Finite Group Databasebeta

← #14541655

Subgroups — #14541655

Order 2176.

338 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 133 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#11763519C211yesmaximal, characteristic
2#11763208C212yesmaximal, complemented
3#11763205C211yesmaximal, characteristic, complemented
4#11763116C212yesmaximal, complemented
5#11763106C212yesmaximal
6#11763104C211yesmaximal, characteristic, complemented
7#11763086C211yesmaximal, characteristic
8Q8×#3622C211yesmaximal, characteristic, complemented
9C4×#63230C211yesmaximal, characteristic
10C4×#63230C212yesmaximal
11C17×#557C211yesmaximal, characteristic, complemented
12C17×C4⋊Q8—21no—
13C17×C4⋊Q8C2211yescharacteristic
14C17×#125—22no—
15C17×#125C2212yes—
16C4×Q8×C17—21no—
17C4×Q8×C17C2211yescharacteristic
18C4×Q8×C17C2212yes—
19C4×Q8×C17C2211yescharacteristic
20#10587632—21no—
21C2×#63230C2211yescharacteristic
22#10587582—22no—
23C4⋊C4×D17—21no—
24C4×D68C2211yescharacteristic
25C4×D68C2212yes—
26#10587556C2212yes—
27#10587555C2211yescharacteristic
28#10587506C2212yes—
29#10587506C2211yescharacteristic
30#10587480C2212yes—
31#10587480C2211yescharacteristic
32#10587479C2212yes—
33#10587479C2212yes—
34#10587479C2211yescharacteristic
35#10587478C2212yes—
36#10587478C2212yes—
37C4⋊#3622C2211yescharacteristic
38C4⋊#3622C2212yes—
39C4×#3622C2211yescharacteristic
40C4×#3622C2212yes—
41C2×Q8×C17—21no—
42C2×Q8×C17D411yescharacteristic
43C2×Q8×C17C2311yescharacteristic
44C2×D68C2311yescharacteristic
45C2×C4×D17—21no—
46C2×C4×D17—22no—
47C2×Dic34—21no—
48C4⋊C4×C17—21no—
49C4⋊C4×C17—21no—
50C4⋊C4×C17—22no—
51C4⋊C4×C17—22no—
52C4⋊C4×C17—22no—
53C4⋊C4×C17D412yes—
54C4⋊C4×C17D411yescharacteristic
55C4⋊C4×C17C2311yescharacteristic
56C4⋊C4×C17C2312yes—
57C42×C17—22no—
58C42×C17—21no—
59C42×C17C2312yes—
60C42×C17C2311yescharacteristic
61#63230Q814yescomplemented
62#63227—21no—
63#63227—22no—
64C4⋊Dic17—21no—
65C4⋊Dic17C2311yescharacteristic
66C4⋊Dic17C2312yes—
67#63225—22no—
68#63225—22no—
69C2×#3622C2312yes—
70C2×#3622C2311yescharacteristic
71C2×#3622C2311yescharacteristic
72C22×D17—21no—
73Q8×C17—22no—
74Q8×C17—21no—
75Q8×C17—42no—
76Q8×C17C2×Q812yes—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17C2×D412yes—
81C2×C4×C17C2×D411yescharacteristic
82C2×C4×C17C2×D412yes—
83C2×C4×C17C2×D411yescharacteristic
84C2×C4×C17C2411yescharacteristic
85C2×Dic17—22no—
86C2×Dic17—21no—
87D68—21no—
88D68C2×Q812yes—
89C4×D17—42no—
90C4×D17—82no—
91Dic34—81no—
92#3622—22no—
93#3622—21no—
94#3622C2×Q812yes—
95#3381—171nomaximal, Sylow
96C22×C17C22×D411yescharacteristic
97C2×D17—22no—
98C2×D17—41no—
99C2×D17—41no—
100C4×C17—22no—
101C4×C17—22no—
102C4×C17—21no—
103C4×C17—22no—
104C4×C17—42no—
105C4×C17—41no—
106C4×C17#13512yes—
107C4×C17C22×Q811yescharacteristic
108C4×C172−1+411yescharacteristic
109Dic17—41no—
110Dic17—42no—
111#557D1711yescharacteristic, complemented
112#554—171no—
113#499—172no—
114#498—171no—
115#479—172no—
116#478—172no—
117#477—171no—
118#474—171no—
119C8×Q8—171no—
120C4×SD16—171no—
121C4×SD16—172no—
122C2×C17D4×Q811yescharacteristic
123C2×C17C2×#13511yescharacteristic
124C2×C17#57711yescharacteristic
125D17—42no—
126C2×SD16—171no—
127C4⋊Q8—21no—
128C4⋊Q8—171no—
129C4⋊Q8C2×D1711yescharacteristic
130#125—22no—
131#125—172no—
132#125C2×D1712yes—
133C22⋊Q8—341no—
134C22⋊Q8—342no—
135C4×Q8—21no—
136C4×Q8C2×D1711yescharacteristic
137C4×Q8C2×D1712yes—
138C4×Q8C2×D1711yescharacteristic
139C4×D4—171no—
140C4×D4—172no—
141C2×C4⋊C4—341no—
142#107—172no—
143#107—172no—
144#106—172no—
145#106—171no—
146#106—172no—
147C4⋊C8—171no—
148C4⋊C8—172no—
149#103—172no—
150#103—171no—
151#102—172no—
152#102—171no—
153C4×C8—171no—
154C4×C8—172no—
155C17#338111yescharacteristic, complemented, Sylow
156C2×Q8—21no—
157C2×Q8—341no—
158C2×Q8C22⋊D1711yescharacteristic
159C2×Q8C22×D1711yescharacteristic
160C2×D4—171no—
161C22×C4—341no—
162C22×C4—342no—
163SD16—174no—
164C2×C8—171no—
165C2×C8—171no—
166C2×C8—172no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—172no—
173C4⋊C4—171no—
174C4⋊C4—342no—
175C4⋊C4—341no—
176C4⋊C4—342no—
177C4⋊C4C22⋊D1711yescharacteristic
178C4⋊C4C22⋊D1712yes—
179C4⋊C4C22×D1712yes—
180C4⋊C4C22×D1711yescharacteristic
181C22⋊C4—341no—
182C22⋊C4—342no—
183C42—22no—
184C42—21no—
185C42C22×D1712yes—
186C42C22×D1711yescharacteristic
187C23—341no—
188Q8—21no—
189Q8—22no—
190Q8—42no—
191Q8—1361no—
192Q8Q8×D1712yes—
193D4—172no—
194D4—341no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—341no—
199C2×C4—342no—
200C2×C4—682no—
201C2×C4—1362no—
202C2×C4C2×C22⋊D1712yes—
203C2×C4C2×C22⋊D1712yes—
204C2×C4C2×C22⋊D1711yescharacteristic
205C2×C4C2×C22⋊D1711yescharacteristic
206C2×C4C23×D1711yescharacteristic
207C8—172no—
208C8—341no—
209C8—342no—
210C22—342no—
211C22—681no—
212C22—681no—
213C22C22×C22⋊D1711yescharacteristic
214C4—22no—
215C4—22no—
216C4—22no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—681no—
221C4—682no—
222C4#1058763612yes—
223C4C2×Q8×D1711yescharacteristic
224C4#1058769411yescharacteristic
225C2—682no—
226C2Q8×C22⋊D1711yescharacteristic
227C2C2×#1058763611yescharacteristic
228C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#557C2
socleC22×C17C22×D4
prime radicalC2×D68C23
hypercenter#557D17
hypocenterC17#3381
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
2. #11763208C2semidirect
3. #11763205C2semidirect
4. #11763116C2semidirect
6. #11763104C2semidirect
8. Q8×#3622C2semidirect
11. C17×#557C2semidirect
61. #63230Q8semidirect
111. #557D17semidirect
155. C17#3381semidirect

Maximal subgroups

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

subgroupindex
#117635192
#117632082
#117632052
#117631162
#117631062
#117631042
#117630862
Q8×#36222
C4×#632302
C17×#5572
#338117