Finite Group Databasebeta

← #16945519

Subgroups — #16945519

Order 4352.

274 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 61 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#14539163C211yesmaximal, characteristic
2#14530594C212yesmaximal
3#14526819C211yesmaximal, characteristic, complemented
4#14526818C212yesmaximal, complemented
5C17×#1356C211yesmaximal, characteristic, complemented
6#11763365C411yescharacteristic
7C4⋊Q8×D17C411yescharacteristic
8#11763179C411yescharacteristic
9#11763178C2211yescharacteristic
10#11763177C411yescharacteristic
11#11763158—22no—
12#11762816—22no—
13#11762761C2211yescharacteristic
14#11762629—22no—
15#11762590—22no—
16#11762587C2212yes—
17C17×#491C2211yescharacteristic
18C17×#347C2212yes—
19C2×Q8×D17—21no—
20C17×C4⋊Q8C2×C411yescharacteristic
21C17×#125C2×C411yescharacteristic
22#10587630—21no—
23C2×#63218D412yes—
24#10587582—41no—
25#10587578—21no—
26C4⋊C4×D17—41no—
27#10587574—21no—
28#10587574—41no—
29#10587573—41no—
30C42×D17C2×C411yescharacteristic
31#10587556C2×C411yescharacteristic
32#10587555C2×C411yescharacteristic
33C17×M32—22no—
34C17×#103—22no—
35C17×#97C2311yescharacteristic
36#10587506—22no—
37#10587487—22no—
38C8×Dic17D412yes—
39#10587483—22no—
40#10587481—22no—
41#10587474C2×C411yescharacteristic
42C2×Q8×C17—21no—
43Q8×D17—44no—
44C2×C4×D17—21no—
45C2×C4×D17—41no—
46C2×C4×D17C22⋊C411yescharacteristic
47C2×Dic34—21no—
48C2×Dic34—41no—
49C2×Dic34—41no—
50C2×C8×C17C2×D412yes—
51C4⋊C4×C17—21no—
52C4⋊C4×C17—41no—
53C4⋊C4×C17—41no—
54C42×C17C22×C411yescharacteristic
55#63227—41no—
56C4⋊Dic17—21no—
57C4⋊Dic17—41no—
58#63225—41no—
59#63225—42no—
60#63225—41no—
61C4×Dic17—21no—
62C4×Dic17—41no—
63C4×Dic17C22⋊C411yescharacteristic
64C2×#3622C22⋊C412yes—
65#63218—42no—
66C16×C17—42no—
67#63214—42no—
68#11916—171nomaximal, Sylow
69C22×D17#10011yescharacteristic
70Q8×C17—42no—
71C2×C4×C17—21no—
72C2×C4×C17—41no—
73C2×C4×C17—41no—
74C2×C4×C17C2×C22⋊C411yescharacteristic
75C2×Dic17—21no—
76C2×Dic17—41no—
77C2×Dic17—41no—
78C2×Dic17#10011yescharacteristic
79C4×D17—22no—
80C4×D17—42no—
81C4×D17—82no—
82Dic34—42no—
83Dic34—82no—
84Dic34—82no—
85C8×C17—22no—
86#3622—22no—
87#3139—171no—
88#2165—172no—
89#1356—171no—
90#1356—172no—
91#1356D1711yescharacteristic, complemented
92C22×C17C2×#10011yescharacteristic
93C2×D17—21no—
94C2×D17—41no—
95C4×C17—41no—
96C4×C17—81no—
97C4×C17—81no—
98C4×C17#42112yes—
99Dic17—22no—
100Dic17—21no—
101Dic17—81no—
102Dic17—81no—
103#533—171no—
104C2×C4⋊Q8—171no—
105#491—171no—
106#491—171no—
107#491—171no—
108#491C2×D1711yescharacteristic
109#487—342no—
110#419—342no—
111C4×M16—171no—
112#364—342no—
113#349—342no—
114#347—172no—
115#347C2×D1712yes—
116C2×C17—21no—
117C2×C17#153311yescharacteristic
118D17—41no—
119C22×Q8—341no—
120C2×M16—172no—
121C4⋊Q8—171no—
122C4⋊Q8—341no—
123C4⋊Q8—341no—
124C4⋊Q8—681no—
125C4⋊Q8C4×D1711yescharacteristic
126#125—171no—
127#125C4×D1711yescharacteristic
128C22⋊Q8—681no—
129C4×Q8—681no—
130#117—341no—
131C2×C4⋊C4—681no—
132C2×C42—171no—
133M32—22no—
134M32—342no—
135#103—22no—
136#103—342no—
137#103—342no—
138#103—342no—
139#97—171no—
140#97C22×D1711yescharacteristic
141C4×C8—172no—
142C17#1191611yescharacteristic, complemented, Sylow
143C2×Q8—21no—
144C2×Q8—341no—
145C2×Q8—681no—
146C2×Q8—681no—
147C2×Q8—684no—
148C22×C4—171no—
149C22×C4—341no—
150C22×C4—681no—
151M16—682no—
152C2×C8—172no—
153C2×C8D4×D1712yes—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—341no—
158C4⋊C4—681no—
159C4⋊C4—681no—
160C4⋊C4—682no—
161C4⋊C4—681no—
162C22⋊C4—681no—
163C42—171no—
164C42—341no—
165C42—681no—
166C42C2×C4×D1711yescharacteristic
167C16—42no—
168C16—682no—
169C23—171no—
170Q8—42no—
171Q8—682no—
172Q8—1362no—
173Q8—1362no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—171no—
178C2×C4—341no—
179C2×C4—342no—
180C2×C4—682no—
181C2×C4—681no—
182C2×C4—681no—
183C2×C4—1362no—
184C2×C4C22⋊C4×D1711yescharacteristic
185C8—22no—
186C8—342no—
187C22—341no—
188C22—681no—
189C22#100×D1711yescharacteristic
190C4—41no—
191C4—81no—
192C4—81no—
193C4—342no—
194C4—341no—
195C4—1361no—
196C4—1361no—
197C4#1176282412yes—
198C2—21no—
199C2—681no—
200C2#1452740611yescharacteristic
subgroupG/H
centerC2#14527406
commutatorC42×C17C22×C4
Frattini#97C22×D17
FittingC17×#1356C2
socleC2×C17#1533
prime radicalC22×D17#100
cosocleC17×#97C23
hypercenter#1356D17
hypocenterC17#11916
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1356

Complements

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

NH ≅ G/Ntype
3. #14526819C2semidirect
4. #14526818C2semidirect
5. C17×#1356C2semidirect
91. #1356D17semidirect
142. C17#11916semidirect

Maximal subgroups

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

subgroupindex
#145391632
#145305942
#145268192
#145268182
C17×#13562
#1191617