Finite Group Databasebeta

← #11725949

Subgroups — #11725949

Order 896.

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×D28C211yesmaximal, characteristic
2#89780C211yesmaximal, characteristic, complemented
3#89614C211yesmaximal, characteristic, complemented
4#89465C211yesmaximal, characteristic, complemented
5C7×D4⋊C8C211yesmaximal, characteristic, complemented
6#89280C211yesmaximal, characteristic, complemented
7#89254C211yesmaximal, characteristic, complemented
8C2×D4×D7—21no—
9C22×D28—21no—
10C4×C7×D4C2211yescharacteristic
11#6493—21no—
12C2×#1132—22no—
13#6458—21no—
14#6445—22no—
15#6437C2211yescharacteristic
16C4×D28C2211yescharacteristic
17#6432C2211yescharacteristic, complemented
18C7×C4⋊C8C2211yescharacteristic, complemented
19C4×C7×C8C2211yescharacteristic, complemented
20#6406—21no—
21#6395—21no—
22#6391—21no—
23#6382—21no—
24C4⋊(C7⋊C8)C2211yescharacteristic, complemented
25#1629—71nomaximal, Sylow
26C23×D7—22no—
27C2×C7×D4D411yescharacteristic
28C22×C4×C7—21no—
29C2×C22⋊D7—22no—
30D4×D7—42no—
31D4×D7—42no—
32C2×D28—22no—
33C2×D28—21no—
34C2×D28—21no—
35C2×D28—42no—
36C2×D28—42no—
37C2×D28D411yescharacteristic
38C2×D28D412yes—
39C2×C4×D7—21no—
40C2×Dic14—21no—
41C2×C7×C8—21no—
42C2×C7×C8—21no—
43C7×C4⋊C4D411yescharacteristic
44C7×C22⋊C4—21no—
45C42×C7C2311yescharacteristic
46(C2×D7)⋊C4—22no—
47(C2×D7)⋊C4—21no—
48C4⋊Dic7—21no—
49C4⋊Dic7—21no—
50C4⋊Dic7D411yescharacteristic
51C2×C7⋊C8—21no—
52#1132—44no—
53D42—71no—
54#492—71no—
55#476—71no—
56#476—71no—
57#331—71no—
58D4⋊C8—71no—
59D4⋊C8D711yescharacteristic, complemented
60C23×C7—21no—
61C22×D7—21no—
62C22×D7—22no—
63C22×D7—44no—
64C22×D7—44no—
65C22×D7—82no—
66C7×D4—21no—
67C7×D4SD1612yescomplemented
68C2×C4×C7—21no—
69C2×C4×C7—41no—
70C2×C4×C7—41no—
71C2×C4×C7C2×D411yescharacteristic
72C2×C4×C7C2×D411yescharacteristic
73C2×C4×C7C2×D411yescharacteristic
74C22⋊D7—44no—
75C2×Dic7—21no—
76C2×Dic7—41no—
77D28—21no—
78D28—42no—
79D28—44no—
80D28—41no—
81D28—41no—
82D28—41no—
83D28—41no—
84D28SD1612yescomplemented
85C4×D7—41no—
86C4×D7—41no—
87Dic14—42no—
88C7×C8—22no—
89C7×C8—41no—
90C7⋊C8—41no—
91C22×D4—141no—
92C22×D4—141no—
93C2×SD16—142no—
94C4⋊Q8—71no—
95#127—71no—
96#121—141no—
97#121—141no—
98#120—142no—
99C4×D4—71no—
100C4×D4C2×D711yescharacteristic
101#106—141no—
102#106—141no—
103C4⋊C8—71no—
104C4⋊C8C2×D711yescharacteristic, complemented
105#102—141no—
106#102—141no—
107C4×C8C2×D711yescharacteristic, complemented
108C22×C7—22no—
109C22×C7—41no—
110C22×C7—41no—
111C22×C7#12011yescharacteristic
112C2×D7—22no—
113C2×D7—41no—
114C2×D7—41no—
115C2×D7—42no—
116C2×D7—42no—
117C2×D7—42no—
118C2×D7—88no—
119C4×C7—21no—
120C4×C7—41no—
121C4×C7C2×SD1611yescharacteristic
122C4×C7C2×SD1611yescharacteristic
123C4×C7AGL(1,ℤ/8)11yescharacteristic
124C4×C7AGL(1,ℤ/8)11yescharacteristic
125Dic7—41no—
126Dic7—81no—
127C24—142no—
128C2×Q8—141no—
129C2×D4—71no—
130C2×D4—72no—
131C2×D4—142no—
132C2×D4—142no—
133C2×D4—141no—
134C2×D4—141no—
135C2×D4—282no—
136C2×D4—282no—
137C2×D4—282no—
138C2×D4—282no—
139C2×D4D2811yescharacteristic
140C22×C4—21no—
141C22×C4—141no—
142SD16—284no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—141no—
146C4⋊C4—71no—
147C4⋊C4—141no—
148C4⋊C4—141no—
149C4⋊C4D2811yescharacteristic
150C22⋊C4—21no—
151C22⋊C4—142no—
152C22⋊C4—141no—
153C42C22×D711yescharacteristic
154C2×C7—42no—
155C2×C7#45011yescharacteristic
156C2×C7#45011yescharacteristic
157C2×C7#45311yescharacteristic
158D7—42no—
159D7—82no—
160C23—21no—
161C23—141no—
162C23—142no—
163C23—284no—
164C23—284no—
165C23—562no—
166Q8—282no—
167D4—21no—
168D4—72no—
169D4—141no—
170D4—282no—
171D4—284no—
172D4—284no—
173D4—281no—
174D4—281no—
175D4—281no—
176D4—281no—
177D4#113212yescomplemented
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—141no—
182C2×C4—281no—
183C2×C4—281no—
184C2×C4—281no—
185C2×C4C2×D2811yescharacteristic
186C2×C4D4×D711yescharacteristic
187C2×C4D4×D711yescharacteristic
188C8—22no—
189C8—41no—
190C8—281no—
191C7#162911yescharacteristic, complemented, Sylow
192C22—22no—
193C22—41no—
194C22—41no—
195C22—142no—
196C22—281no—
197C22—281no—
198C22—282no—
199C22—282no—
200C22—282no—
201C22—568no—
202C22#644511yescharacteristic
203C4—21no—
204C4—41no—
205C4—281no—
206C4—561no—
207C4C2×#113211yescharacteristic
208C4#647111yescharacteristic
209C4#647411yescharacteristic
210C4D7×SD1611yescharacteristic
211C2—42no—
212C2—282no—
213C2—562no—
214C2#8950311yescharacteristic
215C2#8954811yescharacteristic
216C2#8959411yescharacteristic
subgroupG/H
centerC22#6445
commutator, cosocleC42×C7C23
FrattiniC42C22×D7
FittingC7×D4⋊C8C2
socleC22×C7#120
prime radicalD4×D28C2
hypercenterD4⋊C8D7
hypocenterC7#1629
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #89780C2semidirect
3. #89614C2semidirect
4. #89465C2semidirect
5. C7×D4⋊C8C2semidirect
6. #89280C2semidirect
7. #89254C2semidirect
17. #6432C22semidirect
18. C7×C4⋊C8C22semidirect
19. C4×C7×C8C22semidirect
24. C4⋊(C7⋊C8)C22semidirect
59. D4⋊C8D7semidirect
67. C7×D4SD16semidirect
84. D28SD16semidirect
104. C4⋊C8C2×D7semidirect
107. C4×C8C2×D7semidirect
177. D4#1132semidirect
191. C7#1629semidirect

Maximal subgroups

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

subgroupindex
D4×D282
#897802
#896142
#894652
C7×D4⋊C82
#892802
#892542
#16297