Finite Group Databasebeta

← #17002075

Subgroups — #17002075

Order 4352.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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
1#14542594C211yesmaximal, characteristic
2#14537865C211yesmaximal, characteristic
3#14527347C216yesmaximal
4#14527282C216yesmaximal
5C17×#2351C211yesmaximal, characteristic
6Q8×Dic34C414yescomplemented
7#11763538C414yescomplemented
8#11763535—24no—
9#11763440—23no—
10#11763366C2216yes—
11#11763287—21no—
12Q8×#3622—24no—
13#11762862C2216yes—
14#11762545C2214yes—
15#11762545C22112yes—
16C2×C4×Q8×C17—21no—
17C2×C4×Q8×C17C2211yescharacteristic
18C17×#389—23no—
19C17×#382C2213yes—
20C4×C4⋊C4×C17C2213yes—
21C22×Q8×C17D411yescharacteristic
22C4×Q8×C17—44no—
23C4×Q8×C17C2×C414yes—
24C4×Q8×C17D414yes—
25C2×C4⋊C4×C17—23no—
26C2×C4⋊C4×C17—23no—
27C2×C4⋊C4×C17D413yes—
28C2×C42×C17—23no—
29C2×C42×C17C2313yes—
30Q8×Dic17—24no—
31Q8×Dic17—24no—
32#10587612—26no—
33#10587610—26no—
34#10587610C2314yes—
35#10587576—212no—
36#10587574—212no—
37#10587556—26no—
38#10587555C2×C416yes—
39#10587555C2×C416yes—
40C4×Dic34—26no—
41C4×Dic34C2×C4112yes—
42C17×#95—23no—
43C17×#95C2311yescharacteristic
44C17×#95C2313yes—
45C22⋊#3622C2314yes—
46C4⋊#3622—26no—
47C4×#3622—26no—
48C2×Q8×C17C22⋊C414yes—
49C2×Q8×C17SD1614yes—
50C2×Q8×C17Q1614yes—
51C22×C4×C17—21no—
52C22×C4×C17—23no—
53C22×C4×C17C2×D413yes—
54C22×C4×C17C2×D413yes—
55C22×C4×C17C2411yescharacteristic
56C2×Dic34—26no—
57C2×Dic34C22×C414yes—
58C4⋊C4×C17—46no—
59C4⋊C4×C17—46no—
60C4⋊C4×C17C22⋊C4112yes—
61C42×C17—46no—
62C42×C17C22×C416yes—
63C42×C17C2×D416yes—
64C22⋊Dic17—24no—
65C4⋊Dic17—26no—
66C4⋊Dic17—26no—
67C4⋊Dic17—26no—
68C4⋊Dic17C22×C414yes—
69#63225—212no—
70#63225—24no—
71C4×Dic17—212no—
72C2×#3622—24no—
73#17619—171nomaximal, Sylow
74C23×C17C22×D411yescharacteristic
75Q8×C17#103116yes—
76C2×C4×C17—22no—
77C2×C4×C17—43no—
78C2×C4×C17—41no—
79C2×C4×C17—41no—
80C2×C4×C17—41no—
81C2×C4×C17—43no—
82C2×C4×C17—43no—
83C2×C4×C17—43no—
84C2×C4×C17C2×C22⋊C416yes—
85C2×C4×C17C2×C22⋊C416yes—
86C2×C4×C17C2×SD1616yes—
87C2×C4×C17C2×Q1616yes—
88C2×C4×C17C23×C411yescharacteristic
89C2×C4×C17C22×D411yescharacteristic
90C2×Dic17—24no—
91C2×Dic17—24no—
92Dic34—44no—
93Dic34—86no—
94#3622—44no—
95#3502—171no—
96#2982—171no—
97#2351D1711yescharacteristic
98#1518—176no—
99#1503—176no—
100C22×C17—21no—
101C22×C17—21no—
102C22×C17C22×C22⋊C411yescharacteristic
103C22×C17C22×SD1611yescharacteristic
104C22×C17C22×Q1611yescharacteristic
105C4×C17—21no—
106C4×C17—21no—
107C4×C17—26no—
108C4×C17—42no—
109C4×C17—83no—
110C4×C17C2×#103112yes—
111Dic17—44no—
112Dic17—44no—
113Q82—174no—
114#557—174no—
115#557—344no—
116#544—343no—
117#533—176no—
118#519—341no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2×D1711yescharacteristic
121C8×Q8—344no—
122#431—176no—
123#389—23no—
124#382C2×D1713yes—
125C4×C4⋊C4C2×D1713yes—
126#328—174no—
127#328—1712no—
128C2×C17—21no—
129C2×C17—21no—
130C2×C17#288411yescharacteristic
131C2×C17C22×#10311yescharacteristic
132C2×C17#290111yescharacteristic
133C22×Q8C22⋊D1711yescharacteristic
134C4⋊Q8—176no—
135C4⋊Q8—176no—
136C4⋊Q8—3412no—
137#125—3412no—
138#125—346no—
139C22⋊Q8—174no—
140C22⋊Q8—346no—
141C4×Q8—44no—
142C4×Q8—1712no—
143C4×Q8—344no—
144C4×Q8—344no—
145C4×Q8—346no—
146C4×Q8C4×D1714yes—
147C4×Q8D6814yes—
148#117—346no—
149C2×C4⋊C4—23no—
150C2×C4⋊C4—23no—
151C2×C4⋊C4C22⋊D1713yes—
152C2×C42—23no—
153C2×C42C22×D1713yes—
154C4⋊C8—346no—
155C22⋊C8—174no—
156C4×C8—346no—
157#95—23no—
158#95C22×D1711yescharacteristic
159#95C22×D1713yes—
160C17#1761911yescharacteristic, complemented, Sylow
161C2×Q8—174no—
162C2×Q8—346no—
163C2×Q8#6322714yes—
164C2×Q8#6322914yes—
165C2×Q8#6323114yes—
166C22×C4—21no—
167C22×C4—23no—
168C22×C4C2×C22⋊D1713yes—
169C22×C4C2×C22⋊D1713yes—
170C22×C4C23×D1711yescharacteristic
171C2×C8—344no—
172C4⋊C4—46no—
173C4⋊C4—46no—
174C4⋊C4—174no—
175C4⋊C4—344no—
176C4⋊C4—346no—
177C4⋊C4—346no—
178C4⋊C4—3412no—
179C4⋊C4—346no—
180C4⋊C4#63227112yes—
181C22⋊C4—344no—
182C42—46no—
183C42—3412no—
184C42C2×C4×D1716yes—
185C42C2×D6816yes—
186C23C22×C22⋊D1711yescharacteristic
187Q8—684no—
188Q8—1366no—
189Q8#10587481116yes—
190C2×C4—22no—
191C2×C4—43no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—43no—
196C2×C4—43no—
197C2×C4—43no—
198C2×C4—344no—
199C2×C4—344no—
200C2×C4C2×#6322716yes—
201C2×C4C2×#6322716yes—
202C2×C4C2×#6322916yes—
203C2×C4C2×#6323116yes—
204C2×C4C22×C4×D1711yescharacteristic
205C2×C4C22×D6811yescharacteristic
206C8—684no—
207C22—21no—
208C22—21no—
209C22C22×#6322711yescharacteristic
210C22C22×#6322911yescharacteristic
211C22C22×#6323111yescharacteristic
212C4—21no—
213C4—21no—
214C4—26no—
215C4—42no—
216C4—83no—
217C4—684no—
218C4—684no—
219C4C2×#10587481112yes—
220C2—21no—
221C2—21no—
222C2#1453724711yescharacteristic
223C2C22×#1058748111yescharacteristic
224C2#1453733211yescharacteristic
subgroupG/H
centerC22C22×#63227
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2351C2
socleC22×C17C22×C22⋊C4
prime radicalC23×C17C22×D4
cosocleC22×C4×C17C24
hypercenter#2351D17
hypocenterC17#17619
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2351

Complements

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

NH ≅ G/Ntype
6. Q8×Dic34C4semidirect
7. #11763538C4semidirect
160. C17#17619semidirect

Maximal subgroups

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

subgroupindex
#145425942
#145378652
#145273472
#145272822
C17×#23512
#1761917