Finite Group Databasebeta

← #11739999

Subgroups — #11739999

Order 896.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C7×#519C211yesmaximal, characteristic, complemented
2C22×#1144C211yesmaximal, characteristic, complemented
3C2×#6505C211yesmaximal, characteristic
4#90217C211yesmaximal, characteristic
5#89803C214yesmaximal, complemented
6#89800C214yesmaximal, complemented
7#89776C211yesmaximal, characteristic, complemented
8#89774C211yesmaximal, characteristic
9#89771C211yesmaximal, characteristic, complemented
10C22×C7×Q8C2211yescharacteristic
11C2×#1157C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C4×C7×Q8—22no—
14C4×C7×Q8C2214yes—
15C7×#117—22no—
16C7×#117C2211yescharacteristic
17C7×#117C2211yescharacteristic
18C2×#1144—24no—
19C2×#1144C2212yes—
20C2×#1144C2214yes—
21#6505—24no—
22C2×#1143C2212yes—
23C2×#1136C2211yescharacteristic
24C22×C7⋊C8C2211yescharacteristic
25#6460—22no—
26#6449—22no—
27#6438C2212yes—
28#6432C2212yes—
29#6384C2212yes—
30#6384C2214yes—
31#6383C2212yes—
32C4⋊(C7⋊C8)C2214yes—
33#3044—71nomaximal, Sylow
34C2×C7×Q8—21no—
35C2×C7×Q8—22no—
36C2×C7×Q8D414yes—
37C2×C7×Q8C2312yes—
38C22×C4×C7—21no—
39C22×C4×C7C2311yescharacteristic
40C2×C22⋊D7—21no—
41C22×Dic7—21no—
42#1157—22no—
43#1157—22no—
44C2×D28C2311yescharacteristic
45C2×C4×D7—21no—
46C2×Dic14—22no—
47C2×Dic14—22no—
48C2×Dic14—21no—
49C2×Dic14C2311yescharacteristic
50C2×Dic14C2312yes—
51C7×C4⋊C4—24no—
52C7×C4⋊C4D412yes—
53C7×C4⋊C4C2312yes—
54C7×C22⋊C4—21no—
55C7×C22⋊C4D412yes—
56C42×C7—24no—
57C42×C7—21no—
58C42×C7C2312yes—
59#1144—24no—
60#1144—28no—
61#1143—24no—
62(C2×D7)⋊C4—24no—
63C4⋊Dic7—24no—
64#1136—22no—
65C2×C7⋊C8—22no—
66C2×C7⋊C8C2312yes—
67C2×C7⋊C8C2312yes—
68C2×#137—71no—
69C22×Q16—71no—
70#536—71no—
71#519D711yescharacteristic, complemented
72#464—74no—
73#462—74no—
74#424—71no—
75#419—71no—
76#416—71no—
77C23×C7C2×D411yescharacteristic
78C22×D7—21no—
79C7×Q8—24no—
80C7×Q8—22no—
81C7×Q8C2×D414yes—
82C2×C4×C7—24no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7C2×D412yes—
86C2×C4×C7C2×D414yes—
87C2×C4×C7C2×D411yescharacteristic
88C2×C4×C7#4112yes—
89C2×C4×C7C2411yescharacteristic
90C22⋊D7—42no—
91C2×Dic7—21no—
92C2×Dic7—22no—
93C2×Dic7—42no—
94D28—21no—
95D28—21no—
96C4×D7—42no—
97Dic14—22no—
98Dic14—21no—
99Dic14—22no—
100Dic14—21no—
101Dic14—42no—
102C7⋊C8—22no—
103C7⋊C8—22no—
104C2×#41—71no—
105C22×Q8—71no—
106C22×Q8C2×D711yescharacteristic
107#137—144no—
108C2×Q16—74no—
109C2×Q16—72no—
110C2×Q16—144no—
111C2×SD16—72no—
112C2×M16—71no—
113C22×C8—71no—
114C4⋊Q8—72no—
115#124—72no—
116#123—142no—
117C22⋊Q8—142no—
118C4×Q8—22no—
119C4×Q8C2×D714yes—
120#117—22no—
121#117C2×D711yescharacteristic
122#117C2×D711yescharacteristic
123C4⋊C8—74no—
124#103—72no—
125#103—74no—
126#102—72no—
127C22×C7—22no—
128C22×C7#12112yes—
129C22×C7C22×D411yescharacteristic
130C2×D7—41no—
131C2×D7—42no—
132C4×C7—24no—
133C4×C7—24no—
134C4×C7—42no—
135C4×C7#12112yes—
136C4×C7C22×D411yescharacteristic
137C4×C7C2×#4111yescharacteristic
138Dic7—42no—
139Dic7—41no—
140#41—142no—
141#41—142no—
142C2×Q8—22no—
143C2×Q8—21no—
144C2×Q8—72no—
145C2×Q8—71no—
146C2×Q8—141no—
147C2×Q8—142no—
148C2×Q8—142no—
149C2×Q8D2814yes—
150C2×Q8C22×D712yes—
151C2×D4—71no—
152C2×D4—141no—
153C22×C4—21no—
154C22×C4—141no—
155C22×C4—141no—
156C22×C4C22×D711yescharacteristic
157Q16—148no—
158Q16—144no—
159SD16—144no—
160M16—142no—
161C2×C8—72no—
162C2×C8—72no—
163C2×C8—142no—
164C4⋊C4—24no—
165C4⋊C4—144no—
166C4⋊C4C22⋊D712yes—
167C4⋊C4C22×D712yes—
168C22⋊C4—21no—
169C22⋊C4—144no—
170C22⋊C4C22⋊D712yes—
171C42—24no—
172C42—21no—
173C42C22×D712yes—
174C2×C7—22no—
175C2×C7C2×#12111yescharacteristic
176C2×C7#57812yes—
177D7—81no—
178C23—141no—
179C23C2×C22⋊D711yescharacteristic
180Q8—24no—
181Q8—22no—
182Q8—142no—
183Q8—141no—
184Q8—142no—
185Q8—141no—
186Q8—282no—
187Q8C2×D2814yes—
188D4—141no—
189D4—141no—
190D4—282no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—142no—
195C2×C4—141no—
196C2×C4—282no—
197C2×C4—282no—
198C2×C4C2×D2812yes—
199C2×C4#115712yes—
200C2×C4C2×C22⋊D714yes—
201C2×C4C2×C22⋊D711yescharacteristic
202C2×C4C23×D711yescharacteristic
203C8—142no—
204C8—142no—
205C7#304411yescharacteristic, complemented, Sylow
206C22—22no—
207C22—281no—
208C22—282no—
209C22#649312yes—
210C22C22×C22⋊D711yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—281no—
215C4—282no—
216C4#649312yes—
217C4C22×D2811yescharacteristic
218C4C2×#115711yescharacteristic
219C2—22no—
220C2—561no—
221C2C2×#649311yescharacteristic
222C2#9052912yes—
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#519C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#519D7
hypocenterC7#3044
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C7×#519C2semidirect
2. C22×#1144C2semidirect
5. #89803C2semidirect
6. #89800C2semidirect
7. #89776C2semidirect
9. #89771C2semidirect
71. #519D7semidirect
205. C7#3044semidirect

Maximal subgroups

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

subgroupindex
C7×#5192
C22×#11442
C2×#65052
#902172
#898032
#898002
#897762
#897742
#897712
#30447