Finite Group Databasebeta

← #14539604

Subgroups — #14539604

Order 2176.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22⋊Q8×D17C212yesmaximal
2#11763141C212yesmaximal, complemented
3#11763138C212yesmaximal, complemented
4#11763067C211yesmaximal, characteristic, complemented
5#11763065C212yesmaximal, complemented
6#11762966C212yesmaximal, complemented
7#11762963C212yesmaximal, complemented
8#11762772C211yesmaximal, characteristic
9C17×#470C211yesmaximal, characteristic, complemented
10C2×Q8×D17C2212yes—
11C22×C4×D17C2211yescharacteristic
12C2×Q16×C17C2211yescharacteristic, complemented
13C2×M16×C17C2211yescharacteristic
14C17×C22⋊Q8C2212yes—
15#10587632C2212yes—
16C2×#63231C2212yescomplemented
17#10587610C2212yes—
18C2×#63223C2211yescharacteristic
19C2×Dic68C2211yescharacteristic, complemented
20#10587583C2212yes—
21#10587582—22no—
22C4⋊C4×D17—22no—
23C4⋊C4×D17C2212yes—
24C22⋊C4×D17—22no—
25#10587564—22no—
26C17×#106C2211yescharacteristic, complemented
27C17×#103C2212yescomplemented
28#10587506C2212yescomplemented
29#10587491C2212yes—
30#10587491C2212yes—
31#10587488C2211yescharacteristic, complemented
32#10587487C2212yescomplemented
33#10587481C2212yescomplemented
34#10587479C2212yescomplemented
35C23×D17D411yescharacteristic
36C2×Q8×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17D411yescharacteristic
39Q8×D17—42no—
40Q8×D17—42no—
41C2×C4×D17—22no—
42C2×C4×D17—22no—
43C2×C4×D17—41no—
44C2×C4×D17—41no—
45C2×C4×D17D412yes—
46C2×C4×D17C2312yes—
47C2×Dic34—22no—
48C2×Dic34C2312yes—
49Q16×C17—41no—
50M16×C17—41no—
51C2×C8×C17C2311yescharacteristic
52C2×C8×C17C2311yescharacteristic
53C4⋊C4×C17—22no—
54C4⋊C4×C17C2312yes—
55C22⋊C4×C17—22no—
56C22⋊Dic17—22no—
57#63231—42no—
58#63227—22no—
59#63227—22no—
60C4⋊Dic17—22no—
61C4⋊Dic17C2312yes—
62#63225—22no—
63#63225—22no—
64#63225—22no—
65#63223—41no—
66C2×#3622C2312yes—
67Dic68—41no—
68C23×C17C2×D411yescharacteristic
69C22×D17—24no—
70C22×D17—41no—
71C22×D17C2×D412yes—
72Q8×C17—22no—
73Q8×C17—22no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2411yescharacteristic
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—41no—
83C2×Dic17C2×D412yes—
84C4×D17—22no—
85C4×D17—22no—
86C4×D17—42no—
87C4×D17—44no—
88C4×D17—44no—
89Dic34—22no—
90Dic34—22no—
91Dic34—44no—
92C8×C17—21no—
93C8×C17—21no—
94#3622—22no—
95#3195—171nomaximal, Sylow
96C22×C17—21no—
97C22×C17—22no—
98C22×C17C22×D411yescharacteristic
99C2×D17—22no—
100C2×D17—22no—
101C2×D17—44no—
102C2×D17#13714yes—
103C4×C17—21no—
104C4×C17—42no—
105C4×C17—42no—
106C4×C17H(5,𝔽2)11yescharacteristic
107C4×C17H(5,𝔽2)11yescharacteristic
108Dic17—22no—
109Dic17—42no—
110Dic17—42no—
111C2×C22⋊Q8—172no—
112#483—172no—
113#483—172no—
114#470—171no—
115#470—172no—
116#470D1711yescharacteristic, complemented
117#451—172no—
118#451—172no—
119#407—171no—
120C2×C17—41no—
121C2×C17#53411yescharacteristic
122C2×C17C2×#13712yes—
123D17—24no—
124C22×Q8—172no—
125C23×C4—171no—
126C2×Q16—172no—
127C2×Q16—171no—
128C2×Q16C2×D1711yescharacteristic, complemented
129C2×M16—171no—
130C2×M16C2×D1711yescharacteristic
131C22⋊Q8—172no—
132C22⋊Q8—172no—
133C22⋊Q8—172no—
134C22⋊Q8—342no—
135C22⋊Q8—342no—
136C22⋊Q8C2×D1712yes—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—342no—
139C2×C22⋊C4—342no—
140#106—171no—
141#106—172no—
142#106C2×D1711yescharacteristic, complemented
143#103—172no—
144#103—172no—
145#103—172no—
146#103C2×D1712yescomplemented
147C22⋊C8—172no—
148C22⋊C8—172no—
149C17#319511yescharacteristic, complemented, Sylow
150C24—171no—
151C2×Q8—172no—
152C2×Q8—342no—
153C2×Q8—682no—
154C2×Q8—682no—
155C2×Q8C22×D1712yes—
156C22×C4—171no—
157C22×C4—172no—
158C22×C4—172no—
159C22×C4—342no—
160C22×C4—342no—
161C22×C4—681no—
162C22×C4—681no—
163C22×C4C22×D1711yescharacteristic
164Q16—41no—
165Q16—681no—
166Q16—682no—
167M16—41no—
168M16—681no—
169C2×C8—172no—
170C2×C8C22×D1711yescharacteristic
171C2×C8C22×D1711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—172no—
174C4⋊C4—342no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4C22×D1712yes—
179C22⋊C4—22no—
180C22⋊C4—342no—
181C22⋊C4—342no—
182C22⋊C4—342no—
183C23—172no—
184C23—344no—
185C23—681no—
186C23D4×D1711yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—342no—
190Q8—342no—
191Q8—684no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—172no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—682no—
202C2×C4—684no—
203C2×C4—684no—
204C2×C4—681no—
205C2×C4D4×D1711yescharacteristic
206C2×C4C23×D1711yescharacteristic
207C8—21no—
208C8—21no—
209C8—342no—
210C22—22no—
211C22—21no—
212C22—174no—
213C22—342no—
214C22—342no—
215C22—684no—
216C22C2×D4×D1711yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—342no—
221C4—682no—
222C4—682no—
223C4#1058769111yescharacteristic
224C4#1058769611yescharacteristic
225C2—41no—
226C2—344no—
227C2#1176336711yescharacteristic
228C2#137×D1712yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#470C2
socleC22×C17C22×D4
prime radicalC23×D17D4
hypercenter#470D17
hypocenterC17#3195
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #11763141C2semidirect
3. #11763138C2semidirect
4. #11763067C2semidirect
5. #11763065C2semidirect
6. #11762966C2semidirect
7. #11762963C2semidirect
9. C17×#470C2semidirect
12. C2×Q16×C17C22semidirect
16. C2×#63231C22semidirect
19. C2×Dic68C22semidirect
26. C17×#106C22semidirect
27. C17×#103C22semidirect
28. #10587506C22semidirect
31. #10587488C22semidirect
32. #10587487C22semidirect
33. #10587481C22semidirect
34. #10587479C22semidirect
116. #470D17semidirect
128. C2×Q16C2×D17semidirect
142. #106C2×D17semidirect
146. #103C2×D17semidirect
149. C17#3195semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D172
#117631412
#117631382
#117630672
#117630652
#117629662
#117629632
#117627722
C17×#4702
#319517