Finite Group Databasebeta

← #14528138

Subgroups — #14528138

Order 2176.

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

normal subgroups only

This group has 204 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#11763185C211yesmaximal, characteristic, complemented
2#11763005C211yesmaximal, characteristic, complemented
3#11763001C211yesmaximal, characteristic, complemented
4#11762874C211yesmaximal, characteristic, complemented
5#11762561C211yesmaximal, characteristic, complemented
6#11762552C211yesmaximal, characteristic, complemented
7C17×#331C211yesmaximal, characteristic, complemented
8C17×#127C2211yescharacteristic
9C2×#63228—21no—
10C2×#63228—21no—
11C2×#63228—21no—
12C2×D136—21no—
13C2×C8×D17—21no—
14#10587560C2211yescharacteristic
15C4×D68C2211yescharacteristic
16C17×C4⋊C8C2211yescharacteristic
17C17×C4⋊C8C2211yescharacteristic
18#10587503—21no—
19#10587492—21no—
20#10587491—21no—
21C4⋊#3622C2211yescharacteristic
22C4×#3622C2211yescharacteristic
23C2×D4×C17—21no—
24C2×D4×C17—21no—
25C2×D4×C17—21no—
26C2×D68—21no—
27C2×D68—21no—
28C2×D68—21no—
29C2×D68D411yescharacteristic
30C2×C4×D17—21no—
31C2×C8×C17—21no—
32C2×C8×C17—21no—
33C42×C17C2311yescharacteristic
34#63228—42no—
35#63228—42no—
36#63228—42no—
37#63227—21no—
38C4⋊Dic17D411yescharacteristic
39C2×#3622—21no—
40C2×#3622D411yescharacteristic
41C2×#3622D411yescharacteristic
42D136—42no—
43C8×D17—42no—
44C23×C17—41no—
45C22×D17—21no—
46C22×D17—41no—
47D4×C17—41no—
48D4×C17—41no—
49D4×C17—41no—
50D4×C17—41no—
51D4×C17—81no—
52C2×C4×C17C2×D411yescharacteristic
53C2×C4×C17C2×D411yescharacteristic
54C2×C4×C17#4111yescharacteristic
55C2×Dic17—21no—
56D68—21no—
57D68—41no—
58D68—41no—
59D68—41no—
60D68—41no—
61D68—81no—
62D68D812yescomplemented
63C4×D17—41no—
64C4×D17—41no—
65C8×C17—41no—
66C8×C17—41no—
67#3622—21no—
68#3622—41no—
69#3622D812yescomplemented
70#1667—171nomaximal, Sylow
71C22×C17—81no—
72C22×C17—81no—
73C22×C17—81no—
74C22×C17#12111yescharacteristic
75C2×D17—22no—
76C2×D17—41no—
77C2×D17—41no—
78C2×D17—81no—
79C2×D17—81no—
80C2×D17—81no—
81C4×C17—21no—
82C4×C17C2×D811yescharacteristic
83C4×C17C2×D811yescharacteristic
84C4×C17#13511yescharacteristic
85C4×C17AGL(1,ℤ/8)11yescharacteristic
86Dic17—41no—
87#493—171no—
88#459—171no—
89#459—171no—
90C8×D4—171no—
91#335—171no—
92#331—171no—
93#331D1711yescharacteristic, complemented
94C2×C17—161no—
95C2×C17#45911yescharacteristic
96C2×C17#46611yescharacteristic
97C2×C17#47211yescharacteristic
98D17—42no—
99D17—161no—
100C2×D8—341no—
101C2×D8—341no—
102C2×D8—341no—
103C2×D8—341no—
104C22×C8—341no—
105#127—171no—
106#127C2×D1711yescharacteristic
107C4×D4—171no—
108C4⋊C8—171no—
109C4⋊C8C2×D1711yescharacteristic
110C4⋊C8C2×D1711yescharacteristic
111#102—341no—
112#102—341no—
113C22⋊C8—341no—
114C4×C8—171no—
115C17#166711yescharacteristic, complemented, Sylow
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—171no—
120C2×D4—341no—
121C2×D4—341no—
122C2×D4—341no—
123C22×C4—341no—
124D8—682no—
125D8—682no—
126D8—682no—
127D8—682no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—171no—
131C2×C8—171no—
132C2×C8—341no—
133C2×C8—682no—
134C4⋊C4—171no—
135C22⋊C4—341no—
136C42C22×D1711yescharacteristic
137C23—41no—
138C23—341no—
139C23—681no—
140D4—41no—
141D4—41no—
142D4—41no—
143D4—41no—
144D4—81no—
145D4—172no—
146D4—341no—
147D4—681no—
148D4—681no—
149D4—681no—
150D4—681no—
151D4—1361no—
152C2×C4—341no—
153C2×C4—681no—
154C2×C4—681no—
155C2×C4#6325211yescharacteristic
156C2×C4D4×D1711yescharacteristic
157C2×C4D4×D1711yescharacteristic
158C8—41no—
159C8—41no—
160C8—172no—
161C8—341no—
162C8—681no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—342no—
167C22—681no—
168C22—681no—
169C22—1361no—
170C22—1361no—
171C22—1361no—
172C22#1058757011yescharacteristic
173C4—21no—
174C4—681no—
175C4D8×D1711yescharacteristic
176C4D8×D1711yescharacteristic
177C4#1058760011yescharacteristic
178C4#1058760211yescharacteristic
179C2—161no—
180C2—2721no—
181C2—682no—
182C2#1176300311yescharacteristic
183C2#1176304411yescharacteristic
184C2#1176307811yescharacteristic
subgroupG/H
centerC22#10587570
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#331C2
socleC22×C17#121
hypercenter#331D17
hypocenterC17#1667
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #11763185C2semidirect
2. #11763005C2semidirect
3. #11763001C2semidirect
4. #11762874C2semidirect
5. #11762561C2semidirect
6. #11762552C2semidirect
7. C17×#331C2semidirect
62. D68D8semidirect
69. #3622D8semidirect
93. #331D17semidirect
115. C17#1667semidirect

Maximal subgroups

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

subgroupindex
#117631852
#117630052
#117630012
#117628742
#117625612
#117625522
C17×#3312
#166717