Finite Group Databasebeta

← #14527902

Subgroups — #14527902

Order 2176.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D68C211yesmaximal, characteristic
2#11763182C211yesmaximal, characteristic, complemented
3#11763100C211yesmaximal, characteristic, complemented
4#11763099C211yesmaximal, characteristic, complemented
5#11762552C211yesmaximal, characteristic, complemented
6#11762538C211yesmaximal, characteristic, complemented
7C17×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×D4×D17—21no—
9C22×D68—21no—
10C4×D4×C17C2211yescharacteristic
11#10587616—21no—
12C2×#63219—22no—
13#10587581—21no—
14#10587568—22no—
15#10587560C2211yescharacteristic
16C4×D68C2211yescharacteristic
17#10587555C2211yescharacteristic, complemented
18C17×C4⋊C8C2211yescharacteristic, complemented
19C4×C8×C17C2211yescharacteristic, complemented
20#10587503—21no—
21#10587492—21no—
22#10587488—21no—
23#10587479—21no—
24C4⋊#3622C2211yescharacteristic, complemented
25C23×D17—22no—
26C2×D4×C17D411yescharacteristic
27C22×C4×C17—21no—
28C2×C22⋊D17—22no—
29D4×D17—42no—
30D4×D17—42no—
31C2×D68—22no—
32C2×D68—21no—
33C2×D68—21no—
34C2×D68—42no—
35C2×D68—42no—
36C2×D68D411yescharacteristic
37C2×D68D412yes—
38C2×C4×D17—21no—
39C2×Dic34—21no—
40C2×C8×C17—21no—
41C2×C8×C17—21no—
42C4⋊C4×C17D411yescharacteristic
43C22⋊C4×C17—21no—
44C42×C17C2311yescharacteristic
45#63227—22no—
46#63227—21no—
47C4⋊Dic17—21no—
48C4⋊Dic17—21no—
49C4⋊Dic17D411yescharacteristic
50C2×#3622—21no—
51#63219—44no—
52C23×C17—21no—
53C22×D17—21no—
54C22×D17—22no—
55C22×D17—44no—
56C22×D17—44no—
57C22×D17—82no—
58D4×C17—21no—
59D4×C17SD1612yescomplemented
60C2×C4×C17—21no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×C4×C17C2×D411yescharacteristic
64C2×C4×C17C2×D411yescharacteristic
65C2×C4×C17C2×D411yescharacteristic
66C22⋊D17—44no—
67C2×Dic17—21no—
68C2×Dic17—41no—
69D68—21no—
70D68—42no—
71D68—44no—
72D68—41no—
73D68—41no—
74D68—41no—
75D68—41no—
76D68SD1612yescomplemented
77C4×D17—41no—
78C4×D17—41no—
79Dic34—42no—
80C8×C17—22no—
81C8×C17—41no—
82#3622—41no—
83#1629—171nomaximal, Sylow
84C22×C17—22no—
85C22×C17—41no—
86C22×C17—41no—
87C22×C17#12011yescharacteristic
88C2×D17—22no—
89C2×D17—41no—
90C2×D17—41no—
91C2×D17—42no—
92C2×D17—42no—
93C2×D17—42no—
94C2×D17—88no—
95C4×C17—21no—
96C4×C17—41no—
97C4×C17C2×SD1611yescharacteristic
98C4×C17C2×SD1611yescharacteristic
99C4×C17AGL(1,ℤ/8)11yescharacteristic
100C4×C17AGL(1,ℤ/8)11yescharacteristic
101Dic17—41no—
102Dic17—81no—
103D42—171no—
104#492—171no—
105#476—171no—
106#476—171no—
107#331—171no—
108D4⋊C8—171no—
109D4⋊C8D1711yescharacteristic, complemented
110C2×C17—42no—
111C2×C17#45011yescharacteristic
112C2×C17#45011yescharacteristic
113C2×C17#45311yescharacteristic
114D17—42no—
115D17—82no—
116C22×D4—341no—
117C22×D4—341no—
118C2×SD16—342no—
119C4⋊Q8—171no—
120#127—171no—
121#121—341no—
122#121—341no—
123#120—342no—
124C4×D4—171no—
125C4×D4C2×D1711yescharacteristic
126#106—341no—
127#106—341no—
128C4⋊C8—171no—
129C4⋊C8C2×D1711yescharacteristic, complemented
130#102—341no—
131#102—341no—
132C4×C8C2×D1711yescharacteristic, complemented
133C17#162911yescharacteristic, complemented, Sylow
134C24—342no—
135C2×Q8—341no—
136C2×D4—171no—
137C2×D4—172no—
138C2×D4—342no—
139C2×D4—342no—
140C2×D4—341no—
141C2×D4—341no—
142C2×D4—682no—
143C2×D4—682no—
144C2×D4—682no—
145C2×D4—682no—
146C2×D4D6811yescharacteristic
147C22×C4—21no—
148C22×C4—341no—
149SD16—684no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—341no—
153C4⋊C4—171no—
154C4⋊C4—341no—
155C4⋊C4—341no—
156C4⋊C4D6811yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—342no—
159C22⋊C4—341no—
160C42C22×D1711yescharacteristic
161C23—21no—
162C23—341no—
163C23—342no—
164C23—684no—
165C23—684no—
166C23—1362no—
167Q8—682no—
168D4—21no—
169D4—172no—
170D4—341no—
171D4—682no—
172D4—684no—
173D4—684no—
174D4—681no—
175D4—681no—
176D4—681no—
177D4—681no—
178D4#6321912yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—341no—
183C2×C4—681no—
184C2×C4—681no—
185C2×C4—681no—
186C2×C4C2×D6811yescharacteristic
187C2×C4D4×D1711yescharacteristic
188C2×C4D4×D1711yescharacteristic
189C8—22no—
190C8—41no—
191C8—681no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—342no—
196C22—681no—
197C22—681no—
198C22—682no—
199C22—682no—
200C22—682no—
201C22—1368no—
202C22#1058756811yescharacteristic
203C4—21no—
204C4—41no—
205C4—681no—
206C4—1361no—
207C4C2×#6321911yescharacteristic
208C4#1058759411yescharacteristic
209C4#1058759711yescharacteristic
210C4SD16×D1711yescharacteristic
211C2—42no—
212C2—682no—
213C2—1362no—
214C2#1176296111yescharacteristic
215C2#1176296211yescharacteristic
216C2#1176297811yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×D4⋊C8C2
socleC22×C17#120
prime radicalD4×D68C2
hypercenterD4⋊C8D17
hypocenterC17#1629
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #11763182C2semidirect
3. #11763100C2semidirect
4. #11763099C2semidirect
5. #11762552C2semidirect
6. #11762538C2semidirect
7. C17×D4⋊C8C2semidirect
17. #10587555C22semidirect
18. C17×C4⋊C8C22semidirect
19. C4×C8×C17C22semidirect
24. C4⋊#3622C22semidirect
59. D4×C17SD16semidirect
76. D68SD16semidirect
109. D4⋊C8D17semidirect
129. C4⋊C8C2×D17semidirect
132. C4×C8C2×D17semidirect
133. C17#1629semidirect
178. D4#63219semidirect

Maximal subgroups

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

subgroupindex
D4×D682
#117631822
#117631002
#117630992
#117625522
#117625382
C17×D4⋊C82
#162917