Finite Group Databasebeta

← #16937428

Subgroups — #16937428

Order 4352.

400 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 97 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#14537147C211yesmaximal, characteristic
2#14529761C211yesmaximal, characteristic
3#14527349C212yesmaximal
4#14526643C212yesmaximal
5C17×#1276C211yesmaximal, characteristic
6#11763366C2211yescharacteristic
7C2×#10587556—21no—
8#11762833C2211yescharacteristic
9#11762707—24no—
10#11762679—21no—
11#11762675C2212yes—
12#11762569—21no—
13#11762569—21no—
14#11762547—24no—
15C17×#431C2212yes—
16C43×C17C2211yescharacteristic
17C2×C42×C17—22no—
18C2×C42×C17D411yescharacteristic
19C2×C42×C17D411yescharacteristic
20C2×C42×C17C2311yescharacteristic
21#10587612—21no—
22C2×C4⋊Dic17—24no—
23C2×C4⋊Dic17—22no—
24#10587610—22no—
25C2×#63225—24no—
26C2×#63223—21no—
27#10587556—21no—
28#10587556—42no—
29#10587556—42no—
30#10587555Q812yes—
31C4×Dic34—22no—
32C17×C4⋊C8—22no—
33C17×C22⋊C8—22no—
34C4×C8×C17—22no—
35#10587502—22no—
36#10587502D414yes—
37C4⋊#3622—22no—
38C22×C4×C17—21no—
39C22×C4×C17—21no—
40C22×C4×C17C2×D411yescharacteristic
41C22×C4×C17C2×D412yes—
42C22×Dic17—24no—
43C2×Dic34—22no—
44C2×C8×C17—42no—
45C42×C17—21no—
46C42×C17—21no—
47C42×C17—44no—
48C42×C17SD1612yes—
49C42×C17Q1612yes—
50C42×C17C2×Q811yescharacteristic
51C42×C17#4112yes—
52C42×C17#4111yescharacteristic
53C22⋊Dic17—41no—
54C4⋊Dic17—22no—
55C4⋊Dic17—22no—
56C4⋊Dic17—48no—
57C4⋊Dic17—42no—
58C4⋊Dic17—42no—
59#63225—48no—
60#63225—42no—
61C4×Dic17—41no—
62#63223—41no—
63#63223—41no—
64C2×#3622—41no—
65#10365—171nomaximal, Sylow
66C23×C17#12011yescharacteristic
67C2×C4×C17—24no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—42no—
71C2×C4×C17—42no—
72C2×C4×C17C22⋊Q812yes—
73C2×C4×C17C22⋊Q811yescharacteristic
74C2×C4×C17#12411yescharacteristic
75C2×C4×C17#12412yes—
76C2×C4×C17C2×SD1611yescharacteristic
77C2×C4×C17C2×Q1611yescharacteristic
78C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
79C2×C4×C17#13711yescharacteristic
80C2×Dic17—48no—
81C2×Dic17—42no—
82C2×Dic17—44no—
83Dic34—82no—
84C8×C17—82no—
85#3622—81no—
86#2859—171no—
87#1994—171no—
88#1518—172no—
89#1286—172no—
90#1276D1711yescharacteristic
91C22×C17—22no—
92C22×C17#39311yescharacteristic
93C22×C17#45011yescharacteristic
94C22×C17#45111yescharacteristic
95C4×C17—24no—
96C4×C17—22no—
97C4×C17—22no—
98C4×C17—42no—
99C4×C17#47511yescharacteristic
100C4×C17#47711yescharacteristic
101C4×C17#48711yescharacteristic
102C4×C17#48911yescharacteristic
103Dic17—84no—
104Dic17—82no—
105#533—171no—
106C2×#125—341no—
107#431C2×D1712yes—
108#423—171no—
109#389—344no—
110#384—341no—
111#382—172no—
112C43C2×D1711yescharacteristic
113#339—341no—
114#339—341no—
115#328—344no—
116C2×C17—21no—
117C2×C17—21no—
118C2×C17#202411yescharacteristic
119C2×C17#203112yes—
120C2×M16—341no—
121C4⋊Q8—172no—
122#125—341no—
123#125—682no—
124#125—682no—
125C22⋊Q8—342no—
126C4×Q8—342no—
127#117—341no—
128C2×C4⋊C4—344no—
129C2×C4⋊C4—344no—
130C2×C4⋊C4—342no—
131C2×C42—22no—
132C2×C42D6811yescharacteristic
133C2×C42D6811yescharacteristic
134C2×C42C22×D1711yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—342no—
137C22⋊C8—22no—
138C4×C8—22no—
139#95—174no—
140#95—342no—
141C17#1036511yescharacteristic, complemented, Sylow
142C2×Q8—342no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—344no—
146C22×C4C2×D6811yescharacteristic
147C22×C4D4×D1712yes—
148M16—681no—
149M16—681no—
150C2×C8—42no—
151C2×C8—681no—
152C4⋊C4—342no—
153C4⋊C4—342no—
154C4⋊C4—688no—
155C4⋊C4—688no—
156C4⋊C4—682no—
157C4⋊C4—682no—
158C4⋊C4—682no—
159C22⋊C4—681no—
160C42—21no—
161C42—21no—
162C42—44no—
163C42—681no—
164C42#6321912yes—
165C42Dic6812yes—
166C42#6325212yes—
167C42#6325411yescharacteristic
168C42Q8×D1711yescharacteristic
169C23#1058756811yescharacteristic
170Q8—1362no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—688no—
177C2×C4—684no—
178C2×C4—682no—
179C2×C4#1058756111yescharacteristic
180C2×C4#1058757112yes—
181C2×C4#1058758212yes—
182C2×C4#1058758311yescharacteristic
183C2×C4C2×#6321911yescharacteristic
184C2×C4C2×Dic6811yescharacteristic
185C2×C4#1058759411yescharacteristic
186C2×C4#1058759511yescharacteristic
187C8—82no—
188C8—1361no—
189C22—22no—
190C22#1176272211yescharacteristic
191C22#1176296211yescharacteristic
192C22#1176296711yescharacteristic
193C4—24no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—1364no—
198C4—1362no—
199C4#1176309511yescharacteristic
200C4#1176310511yescharacteristic
201C4#1176316011yescharacteristic
202C4#1176316911yescharacteristic
203C2—21no—
204C2—21no—
205C2#1452992611yescharacteristic
206C2#1452995712yes—
subgroupG/H
centerC22#11762722
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1276C2
socleC22×C17#393
prime radicalC23×C17#120
hypercenter#1276D17
hypocenterC17#10365
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
141. C17#10365semidirect

Maximal subgroups

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

subgroupindex
#145371472
#145297612
#145273492
#145266432
C17×#12762
#1036517