Finite Group Databasebeta

← #14536853

Subgroups — #14536853

Order 2176.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#10587576C211yesmaximal, characteristic
2C2×#10587575C211yesmaximal, characteristic
3#11762750C212yesmaximal
4#11762747C211yesmaximal, characteristic
5#11762743C212yesmaximal
6#11762743C212yesmaximal
7#11762741C212yesmaximal
8#11762741C211yesmaximal, characteristic
9#11762739C211yesmaximal, characteristic
10#11762673C211yesmaximal, characteristic
11C17×#400C211yesmaximal, characteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊Dic17C2211yescharacteristic
15C2×C4⋊Dic17C2211yescharacteristic
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2212yes—
18C2×#63225C2211yescharacteristic
19C2×#63225C2211yescharacteristic
20C2×#63225C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2211yescharacteristic
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×C4×Dic17C2211yescharacteristic
26#10587576—24no—
27#10587575—24no—
28C17×#95C2212yes—
29C17×#95C2211yescharacteristic
30C17×#95C2212yes—
31#10587502C2212yes—
32#10587502C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×Dic17C2312yes—
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C4⋊C4×C17—22no—
49C4⋊C4×C17—22no—
50C4⋊Dic17—22no—
51C4⋊Dic17—22no—
52C4⋊Dic17—24no—
53C4⋊Dic17—42no—
54#63225—22no—
55#63225—22no—
56#63225—22no—
57#63225—22no—
58#63225—24no—
59#63225—42no—
60#63225—42no—
61C4×Dic17D414yes—
62C23×C17C2411yescharacteristic
63C2×C4×C17—22no—
64C2×C4×C17—21no—
65C2×C4×C17—21no—
66C2×C4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—22no—
69C2×C4×C17—21no—
70C2×C4×C17—24no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17C2×D412yes—
75C2×Dic17—21no—
76C2×Dic17—21no—
77C2×Dic17—22no—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—21no—
84C2×Dic17—21no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17C2×D412yes—
90C2×Dic17C2×D412yes—
91#2822—171nomaximal, Sylow
92C22×C17C22×D411yescharacteristic
93C22×C17H(5,𝔽2)11yescharacteristic
94C22×C172−1+411yescharacteristic
95C22×C172−1+412yes—
96C22×C172−1+412yes—
97C4×C17—22no—
98C4×C17—42no—
99C4×C17—41no—
100C4×C17—41no—
101C4×C17—42no—
102Dic17—22no—
103Dic17—22no—
104Dic17—41no—
105Dic17—41no—
106Dic17—42no—
107Dic17—42no—
108C2×#125—171no—
109C2×#125—171no—
110#400—171no—
111#400—172no—
112#400D1711yescharacteristic
113#398—171no—
114#398—171no—
115#398—172no—
116#398—172no—
117#398—172no—
118#382—171no—
119C2×C17#53611yescharacteristic
120C2×C17#53611yescharacteristic
121C2×C17#53711yescharacteristic
122C2×C17#56312yes—
123C2×C17#56412yes—
124#125—344no—
125#125—344no—
126C2×C4⋊C4—171no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4—171no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4—171no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—171no—
133C2×C4⋊C4—171no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4—172no—
136C2×C4⋊C4—172no—
137C2×C4⋊C4C2×D1711yescharacteristic
138C2×C4⋊C4C2×D1711yescharacteristic
139C2×C42—171no—
140#95—172no—
141#95—171no—
142#95—172no—
143#95—172no—
144#95—172no—
145#95—172no—
146#95C2×D1711yescharacteristic
147#95C2×D1712yes—
148#95C2×D1712yes—
149C17#282211yescharacteristic, complemented, Sylow
150C22×C4—171no—
151C22×C4—171no—
152C22×C4—172no—
153C22×C4—171no—
154C22×C4—171no—
155C22×C4—172no—
156C22×C4C22×D1712yes—
157C22×C4C22×D1711yescharacteristic
158C22×C4C22×D1712yes—
159C22×C4C22×D1711yescharacteristic
160C22×C4C22×D1711yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—342no—
164C4⋊C4—342no—
165C4⋊C4—344no—
166C4⋊C4—342no—
167C4⋊C4—342no—
168C4⋊C4—344no—
169C4⋊C4—342no—
170C4⋊C4—342no—
171C4⋊C4—682no—
172C4⋊C4—682no—
173C4⋊C4—682no—
174C42—174no—
175C23C23×D1711yescharacteristic
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—172no—
188C2×C4—172no—
189C2×C4—341no—
190C2×C4—341no—
191C2×C4—342no—
192C2×C4—341no—
193C2×C4—341no—
194C2×C4—342no—
195C2×C4—341no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—341no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4D4×D1712yes—
204C22C2×D4×D1711yescharacteristic
205C22#1058769411yescharacteristic
206C22#1058769611yescharacteristic
207C22#1058769712yes—
208C22#1058769712yes—
209C4—22no—
210C4—42no—
211C4—42no—
212C4—41no—
213C4—41no—
214C4—342no—
215C4—342no—
216C4—682no—
217C4—681no—
218C4—681no—
219C4—682no—
220C2#1176337911yescharacteristic
221C2#1176338111yescharacteristic
222C2#1176338911yescharacteristic
223C2#1176356112yes—
224C2C17⋊#56412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#400C2
hypercenter#400D17
hypocenterC17#2822
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
149. C17#2822semidirect

Maximal subgroups

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

subgroupindex
C2×#105875762
C2×#105875752
#117627502
#117627472
#117627432
#117627412
#117627392
#117626732
C17×#4002
#282217