Finite Group Databasebeta

← #14532115

Subgroups — #14532115

Order 2176.

864 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 185 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×#10587621C212yesmaximal, complemented
2C22×#63225C211yesmaximal, characteristic, complemented
3#11762728C212yesmaximal, complemented
4#11762723C212yesmaximal, complemented
5#11762689C212yesmaximal, complemented
6#11762682C214yesmaximal, complemented
7#11762672C211yesmaximal, characteristic, complemented
8C22×#41×C17C211yesmaximal, characteristic
9C2×#41×C17—28no—
10C2×#41×C17—24no—
11C22×Q8×C17C2211yescharacteristic
12C22×D4×C17C2211yescharacteristic
13C22×D4×C17C2212yes—
14C23×C4×C17C2212yes—
15C23×C4×C17C2211yescharacteristic
16C23×Dic17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C2×C22⋊Dic17C2214yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2214yes—
21#10587621—28no—
22C2×#63225—24no—
23C2×#63225C2212yes—
24C2×#63225C2214yes—
25C2×#63225C2212yes—
26#10587502C2214yes—
27#10587502C2212yes—
28#10587502C2212yes—
29C24×C17C2312yes—
30C24×C17C2311yescharacteristic
31#41×C17—232no—
32C2×Q8×C17—24no—
33C2×Q8×C17—22no—
34C2×D4×C17—24no—
35C2×D4×C17—24no—
36C2×D4×C17—24no—
37C2×D4×C17—22no—
38C2×D4×C17D418yes—
39C22×C4×C17—24no—
40C22×C4×C17—24no—
41C22×C4×C17—24no—
42C22×C4×C17—24no—
43C22×C4×C17D414yes—
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C22×Dic17—22no—
48C22×Dic17—24no—
49C22×Dic17—22no—
50C22×Dic17—24no—
51C22×Dic17C2312yes—
52C22×Dic17C2314yes—
53C22×Dic17C2312yes—
54C22⋊Dic17—24no—
55C22⋊Dic17—28no—
56C22⋊Dic17—28no—
57#63225—24no—
58#63225—24no—
59#63225—28no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17—24no—
65C23×C17—24no—
66C23×C17C2×D414yes—
67C23×C17C2×D414yes—
68C23×C17C2×D412yes—
69C23×C17C2411yescharacteristic
70Q8×C17—28no—
71D4×C17—28no—
72D4×C17—216no—
73C2×C4×C17—216no—
74C2×C4×C17—28no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17C2×D412yes—
80C2×C4×C17C2×D414yes—
81C2×C4×C17C2×D412yes—
82C2×Dic17—28no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—24no—
86C2×Dic17—24no—
87C2×Dic17—22no—
88C2×Dic17—24no—
89C2×Dic17—24no—
90C2×Dic17—44no—
91C2×Dic17#4114yescomplemented
92#2406—171nomaximal, Sylow
93C22×C17—22no—
94C22×C17—22no—
95C22×C17—28no—
96C22×C17—24no—
97C22×C17—28no—
98C22×C17—42no—
99C22×C17—42no—
100C22×C17—44no—
101C22×C17#12014yes—
102C22×C17C22×D411yescharacteristic
103C22×C17C22×D412yes—
104C22×C17C2×#4112yes—
105C22×C172−1+412yes—
106C4×C17—22no—
107C4×C17—24no—
108C4×C17—22no—
109Dic17—24no—
110Dic17—42no—
111Dic17—44no—
112C22×#41D1711yescharacteristic
113C2×#123—172no—
114C22×C4⋊C4—171no—
115#395—172no—
116#394—172no—
117#386—172no—
118#385—174no—
119#381—171no—
120C2×C17—24no—
121C2×C17—44no—
122C2×C17C2×#12011yescharacteristic
123C2×C17#52511yescharacteristic
124C2×C17#53611yescharacteristic
125C2×C17#54714yes—
126C2×#41—28no—
127C2×#41—24no—
128C22×Q8C2×D1711yescharacteristic
129C22×D4C2×D1711yescharacteristic
130C22×D4C2×D1712yes—
131C23×C4—171no—
132C23×C4—171no—
133C23×C4C2×D1712yes—
134C23×C4C2×D1711yescharacteristic
135#123—348no—
136C2×C4⋊C4—172no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—174no—
139C2×C4⋊C4—344no—
140C2×C22⋊C4—172no—
141C2×C22⋊C4—174no—
142C2×C22⋊C4—174no—
143#95—172no—
144#95—172no—
145#95—174no—
146C17#240611yescharacteristic, complemented, Sylow
147C24C22×D1712yes—
148C24C22×D1711yescharacteristic
149#41—232no—
150C2×Q8—24no—
151C2×Q8—22no—
152C2×D4—24no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4C22⋊D1718yes—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—172no—
162C22×C4—174no—
163C22×C4—172no—
164C22×C4—344no—
165C22×C4—342no—
166C22×C4—342no—
167C22×C4—344no—
168C22×C4C22⋊D1714yes—
169C22×C4C22×D1711yescharacteristic
170C22×C4C22×D1712yes—
171C22×C4C22×D1711yescharacteristic
172C4⋊C4—344no—
173C4⋊C4—344no—
174C4⋊C4—348no—
175C22⋊C4—348no—
176C22⋊C4—344no—
177C22⋊C4—348no—
178C23—22no—
179C23—22no—
180C23—24no—
181C23—24no—
182C23—22no—
183C23—22no—
184C23C2×C22⋊D1714yes—
185C23C2×C22⋊D1714yes—
186C23C2×C22⋊D1712yes—
187C23C23×D1711yescharacteristic
188Q8—28no—
189D4—28no—
190D4—216no—
191C2×C4—216no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—28no—
197C2×C4—174no—
198C2×C4—348no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—344no—
202C2×C4—344no—
203C2×C4—344no—
204C2×C4—344no—
205C2×C4—342no—
206C2×C4—684no—
207C2×C4C2×C22⋊D1712yes—
208C2×C4C2×C22⋊D1714yes—
209C2×C4C2×C22⋊D1712yes—
210C22—28no—
211C22—22no—
212C22—22no—
213C22—28no—
214C22—24no—
215C22—44no—
216C22—42no—
217C22—42no—
218C22#1058763914yes—
219C22#41×D1712yes—
220C22#1058769712yes—
221C22C22×C22⋊D1712yes—
222C22C22×C22⋊D1711yescharacteristic
223C4—22no—
224C4—24no—
225C4—22no—
226C4—344no—
227C4—684no—
228C4—682no—
229C2—24no—
230C2—44no—
231C2C2×#1058763911yescharacteristic
232C2#1176331611yescharacteristic
233C2#1176338011yescharacteristic
234C2#1176345814yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC22×#41×C17C2
hypercenterC22×#41D17
hypocenterC17#2406
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#10587621C2semidirect
2. C22×#63225C2semidirect
3. #11762728C2semidirect
4. #11762723C2semidirect
5. #11762689C2semidirect
6. #11762682C2semidirect
7. #11762672C2semidirect
91. C2×Dic17#41semidirect
146. C17#2406semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
C22×#632252
#117627282
#117627232
#117626892
#117626822
#117626722
C22×#41×C172
#240617