Finite Group Databasebeta

← C22×#11762858 #17898017

Subgroups — C22×#11762858 #17898017

Order 4352.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#63223C212yesmaximal
2C2×#11762858C2124yesmaximal, direct product factor
3C22×#10587490C214yesmaximal, complemented
4C23×M16×C17C211yesmaximal, characteristic
5C22×#63223—22no—
6C22×#63223—212no—
7C22×#63223C2214yes—
8C22×#63223C22124yes—
9C23×#3622—22no—
10#11762858C22164yesdirect product factor
11C2×#10587490C22148yescomplemented
12C24×C4×C17C2211yescharacteristic
13C22×M16×C17C414yes—
14C22×M16×C17C4112yes—
15C22×M16×C17C22112yes—
16C23×C8×C17C2212yes—
17C25×C17Q811yescharacteristic
18C23×C4×C17C2×C412yes—
19C23×C4×C17C2×C416yes—
20C23×C4×C17D412yes—
21C23×C4×C17D416yes—
22C23×C4×C17Q811yescharacteristic
23C23×C4×C17Q816yes—
24C23×C4×C17C2311yescharacteristic
25C23×C4×C17C2316yes—
26C2×M16×C17C2×C4148yes—
27C2×M16×C17C2×C4148yes—
28C2×M16×C17C23116yes—
29C22×C8×C17C2×C4112yes—
30C22×C8×C17C2×C414yes—
31C22×C8×C17C23112yes—
32C2×#63223—224no—
33C2×#63223—212no—
34C2×#63223—248no—
35C2×#63223C23124yes—
36C2×#63223C23132yes—
37C22×#3622—224no—
38C22×#3622—24no—
39#10587490C23164yescomplemented
40C24×C17—44no—
41C24×C17C4⋊C412yes—
42C24×C17C4⋊C416yes—
43C24×C17C2×Q811yescharacteristic
44C24×C17C2×Q816yes—
45C22×C4×C17C4⋊C4112yes—
46C22×C4×C17C4⋊C4112yes—
47C22×C4×C17C4⋊C416yes—
48C22×C4×C17C4⋊C4124yes—
49C22×C4×C17C4⋊C412yes—
50C22×C4×C17C22×C413yes—
51C22×C4×C17C22×C411yescharacteristic
52C22×C4×C17C22×C4112yes—
53C22×C4×C17C22×C4112yes—
54C22×C4×C17C2×D4112yes—
55C22×C4×C17C2×D413yes—
56C22×C4×C17C2×D4112yes—
57C22×C4×C17C2×D411yescharacteristic
58C22×C4×C17C2×Q813yes—
59C22×C4×C17C2×Q816yes—
60C22×C4×C17C2×Q8112yes—
61C22×C4×C17C2413yes—
62C22×C4×C17C2414yes—
63M16×C17C22×C4164yes—
64C2×C8×C17C22×C4124yes—
65C2×C8×C17C22×C4124yes—
66C2×C8×C17C2418yes—
67#63223—232no—
68#63223—224no—
69#63223C24116yes—
70C2×#3622—224no—
71C2×#3622—232no—
72C22×#430—171nomaximal, Sylow
73C23×C17—28no—
74C23×C17—24no—
75C23×C17—412no—
76C23×C17—412no—
77C23×C17C2×C4⋊C413yes—
78C23×C17C2×C4⋊C4112yes—
79C23×C17C2×C4⋊C4112yes—
80C23×C17C2×C4⋊C411yescharacteristic
81C23×C17C22×Q813yes—
82C23×C17C22×Q814yes—
83C2×C4×C17C2×C4⋊C416yes—
84C2×C4×C17C2×C4⋊C4124yes—
85C2×C4×C17C2×C4⋊C4124yes—
86C2×C4×C17C2×C4⋊C4112yes—
87C2×C4×C17C2×C4⋊C413yes—
88C2×C4×C17C2×C4⋊C4112yes—
89C2×C4×C17C2×C4⋊C413yes—
90C2×C4×C17C23×C413yes—
91C2×C4×C17C23×C413yes—
92C2×C4×C17C23×C418yes—
93C2×C4×C17C22×D413yes—
94C2×C4×C17C22×D418yes—
95C2×C4×C17C22×D413yes—
96C2×C4×C17C22×Q813yes—
97C2×C4×C17C22×Q814yes—
98C2×C4×C17C2511yescharacteristic
99C8×C17C23×C4116yes—
100#3622—216no—
101C23×M16—172no—
102C23×M16D1711yescharacteristic
103C2×#430—1724no—
104C22×#108—174no—
105C22×C17—224no—
106C22×C17—26no—
107C22×C17—26no—
108C22×C17—412no—
109C22×C17—44no—
110C22×C17#43014yescomplemented
111C22×C17C22×C4⋊C413yes—
112C22×C17C22×C4⋊C418yes—
113C22×C17C22×C4⋊C413yes—
114C22×C17C23×Q811yescharacteristic
115C4×C17C22×C4⋊C413yes—
116C4×C17C22×C4⋊C413yes—
117C4×C17C22×C4⋊C418yes—
118C4×C17C24×C411yescharacteristic
119C4×C17C23×D411yescharacteristic
120C24×C4C2×D1711yescharacteristic
121C22×M16—174no—
122C22×M16—1724no—
123C22×M16—3412no—
124C22×M16—342no—
125C22×M16Dic1714yes—
126C22×M16Dic17112yes—
127C22×M16C2×D17112yes—
128C23×C8—342no—
129C23×C8C2×D1712yes—
130#430—1764no—
131C2×#108—1748no—
132C2×C17—28no—
133C2×C17—23no—
134C2×C17—21no—
135C2×C17C2×#43016yescomplemented
136C2×C17C23×C4⋊C411yescharacteristic
137C25Dic3411yescharacteristic
138C23×C4Dic3411yescharacteristic
139C23×C4Dic3416yes—
140C23×C4D6816yes—
141C23×C4D6812yes—
142C23×C4C2×Dic1712yes—
143C23×C4C2×Dic1716yes—
144C23×C4C22×D1716yes—
145C23×C4C22×D1711yescharacteristic
146C2×M16—1724no—
147C2×M16—1732no—
148C2×M16—3424no—
149C2×M16—3412no—
150C2×M16—3448no—
151C2×M16C2×Dic17148yes—
152C2×M16C2×Dic17148yes—
153C2×M16C22×D17116yes—
154C22×C8—3424no—
155C22×C8—344no—
156C22×C8C2×Dic1714yes—
157C22×C8C2×Dic17112yes—
158C22×C8C22×D17112yes—
159#108—1764no—
160C17C22×#43011yescharacteristic, complemented, Sylow
161C24—44no—
162C24C4⋊Dic1712yes—
163C24C4⋊Dic1716yes—
164C24C2×Dic3411yescharacteristic
165C24C2×Dic3416yes—
166C22×C4C4⋊Dic1716yes—
167C22×C4C4⋊Dic17112yes—
168C22×C4C4⋊Dic17124yes—
169C22×C4C4⋊Dic1712yes—
170C22×C4C4⋊Dic17112yes—
171C22×C4C2×Dic3416yes—
172C22×C4C2×Dic34112yes—
173C22×C4C2×Dic3413yes—
174C22×C4C2×D68112yes—
175C22×C4C2×D6813yes—
176C22×C4C2×D68112yes—
177C22×C4C2×D6811yescharacteristic
178C22×C4C22×Dic1711yescharacteristic
179C22×C4C22×Dic1713yes—
180C22×C4C22×Dic17112yes—
181C22×C4C22×Dic17112yes—
182C22×C4C23×D1713yes—
183C22×C4C23×D1714yes—
184M16—1716no—
185M16—3432no—
186M16—3424no—
187M16C22×Dic17164yes—
188C2×C8—3424no—
189C2×C8—3432no—
190C2×C8C22×Dic17124yes—
191C2×C8C22×Dic17124yes—
192C2×C8C23×D1718yes—
193C23—24no—
194C23—28no—
195C23—412no—
196C23—412no—
197C23C2×C4⋊Dic17112yes—
198C23C2×C4⋊Dic1713yes—
199C23C2×C4⋊Dic17112yes—
200C23C2×C4⋊Dic1711yescharacteristic
201C23C22×Dic3413yes—
202C23C22×Dic3414yes—
203C2×C4C2×C4⋊Dic17124yes—
204C2×C4C2×C4⋊Dic1716yes—
205C2×C4C2×C4⋊Dic17112yes—
206C2×C4C2×C4⋊Dic17124yes—
207C2×C4C2×C4⋊Dic17112yes—
208C2×C4C2×C4⋊Dic1713yes—
209C2×C4C2×C4⋊Dic1713yes—
210C2×C4C22×Dic3413yes—
211C2×C4C22×Dic3414yes—
212C2×C4C22×D6813yes—
213C2×C4C22×D6818yes—
214C2×C4C22×D6813yes—
215C2×C4C23×Dic1713yes—
216C2×C4C23×Dic1713yes—
217C2×C4C23×Dic1718yes—
218C2×C4C24×D1711yescharacteristic
219C8—3416no—
220C8C23×Dic17116yes—
221C22—26no—
222C22—26no—
223C22—224no—
224C22—44no—
225C22—412no—
226C22#1176285814yesdirect product factor
227C22C22×C4⋊Dic1718yes—
228C22C22×C4⋊Dic1713yes—
229C22C22×C4⋊Dic1713yes—
230C22C23×Dic3411yescharacteristic
231C4C22×C4⋊Dic1713yes—
232C4C22×C4⋊Dic1718yes—
233C4C22×C4⋊Dic1713yes—
234C4C24×Dic1711yescharacteristic
235C4C23×D6811yescharacteristic
236C2—23no—
237C2—28no—
238C2—21no—
239C2C2×#1176285816yesdirect product factor
240C2C23×C4⋊Dic1711yescharacteristic
subgroupG/H
centerC22×C4C2×D68
commutatorC4×C17C24×C4
FrattiniC2×C4C24×D17
FittingC23×M16×C17C2
socleC23×C17C2×C4⋊C4
prime radicalC25×C17Q8
cosocleC2×C4×C17C25
hypercenterC23×M16D17
hypocenterC17C22×#430
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
2. C2×#11762858C2direct (N × H)
3. C22×#10587490C2semidirect
10. #11762858C22direct (N × H)
11. C2×#10587490C22semidirect
39. #10587490C23semidirect
110. C22×C17#430semidirect
135. C2×C17C2×#430semidirect
160. C17C22×#430semidirect
226. C22#11762858direct (N × H)
239. C2C2×#11762858direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#632232
C2×#117628582
C22×#105874902
C23×M16×C172
C22×#43017