Finite Group Databasebeta

← #14539861

Subgroups — #14539861

Order 2176.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#11763385C211yesmaximal, characteristic
2#11763201C211yesmaximal, characteristic, complemented
3#11763188C211yesmaximal, characteristic, complemented
4#11763179C211yesmaximal, characteristic, complemented
5#11763166C211yesmaximal, characteristic, complemented
6#11763031C212yesmaximal, complemented
7#11763019C212yesmaximal, complemented
8#11762969C212yesmaximal
9#11762964C212yesmaximal, complemented
10#11762870C211yesmaximal, characteristic, complemented
11C17×#541C211yesmaximal, characteristic, complemented
12C2×#63252C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C17×C4⋊Q8C2211yescharacteristic
15C17×#126—22no—
16C17×#125—22no—
17C17×#125C2211yescharacteristic
18C17×C22⋊Q8C2212yes—
19C4×Q8×C17C2212yes—
20C17×#117C2211yescharacteristic
21C2×#63231C2212yes—
22C2×#63231C2212yes—
23C2×#63231C2212yes—
24C2×#63230C2212yes—
25#10587583—22no—
26#10587572—22no—
27#10587561C2211yescharacteristic
28#10587561C2211yescharacteristic
29#10587555C2211yescharacteristic
30#10587555C2211yescharacteristic
31C22⋊#3622C2212yes—
32#10587481C2212yes—
33#10587481C2212yes—
34#10587481C2212yes—
35#10587480C2212yes—
36C4⋊#3622C2212yes—
37#10587474C2211yescharacteristic
38C4×#3622C2211yescharacteristic
39C2×Q8×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C2×C22⋊D17—21no—
42C22×Dic17—21no—
43#63252—42no—
44C2×D68C2311yescharacteristic
45C2×C4×D17—21no—
46C2×Dic34—21no—
47C2×Dic34—42no—
48C2×Dic34C2311yescharacteristic
49C2×Dic34C2311yescharacteristic
50C2×Dic34C2311yescharacteristic
51C4⋊C4×C17—22no—
52C4⋊C4×C17—22no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17—22no—
55C4⋊C4×C17D412yes—
56C4⋊C4×C17C2312yes—
57C22⋊C4×C17—22no—
58C22⋊C4×C17D412yes—
59C42×C17—22no—
60C42×C17C2311yescharacteristic
61C42×C17C2311yescharacteristic
62#63231—42no—
63#63231—42no—
64#63231—42no—
65#63230—42no—
66#63227—22no—
67#63227—22no—
68C4⋊Dic17—22no—
69C4⋊Dic17—22no—
70C2×#3622C2312yes—
71C2×#3622C2312yes—
72C23×C17C2×D411yescharacteristic
73C22×D17—21no—
74Q8×C17—22no—
75Q8×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17C2×D412yes—
80C2×C4×C17C2×D412yes—
81C2×C4×C17C2×D411yescharacteristic
82C2×C4×C17C2411yescharacteristic
83C22⋊D17—42no—
84C2×Dic17—21no—
85C2×Dic17—21no—
86C2×Dic17—21no—
87C2×Dic17—42no—
88D68—22no—
89C4×D17—42no—
90Dic34—22no—
91Dic34—22no—
92Dic34—22no—
93Dic34—42no—
94#3622—22no—
95#3622—22no—
96#3223—171nomaximal, Sylow
97C22×C17—21no—
98C22×C17—22no—
99C22×C17C22×D411yescharacteristic
100C2×D17—41no—
101C2×D17—42no—
102C4×C17—21no—
103C4×C17—22no—
104C4×C17—22no—
105C4×C17—42no—
106C4×C17—42no—
107C4×C17H(5,𝔽2)12yes—
108Dic17—41no—
109Dic17—41no—
110Dic17—41no—
111#541D1711yescharacteristic, complemented
112#536—171no—
113#497—171no—
114#494—171no—
115#491—171no—
116#488—171no—
117#464—172no—
118#462—172no—
119#452—172no—
120#451—172no—
121#433—171no—
122C2×C17—41no—
123C2×C17#53511yescharacteristic
124C2×C17#57812yes—
125D17—81no—
126C2×#41—171no—
127C22×Q8—171no—
128C2×Q16—172no—
129C2×Q16—172no—
130C2×Q16—172no—
131C2×SD16—172no—
132C4⋊Q8—171no—
133C4⋊Q8—171no—
134C4⋊Q8C2×D1711yescharacteristic
135#126—22no—
136#125—22no—
137#125C2×D1711yescharacteristic
138#124—171no—
139#124—171no—
140#123—342no—
141C22⋊Q8—342no—
142C22⋊Q8C2×D1712yes—
143C4×Q8C2×D1712yes—
144#117C2×D1711yescharacteristic
145C4⋊C8—172no—
146#103—172no—
147#103—172no—
148#103—172no—
149#102—172no—
150C22⋊C8—172no—
151#97—171no—
152C4×C8—171no—
153C17#322311yescharacteristic, complemented, Sylow
154#41—682no—
155C2×Q8—171no—
156C2×Q8—171no—
157C2×Q8—171no—
158C2×Q8—341no—
159C2×Q8—682no—
160C2×Q8C22×D1712yes—
161C2×D4—171no—
162C2×D4—341no—
163C22×C4—341no—
164C22×C4—341no—
165C22×C4C22×D1711yescharacteristic
166Q16—682no—
167Q16—682no—
168Q16—682no—
169SD16—682no—
170C2×C8—172no—
171C2×C8—172no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4C22⋊D1712yes—
179C4⋊C4C22×D1712yes—
180C22⋊C4—22no—
181C22⋊C4—342no—
182C22⋊C4—342no—
183C22⋊C4C22⋊D1712yes—
184C42—22no—
185C42C22×D1711yescharacteristic
186C42C22×D1711yescharacteristic
187C23—341no—
188C23C2×C22⋊D1711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—342no—
192Q8—342no—
193Q8—342no—
194Q8—682no—
195D4—342no—
196D4—682no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—341no—
201C2×C4—341no—
202C2×C4—341no—
203C2×C4—682no—
204C2×C4—682no—
205C2×C4C2×C22⋊D1712yes—
206C2×C4C2×C22⋊D1712yes—
207C2×C4C2×C22⋊D1711yescharacteristic
208C2×C4C23×D1711yescharacteristic
209C8—342no—
210C8—342no—
211C22—22no—
212C22—21no—
213C22—682no—
214C22—681no—
215C22C22×C22⋊D1711yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—681no—
222C4—681no—
223C4—681no—
224C4#1058769612yes—
225C2—41no—
226C2—1361no—
227C2#1176337311yescharacteristic
228C2#1176362412yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#541C2
socleC22×C17C22×D4
prime radicalC2×#63252C22
hypercenter#541D17
hypocenterC17#3223
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #11763201C2semidirect
3. #11763188C2semidirect
4. #11763179C2semidirect
5. #11763166C2semidirect
6. #11763031C2semidirect
7. #11763019C2semidirect
9. #11762964C2semidirect
10. #11762870C2semidirect
11. C17×#541C2semidirect
111. #541D17semidirect
153. C17#3223semidirect

Maximal subgroups

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

subgroupindex
#117633852
#117632012
#117631882
#117631792
#117631662
#117630312
#117630192
#117629692
#117629642
#117628702
C17×#5412
#322317