Finite Group Databasebeta

← #14532425

Subgroups — #14532425

Order 2176.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×Q8×D17C211yesmaximal, characteristic
2C2×#10587630C211yesmaximal, characteristic, complemented
3C2×#10587632C211yesmaximal, characteristic
4#11762729C214yesmaximal, complemented
5#11762716C211yesmaximal, characteristic, complemented
6#11762715C211yesmaximal, characteristic, complemented
7#11762680C214yesmaximal
8C4×#63227C211yesmaximal, characteristic
9C2×C17×C4⋊Q8C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×Q8×C17C2211yescharacteristic
12C2×Q8×D17—24no—
13C2×Q8×D17—24no—
14C2×Q8×D17—24no—
15C22×C4×D17C2212yes—
16C22×C4×D17C2211yescharacteristic
17C22×Dic34C2212yes—
18C22×Dic34C2211yescharacteristic
19C17×C4⋊Q8—24no—
20C2×C4⋊C4×C17C2214yes—
21C2×C42×C17C2211yescharacteristic
22#10587632—24no—
23#10587630—24no—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2212yes—
28C2×#63225C2214yes—
29C2×C4×Dic17C2211yescharacteristic
30#10587502C2214yes—
31#10587502C2214yes—
32#10587502C2212yes—
33C23×D17C2311yescharacteristic
34C2×Q8×C17—22no—
35C2×Q8×C17—22no—
36C2×Q8×C17—22no—
37C2×Q8×C17—22no—
38C2×Q8×C17—22no—
39C2×Q8×C17D414yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48Q8×D17—232no—
49C2×C4×D17—24no—
50C2×C4×D17—22no—
51C2×C4×D17—24no—
52C2×C4×D17—22no—
53C2×C4×D17—24no—
54C2×C4×D17D414yes—
55C2×Dic34—22no—
56C2×Dic34—24no—
57C2×Dic34—22no—
58C2×Dic34—24no—
59C2×Dic34—24no—
60C2×Dic34D414yes—
61C4⋊C4×C17—28no—
62C42×C17—22no—
63#63227—24no—
64#63227Q818yescomplemented
65C4⋊Dic17—22no—
66#63225—28no—
67#63225—24no—
68C4×Dic17—22no—
69C23×C17C2411yescharacteristic
70C22×D17—21no—
71C22×D17—21no—
72C22×D17—22no—
73C22×D17C2×D412yes—
74C22×D17C2×Q814yes—
75Q8×C17—28no—
76Q8×C17—44no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17C2×D412yes—
85C2×C4×C17C2×D412yes—
86C2×C4×C17C2×D414yes—
87C2×C4×C17C2×D412yes—
88C2×C4×C17C2×Q814yescomplemented
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—21no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—24no—
97C2×Dic17C2×D412yes—
98C2×Dic17C2×D414yes—
99C2×Dic17C2×Q814yescomplemented
100C4×D17—216no—
101C4×D17—28no—
102Dic34—216no—
103Dic34—28no—
104#2434—171nomaximal, Sylow
105C22×C17C22×D411yescharacteristic
106C22×C17C22×D411yescharacteristic
107C22×C17C22×D411yescharacteristic
108C22×C17C22×Q812yes—
109C22×C172−1+412yes—
110C2×D17—22no—
111C2×D17—22no—
112C2×D17—24no—
113C2×D17—24no—
114C2×D17C4⋊Q814yescomplemented
115C4×C17—24no—
116C4×C17—24no—
117C4×C17—42no—
118C4×C17#12014yes—
119Dic17—22no—
120Dic17—24no—
121Dic17—24no—
122Dic17—42no—
123C23×Q8—171no—
124C2×C4⋊Q8—171no—
125C2×C4⋊Q8D1711yescharacteristic, complemented
126C2×C22⋊Q8—171no—
127#395—174no—
128#391—171no—
129#391—171no—
130#385—174no—
131C4×C22⋊C4—171no—
132C2×C17C2×#12011yescharacteristic
133C2×C17C2×C4⋊Q811yescharacteristic
134C2×C17#53611yescharacteristic
135C2×C17D4×Q812yes—
136C2×C17D4×Q812yes—
137D17—24no—
138C22×Q8—171no—
139C22×Q8—172no—
140C22×Q8—344no—
141C22×Q8—344no—
142C22×Q8—344no—
143C22×Q8C2×D1711yescharacteristic
144C22×Q8C2×D1711yescharacteristic
145C23×C4—171no—
146C23×C4—172no—
147C4⋊Q8—24no—
148C4⋊Q8—344no—
149C22⋊Q8—344no—
150C2×C4⋊C4—171no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—174no—
153C2×C4⋊C4C2×D1714yes—
154C2×C22⋊C4—172no—
155C2×C22⋊C4—172no—
156C2×C42—171no—
157C2×C42C2×D1711yescharacteristic
158#95—172no—
159#95—174no—
160#95—174no—
161C17#243411yescharacteristic, complemented, Sylow
162C24—171no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—174no—
169C2×Q8—344no—
170C2×Q8—344no—
171C2×Q8—3432no—
172C2×Q8—342no—
173C2×Q8—344no—
174C2×Q8—342no—
175C2×Q8C22⋊D1714yes—
176C22×C4—172no—
177C22×C4—171no—
178C22×C4—172no—
179C22×C4—174no—
180C22×C4—172no—
181C22×C4—344no—
182C22×C4—344no—
183C22×C4—342no—
184C22×C4—344no—
185C22×C4—342no—
186C22×C4C22×D1712yes—
187C22×C4C22×D1712yes—
188C22×C4C22×D1712yes—
189C22×C4C22×D1711yescharacteristic
190C4⋊C4—28no—
191C4⋊C4—344no—
192C4⋊C4—348no—
193C4⋊C4—342no—
194C22⋊C4—178no—
195C22⋊C4—344no—
196C42—22no—
197C42—342no—
198C23—172no—
199C23—174no—
200C23—341no—
201C23—342no—
202C23—341no—
203C23C23×D1711yescharacteristic
204Q8—28no—
205Q8—44no—
206Q8—3416no—
207Q8—348no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—172no—
216C2×C4—174no—
217C2×C4—174no—
218C2×C4—3416no—
219C2×C4—342no—
220C2×C4—341no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—341no—
224C2×C4—342no—
225C2×C4—348no—
226C2×C4—342no—
227C2×C4—344no—
228C2×C4D4×D1712yes—
229C2×C4D4×D1712yes—
230C2×C4Q8×D1714yescomplemented
231C2×C4C2×C22⋊D1714yes—
232C2×C4C2×C22⋊D1712yes—
233C22—174no—
234C22—344no—
235C22—344no—
236C22—342no—
237C22—342no—
238C22C2×D4×D1711yescharacteristic
239C22C2×D4×D1711yescharacteristic
240C22C2×Q8×D1712yes—
241C22#1058769412yes—
242C22C22×C22⋊D1711yescharacteristic
243C4—24no—
244C4—24no—
245C4—42no—
246C4—344no—
247C4—342no—
248C4—344no—
249C4—682no—
250C4#1058762314yes—
251C2—344no—
252C2C2×#1058762311yescharacteristic
253C2C4⋊Q8×D1711yescharacteristic
254C2#1176338411yescharacteristic
255C2Q8×C22⋊D1712yes—
256C2#1176347412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×C4⋊Q8C2
prime radicalC23×D17C23
hypercenterC2×C4⋊Q8D17
hypocenterC17#2434
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#10587630C2semidirect
4. #11762729C2semidirect
5. #11762716C2semidirect
6. #11762715C2semidirect
9. C2×C17×C4⋊Q8C2semidirect
64. #63227Q8semidirect
88. C2×C4×C17C2×Q8semidirect
99. C2×Dic17C2×Q8semidirect
114. C2×D17C4⋊Q8semidirect
125. C2×C4⋊Q8D17semidirect
161. C17#2434semidirect
230. C2×C4Q8×D17semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D172
C2×#105876302
C2×#105876322
#117627292
#117627162
#117627152
#117626802
C4×#632272
C2×C17×C4⋊Q82
#243417