Finite Group Databasebeta

← #16939354

Subgroups — #16939354

Order 4352.

296 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 89 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#14529764C211yesmaximal, characteristic
2#14528903C211yesmaximal, characteristic
3#14527556C211yesmaximal, characteristic
4#14527188C211yesmaximal, characteristic
5#14527186C211yesmaximal, characteristic
6#14526612C211yesmaximal, characteristic
7C17×#1571C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11762862C2211yescharacteristic
10C2×C4⋊#3622C2211yescharacteristic
11#11762655C2211yescharacteristic
12#11762570—22no—
13#11762563—22no—
14#11762559—22no—
15#11762541—22no—
16#11762539—22no—
17C17×#431C2211yescharacteristic
18C17×#423C2211yescharacteristic
19C2×C17×C4⋊C8C2211yescharacteristic
20C17×#335—22no—
21C2×M16×C17—21no—
22C22×C8×C17—21no—
23C2×C42×C17C2311yescharacteristic
24#10587612—21no—
25#10587610—21no—
26#10587610—21no—
27C2×C4×Dic17D412yes—
28C22×#3622—21no—
29#10587556—21no—
30#10587555Q811yescharacteristic
31#10587555Q811yescharacteristic
32C4×Dic34—21no—
33C4×Dic34—21no—
34C17×C4⋊C8—21no—
35C17×C4⋊C8—21no—
36C17×C4⋊C8—21no—
37C17×C4⋊C8—21no—
38C17×C4⋊C8—21no—
39C17×C22⋊C8—21no—
40C4×C8×C17D412yes—
41#10587502—21no—
42#10587502—21no—
43C22⋊#3622—21no—
44C4⋊#3622—21no—
45C4⋊#3622—21no—
46C4⋊#3622—21no—
47C4×#3622D412yes—
48C22×C4×C17C2×D411yescharacteristic
49C22×C4×C17#4111yescharacteristic
50C22×C4×C17#4111yescharacteristic
51C22×Dic17—22no—
52C2×Dic34—21no—
53C2×Dic34—21no—
54M16×C17—42no—
55C2×C8×C17—22no—
56C2×C8×C17—41no—
57C2×C8×C17—41no—
58C2×C8×C17—42no—
59C42×C17C2×D411yescharacteristic
60C42×C17C2×D411yescharacteristic
61C42×C17C2×Q811yescharacteristic
62C42×C17#4111yescharacteristic
63C22⋊Dic17—41no—
64C4⋊Dic17—21no—
65C4⋊Dic17—21no—
66C4⋊Dic17—21no—
67C4⋊Dic17—21no—
68#63225—41no—
69#63225—41no—
70C4×Dic17—22no—
71C4×Dic17—22no—
72C4×Dic17—41no—
73C2×#3622—22no—
74C2×#3622—41no—
75C2×#3622—42no—
76#10669—171nomaximal, Sylow
77C23×C17#12311yescharacteristic
78C2×C4×C17—41no—
79C2×C4×C17—41no—
80C2×C4×C17C22⋊Q811yescharacteristic
81C2×C4×C17C22⋊Q811yescharacteristic
82C2×C4×C17C22⋊Q811yescharacteristic
83C2×C4×C17#12311yescharacteristic
84C2×C4×C17#12311yescharacteristic
85C2×C4×C17#12711yescharacteristic
86C2×C4×C17#13512yes—
87C2×C4×C17#13712yes—
88C2×Dic17—42no—
89C2×Dic17—42no—
90C2×Dic17—41no—
91C2×Dic17—41no—
92C2×Dic17—42no—
93Dic34—42no—
94Dic34—42no—
95C8×C17—42no—
96C8×C17—81no—
97C8×C17—81no—
98#3622—42no—
99#3622—81no—
100#1995—171no—
101#1832—171no—
102#1571D1711yescharacteristic
103#1570—171no—
104#1481—171no—
105#1481—171no—
106#1274—171no—
107C22×C17—21no—
108C22×C17—21no—
109C22×C17#39911yescharacteristic
110C22×C17#48512yes—
111C4×C17—21no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—21no—
115C4×C17#47912yes—
116C4×C17#49511yescharacteristic
117C4×C17#49711yescharacteristic
118Dic17—82no—
119Dic17—81no—
120Dic17—81no—
121#533—171no—
122#431—171no—
123#431C2×D1711yescharacteristic
124#423C2×D1711yescharacteristic
125C2×C4⋊C8—171no—
126C2×C4⋊C8C2×D1711yescharacteristic
127#376—171no—
128#339—342no—
129#336—342no—
130#335—22no—
131#334—342no—
132Q8⋊C8—342no—
133Q8⋊C8—342no—
134C2×C17—22no—
135C2×C17#209711yescharacteristic
136C2×C17#209811yescharacteristic
137C2×C17#209811yescharacteristic
138C2×M16—21no—
139C22×C8—21no—
140C22×C8—341no—
141C4⋊Q8—171no—
142C4⋊Q8—171no—
143#125—341no—
144C22⋊Q8—341no—
145C22⋊Q8—341no—
146C4×Q8—341no—
147C4×Q8—341no—
148#117—341no—
149C2×C42—172no—
150C2×C42C22×D1711yescharacteristic
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—341no—
157C4⋊C8—341no—
158C4⋊C8—341no—
159C22⋊C8—21no—
160C22⋊C8—341no—
161C4×C8—172no—
162C4×C8C22⋊D1712yes—
163#95—341no—
164#95—341no—
165C17#1066911yescharacteristic, complemented, Sylow
166C2×Q8—341no—
167C2×Q8—341no—
168C22×C4—342no—
169C22×C4#6325211yescharacteristic
170C22×C4D4×D1711yescharacteristic
171C22×C4#6325611yescharacteristic
172M16—42no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—342no—
178C2×C8—682no—
179C2×C8—681no—
180C4⋊C4—341no—
181C4⋊C4—341no—
182C4⋊C4—341no—
183C4⋊C4—341no—
184C4⋊C4—681no—
185C4⋊C4—681no—
186C22⋊C4—681no—
187C42—342no—
188C42—342no—
189C42—681no—
190C42#6325211yescharacteristic
191C42D4×D1711yescharacteristic
192C42Q8×D1711yescharacteristic
193C42C2×C22⋊D1711yescharacteristic
194C23#1058758011yescharacteristic
195Q8—682no—
196Q8—682no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—682no—
200C2×C4—682no—
201C2×C4—681no—
202C2×C4—681no—
203C2×C4—682no—
204C2×C4#1058758011yescharacteristic
205C2×C4#1058758211yescharacteristic
206C2×C4#1058758211yescharacteristic
207C2×C4#1058759812yes—
208C2×C4#1058760412yes—
209C2×C4#1058761511yescharacteristic
210C2×C4#1058762611yescharacteristic
211C2×C4#1058763211yescharacteristic
212C8—42no—
213C8—81no—
214C8—81no—
215C8—682no—
216C8—1361no—
217C22—21no—
218C22—21no—
219C22#1176274411yescharacteristic
220C22#1176315212yes—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—1362no—
226C4—1361no—
227C4—1361no—
228C4#1176311512yes—
229C4#1176319211yescharacteristic
230C4#1176320211yescharacteristic
231C2—22no—
232C2#1453032911yescharacteristic
233C2#1453033611yescharacteristic
234C2#1453034011yescharacteristic
subgroupG/H
centerC2×C4#10587626
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1571C2
socleC22×C17#399
prime radicalC23×C17#123
hypercenter#1571D17
hypocenterC17#10669
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
165. C17#10669semidirect

Maximal subgroups

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

subgroupindex
#145297642
#145289032
#145275562
#145271882
#145271862
#145266122
C17×#15712
#1066917