Finite Group Databasebeta

← #14534416

Subgroups — #14534416

Order 2176.

470 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 145 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×#10587562C211yesmaximal, characteristic
2#11762747C211yesmaximal, characteristic, complemented
3#11762744C211yesmaximal, characteristic
4#11762728C211yesmaximal, characteristic, complemented
5#11762720C211yesmaximal, characteristic
6#11762704C212yesmaximal, complemented
7#11762700C212yesmaximal
8#11762699C212yesmaximal
9#11762693C212yesmaximal, complemented
10#11762655C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C17C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C4×Q8×C17—24no—
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C42×C17C2212yes—
17C2×C42×C17C2211yescharacteristic
18C2×#63227C2211yescharacteristic
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2211yescharacteristic
22C2×#63227C2211yescharacteristic
23C2×#63225C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×C4×Dic17C2212yes—
29#10587562—24no—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C23×D17C2311yescharacteristic
38C2×Q8×C17—22no—
39C2×Q8×C17—24no—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C4⋊C4×C17—24no—
51C4⋊C4×C17—22no—
52C42×C17—24no—
53C42×C17D414yes—
54#63227—22no—
55#63227—22no—
56#63227—22no—
57#63227—24no—
58#63227—24no—
59#63225—22no—
60#63225—22no—
61#63225—22no—
62#63225—24no—
63#63225—24no—
64C4×Dic17—24no—
65C23×C17C2411yescharacteristic
66C22×D17—21no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—21no—
70C22×D17—21no—
71Q8×C17—28no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17C2×D412yes—
82C2×C4×C17C2×D412yes—
83C2×C4×C17C2×D412yes—
84C2×C4×C17#4114yes—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—22no—
97C2×Dic17#4114yes—
98#2604—171nomaximal, Sylow
99C22×C17C22×D411yescharacteristic
100C22×C17C2×#4112yes—
101C22×C17C2×#4112yes—
102C22×C172−1+411yescharacteristic
103C22×C172−1+411yescharacteristic
104C2×D17—41no—
105C2×D17—41no—
106C2×D17—41no—
107C2×D17—42no—
108C2×D17—42no—
109C4×C17—22no—
110C4×C17—24no—
111C4×C17—22no—
112C4×C17—22no—
113C4×C17—42no—
114Dic17—24no—
115Dic17—41no—
116Dic17—41no—
117Dic17—41no—
118Dic17—42no—
119C2×#126—171no—
120C2×C4×Q8D1711yescharacteristic, complemented
121#400—171no—
122#399—171no—
123#395—171no—
124#393—171no—
125#389—172no—
126#388—172no—
127#388—172no—
128#387—172no—
129#376—171no—
130C2×C17#52511yescharacteristic
131C2×C17#53211yescharacteristic
132C2×C17#53611yescharacteristic
133C2×C17#55212yes—
134C2×C17#55612yes—
135D17—81no—
136C22×Q8C2×D1711yescharacteristic
137#126—344no—
138C4×Q8—24no—
139C2×C4⋊C4—171no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4C2×D1712yes—
145C2×C4⋊C4C2×D1711yescharacteristic
146C2×C22⋊C4—171no—
147C2×C22⋊C4—171no—
148C2×C22⋊C4—171no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4—172no—
151C2×C42—172no—
152C2×C42C2×D1712yes—
153C2×C42C2×D1711yescharacteristic
154#95—172no—
155#95—171no—
156#95—171no—
157#95—172no—
158#95—172no—
159#95—172no—
160#95—172no—
161C17#260411yescharacteristic, complemented, Sylow
162C24—171no—
163C2×Q8—24no—
164C2×Q8—22no—
165C22×C4—172no—
166C22×C4—172no—
167C22×C4—171no—
168C22×C4—171no—
169C22×C4—171no—
170C22×C4C22×D1712yes—
171C22×C4C22×D1712yes—
172C22×C4C22×D1711yescharacteristic
173C22×C4C22×D1711yescharacteristic
174C22×C4C22×D1711yescharacteristic
175C4⋊C4—24no—
176C4⋊C4—22no—
177C4⋊C4—344no—
178C4⋊C4—344no—
179C4⋊C4—342no—
180C4⋊C4—342no—
181C4⋊C4—342no—
182C22⋊C4—344no—
183C22⋊C4—344no—
184C22⋊C4—342no—
185C22⋊C4—342no—
186C22⋊C4—342no—
187C42—24no—
188C42—344no—
189C42C22⋊D1714yes—
190C23—342no—
191C23—341no—
192C23—341no—
193C23—342no—
194C23—341no—
195C23C23×D1711yescharacteristic
196Q8—28no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—174no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—341no—
213C2×C4—341no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—341no—
217C2×C4—341no—
218C2×C4—341no—
219C2×C4#6325214yes—
220C2×C4C2×C22⋊D1712yes—
221C2×C4C2×C22⋊D1712yes—
222C2×C4C2×C22⋊D1712yes—
223C22—682no—
224C22—682no—
225C22—681no—
226C22—681no—
227C22—681no—
228C22C2×#6325212yes—
229C22#1058769411yescharacteristic
230C22#1058769411yescharacteristic
231C22#41×D1712yes—
232C22C22×C22⋊D1711yescharacteristic
233C4—24no—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—344no—
239C4—682no—
240C4—681no—
241C4—681no—
242C4—681no—
243C2—1361no—
244C2#1176332011yescharacteristic
245C2#1176335511yescharacteristic
246C2#1176338311yescharacteristic
247C2#1176350012yes—
248C2#1176353112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C4×Q8×C17C2
prime radicalC23×D17C23
hypercenterC2×C4×Q8D17
hypocenterC17#2604
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #11762747C2semidirect
4. #11762728C2semidirect
6. #11762704C2semidirect
9. #11762693C2semidirect
10. #11762655C2semidirect
11. C2×C4×Q8×C17C2semidirect
120. C2×C4×Q8D17semidirect
161. C17#2604semidirect

Maximal subgroups

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

subgroupindex
C2×#105875622
#117627472
#117627442
#117627282
#117627202
#117627042
#117627002
#117626992
#117626932
#117626552
C2×C4×Q8×C172
#260417