Finite Group Databasebeta

← #16955199

Subgroups — #16955199

Order 4352.

184 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×#11762648C211yesmaximal, characteristic, complemented
2#14530909C211yesmaximal, characteristic, complemented
3#14530904C211yesmaximal, characteristic
4#14526988C211yesmaximal, characteristic, complemented
5#14526978C211yesmaximal, characteristic, complemented
6#14526975C211yesmaximal, characteristic, complemented
7C17×#2258C211yesmaximal, characteristic, complemented
8C2×Dic136C2211yescharacteristic
9C2×D272C2211yescharacteristic
10#11763084—21no—
11#11763079—21no—
12#11762648—21no—
13#11762648—21no—
14#11762639C2211yescharacteristic
15#11762639C2211yescharacteristic
16#11762626—21no—
17#11762623—21no—
18C17×#504C2211yescharacteristic
19C17×M64—21no—
20C17×M64C2211yescharacteristic
21C2×C17×C32—21no—
22C2×C17×C32C2211yescharacteristic
23C17×#131D411yescharacteristic
24#10587637—41no—
25#10587635—41no—
26C2×Dic68—21no—
27C2×D136—21no—
28C17×M32—21no—
29C17×M32D411yescharacteristic
30C2×C16×C17—21no—
31C2×C16×C17C2311yescharacteristic
32#10587495—41no—
33#10587494—41no—
34#10587490—21no—
35#10587490—21no—
36Dic136—21no—
37Dic136—21no—
38D272—21no—
39D272—21no—
40C17×C32—21no—
41C17×C32D411yescharacteristic
42C17×C32D411yescharacteristic
43#41×C17D811yescharacteristic
44C2×D68—41no—
45C2×Dic34—41no—
46M16×C17—21no—
47M16×C17D811yescharacteristic
48C2×C8×C17—21no—
49C2×C8×C17C2×D411yescharacteristic
50#63231—81no—
51#63230—81no—
52#63229—81no—
53#63228—81no—
54#63223—41no—
55#63223—41no—
56Dic68—41no—
57Dic68—41no—
58D136—41no—
59D136—41no—
60C16×C17—21no—
61C16×C17C2×D411yescharacteristic
62C16×C17#4111yescharacteristic
63#13702—171nomaximal, Sylow
64C22×D17—81no—
65Q8×C17D1611yescharacteristic
66D4×C17—21no—
67D4×C17D1611yescharacteristic
68C2×C4×C17—21no—
69C2×C4×C17C2×D811yescharacteristic
70C2×Dic17—81no—
71D68—81no—
72D68—81no—
73Dic34—81no—
74Dic34—81no—
75C8×C17—21no—
76C8×C17#12111yescharacteristic
77C8×C17#13511yescharacteristic
78#3622—81no—
79#3622—81no—
80C2×SD64—171no—
81#2258D1711yescharacteristic, complemented
82#2222—171no—
83#2221—171no—
84#1425—171no—
85#1420—171no—
86#1419—171no—
87C22×C17—21no—
88C22×C17#50811yescharacteristic
89C2×D17—161no—
90C2×D17—321no—
91C4×C17—21no—
92C4×C17#46611yescharacteristic
93C4×C17C2×D1611yescharacteristic
94Dic17—161no—
95C2×Q32—171no—
96C2×D16—171no—
97#504C2×D1711yescharacteristic
98#473—341no—
99#472—341no—
100SD64—341no—
101SD64—341no—
102M64—21no—
103M64C2×D1711yescharacteristic
104C2×C32—21no—
105C2×C32C2×D1711yescharacteristic
106#368—171no—
107#368—171no—
108#362—341no—
109#361—341no—
110C2×C17—21no—
111C2×C17—41no—
112C2×C17#221511yescharacteristic
113D17—321no—
114#137—681no—
115AGL(1,ℤ/8)—681no—
116C2×Q16—341no—
117C2×D8—341no—
118#131D6811yescharacteristic
119Q32—341no—
120Q32—341no—
121D16—341no—
122D16—341no—
123M32—21no—
124M32D6811yescharacteristic
125C2×C16—21no—
126C2×C16C22×D1711yescharacteristic
127#108—341no—
128#108—341no—
129#101—681no—
130#100—681no—
131C32—21no—
132C32C22⋊D1711yescharacteristic
133C32C22⋊D1711yescharacteristic
134C17#1370211yescharacteristic, complemented, Sylow
135#41D13611yescharacteristic
136C2×Q8—681no—
137C2×D4—681no—
138Q16—681no—
139Q16—681no—
140Q16—1361no—
141SD16—1361no—
142SD16—1361no—
143D8—681no—
144D8—681no—
145D8—1361no—
146M16—21no—
147M16—681no—
148M16—681no—
149M16D13611yescharacteristic
150C2×C8—21no—
151C2×C8C2×D6811yescharacteristic
152C16—21no—
153C16#6325211yescharacteristic
154C16C2×C22⋊D1711yescharacteristic
155C23—1361no—
156Q8—1361no—
157Q8—1361no—
158Q8D27211yescharacteristic
159D4—21no—
160D4—1361no—
161D4—1361no—
162D4D27211yescharacteristic
163C2×C4—21no—
164C2×C4—1361no—
165C2×C4C2×D13611yescharacteristic
166C8—21no—
167C8—1361no—
168C8—1361no—
169C8#1058759011yescharacteristic
170C8#1058761611yescharacteristic
171C22—21no—
172C22—2721no—
173C22—5441no—
174C22#1176324111yescharacteristic
175C4—21no—
176C4—2721no—
177C4#1176304511yescharacteristic
178C4C2×D27211yescharacteristic
179C2—21no—
180C2—41no—
181C2—5441no—
182C2#1453087011yescharacteristic
subgroupG/H
centerC2#14530870
commutator, cosocleC2×C16×C17C23
FrattiniC2×C16C22×D17
FittingC17×#2258C2
socleC2×C17#2215
prime radical#14530904C2
hypercenter#2258D17
hypocenterC17#13702
LCSG ▷ C2×C16×C17 ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #504 ◁ #2258

Complements

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

NH ≅ G/Ntype
1. C2×#11762648C2semidirect
2. #14530909C2semidirect
4. #14526988C2semidirect
5. #14526978C2semidirect
6. #14526975C2semidirect
7. C17×#2258C2semidirect
81. #2258D17semidirect
134. C17#13702semidirect

Maximal subgroups

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

subgroupindex
C2×#117626482
#145309092
#145309042
#145269882
#145269782
#145269752
C17×#22582
#1370217