Finite Group Databasebeta

← #14527872

Subgroups — #14527872

Order 2176.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#11763464C211yesmaximal, characteristic, complemented
2#11763155C211yesmaximal, characteristic, complemented
3#11763001C211yesmaximal, characteristic, complemented
4#11762554C211yesmaximal, characteristic, complemented
5D4⋊#3622C211yesmaximal, characteristic, complemented
6#11762536C211yesmaximal, characteristic, complemented
7C17×#459C211yesmaximal, characteristic, complemented
8C2×#63252—21no—
9C2×D8×C17—21no—
10C17×#127C2211yescharacteristic
11C4×D4×C17C2211yescharacteristic
12#10587624—21no—
13C2×#63228—21no—
14#10587610—21no—
15C4⋊C4×D17—21no—
16#10587569—22no—
17C4×D68C2211yescharacteristic
18#10587555C2211yescharacteristic
19C17×C4⋊C8C2211yescharacteristic
20C17×#102—21no—
21#10587503—21no—
22#10587503—22no—
23C4⋊#3622C2211yescharacteristic
24C4×#3622C2211yescharacteristic
25C2×D4×C17—21no—
26C2×D4×C17—21no—
27C2×D4×C17—21no—
28C2×D4×C17D411yescharacteristic
29C22×C4×C17—21no—
30C2×C22⋊D17—22no—
31#63252—42no—
32#63252—42no—
33C2×D68D411yescharacteristic
34C2×C4×D17—22no—
35C2×C4×D17—21no—
36C2×Dic34—21no—
37D8×C17—42no—
38C2×C8×C17—21no—
39C4⋊C4×C17D411yescharacteristic
40C22⋊C4×C17—21no—
41C42×C17C2311yescharacteristic
42C22⋊Dic17—22no—
43#63228—42no—
44#63227—21no—
45C4⋊Dic17—21no—
46C4⋊Dic17D412yes—
47C4⋊Dic17D411yescharacteristic
48#63225—22no—
49C2×#3622—21no—
50C2×#3622—21no—
51C23×C17—21no—
52C23×C17—41no—
53C22×D17—21no—
54D4×C17—21no—
55D4×C17—21no—
56D4×C17—42no—
57D4×C17—41no—
58D4×C17—41no—
59D4×C17—41no—
60D4×C17—41no—
61C2×C4×C17—21no—
62C2×C4×C17—41no—
63C2×C4×C17—41no—
64C2×C4×C17C2×D411yescharacteristic
65C2×C4×C17C2×D411yescharacteristic
66C2×C4×C17C2×D411yescharacteristic
67C22⋊D17—44no—
68C2×Dic17—21no—
69C2×Dic17—22no—
70D68—21no—
71D68D812yescomplemented
72C4×D17—44no—
73C4×D17—41no—
74C4×D17—41no—
75Dic34—41no—
76Dic34—41no—
77C8×C17—41no—
78#3622—41no—
79#3622—41no—
80#1625—171nomaximal, Sylow
81C22×C17—41no—
82C22×C17—41no—
83C22×C17—41no—
84C22×C17—81no—
85C22×C17—81no—
86C22×C17—81no—
87C22×C17#12011yescharacteristic
88C2×D17—22no—
89C2×D17—41no—
90C2×D17—41no—
91C4×C17—21no—
92C4×C17—41no—
93C4×C17C2×D811yescharacteristic
94C4×C17#13511yescharacteristic
95C4×C17AGL(1,ℤ/8)11yescharacteristic
96C4×C17AGL(1,ℤ/8)11yescharacteristic
97Dic17—42no—
98Dic17—41no—
99#547—171no—
100#486—171no—
101#459—171no—
102#459D1711yescharacteristic, complemented
103#332—171no—
104D4⋊C8—171no—
105D4⋊C8—171no—
106C2×C17—81no—
107C2×C17—161no—
108C2×C17#44711yescharacteristic
109C2×C17#44911yescharacteristic
110C2×C17#45411yescharacteristic
111D17—42no—
112C2×#41—341no—
113C2×D8—21no—
114C2×D8—341no—
115C4⋊Q8—171no—
116#127C2×D1711yescharacteristic
117#123—342no—
118C22⋊Q8—341no—
119#121—341no—
120C4×D4—171no—
121C4×D4C2×D1711yescharacteristic
122C2×C4⋊C4—341no—
123C4⋊C8—171no—
124C4⋊C8C2×D1711yescharacteristic
125#102—21no—
126#102—341no—
127#102—342no—
128C4×C8—171no—
129C17#162511yescharacteristic, complemented, Sylow
130#41—682no—
131#41—682no—
132C2×Q8—341no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—171no—
137C2×D4—342no—
138C2×D4C22⋊D1711yescharacteristic
139C22×C4—21no—
140C22×C4—342no—
141C22×C4—341no—
142D8—42no—
143D8—682no—
144C2×C8—21no—
145C2×C8—341no—
146C2×C8—341no—
147C4⋊C4—171no—
148C4⋊C4—172no—
149C4⋊C4—342no—
150C4⋊C4—341no—
151C4⋊C4C22⋊D1711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—342no—
154C22⋊C4—341no—
155C42C22×D1711yescharacteristic
156C23—21no—
157C23—41no—
158C23—341no—
159Q8—681no—
160Q8—681no—
161D4—21no—
162D4—21no—
163D4—42no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168D4—172no—
169D4—341no—
170D4—684no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—341no—
175C2×C4—342no—
176C2×C4—684no—
177C2×C4—681no—
178C2×C4—681no—
179C2×C4D4×D1711yescharacteristic
180C2×C4D4×D1711yescharacteristic
181C2×C4C2×C22⋊D1711yescharacteristic
182C8—41no—
183C8—681no—
184C8—681no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—342no—
192C22—681no—
193C22—681no—
194C22#1058762311yescharacteristic
195C4—21no—
196C4—41no—
197C4—681no—
198C4—682no—
199C4D8×D1711yescharacteristic
200C4#1058759711yescharacteristic
201C4#1058761811yescharacteristic
202C4#1058763611yescharacteristic
203C2—81no—
204C2—161no—
205C2—682no—
206C2#1176294111yescharacteristic
207C2#1176295211yescharacteristic
208C2#1176298011yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#459C2
socleC22×C17#120
hypercenter#459D17
hypocenterC17#1625
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #11763464C2semidirect
2. #11763155C2semidirect
3. #11763001C2semidirect
4. #11762554C2semidirect
5. D4⋊#3622C2semidirect
6. #11762536C2semidirect
7. C17×#459C2semidirect
71. D68D8semidirect
102. #459D17semidirect
129. C17#1625semidirect

Maximal subgroups

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

subgroupindex
#117634642
#117631552
#117630012
#117625542
D4⋊#36222
#117625362
C17×#4592
#162517