Finite Group Databasebeta

← #14536727

Subgroups — #14536727

Order 2176.

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

normal subgroups only

This group has 480 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#11762751C212yesmaximal, complemented
2#11762721C212yesmaximal, complemented
3#11762721C212yesmaximal, complemented
4#11762713C212yesmaximal, complemented
5#11762712C211yesmaximal, characteristic
6#11762689C211yesmaximal, characteristic
7#11762682C212yesmaximal, complemented
8#11762681C212yesmaximal, complemented
9C17×#386C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C23×Dic17C2211yescharacteristic
13C22×Dic34C2212yes—
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×C4×Dic17C2211yescharacteristic
24C17×#95C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C24×C17C2311yescharacteristic
33C2×D4×C17—41no—
34C2×D4×C17—41no—
35C2×D4×C17Q814yescomplemented
36C22×C4×C17—42no—
37C22×C4×C17—41no—
38C22×C4×C17C2312yes—
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×Dic17—42no—
42C22×Dic17—41no—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C2×Dic34—24no—
48C2×Dic34—42no—
49C2×Dic34—42no—
50C22⋊C4×C17—41no—
51C22⋊C4×C17—42no—
52C22⋊Dic17—24no—
53C22⋊Dic17—24no—
54C22⋊Dic17—41no—
55C22⋊Dic17—42no—
56C4⋊Dic17—41no—
57#63225—42no—
58#63225—42no—
59C4×Dic17—41no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—21no—
64C23×C17—22no—
65C23×C17—21no—
66C23×C17—21no—
67C23×C17—21no—
68C23×C17C2×Q812yes—
69C23×C17C2×Q812yes—
70C23×C17C2411yescharacteristic
71D4×C17—44no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—44no—
81C2×C4×C17C2×Q812yes—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—44no—
94C2×Dic17#4114yes—
95Dic34—48no—
96#2804—171nomaximal, Sylow
97C22×C17—22no—
98C22×C17—22no—
99C22×C17—24no—
100C22×C17—24no—
101C22×C17—42no—
102C22×C17—42no—
103C22×C17—41no—
104C22×C17—41no—
105C22×C17—41no—
106C22×C17—41no—
107C22×C17C22×Q811yescharacteristic
108C22×C17C2×#4112yes—
109C22×C17H(5,𝔽2)11yescharacteristic
110C22×C17H(5,𝔽2)12yes—
111C22×C17H(5,𝔽2)11yescharacteristic
112C4×C17—22no—
113C4×C17—42no—
114C4×C17—42no—
115Dic17—24no—
116Dic17—42no—
117Dic17—42no—
118Dic17—42no—
119#400—172no—
120#393—172no—
121#393—172no—
122#391—172no—
123#390—171no—
124#386—171no—
125#386D1711yescharacteristic
126#385—172no—
127#385—172no—
128C2×C17—42no—
129C2×C17—42no—
130C2×C17#53812yes—
131C2×C17#54311yescharacteristic
132C2×C17#55412yes—
133C2×C17#55911yescharacteristic
134C2×C17H(3,𝔽4)11yescharacteristic
135C22×Q8—172no—
136C22×D4C2×D1711yescharacteristic
137C23×C4—171no—
138C23×C4C2×D1711yescharacteristic
139C2×C4⋊C4—172no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—171no—
142C2×C22⋊C4—172no—
143C2×C22⋊C4—172no—
144C2×C22⋊C4—172no—
145C2×C22⋊C4—171no—
146C2×C22⋊C4C2×D1712yes—
147C2×C22⋊C4C2×D1711yescharacteristic
148C2×C42—171no—
149#95—172no—
150#95—172no—
151#95—172no—
152#95—172no—
153#95—172no—
154#95—172no—
155#95C2×D1712yes—
156C17#280411yescharacteristic, complemented, Sylow
157C24C22×D1711yescharacteristic
158C24C22×D1711yescharacteristic
159C2×Q8—344no—
160C2×Q8—682no—
161C2×Q8—682no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4Dic3414yescomplemented
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—172no—
168C22×C4—172no—
169C22×C4—172no—
170C22×C4—172no—
171C22×C4—682no—
172C22×C4—681no—
173C22×C4C22×D1712yes—
174C22×C4C22×D1712yes—
175C22×C4C22×D1711yescharacteristic
176C4⋊C4—682no—
177C4⋊C4—682no—
178C4⋊C4—681no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—344no—
182C22⋊C4—344no—
183C22⋊C4—682no—
184C22⋊C4—681no—
185C42—681no—
186C23—22no—
187C23—21no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23C2×Dic3412yes—
195C23C2×Dic3412yes—
196C23C23×D1711yescharacteristic
197Q8—688no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—174no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—684no—
221C2×C4C2×Dic3412yes—
222C22—24no—
223C22—24no—
224C22—22no—
225C22—22no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22C22×Dic3411yescharacteristic
233C22#1058769112yes—
234C22#1058769111yescharacteristic
235C22#1058769111yescharacteristic
236C22#41×D1712yes—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—344no—
241C4—682no—
242C4—682no—
243C4—682no—
244C2—42no—
245C2—42no—
246C2#1176339112yes—
247C2#1176343511yescharacteristic
248C2#1176351812yes—
249C2#1176354311yescharacteristic
250C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC22×D4×C17C22
hypercenter#386D17
hypocenterC17#2804
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #11762751C2semidirect
2. #11762721C2semidirect
3. #11762721C2semidirect
4. #11762713C2semidirect
7. #11762682C2semidirect
8. #11762681C2semidirect
35. C2×D4×C17Q8semidirect
156. C17#2804semidirect
164. C2×D4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627212
#117627132
#117627122
#117626892
#117626822
#117626812
C17×#3862
#280417