Finite Group Databasebeta

← #14536955

Subgroups — #14536955

Order 2176.

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

normal subgroups only

This group has 416 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#11762751C212yesmaximal, complemented
2#11762734C212yesmaximal, complemented
3#11762729C211yesmaximal, characteristic, complemented
4#11762723C212yesmaximal
5#11762721C212yesmaximal, complemented
6#11762721C212yesmaximal, complemented
7#11762714C211yesmaximal, characteristic, complemented
8#11762702C212yesmaximal, complemented
9C17×#394C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C4⋊Dic17C2212yes—
21C2×#63225C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23C17×#95C2211yescharacteristic
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31C24×C17C2312yes—
32C2×D4×C17—41no—
33C2×D4×C17—42no—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C2×Dic34—22no—
43C2×Dic34—41no—
44C2×Dic34—41no—
45C2×Dic34—41no—
46C2×Dic34—42no—
47C4⋊C4×C17—41no—
48C22⋊C4×C17—24no—
49C22⋊C4×C17—42no—
50C22⋊Dic17—42no—
51C22⋊Dic17—42no—
52C22⋊Dic17—42no—
53C22⋊Dic17—42no—
54C4⋊Dic17—42no—
55#63225—42no—
56C4×Dic17—41no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17C2411yescharacteristic
65D4×C17—44no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17#4112yes—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88Dic34—44no—
89Dic34—44no—
90#2832—171nomaximal, Sylow
91C22×C17—42no—
92C22×C17—42no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17—42no—
96C22×C17—42no—
97C22×C17—42no—
98C22×C17C2×#4111yescharacteristic
99C22×C17H(5,𝔽2)11yescharacteristic
100C22×C17H(5,𝔽2)12yes—
101C22×C17H(5,𝔽2)11yescharacteristic
102C22×C172−1+412yes—
103C4×C17—22no—
104C4×C17—42no—
105C4×C17—41no—
106C4×C17—41no—
107Dic17—42no—
108Dic17—42no—
109Dic17—42no—
110Dic17—42no—
111#400—172no—
112#397—172no—
113#395—171no—
114#394—172no—
115#394D1711yescharacteristic
116#393—172no—
117#393—172no—
118#391—171no—
119#388—172no—
120C2×C17—82no—
121C2×C17#53811yescharacteristic
122C2×C17#54212yes—
123C2×C17H(3,𝔽4)11yescharacteristic
124C2×C17#56212yes—
125C2×C17#56311yescharacteristic
126C22×Q8—171no—
127C22×Q8—171no—
128C22×D4C2×D1711yescharacteristic
129C2×C4⋊C4—172no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4C2×D1711yescharacteristic
132C2×C22⋊C4—172no—
133C2×C22⋊C4—172no—
134C2×C22⋊C4—172no—
135C2×C22⋊C4—172no—
136C2×C22⋊C4C2×D1712yes—
137C2×C22⋊C4C2×D1712yes—
138C2×C42—171no—
139#95—172no—
140#95—172no—
141#95—172no—
142#95—172no—
143#95—172no—
144#95—172no—
145#95—171no—
146#95C2×D1711yescharacteristic
147C17#283211yescharacteristic, complemented, Sylow
148C24C22×D1712yes—
149C2×Q8—342no—
150C2×Q8—681no—
151C2×Q8—681no—
152C2×Q8—682no—
153C2×Q8—681no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—172no—
157C22×C4—172no—
158C22×C4—172no—
159C22×C4—172no—
160C22×C4C22×D1712yes—
161C22×C4C22×D1711yescharacteristic
162C22×C4C22×D1711yescharacteristic
163C22×C4C22×D1711yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—682no—
166C4⋊C4—682no—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—682no—
170C22⋊C4—682no—
171C22⋊C4—682no—
172C22⋊C4—682no—
173C42—681no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23C23×D1711yescharacteristic
182Q8—684no—
183Q8—684no—
184D4—44no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4#6325412yes—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×#6325411yescharacteristic
215C22#1058769112yes—
216C22#1058769111yescharacteristic
217C22#1058769611yescharacteristic
218C22#1058769712yes—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—682no—
224C4—682no—
225C4—682no—
226C4—682no—
227C2—82no—
228C2#1176339611yescharacteristic
229C2#1176342612yes—
230C2#1176355211yescharacteristic
231C2#1176355612yes—
232C2#1176355911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2832
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #11762751C2semidirect
2. #11762734C2semidirect
3. #11762729C2semidirect
5. #11762721C2semidirect
6. #11762721C2semidirect
7. #11762714C2semidirect
8. #11762702C2semidirect
147. C17#2832semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627342
#117627292
#117627232
#117627212
#117627142
#117627022
C17×#3942
#283217