Finite Group Databasebeta

← #14536485

Subgroups — #14536485

Order 2176.

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

normal subgroups only

This group has 472 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#11762748C211yesmaximal, characteristic, complemented
2#11762732C212yesmaximal, complemented
3#11762729C211yesmaximal, characteristic, complemented
4#11762721C212yesmaximal, complemented
5#11762693C212yesmaximal, complemented
6#11762689C212yesmaximal
7#11762681C212yesmaximal, complemented
8#11762667C212yesmaximal, complemented
9C17×#390C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×Dic17C2212yes—
12C22×Dic34C2211yescharacteristic
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×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2211yescharacteristic
31C24×C17C2312yes—
32C2×D4×C17—41no—
33C2×D4×C17—41no—
34C2×D4×C17Q814yescomplemented
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×Dic17—24no—
40C22×Dic17—42no—
41C22×Dic17—42no—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C2×Dic34—41no—
47C2×Dic34—42no—
48C4⋊C4×C17—41no—
49C22⋊C4×C17—42no—
50C22⋊C4×C17—42no—
51C42×C17—41no—
52C22⋊Dic17—24no—
53C22⋊Dic17—42no—
54C22⋊Dic17—42no—
55C4⋊Dic17—41no—
56#63225—24no—
57#63225—42no—
58#63225—42no—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17—22no—
65C23×C17C2×Q814yes—
66C23×C17C2411yescharacteristic
67D4×C17—44no—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17C2×Q812yes—
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—
88C2×Dic17—22no—
89C2×Dic17—48no—
90C2×Dic17#4114yes—
91Dic34—44no—
92#2784—171nomaximal, Sylow
93C22×C17—24no—
94C22×C17—24no—
95C22×C17—24no—
96C22×C17—42no—
97C22×C17—42no—
98C22×C17—42no—
99C22×C17—42no—
100C22×C17C22×Q811yescharacteristic
101C22×C17C2×#4112yes—
102C22×C17H(5,𝔽2)11yescharacteristic
103C22×C17H(5,𝔽2)12yes—
104C22×C17H(5,𝔽2)11yescharacteristic
105C4×C17—22no—
106C4×C17—42no—
107C4×C17—41no—
108C4×C17—41no—
109Dic17—24no—
110Dic17—42no—
111Dic17—42no—
112Dic17—42no—
113#400—171no—
114#396—172no—
115#395—171no—
116#393—172no—
117#390D1711yescharacteristic
118#387—172no—
119#386—172no—
120#385—172no—
121#380—172no—
122C2×C17—44no—
123C2×C17#53812yes—
124C2×C17#54311yescharacteristic
125C2×C17#55412yes—
126C2×C17#55511yescharacteristic
127C2×C17#56011yescharacteristic
128C22×Q8—171no—
129C22×D4C2×D1711yescharacteristic
130C23×C4—172no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—172no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—171no—
135C2×C4⋊C4C2×D1711yescharacteristic
136C2×C22⋊C4—172no—
137C2×C22⋊C4—172no—
138C2×C22⋊C4—172no—
139C2×C22⋊C4C2×D1712yes—
140C2×C22⋊C4C2×D1712yes—
141C2×C42C2×D1711yescharacteristic
142#95—171no—
143#95—172no—
144#95—171no—
145#95—172no—
146#95—172no—
147#95—172no—
148#95—172no—
149C17#278411yescharacteristic, complemented, Sylow
150C24C22×D1712yes—
151C2×Q8—682no—
152C2×Q8—681no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4Dic3414yescomplemented
156C22×C4—172no—
157C22×C4—172no—
158C22×C4—172no—
159C22×C4—172no—
160C22×C4—344no—
161C22×C4—682no—
162C22×C4—682no—
163C22×C4C22×D1711yescharacteristic
164C22×C4C22×D1712yes—
165C22×C4C22×D1711yescharacteristic
166C22×C4C22×D1711yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—344no—
169C4⋊C4—682no—
170C4⋊C4—682no—
171C4⋊C4—681no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—344no—
175C22⋊C4—682no—
176C22⋊C4—682no—
177C42—41no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23C2×Dic3414yes—
185C23C23×D1711yescharacteristic
186Q8—684no—
187D4—44no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—174no—
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—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—688no—
210C2×C4C2×Dic3412yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22C22×Dic3411yescharacteristic
219C22#1058769112yes—
220C22#1058769111yescharacteristic
221C22#1058769111yescharacteristic
222C22#41×D1712yes—
223C4—22no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—344no—
228C4—682no—
229C4—682no—
230C4—682no—
231C2—44no—
232C2#1176339112yes—
233C2#1176343511yescharacteristic
234C2#1176351812yes—
235C2#1176352111yescharacteristic
236C2#1176354611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC22×D4×C17C22
hypercenter#390D17
hypocenterC17#2784
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #11762748C2semidirect
2. #11762732C2semidirect
3. #11762729C2semidirect
4. #11762721C2semidirect
5. #11762693C2semidirect
7. #11762681C2semidirect
8. #11762667C2semidirect
34. C2×D4×C17Q8semidirect
149. C17#2784semidirect
155. C2×D4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627482
#117627322
#117627292
#117627212
#117626932
#117626892
#117626812
#117626672
C17×#3902
#278417