Finite Group Databasebeta

← #11741293

Subgroups — #11741293

Order 896.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×C7×SD16C211yesmaximal, characteristic, complemented
2#90507C211yesmaximal, characteristic, complemented
3#90489C211yesmaximal, characteristic, complemented
4C2×#6495C211yesmaximal, characteristic
5#89997C211yesmaximal, characteristic, complemented
6#89937C214yesmaximal, complemented
7#89934C214yesmaximal, complemented
8#89885C211yesmaximal, characteristic, complemented
9#89874C211yesmaximal, characteristic, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C7×D4C2211yescharacteristic
12C2×#1157C2211yescharacteristic
13C2×C7×SD16—24no—
14C2×C7×SD16C2214yes—
15C22×C7×C8C2211yescharacteristic
16C2×C22⋊Dic7—22no—
17#6510C2212yescomplemented
18#6509—22no—
19#6507C2212yescomplemented
20D4×Dic7—22no—
21D4×Dic7C2214yes—
22C2×#1142C2212yes—
23#6495—24no—
24C2×#1141C2212yes—
25#6492—22no—
26#6489C2211yescharacteristic
27#6489C2211yescharacteristic
28C2×#1136C2211yescharacteristic
29#6409C2214yescomplemented
30#6395C2212yescomplemented
31#6390C2212yescomplemented
32Dic7⋊C8C2214yescomplemented
33#3042—71nomaximal, Sylow
34C24×C7—21no—
35C2×C7×Q8—21no—
36C2×C7×Q8—22no—
37C2×C7×Q8—22no—
38C2×C7×Q8C2312yes—
39C2×C7×D4—21no—
40C2×C7×D4—22no—
41C2×C7×D4D414yes—
42C2×C7×D4C2312yes—
43C22×C4×C7—21no—
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—21no—
46C22×Dic7—24no—
47#1157—22no—
48#1157—22no—
49C2×D28C2311yescharacteristic
50C2×C4×D7—21no—
51C2×Dic14C2311yescharacteristic
52C7×SD16—28no—
53C2×C7×C8—22no—
54C2×C7×C8C2312yes—
55C22⋊Dic7—21no—
56C22⋊Dic7—24no—
57C22⋊Dic7D412yes—
58#1142—24no—
59#1141—24no—
60(C2×D7)⋊C4—24no—
61C4⋊Dic7D412yes—
62C4⋊Dic7C2312yes—
63Dic7⋊C4—24no—
64C4×Dic7—21no—
65C4×Dic7C2312yes—
66#1136—22no—
67C2×C7⋊C8C2312yes—
68C2×AGL(1,ℤ/8)—71no—
69C22×SD16D711yescharacteristic, complemented
70#536—71no—
71#518—71no—
72#463—74no—
73#461—74no—
74#424—71no—
75#419—71no—
76#416—71no—
77C23×C7—22no—
78C23×C7—24no—
79C23×C7—82no—
80C23×C7C2×D411yescharacteristic
81C22×D7—21no—
82C7×Q8—22no—
83C7×Q8—22no—
84C7×Q8—42no—
85C7×D4—24no—
86C7×D4—22no—
87C7×D4C2×D414yes—
88C2×C4×C7—22no—
89C2×C4×C7—42no—
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D412yes—
92C2×C4×C7#4112yes—
93C2×C4×C7C2411yescharacteristic
94C22⋊D7—42no—
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7—84no—
98C2×Dic7C2×D414yes—
99D28—21no—
100D28—21no—
101C4×D7—42no—
102Dic14—21no—
103Dic14—21no—
104C7×C8—22no—
105C7⋊C8—22no—
106C2×#41—71no—
107C22×Q8C2×D711yescharacteristic
108C22×D4C2×D711yescharacteristic
109AGL(1,ℤ/8)—144no—
110C2×SD16—24no—
111C2×SD16—72no—
112C2×SD16C2×D714yes—
113C2×D8—72no—
114C2×M16—71no—
115C22×C8C2×D711yescharacteristic
116C4⋊Q8—72no—
117#124—72no—
118#123—142no—
119C22⋊Q8—142no—
120C4×D4—74no—
121C4×D4—142no—
122#117—71no—
123#117—71no—
124C2×C22⋊C4—142no—
125C4⋊C8—74no—
126#103—74no—
127#103—72no—
128#102—72no—
129C22×C7—22no—
130C22×C7—24no—
131C22×C7—44no—
132C22×C7—84no—
133C22×C7#12112yes—
134C22×C7C22×D411yescharacteristic
135C2×D7—41no—
136C2×D7—42no—
137C4×C7—42no—
138C4×C7#12112yes—
139C4×C7C22×D411yescharacteristic
140C4×C7C2×#4111yescharacteristic
141Dic7—24no—
142Dic7—41no—
143Dic7—42no—
144C24—21no—
145#41—142no—
146#41—142no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—71no—
151C2×Q8C22×D712yes—
152C2×D4—21no—
153C2×D4—22no—
154C2×D4—71no—
155C2×D4—141no—
156C2×D4C22⋊D714yes—
157C2×D4C22×D712yes—
158C22×C4—21no—
159C22×C4—144no—
160C22×C4—141no—
161C22×C4C22×D711yescharacteristic
162SD16—28no—
163SD16—144no—
164D8—144no—
165M16—142no—
166C2×C8—22no—
167C2×C8—72no—
168C2×C8C22×D712yes—
169C4⋊C4—72no—
170C4⋊C4—72no—
171C4⋊C4—144no—
172C22⋊C4—72no—
173C22⋊C4—144no—
174C22⋊C4—144no—
175C22⋊C4—141no—
176C42—72no—
177C42—141no—
178C2×C7—22no—
179C2×C7—44no—
180C2×C7C2×#12111yescharacteristic
181C2×C7#57712yes—
182D7—81no—
183C23—22no—
184C23—24no—
185C23—82no—
186C23—141no—
187C23D4×D711yescharacteristic
188Q8—22no—
189Q8—22no—
190Q8—42no—
191Q8—141no—
192Q8—141no—
193D4—24no—
194D4—22no—
195D4—141no—
196D4—141no—
197D4—282no—
198D4C2×C22⋊D714yes—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—74no—
202C2×C4—142no—
203C2×C4—141no—
204C2×C4—282no—
205C2×C4—564no—
206C2×C4D4×D711yescharacteristic
207C2×C4#115912yes—
208C2×C4C2×C22⋊D712yes—
209C2×C4C23×D711yescharacteristic
210C8—22no—
211C8—142no—
212C7#304211yescharacteristic, complemented, Sylow
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—282no—
218C22—281no—
219C22#650212yes—
220C22C2×D4×D711yescharacteristic
221C4—42no—
222C4—144no—
223C4—282no—
224C4—281no—
225C4#650212yes—
226C4C2×#115911yescharacteristic
227C4C22×C22⋊D711yescharacteristic
228C2—22no—
229C2—44no—
230C2—561no—
231C2#9045312yes—
232C2C2×#650211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC22×C7×SD16C2
socleC22×C7C22×D4
prime radicalC2×#6495C2
hypercenterC22×SD16D7
hypocenterC7#3042
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×C7×SD16C2semidirect
2. #90507C2semidirect
3. #90489C2semidirect
5. #89997C2semidirect
6. #89937C2semidirect
7. #89934C2semidirect
8. #89885C2semidirect
9. #89874C2semidirect
17. #6510C22semidirect
19. #6507C22semidirect
29. #6409C22semidirect
30. #6395C22semidirect
31. #6390C22semidirect
32. Dic7⋊C8C22semidirect
69. C22×SD16D7semidirect
212. C7#3042semidirect

Maximal subgroups

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

subgroupindex
C22×C7×SD162
#905072
#904892
C2×#64952
#899972
#899372
#899342
#898852
#898742
#30427