C2212yes—
22#10587576C2212yes—
23#10587574—21no—
24#10587574C2212yes—
25#10587574C2212yes—
26#10587564—22no—
27#10587556—21no—
28#10587555—21no—
29C17×#107C2212yes—
30C17×#106C2212yes—
31#10587489C2212yes—
32#10587488C2212yes—
33#10587486C2212yes—
34C8×Dic17C2212yes—
35#10587479C2211yescharacteristic
36#10587479C2212yes—
37#10587479C2211yescharacteristic
38#10587479C2212yes—
39#10587478C2211yescharacteristic
40#10587478C2211yescharacteristic
41C22×C4×C17—21no—
42C22×C4×C17C2311yescharacteristic
43C22×Dic17—21no—
44C2×Dic34—22no—
45C2×Dic34—22no—
46M16×C17Q814yescomplemented
47C2×C8×C17C2312yes—
48C4⋊C4×C17—21no—
49C4⋊C4×C17C2311yescharacteristic
50C4⋊C4×C17C2311yescharacteristic
51C4⋊C4×C17C2312yes—
52C22⋊C4×C17—21no—
53C42×C17—21no—
54C22⋊Dic17—21no—
55C22⋊Dic17D412yes—
56C4⋊Dic17—21no—
57C4⋊Dic17—22no—
58C4⋊Dic17—22no—
59C4⋊Dic17D411yescharacteristic
60C4⋊Dic17D411yescharacteristic
61C4⋊Dic17C2311yescharacteristic
62C4⋊Dic17C2311yescharacteristic
63C4⋊Dic17C2312yes—
64#63225—22no—
65#63225—22no—
66#63225—22no—
67#63225—22no—
68C4×Dic17—21no—
69C4×Dic17C2312yes—
70C2×#3622Q814yescomplemented
71C2×#3622C2311yescharacteristic
72C2×#3622C2311yescharacteristic
73C23×C17C2×D411yescharacteristic
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—81no—
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2×Q812yes—
80C2×C4×C17C2×Q812yes—
81C2×C4×C17C2411yescharacteristic
82C2×Dic17—21no—
83C2×Dic17—21no—
84C2×Dic17—22no—
85C2×Dic17—81no—
86C2×Dic17C2×D412yes—
87C2×Dic17C2×D412yes—
88Dic34—82no—
89Dic34—82no—
90C8×C17C2×Q814yes—
91#3622C2×Q812yes—
92#3622C2×Q812yes—
93#3163—171nomaximal, Sylow
94C22×C17—21no—
95C22×C17—21no—
96C22×C17C4⋊Q812yes—
97C22×C17C22×D411yescharacteristic
98C4×C17—41no—
99C4×C17—42no—
100C4×C17—41no—
101C4×C17C4⋊Q812yes—
102C4×C17C22×Q811yescharacteristic
103C4×C17C22×Q811yescharacteristic
104Dic17—22no—
105Dic17—22no—
106Dic17—41no—
107Dic17—42no—
108Dic17—41no—
109#544—171no—
110#544—171no—
111#501—172no—
112#501—172no—
113#499—172no—
114#498—172no—
115#428—171no—
116#428D1711yescharacteristic
117#427—171no—
118C2×#106—171no—
119#405—171no—
120C2×C17—22no—
121C2×C17C2×C4⋊Q811yescharacteristic
122C2×C17#57711yescharacteristic
123C2×C17#57711yescharacteristic
124C2×M16C2×D1711yescharacteristic
125C22×C8—171no—
126C4⋊Q8—172no—
127C4⋊Q8—172no—
128C4⋊Q8—341no—
129C4⋊Q8—341no—
130#125—172no—
131#125—172no—
132#125—341no—
133#125—341no—
134C22⋊Q8—342no—
135C22⋊Q8—342no—
136#117—171no—
137#117—171no—
138#117C2×D1711yescharacteristic
139C2×C4⋊C4—171no—
140C2×C4⋊C4C2×D1711yescharacteristic
141#107—172no—
142#107—171no—
143#107—171no—
144#107C2×D1712yes—
145#106—172no—
146#106—172no—
147#106—172no—
148#106—171no—
149#106—171no—
150#106C2×D1712yes—
151#97—172no—
152C4×C8—172no—
153C17#316311yescharacteristic, complemented, Sylow
154C2×Q8—342no—
155C2×Q8—342no—
156C22×C4—21no—
157C22×C4—341no—
158C22×C4C22×D1711yescharacteristic
159M16Dic3414yescomplemented
160C2×C8—174no—
161C2×C8—171no—
162C2×C8—171no—
163C2×C8C22×D1712yes—
164C4⋊C4—21no—
165C4⋊C4—171no—
166C4⋊C4—171no—
167C4⋊C4—172no—
168C4⋊C4—171no—
169C4⋊C4—171no—
170C4⋊C4—342no—
171C4⋊C4—342no—
172C4⋊C4—342no—
173C4⋊C4—342no—
174C4⋊C4—342no—
175C4⋊C4—342no—
176C4⋊C4—341no—
177C4⋊C4C22×D1712yes—
178C4⋊C4C22×D1711yescharacteristic
179C4⋊C4C22×D1711yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—172no—
182C22⋊C4—341no—
183C42—21no—
184C42—172no—
185C42—341no—
186C23D4×D1711yescharacteristic
187Q8—1362no—
188Q8—1362no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—172no—
194C2×C4—172no—
195C2×C4—341no—
196C2×C4—341no—
197C2×C4—342no—
198C2×C4—1361no—
199C2×C4C2×Dic3412yes—
200C2×C4D4×D1711yescharacteristic
201C2×C4Q8×D1712yes—
202C2×C4C23×D1711yescharacteristic
203C8—172no—
204C8—172no—
205C8C2×Dic3414yes—
206C22—21no—
207C22—21no—
208C22#1058757412yes—
209C22C2×D4×D1711yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—342no—
214C4—342no—
215C4—682no—
216C4—681no—
217C4—681no—
218C4#1058757412yes—
219C4C22×Dic3411yescharacteristic
220C4C2×Q8×D1711yescharacteristic
221C2—22no—
222C2C2×#1058757411yescharacteristic
223C2#1176361811yescharacteristic
224C2#1176361911yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#428C2
socleC22×C17C22×D4
prime radicalC23×C17C2×D4
hypercenter#428D17
hypocenterC17#3163
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #11763217C2semidirect
4. #11763216C2semidirect
5. #11763210C2semidirect
6. #11763206C2semidirect
46. M16×C17Q8semidirect
70. C2×#3622Q8semidirect
153. C17#3163semidirect
159. M16Dic34semidirect

Maximal subgroups

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

subgroupindex
#117634402
#117634372
#117632172
#117632162
#117632102
#117632062
#117628512
#117628452
C2×#105874792
#117627672
C17×#4282
#316317