Finite Group Databasebeta

← #14530143

Subgroups — #14530143

Order 2176.

228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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×#10587490C211yesmaximal, characteristic, complemented
2#11762852C211yesmaximal, characteristic, complemented
3#11762813C211yesmaximal, characteristic
4C2×#10587492C211yesmaximal, characteristic
5C2×#10587494C211yesmaximal, characteristic
6#11762577C211yesmaximal, characteristic, complemented
7C17×#424C211yesmaximal, characteristic, complemented
8C22×D68C2211yescharacteristic
9C2×M16×C17C2211yescharacteristic
10C22×C8×C17C2211yescharacteristic
11C17×#117C2211yescharacteristic
12C2×C4⋊Dic17C2211yescharacteristic
13C2×#63223C2211yescharacteristic
14C2×#63223C2211yescharacteristic
15C17×C4⋊C8—22no—
16#10587494—22no—
17#10587492—22no—
18#10587490—22no—
19#10587480—21no—
20#10587480—21no—
21#10587479—21no—
22#10587478—21no—
23C23×D17—21no—
24C22×C4×C17C2311yescharacteristic
25C22×Dic17—21no—
26C2×D68—21no—
27C2×D68—21no—
28C2×D68—21no—
29C2×D68—21no—
30C2×D68—21no—
31C2×D68Q812yes—
32M16×C17—21no—
33M16×C17—21no—
34C2×C8×C17—21no—
35C2×C8×C17—21no—
36C2×C8×C17—21no—
37C2×C8×C17D412yes—
38C4⋊C4×C17D411yescharacteristic
39C4⋊C4×C17D411yescharacteristic
40C22⋊C4×C17—21no—
41C42×C17—21no—
42C4⋊Dic17—21no—
43C4⋊Dic17—21no—
44#63223—21no—
45#63223—21no—
46#63223—21no—
47#63223D412yes—
48C2×#3622—21no—
49C2×#3622—21no—
50C23×C17#4111yescharacteristic
51C22×D17—22no—
52C22×D17—41no—
53C22×D17—41no—
54C22×D17—81no—
55C22×D17—81no—
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17C2×D411yescharacteristic
59C2×C4×C17C2×D411yescharacteristic
60C2×C4×C17C2×D411yescharacteristic
61C2×C4×C17C2×Q811yescharacteristic
62C2×C4×C17#4111yescharacteristic
63C2×C4×C17#4111yescharacteristic
64C2×Dic17—41no—
65C2×Dic17—81no—
66D68—22no—
67D68—22no—
68D68—41no—
69D68—41no—
70C8×C17—22no—
71C8×C17—41no—
72#3622—22no—
73#3622—41no—
74#2063—171nomaximal, Sylow
75C22×C17—21no—
76C22×C17—21no—
77C22×C17#12111yescharacteristic
78C22×C17C22⋊Q811yescharacteristic
79C22×C17#12611yescharacteristic
80C2×D17—42no—
81C2×D17—82no—
82C2×D17—81no—
83C2×D17—81no—
84C2×D17—82no—
85C4×C17—41no—
86C4×C17—41no—
87C4×C17#12111yescharacteristic
88C4×C17#12111yescharacteristic
89C4×C17C22⋊Q811yescharacteristic
90C4×C17C22⋊Q811yescharacteristic
91Dic17—81no—
92C2×#108—171no—
93#428—171no—
94#424D1711yescharacteristic, complemented
95#418—171no—
96C2×#102—171no—
97C2×#100—171no—
98#343—171no—
99C2×C17—21no—
100C2×C17—21no—
101C2×C17#39611yescharacteristic
102C2×C17#47111yescharacteristic
103C2×C17#47211yescharacteristic
104D17—82no—
105C22×D4—171no—
106C2×M16—171no—
107C2×M16—171no—
108C2×M16C2×D1711yescharacteristic
109C22×C8C2×D1711yescharacteristic
110#117C2×D1711yescharacteristic
111C2×C4⋊C4—171no—
112#108—342no—
113#107—341no—
114#106—341no—
115C4⋊C8—22no—
116#102—342no—
117#102—341no—
118#102—341no—
119#100—342no—
120C17#206311yescharacteristic, complemented, Sylow
121C24—341no—
122C2×D4—172no—
123C2×D4—341no—
124C2×D4—341no—
125C2×D4—341no—
126C2×D4—341no—
127C2×D4—341no—
128C22×C4—341no—
129C22×C4C22×D1711yescharacteristic
130M16—21no—
131M16—21no—
132M16—172no—
133M16—341no—
134M16—341no—
135M16—341no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—341no—
140C2×C8—341no—
141C2×C8C22⋊D1712yes—
142C4⋊C4—341no—
143C4⋊C4—341no—
144C4⋊C4D6811yescharacteristic
145C4⋊C4D6811yescharacteristic
146C22⋊C4—21no—
147C42—21no—
148C23—342no—
149C23—681no—
150C23—681no—
151C23—1361no—
152C23—1361no—
153C23#6325211yescharacteristic
154D4—342no—
155D4—342no—
156D4—681no—
157D4—681no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—681no—
161C2×C4—1361no—
162C2×C4C2×D6811yescharacteristic
163C2×C4D4×D1711yescharacteristic
164C2×C4#6325411yescharacteristic
165C2×C4Q8×D1711yescharacteristic
166C2×C4#6325611yescharacteristic
167C2×C4C2×C22⋊D1711yescharacteristic
168C8—22no—
169C8—41no—
170C8—342no—
171C8—681no—
172C22—21no—
173C22—21no—
174C22—682no—
175C22—1362no—
176C22—1362no—
177C22—1361no—
178C22—1361no—
179C22#1058758211yescharacteristic
180C22#1058758411yescharacteristic
181C22#1058761611yescharacteristic
182C4—41no—
183C4—41no—
184C4—1361no—
185C4#1058758111yescharacteristic
186C4#1058758311yescharacteristic
187C4#1058762511yescharacteristic
188C4#1058763211yescharacteristic
189C2—21no—
190C2—21no—
191C2—1362no—
192C2#1176273111yescharacteristic
193C2#1176307211yescharacteristic
194C2#1176307911yescharacteristic
subgroupG/H
centerC22#10587616
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#424C2
socleC22×C17#121
prime radicalC22×D68C22
hypercenter#424D17
hypocenterC17#2063
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
1. C2×#10587490C2semidirect
2. #11762852C2semidirect
6. #11762577C2semidirect
7. C17×#424C2semidirect
94. #424D17semidirect
120. C17#2063semidirect

Maximal subgroups

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

subgroupindex
C2×#105874902
#117628522
#117628132
C2×#105874922
C2×#105874942
#117625772
C17×#4242
#206317