Finite Group Databasebeta

← #14527786

Subgroups — #14527786

Order 2176.

254 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 53 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#11763390C211yesmaximal, characteristic
2#11763122C212yesmaximal, complemented
3#11762542C212yesmaximal, complemented
4#11762535C211yesmaximal, characteristic, complemented
5C17×#398C211yesmaximal, characteristic, complemented
6C2×#63252—21no—
7C2×C4⋊C4×C17—21no—
8C2×C4⋊C4×C17C2211yescharacteristic
9C2×C4⋊C4×C17C2211yescharacteristic
10#10587616—21no—
11#10587616C2212yes—
12#10587583—22no—
13#10587581—22no—
14C17×#95—21no—
15C17×#95C2211yescharacteristic
16C22⋊#3622C2212yes—
17#10587480—22no—
18#10587478—22no—
19C22×C4×C17—21no—
20C22×C4×C17—21no—
21C22×C4×C17D411yescharacteristic
22C22×C4×C17D411yescharacteristic
23C22×C4×C17C2311yescharacteristic
24C2×C22⋊D17—22no—
25#63252—42no—
26#63252—42no—
27C2×D68—21no—
28C2×D68—21no—
29C2×D68D412yes—
30C2×C4×D17—22no—
31C2×Dic34—21no—
32C4⋊C4×C17—22no—
33C4⋊C4×C17—22no—
34C4⋊C4×C17—42no—
35#63227—22no—
36#63227—22no—
37C4⋊Dic17—21no—
38C4⋊Dic17D412yes—
39C2×#3622—22no—
40C23×C17C2×D411yescharacteristic
41C22×D17—22no—
42C2×C4×C17—21no—
43C2×C4×C17—21no—
44C2×C4×C17—22no—
45C2×C4×C17—21no—
46C2×C4×C17—41no—
47C2×C4×C17—41no—
48C2×C4×C17—41no—
49C2×C4×C17—41no—
50C2×C4×C17—41no—
51C2×C4×C17—41no—
52C2×C4×C17—41no—
53C2×C4×C17D812yes—
54C2×C4×C17C2×D412yes—
55C22⋊D17—42no—
56C22⋊D17—42no—
57C2×Dic17—22no—
58D68—42no—
59D68—42no—
60D68—81no—
61C4×D17—44no—
62Dic34—42no—
63#3622—42no—
64#1610—171nomaximal, Sylow
65C22×C17—21no—
66C22×C17—21no—
67C22×C17#12011yescharacteristic
68C22×C17C2×D811yescharacteristic
69C22×C17AGL(1,ℤ/8)11yescharacteristic
70C2×D17—42no—
71C2×D17—42no—
72C2×D17—42no—
73C4×C17—22no—
74C4×C17—22no—
75C4×C17—42no—
76C4×C17—41no—
77C4×C17—81no—
78Dic17—42no—
79#537—171no—
80#480—172no—
81#398D1711yescharacteristic, complemented
82#327—172no—
83#324—171no—
84C2×C17—21no—
85C2×C17—21no—
86C2×C17#44711yescharacteristic
87C2×C17#45611yescharacteristic
88C2×C17#45811yescharacteristic
89D17—82no—
90C2×#41—341no—
91C22⋊Q8—342no—
92#121—172no—
93#121—342no—
94#121—341no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2×D1711yescharacteristic
97C2×C4⋊C4C2×D1711yescharacteristic
98#107—342no—
99#102—342no—
100C22⋊C8—172no—
101#95—21no—
102#95C2×D1711yescharacteristic
103C17#161011yescharacteristic, complemented, Sylow
104#41—682no—
105#41—682no—
106C2×Q8—341no—
107C2×D4—172no—
108C2×D4—341no—
109C2×D4—341no—
110C2×D4—342no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—342no—
114C22×C4C22⋊D1711yescharacteristic
115C22×C4C22⋊D1711yescharacteristic
116C22×C4C22×D1711yescharacteristic
117C2×C8—342no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—172no—
122C4⋊C4—341no—
123C22⋊C4—342no—
124C22⋊C4—342no—
125C23—342no—
126C23C2×C22⋊D1711yescharacteristic
127Q8—682no—
128D4—682no—
129D4—682no—
130D4—682no—
131D4—682no—
132D4—1361no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—342no—
145C2×C4—684no—
146C2×C4#6322812yes—
147C2×C4D4×D1712yes—
148C8—682no—
149C22—21no—
150C22—21no—
151C22—682no—
152C22—682no—
153C22—682no—
154C22C2×#6322811yescharacteristic
155C22#1058762311yescharacteristic
156C22#1058763511yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—81no—
162C4—682no—
163C2—21no—
164C2—21no—
165C2—1362no—
166C2#1176294211yescharacteristic
167C2#1176299111yescharacteristic
168C2#1176299811yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#398C2
socleC22×C17#120
prime radical#11763390C2
hypercenter#398D17
hypocenterC17#1610
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #11763122C2semidirect
3. #11762542C2semidirect
4. #11762535C2semidirect
5. C17×#398C2semidirect
81. #398D17semidirect
103. C17#1610semidirect

Maximal subgroups

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

subgroupindex
#117633902
#117631222
#117625422
#117625352
C17×#3982
#161017