Finite Group Databasebeta

← #14534745

Subgroups — #14534745

Order 2176.

420 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 133 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#11762734C212yesmaximal, complemented
4#11762732C212yesmaximal, complemented
5#11762712C211yesmaximal, characteristic
6#11762712C211yesmaximal, characteristic
7#11762708C211yesmaximal, characteristic, complemented
8#11762707C211yesmaximal, characteristic, complemented
9#11762705C211yesmaximal, characteristic, complemented
10#11762655C211yesmaximal, 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×C4⋊Dic17C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2211yescharacteristic
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32#10587502C2211yescharacteristic
33#10587502C2211yescharacteristic
34#10587502C2211yescharacteristic
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×Dic17C2311yescharacteristic
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2312yes—
48C22×Dic17C2312yes—
49C4⋊C4×C17—22no—
50C22⋊C4×C17—42no—
51C22⋊C4×C17—42no—
52C42×C17—22no—
53C22⋊Dic17—42no—
54C22⋊Dic17—42no—
55C22⋊Dic17—42no—
56C22⋊Dic17—42no—
57C4⋊Dic17—22no—
58C4⋊Dic17—22no—
59C4⋊Dic17—42no—
60#63225—42no—
61#63225—42no—
62C4×Dic17—22no—
63C4×Dic17—22no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17C2411yescharacteristic
72D4×C17—44no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17#4112yes—
83C2×C4×C17#4112yes—
84C2×C4×C17#4112yes—
85C2×Dic17—21no—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—21no—
102C2×Dic17—21no—
103#2644—171nomaximal, Sylow
104C22×C17—42no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17C2×#4111yescharacteristic
112C22×C17C2×#4111yescharacteristic
113C22×C17C2×#4111yescharacteristic
114C22×C17H(5,𝔽2)11yescharacteristic
115C22×C17H(5,𝔽2)11yescharacteristic
116C22×C17H(5,𝔽2)11yescharacteristic
117C22×C172−1+411yescharacteristic
118C4×C17—41no—
119C4×C17—42no—
120C4×C17—41no—
121C4×C17#12614yes—
122Dic17—41no—
123Dic17—42no—
124Dic17—42no—
125Dic17—41no—
126Dic17—41no—
127Dic17—41no—
128#397—172no—
129#397—172no—
130#397—172no—
131#396—172no—
132#390—171no—
133#390—171no—
134#390D1711yescharacteristic
135#389—171no—
136#389—171no—
137#389—171no—
138#376—171no—
139C2×C17—82no—
140C2×C17C2×#12611yescharacteristic
141C2×C17#54011yescharacteristic
142C2×C17#54011yescharacteristic
143C2×C17#54011yescharacteristic
144C2×C17#54211yescharacteristic
145C2×C17#54211yescharacteristic
146C2×C17#54211yescharacteristic
147C22×D4C2×D1711yescharacteristic
148C2×C4⋊C4—171no—
149C2×C4⋊C4—171no—
150C2×C4⋊C4—172no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4C2×D1711yescharacteristic
154C2×C22⋊C4—172no—
155C2×C22⋊C4—172no—
156C2×C22⋊C4—172no—
157C2×C22⋊C4—172no—
158C2×C22⋊C4C2×D1712yes—
159C2×C22⋊C4C2×D1712yes—
160C2×C42—171no—
161C2×C42—171no—
162C2×C42C2×D1711yescharacteristic
163#95—171no—
164#95—171no—
165#95—171no—
166#95—171no—
167#95—172no—
168#95—172no—
169#95—172no—
170C17#264411yescharacteristic, complemented, Sylow
171C24C22×D1712yes—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C22×C4—171no—
176C22×C4—171no—
177C22×C4—171no—
178C22×C4—171no—
179C22×C4—172no—
180C22×C4—172no—
181C22×C4C22×D1711yescharacteristic
182C22×C4C22×D1711yescharacteristic
183C22×C4C22×D1712yes—
184C22×C4C22×D1711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—342no—
187C4⋊C4—342no—
188C4⋊C4—682no—
189C4⋊C4—682no—
190C4⋊C4—682no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—682no—
194C22⋊C4—682no—
195C22⋊C4—682no—
196C22⋊C4—682no—
197C42—22no—
198C42—342no—
199C42—342no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23C23×D1711yescharacteristic
208D4—44no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—341no—
219C2×C4—341no—
220C2×C4—342no—
221C2×C4—341no—
222C2×C4—341no—
223C2×C4—342no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—341no—
229C2×C4—341no—
230C2×C4—341no—
231C2×C4—341no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—341no—
235C2×C4—341no—
236C2×C4#6325212yes—
237C2×C4#6325412yes—
238C2×C4#6325412yes—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22C2×#6325211yescharacteristic
247C22C2×#6325411yescharacteristic
248C22C2×#6325411yescharacteristic
249C22#1058769111yescharacteristic
250C22#1058769111yescharacteristic
251C22#1058769611yescharacteristic
252C22#1058769711yescharacteristic
253C4—41no—
254C4—42no—
255C4—41no—
256C4—681no—
257C4—682no—
258C4—682no—
259C4—681no—
260C4—681no—
261C4—681no—
262C4#1058756514yes—
263C2—82no—
264C2C2×#1058756511yescharacteristic
265C2#1176341011yescharacteristic
266C2#1176341211yescharacteristic
267C2#1176341311yescharacteristic
268C2#1176342411yescharacteristic
269C2#1176342611yescharacteristic
270C2#1176342811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC22×D4×C17C22
hypercenter#390D17
hypocenterC17#2644
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
3. #11762734C2semidirect
4. #11762732C2semidirect
7. #11762708C2semidirect
8. #11762707C2semidirect
9. #11762705C2semidirect
10. #11762655C2semidirect
170. C17#2644semidirect

Maximal subgroups

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

subgroupindex
#117627372
#117627342
#117627322
#117627122
#117627082
#117627072
#117627052
#117626552
C17×#3902
#264417