Finite Group Databasebeta

← #14533095

Subgroups — #14533095

Order 2176.

584 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 173 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×C4×C22⋊D17C211yesmaximal, characteristic
2#11762747C212yesmaximal, complemented
3#11762744C212yesmaximal, complemented
4#11762705C212yesmaximal, complemented
5#11762688C212yesmaximal
6#11762682C212yesmaximal, complemented
7#11762669C211yesmaximal, characteristic, complemented
8#11762666C212yesmaximal, complemented
9C22×C4⋊C4×C17C211yesmaximal, characteristic, complemented
10C23×C4×C17C2211yescharacteristic
11C23×C4×C17C2212yes—
12C22×C22⋊D17C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C2×C4⋊C4×C17—24no—
15C2×C4⋊C4×C17C2212yes—
16C2×C4⋊C4×C17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C4×C22⋊D17—24no—
20C2×#63227C2211yescharacteristic
21C2×#63227C2214yes—
22C2×C4⋊Dic17C2212yes—
23C2×#63225C2211yescharacteristic
24C2×#63225C2214yes—
25C2×C4×Dic17C2211yescharacteristic
26#10587502C2214yes—
27#10587502C2212yes—
28#10587502C2214yes—
29C24×C17C2311yescharacteristic
30C23×D17C2311yescharacteristic
31C22×C4×C17—24no—
32C22×C4×C17—24no—
33C22×C4×C17—44no—
34C22×C4×C17D414yes—
35C22×C4×C17C2314yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C2×C22⋊D17—22no—
39C2×C22⋊D17—22no—
40C2×C22⋊D17Q814yes—
41C22×Dic17C2311yescharacteristic
42C22×Dic17C2314yes—
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2311yescharacteristic
45C2×C4×D17—22no—
46C2×C4×D17—24no—
47C4⋊C4×C17—24no—
48C4⋊C4×C17—24no—
49C22⋊Dic17—22no—
50C22⋊Dic17—24no—
51#63227—22no—
52#63227—44no—
53C4⋊Dic17—24no—
54#63225—22no—
55#63225—44no—
56C4×Dic17—22no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17C2×D412yes—
60C23×C17C2×Q812yes—
61C23×C17#4112yes—
62C23×C17C2411yescharacteristic
63C22×D17—21no—
64C22×D17—22no—
65C22×D17—21no—
66C22×D17—22no—
67C22×D17C2×Q812yes—
68C2×C4×C17—22no—
69C2×C4×C17—28no—
70C2×C4×C17—24no—
71C2×C4×C17—24no—
72C2×C4×C17—24no—
73C2×C4×C17—48no—
74C2×C4×C17C2×D412yes—
75C2×C4×C17C2×D412yes—
76C2×C4×C17#4114yes—
77C22⋊D17—28no—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17—24no—
83C2×Dic17—22no—
84C2×Dic17—24no—
85C2×Dic17—22no—
86C2×Dic17—24no—
87C2×Dic17C2×Q812yes—
88C4×D17—44no—
89#2487—171nomaximal, Sylow
90C22×C17—22no—
91C22×C17—22no—
92C22×C17—24no—
93C22×C17—24no—
94C22×C17C22⋊Q814yes—
95C22×C17C22×D411yescharacteristic
96C22×C17C22×Q811yescharacteristic
97C22×C17C2×#4112yes—
98C22×C17C2×#4111yescharacteristic
99C22×C17H(5,𝔽2)12yes—
100C2×D17—22no—
101C2×D17—24no—
102C2×D17—41no—
103C2×D17—41no—
104C2×D17—42no—
105C4×C17—22no—
106C4×C17—44no—
107C4×C17#12314yes—
108Dic17—22no—
109Dic17—41no—
110Dic17—41no—
111Dic17—44no—
112C2×C4×D4—171no—
113C22×C4⋊C4D1711yescharacteristic, complemented
114#400—172no—
115#399—172no—
116#389—172no—
117#386—172no—
118#385—172no—
119#380—171no—
120#380—172no—
121C2×C17—24no—
122C2×C17C2×C22⋊Q811yescharacteristic
123C2×C17C2×#12311yescharacteristic
124C2×C17#53511yescharacteristic
125C2×C17#55312yes—
126C2×C17#55412yes—
127D17—42no—
128C22×D4—171no—
129C23×C4—171no—
130C23×C4C2×D1711yescharacteristic
131C23×C4C2×D1712yes—
132C4×D4—344no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—171no—
135C2×C4⋊C4—172no—
136C2×C4⋊C4—174no—
137C2×C4⋊C4C2×D1712yes—
138C2×C4⋊C4C2×D1712yes—
139C2×C22⋊C4—171no—
140C2×C22⋊C4—172no—
141C2×C22⋊C4—171no—
142C2×C22⋊C4—174no—
143C2×C42—171no—
144#95—174no—
145#95—172no—
146#95—174no—
147C17#248711yescharacteristic, complemented, Sylow
148C24—171no—
149C24C22×D1711yescharacteristic
150C2×D4—174no—
151C2×D4—342no—
152C2×D4—342no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—44no—
156C22×C4—171no—
157C22×C4—171no—
158C22×C4—174no—
159C22×C4—171no—
160C22×C4—344no—
161C22×C4—342no—
162C22×C4C22⋊D1714yes—
163C22×C4C22×D1714yes—
164C22×C4C22×D1711yescharacteristic
165C22×C4C22×D1711yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—344no—
169C4⋊C4—342no—
170C4⋊C4—684no—
171C22⋊C4—344no—
172C22⋊C4—342no—
173C22⋊C4—342no—
174C22⋊C4—684no—
175C42—342no—
176C23—22no—
177C23—22no—
178C23—172no—
179C23—342no—
180C23—342no—
181C23—341no—
182C23—341no—
183C23Q8×D1712yes—
184C23#6325612yes—
185C23C2×C22⋊D1712yes—
186C23C23×D1711yescharacteristic
187D4—348no—
188C2×C4—22no—
189C2×C4—28no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—48no—
194C2×C4—172no—
195C2×C4—341no—
196C2×C4—341no—
197C2×C4—344no—
198C2×C4—342no—
199C2×C4—344no—
200C2×C4—342no—
201C2×C4—344no—
202C2×C4—341no—
203C2×C4—341no—
204C2×C4—684no—
205C2×C4#6325214yes—
206C2×C4C2×C22⋊D1712yes—
207C2×C4C2×C22⋊D1712yes—
208C22—24no—
209C22—24no—
210C22—22no—
211C22—22no—
212C22—344no—
213C22—342no—
214C22—682no—
215C22—681no—
216C22—681no—
217C22#1058763214yes—
218C22C2×#6325212yes—
219C22#1058769112yes—
220C22C2×Q8×D1711yescharacteristic
221C22C2×#6325611yescharacteristic
222C22C22×C22⋊D1711yescharacteristic
223C4—22no—
224C4—44no—
225C4—342no—
226C4—684no—
227C4—681no—
228C4—681no—
229C4#1058761514yes—
230C2—24no—
231C2—682no—
232C2C2×#1058763211yescharacteristic
233C2C2×#1058761511yescharacteristic
234C2#1176337611yescharacteristic
235C2#1176350712yes—
236C2#1176351412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C4⋊C4×C17C2
prime radicalC22×C22⋊D17C22
hypercenterC22×C4⋊C4D17
hypocenterC17#2487
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11762747C2semidirect
3. #11762744C2semidirect
4. #11762705C2semidirect
6. #11762682C2semidirect
7. #11762669C2semidirect
8. #11762666C2semidirect
9. C22×C4⋊C4×C17C2semidirect
113. C22×C4⋊C4D17semidirect
147. C17#2487semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D172
#117627472
#117627442
#117627052
#117626882
#117626822
#117626692
#117626662
C22×C4⋊C4×C172
#248717