Finite Group Databasebeta

← #14528010

Subgroups — #14528010

Order 2176.

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

normal subgroups only

This group has 212 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#11763524C211yesmaximal, characteristic
2#11763161C211yesmaximal, characteristic, complemented
3#11763086C211yesmaximal, characteristic, complemented
4#11762557C211yesmaximal, characteristic, complemented
5D4⋊#3622C211yesmaximal, characteristic, complemented
6#11762536C211yesmaximal, characteristic, complemented
7C17×#461C211yesmaximal, characteristic, complemented
8C2×SD16×C17—21no—
9C17×C4⋊Q8C2211yescharacteristic
10C4×D4×C17C2211yescharacteristic
11#10587624—21no—
12#10587616—21no—
13#10587612—21no—
14#10587578—21no—
15#10587571—22no—
16C4×D68C2211yescharacteristic
17#10587555C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19C17×#103—21no—
20#10587481—22no—
21#10587480—21no—
22#10587478—21no—
23C4⋊#3622C2211yescharacteristic
24C4×#3622C2211yescharacteristic
25C2×Q8×C17—21no—
26C2×D4×C17D411yescharacteristic
27C22×C4×C17—21no—
28C2×C22⋊D17—22no—
29C2×D68D411yescharacteristic
30C2×C4×D17—21no—
31C2×Dic34D412yes—
32SD16×C17—42no—
33C2×C8×C17—21no—
34C4⋊C4×C17—21no—
35C4⋊C4×C17—21no—
36C4⋊C4×C17D411yescharacteristic
37C22⋊C4×C17—21no—
38C42×C17C2311yescharacteristic
39C22⋊Dic17—22no—
40#63227—22no—
41#63227—21no—
42C4⋊Dic17—21no—
43C4⋊Dic17—21no—
44C4⋊Dic17D411yescharacteristic
45C4×Dic17—22no—
46C2×#3622—21no—
47C2×#3622—21no—
48C23×C17—21no—
49C22×D17—21no—
50Q8×C17—41no—
51Q8×C17—41no—
52D4×C17—21no—
53D4×C17—21no—
54C2×C4×C17—21no—
55C2×C4×C17—41no—
56C2×C4×C17—41no—
57C2×C4×C17—41no—
58C2×C4×C17C2×D411yescharacteristic
59C2×C4×C17C2×D411yescharacteristic
60C2×C4×C17C2×D411yescharacteristic
61C22⋊D17—82no—
62C2×Dic17—21no—
63C2×Dic17—22no—
64D68—21no—
65D68—21no—
66C4×D17—41no—
67C4×D17—41no—
68Dic34—42no—
69C8×C17—41no—
70#3622—41no—
71#3622—41no—
72#1646—171nomaximal, Sylow
73C22×C17—41no—
74C22×C17—41no—
75C22×C17—41no—
76C22×C17#12011yescharacteristic
77C2×D17—41no—
78C2×D17—41no—
79C2×D17—41no—
80C4×C17—21no—
81C4×C17—41no—
82C4×C17—81no—
83C4×C17#13511yescharacteristic
84C4×C17#13511yescharacteristic
85C4×C17AGL(1,ℤ/8)11yescharacteristic
86C4×C17#13711yescharacteristic
87Dic17—41no—
88Dic17—42no—
89#555—171no—
90#487—171no—
91#474—171no—
92#461D1711yescharacteristic, complemented
93#333—171no—
94D4⋊C8—171no—
95D4⋊C8—171no—
96C2×C17—81no—
97C2×C17#44911yescharacteristic
98C2×C17#45211yescharacteristic
99C2×C17#45511yescharacteristic
100D17—81no—
101C2×SD16—21no—
102C4⋊Q8—171no—
103C4⋊Q8C2×D1711yescharacteristic
104#124—342no—
105#121—341no—
106#121—341no—
107C4×D4—171no—
108C4×D4C2×D1711yescharacteristic
109#117—341no—
110#117—341no—
111#107—341no—
112C4⋊C8—171no—
113C4⋊C8C2×D1711yescharacteristic
114#103—21no—
115#103—342no—
116#102—341no—
117C4×C8—171no—
118C17#164611yescharacteristic, complemented, Sylow
119C2×Q8—21no—
120C2×Q8—172no—
121C2×D4—171no—
122C2×D4—342no—
123C2×D4C22⋊D1711yescharacteristic
124C22×C4—21no—
125C22×C4—341no—
126SD16—42no—
127C2×C8—21no—
128C2×C8—341no—
129C2×C8—341no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—171no—
133C4⋊C4—341no—
134C4⋊C4—341no—
135C4⋊C4C22⋊D1711yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—342no—
138C22⋊C4—342no—
139C22⋊C4—341no—
140C42—342no—
141C42C22×D1711yescharacteristic
142C23—21no—
143C23—341no—
144Q8—41no—
145Q8—41no—
146Q8—682no—
147D4—21no—
148D4—21no—
149D4—341no—
150D4—341no—
151D4—1362no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—341no—
157C2×C4—342no—
158C2×C4—681no—
159C2×C4—681no—
160C2×C4D4×D1711yescharacteristic
161C2×C4D4×D1711yescharacteristic
162C2×C4C2×C22⋊D1711yescharacteristic
163C8—41no—
164C8—681no—
165C8—681no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—681no—
170C22—681no—
171C22—681no—
172C22#1058762311yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—681no—
177C4—682no—
178C4#1058760111yescharacteristic
179C4#1058760211yescharacteristic
180C4#1058761811yescharacteristic
181C4#1058763611yescharacteristic
182C2—81no—
183C2—1361no—
184C2#1176295211yescharacteristic
185C2#1176296811yescharacteristic
186C2#1176298511yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#461C2
socleC22×C17#120
prime radical#11763524C2
hypercenter#461D17
hypocenterC17#1646
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #11763161C2semidirect
3. #11763086C2semidirect
4. #11762557C2semidirect
5. D4⋊#3622C2semidirect
6. #11762536C2semidirect
7. C17×#461C2semidirect
92. #461D17semidirect
118. C17#1646semidirect

Maximal subgroups

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

subgroupindex
#117635242
#117631612
#117630862
#117625572
D4⋊#36222
#117625362
C17×#4612
#164617