Finite Group Databasebeta

← #14529830

Subgroups — #14529830

Order 2176.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#10587628C212yesmaximal, complemented
2#11763390C211yesmaximal, characteristic
3C2×#10587476C212yesmaximal, complemented
4#11762578C211yesmaximal, characteristic, complemented
5C17×#391C211yesmaximal, characteristic, complemented
6C22×Q8×C17C2211yescharacteristic
7C2×#63252C2212yes—
8C2×C4⋊C4×C17C2211yescharacteristic
9C2×C42×C17C2211yescharacteristic
10C2×#63231—22no—
11#10587628—24no—
12#10587628—24no—
13C2×#63230—22no—
14#10587616—22no—
15C2×#63223C2212yes—
16#10587583—22no—
17#10587581—22no—
18C17×#95—22no—
19#10587476—24no—
20C2×Q8×C17—21no—
21C2×Q8×C17—22no—
22C2×Q8×C17—22no—
23C2×Q8×C17D412yescomplemented
24C22×C4×C17—21no—
25C22×C4×C17—21no—
26C22×C4×C17D412yes—
27C22×C4×C17C2311yescharacteristic
28C2×C22⋊D17—22no—
29#63252—22no—
30#63252—22no—
31#63252—22no—
32#63252—22no—
33C2×D68—22no—
34C2×D68D412yes—
35C2×C4×D17—22no—
36C2×Dic34D412yes—
37C4⋊C4×C17—22no—
38C42×C17—22no—
39#63231—28no—
40#63230—28no—
41#63227—24no—
42C4⋊Dic17—22no—
43#63223—22no—
44#63223D414yescomplemented
45C2×#3622—22no—
46C23×C17C2×D411yescharacteristic
47C22×D17—22no—
48Q8×C17—24no—
49Q8×C17—42no—
50C2×C4×C17—22no—
51C2×C4×C17—22no—
52C2×C4×C17—22no—
53C2×C4×C17—41no—
54C2×C4×C17—41no—
55C2×C4×C17—41no—
56C2×C4×C17—41no—
57C2×C4×C17—42no—
58C2×C4×C17C2×D411yescharacteristic
59C2×C4×C17C2×D412yes—
60C2×C4×C17C2×D412yes—
61C2×C4×C17#4111yescharacteristic
62C22⋊D17—42no—
63C22⋊D17—42no—
64C2×Dic17—22no—
65D68—22no—
66D68—22no—
67D68—82no—
68C4×D17—42no—
69C4×D17—42no—
70Dic34—22no—
71Dic34—22no—
72#3622—24no—
73#2007—171nomaximal, Sylow
74C22×C17—21no—
75C22×C17—21no—
76C22×C17#12011yescharacteristic
77C22×C17#12111yescharacteristic
78C22×C17#12711yescharacteristic
79C2×D17—42no—
80C2×D17—42no—
81C2×D17—42no—
82C4×C17—42no—
83C4×C17—42no—
84C4×C17—42no—
85C4×C17#12012yes—
86C4×C17#12112yes—
87Dic17—42no—
88C2×#137—172no—
89#537—171no—
90C2×#104—172no—
91#391D1711yescharacteristic, complemented
92#343—171no—
93C2×C17—21no—
94C2×C17—21no—
95C2×C17#39211yescharacteristic
96C2×C17#45511yescharacteristic
97C2×C17#45611yescharacteristic
98D17—82no—
99C2×#41—172no—
100C22×Q8C2×D1711yescharacteristic
101#137—344no—
102#137—344no—
103C2×Q16—342no—
104C2×SD16—342no—
105C2×M16—172no—
106C22⋊Q8—342no—
107#121—342no—
108#121—342no—
109C2×C4⋊C4C2×D1711yescharacteristic
110C2×C42C2×D1711yescharacteristic
111#104—344no—
112#95—22no—
113C17#200711yescharacteristic, complemented, Sylow
114#41—342no—
115#41—342no—
116#41—342no—
117#41—342no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—172no—
122C2×Q8D6812yescomplemented
123C2×D4—172no—
124C2×D4—342no—
125C2×D4—342no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—342no—
129C22×C4C22⋊D1712yes—
130C22×C4C22×D1711yescharacteristic
131Q16—348no—
132SD16—348no—
133M16—174no—
134M16—342no—
135C2×C8—342no—
136C4⋊C4—22no—
137C4⋊C4—342no—
138C22⋊C4—344no—
139C42—22no—
140C23—342no—
141C23C2×C22⋊D1711yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—342no—
145Q8—342no—
146D4—342no—
147D4—342no—
148D4—682no—
149D4—682no—
150D4—1362no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—342no—
160C2×C4—682no—
161C2×C4—682no—
162C2×C4C2×D6811yescharacteristic
163C2×C4#6325211yescharacteristic
164C2×C4D4×D1712yes—
165C2×C4D4×D1712yes—
166C8—344no—
167C22—21no—
168C22—21no—
169C22—682no—
170C22—682no—
171C22—682no—
172C22#1058761611yescharacteristic
173C22#1058762311yescharacteristic
174C22#1058762611yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—682no—
179C4#1058756812yes—
180C4#1058757012yes—
181C2—21no—
182C2—21no—
183C2—1362no—
184C2#1176271811yescharacteristic
185C2#1176298611yescharacteristic
186C2#1176299111yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#391C2
socleC22×C17#120
prime radical#11763390C2
hypercenter#391D17
hypocenterC17#2007
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#10587628C2semidirect
3. C2×#10587476C2semidirect
4. #11762578C2semidirect
5. C17×#391C2semidirect
23. C2×Q8×C17D4semidirect
44. #63223D4semidirect
91. #391D17semidirect
113. C17#2007semidirect
122. C2×Q8D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105876282
#117633902
C2×#105874762
#117625782
C17×#3912
#200717