Finite Group Databasebeta

← #14530309

Subgroups — #14530309

Order 2176.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#10587478C211yesmaximal, characteristic, complemented
2C2×#10587492C211yesmaximal, characteristic
3C2×#10587480C211yesmaximal, characteristic
4#11762710C211yesmaximal, characteristic
5#11762678C211yesmaximal, characteristic, complemented
6#11762563C211yesmaximal, characteristic, complemented
7C2×C17×#107C211yesmaximal, characteristic, complemented
8C22×D68C2211yescharacteristic
9C22×C8×C17C2211yescharacteristic
10C2×C4⋊C4×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×#63227—22no—
13C2×C4⋊Dic17C2211yescharacteristic
14C2×#63225—22no—
15C2×C4×Dic17C2211yescharacteristic
16C22×#3622C2211yescharacteristic
17C17×#107—22no—
18#10587492—22no—
19#10587480—22no—
20#10587478—22no—
21C23×D17—21no—
22C22×C4×C17—21no—
23C22×C4×C17—21no—
24C22×C4×C17C2311yescharacteristic
25C22×Dic17—21no—
26C22×Dic17D412yes—
27C2×D68—21no—
28C2×D68—21no—
29C2×D68—21no—
30C2×D68—21no—
31C2×D68—21no—
32C2×D68Q812yes—
33C2×C8×C17—21no—
34C2×C8×C17—21no—
35C2×C8×C17D412yes—
36C4⋊C4×C17—21no—
37C4⋊C4×C17—21no—
38C4⋊C4×C17—21no—
39C4⋊C4×C17—21no—
40#63227—44no—
41C4⋊Dic17—21no—
42C4⋊Dic17—21no—
43#63225—44no—
44C4×Dic17—21no—
45C4×Dic17—21no—
46C2×#3622—21no—
47C2×#3622—21no—
48C2×#3622D412yes—
49C23×C17C2×D411yescharacteristic
50C22×D17—22no—
51C22×D17—41no—
52C22×D17—41no—
53C22×D17—42no—
54C22×D17—42no—
55C2×C4×C17—41no—
56C2×C4×C17—41no—
57C2×C4×C17—42no—
58C2×C4×C17—42no—
59C2×C4×C17C2×D411yescharacteristic
60C2×C4×C17C2×D411yescharacteristic
61C2×C4×C17C2×Q811yescharacteristic
62C2×C4×C17#4111yescharacteristic
63C2×C4×C17#4111yescharacteristic
64C2×C4×C17#4111yescharacteristic
65C2×Dic17—22no—
66C2×Dic17—22no—
67C2×Dic17—41no—
68C2×Dic17—42no—
69C2×Dic17D814yes—
70D68—22no—
71D68—22no—
72D68—41no—
73D68—41no—
74C8×C17—22no—
75#3622—22no—
76#2093—171nomaximal, Sylow
77C22×C17C22⋊Q811yescharacteristic
78C22×C17#12311yescharacteristic
79C22×C17#12711yescharacteristic
80C22×C17C2×D812yes—
81C22×C17#13712yes—
82C2×D17—42no—
83C2×D17—44no—
84C2×D17—82no—
85C2×D17—81no—
86C2×D17—81no—
87C4×C17—81no—
88C4×C17—81no—
89C4×C17C22⋊Q811yescharacteristic
90C4×C17C22⋊Q811yescharacteristic
91C4×C17#12311yescharacteristic
92C4×C17#12311yescharacteristic
93Dic17—24no—
94Dic17—81no—
95C2×#107—171no—
96C2×#107D1711yescharacteristic, complemented
97C2×#102—171no—
98C2×#102—171no—
99#390—171no—
100#384—171no—
101#336—171no—
102C2×C17#39911yescharacteristic
103C2×C17#47412yes—
104C2×C17#48012yes—
105C2×C17#49311yescharacteristic
106C2×C17#49711yescharacteristic
107D17—82no—
108C22×D4—171no—
109C22×C8—171no—
110C22×C8C2×D1711yescharacteristic
111C2×C4⋊C4—171no—
112C2×C4⋊C4—342no—
113C2×C4⋊C4C2×D1711yescharacteristic
114C2×C4⋊C4C2×D1711yescharacteristic
115C2×C22⋊C4—342no—
116C2×C42—171no—
117#107—22no—
118#107—342no—
119#102—342no—
120#102—342no—
121C17#209311yescharacteristic, complemented, Sylow
122C24—341no—
123C2×D4—172no—
124C2×D4—341no—
125C2×D4—341no—
126C2×D4—341no—
127C2×D4—341no—
128C2×D4—341no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—172no—
132C22×C4—341no—
133C22×C4C22×D1711yescharacteristic
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—172no—
137C2×C8—341no—
138C2×C8—341no—
139C2×C8C22⋊D1712yes—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—341no—
145C4⋊C4—341no—
146C4⋊C4—684no—
147C22⋊C4—684no—
148C42—341no—
149C42—341no—
150C23—342no—
151C23—681no—
152C23—682no—
153C23—682no—
154C23—681no—
155C23D4×D1711yescharacteristic
156D4—342no—
157D4—342no—
158D4—681no—
159D4—681no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—174no—
165C2×C4—342no—
166C2×C4—342no—
167C2×C4—682no—
168C2×C4—681no—
169C2×C4#6325211yescharacteristic
170C2×C4#6325211yescharacteristic
171C2×C4D4×D1711yescharacteristic
172C2×C4Q8×D1711yescharacteristic
173C2×C4#6325611yescharacteristic
174C2×C4C2×C22⋊D1711yescharacteristic
175C8—22no—
176C8—342no—
177C22—682no—
178C22—684no—
179C22—1361no—
180C22—1362no—
181C22—1361no—
182C22#1058758011yescharacteristic
183C22#1058758211yescharacteristic
184C22D8×D1712yes—
185C22#1058760412yes—
186C22#1058762611yescharacteristic
187C4—81no—
188C4—81no—
189C4—344no—
190C4—1361no—
191C4#1058758011yescharacteristic
192C4#1058758211yescharacteristic
193C4#1058761511yescharacteristic
194C4#1058763211yescharacteristic
195C2—1362no—
196C2#1176274411yescharacteristic
197C2#1176308812yes—
198C2#1176312412yes—
199C2#1176318411yescharacteristic
200C2#1176320211yescharacteristic
subgroupG/H
centerC23D4×D17
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC2×C17×#107C2
socleC23×C17C2×D4
prime radicalC22×D68C22
hypercenterC2×#107D17
hypocenterC17#2093
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. C2×#10587478C2semidirect
5. #11762678C2semidirect
6. #11762563C2semidirect
7. C2×C17×#107C2semidirect
96. C2×#107D17semidirect
121. C17#2093semidirect

Maximal subgroups

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

subgroupindex
C2×#105874782
C2×#105874922
C2×#105874802
#117627102
#117626782
#117625632
C2×C17×#1072
#209317