Finite Group Databasebeta

← #14533383

Subgroups — #14533383

Order 2176.

440 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 137 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#11762737C212yesmaximal, complemented
2#11762737C212yesmaximal, complemented
3#11762712C211yesmaximal, characteristic
4#11762712C211yesmaximal, characteristic
5#11762705C211yesmaximal, characteristic, complemented
6#11762703C212yesmaximal, complemented
7#11762697C212yesmaximal, complemented
8#11762675C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
10C4×#63225C211yesmaximal, characteristic, complemented
11C17×#390C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2212yes—
16C2×C42×C17C2211yescharacteristic
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2211yescharacteristic
23C2×#63225C2211yescharacteristic
24C2×#63225C2211yescharacteristic
25C2×#63225C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2211yescharacteristic
28C2×C4×Dic17C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2211yescharacteristic
34#10587502C2212yes—
35C24×C17C2312yes—
36C2×D4×C17—22no—
37C2×D4×C17—22no—
38C2×D4×C17—22no—
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C4⋊C4×C17—22no—
50C22⋊C4×C17—42no—
51C22⋊C4×C17—42no—
52C42×C17—22no—
53C22⋊Dic17—24no—
54C22⋊Dic17—24no—
55C22⋊Dic17—42no—
56C22⋊Dic17—42no—
57C4⋊Dic17—22no—
58C4⋊Dic17—22no—
59#63225—22no—
60#63225—22no—
61#63225—42no—
62C4×Dic17—22no—
63C4×Dic17—22no—
64C4×Dic17—42no—
65C23×C17—22no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17C2411yescharacteristic
73D4×C17—44no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17#4112yes—
84C2×C4×C17#4112yes—
85C2×C4×C17#4112yes—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17#4112yes—
103C2×Dic17#4112yes—
104#2513—171nomaximal, Sylow
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17—42no—
112C22×C17C2×#4111yescharacteristic
113C22×C17C2×#4111yescharacteristic
114C22×C17C2×#4111yescharacteristic
115C22×C17C2×#4111yescharacteristic
116C22×C17C2×#4111yescharacteristic
117C22×C17H(5,𝔽2)11yescharacteristic
118C22×C17H(5,𝔽2)11yescharacteristic
119C4×C17—42no—
120C4×C17—41no—
121C4×C17—41no—
122C4×C17#12614yes—
123Dic17—22no—
124Dic17—22no—
125Dic17—41no—
126Dic17—41no—
127Dic17—42no—
128Dic17—42no—
129#397—172no—
130#397—172no—
131#390—171no—
132#390—171no—
133#390D1711yescharacteristic
134#389—171no—
135#388—172no—
136#388—172no—
137#382—171no—
138C4×C4⋊C4—171no—
139C4×C4⋊C4—171no—
140C2×C17—82no—
141C2×C17C2×#12611yescharacteristic
142C2×C17#52911yescharacteristic
143C2×C17#54011yescharacteristic
144C2×C17#55311yescharacteristic
145C2×C17#55511yescharacteristic
146C2×C17#55911yescharacteristic
147C2×C17#55911yescharacteristic
148C22×D4C2×D1711yescharacteristic
149C2×C4⋊C4—171no—
150C2×C4⋊C4—171no—
151C2×C4⋊C4—171no—
152C2×C4⋊C4—171no—
153C2×C4⋊C4—172no—
154C2×C4⋊C4C2×D1711yescharacteristic
155C2×C22⋊C4—172no—
156C2×C22⋊C4—172no—
157C2×C22⋊C4—172no—
158C2×C22⋊C4—172no—
159C2×C22⋊C4C2×D1712yes—
160C2×C22⋊C4C2×D1712yes—
161C2×C42—171no—
162C2×C42—171no—
163C2×C42—172no—
164C2×C42C2×D1711yescharacteristic
165#95—171no—
166#95—172no—
167#95—171no—
168#95—172no—
169#95—172no—
170#95—172no—
171C17#251311yescharacteristic, complemented, Sylow
172C24C22×D1712yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C22×C4—171no—
177C22×C4—171no—
178C22×C4—172no—
179C22×C4—171no—
180C22×C4—172no—
181C22×C4—171no—
182C22×C4C22×D1711yescharacteristic
183C22×C4C22×D1711yescharacteristic
184C22×C4C22×D1712yes—
185C22×C4C22×D1711yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—342no—
188C4⋊C4—342no—
189C4⋊C4—342no—
190C4⋊C4—342no—
191C4⋊C4—682no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—344no—
195C22⋊C4—344no—
196C22⋊C4—682no—
197C22⋊C4—682no—
198C42—22no—
199C42—342no—
200C42—342no—
201C42—682no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23C23×D1711yescharacteristic
210D4—44no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—172no—
221C2×C4—172no—
222C2×C4—341no—
223C2×C4—341no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—341no—
229C2×C4—341no—
230C2×C4—342no—
231C2×C4—341no—
232C2×C4—341no—
233C2×C4—342no—
234C2×C4—341no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—341no—
238C2×C4#6325212yes—
239C2×C4#6325412yes—
240C2×C4#6325412yes—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×#6325211yescharacteristic
249C22C2×#6325411yescharacteristic
250C22C2×#6325411yescharacteristic
251C22#1058769111yescharacteristic
252C22#1058769111yescharacteristic
253C22#41×D1711yescharacteristic
254C22#41×D1711yescharacteristic
255C4—41no—
256C4—41no—
257C4—42no—
258C4—342no—
259C4—342no—
260C4—681no—
261C4—682no—
262C4—682no—
263C4—681no—
264C4#1058756514yes—
265C2—82no—
266C2C2×#1058756511yescharacteristic
267C2#1176333811yescharacteristic
268C2#1176341311yescharacteristic
269C2#1176350611yescharacteristic
270C2#1176352311yescharacteristic
271C2#1176354211yescharacteristic
272C2#1176354211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC22×D4×C17C22
hypercenter#390D17
hypocenterC17#2513
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #11762737C2semidirect
2. #11762737C2semidirect
5. #11762705C2semidirect
6. #11762703C2semidirect
7. #11762697C2semidirect
8. #11762675C2semidirect
9. C4⋊C4×Dic17C2semidirect
10. C4×#63225C2semidirect
171. C17#2513semidirect

Maximal subgroups

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

subgroupindex
#117627372
#117627122
#117627052
#117627032
#117626972
#117626752
C4⋊C4×Dic172
C4×#632252
C17×#3902
#251317