Finite Group Databasebeta

← #11736422

Subgroups — #11736422

Order 896.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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#90507C211yesmaximal, characteristic
2C22×#1144C211yesmaximal, characteristic, complemented
3C2×#6472C211yesmaximal, characteristic
4#90207C211yesmaximal, characteristic, complemented
5C7×#419C211yesmaximal, characteristic, complemented
6#89891C211yesmaximal, characteristic, complemented
7#89774C211yesmaximal, characteristic
8#89570C214yesmaximal, complemented
9#89565C214yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C2×#1157C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C7×#117C2211yescharacteristic
15#6510C2212yes—
16#6509—22no—
17#6507C2212yes—
18C2×#1144—24no—
19C2×#1144C2214yes—
20#6492—22no—
21#6489C2211yescharacteristic
22C22×C7⋊C8C2211yescharacteristic
23#6472—24no—
24C2×Dic28C2212yes—
25C2×#1132C2212yes—
26#6450C2214yes—
27#6440—22no—
28C4×Dic14—22no—
29C7×#103C2214yes—
30Dic7⋊C8C2214yes—
31#6384C2212yes—
32#6383C2212yes—
33#3044—71nomaximal, Sylow
34C2×C7×Q8—21no—
35C2×C7×Q8—22no—
36C2×C7×Q8—22no—
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—21no—
47C2×Dic14—22no—
48C2×Dic14D414yes—
49C2×Dic14C2311yescharacteristic
50C2×Dic14C2312yes—
51C7×M16—22no—
52C2×C7×C8C2312yes—
53C7×C4⋊C4C2312yes—
54C7×C22⋊C4—21no—
55C42×C7—21no—
56C22⋊Dic7D412yes—
57#1144—28no—
58(C2×D7)⋊C4—24no—
59C4⋊Dic7D412yes—
60Dic7⋊C4—24no—
61Dic7⋊C4—24no—
62C4×Dic7—24no—
63C4×Dic7C2312yes—
64C2×C7⋊C8—22no—
65C2×C7⋊C8C2312yes—
66Dic28—24no—
67#1132—24no—
68C2×#137—71no—
69C22×Q16—71no—
70#536—71no—
71#519—71no—
72#464—74no—
73#462—74no—
74#424—71no—
75#419D711yescharacteristic, complemented
76#416—71no—
77C23×C7C2×D411yescharacteristic
78C22×D7—21no—
79C7×Q8—22no—
80C7×Q8—22no—
81C7×Q8—42no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—42no—
85C2×C4×C7C2×D411yescharacteristic
86C2×C4×C7C2×D412yes—
87C2×C4×C7#4112yes—
88C2×C4×C7C2411yescharacteristic
89C22⋊D7—42no—
90C2×Dic7—24no—
91C2×Dic7—22no—
92C2×Dic7—21no—
93C2×Dic7C2×D414yes—
94D28—21no—
95D28—21no—
96C4×D7—42no—
97Dic14—24no—
98Dic14—22no—
99Dic14—21no—
100Dic14—21no—
101Dic14C2×D414yes—
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×M16C2×D711yescharacteristic
113C22×C8—71no—
114C4⋊Q8—72no—
115#124—72no—
116#123—142no—
117C22⋊Q8—142no—
118C4×Q8—74no—
119C4×Q8—142no—
120#117—71no—
121#117—142no—
122#117C2×D711yescharacteristic
123C4⋊C8—74no—
124#103—72no—
125#103C2×D714yes—
126#102—72no—
127C22×C7—22no—
128C22×C7#12112yes—
129C22×C7C22×D411yescharacteristic
130C2×D7—41no—
131C2×D7—42no—
132C4×C7—42no—
133C4×C7—42no—
134C4×C7#12112yes—
135C4×C7C22×D411yescharacteristic
136C4×C7C2×#4111yescharacteristic
137Dic7—24no—
138Dic7—24no—
139Dic7—41no—
140#41—142no—
141#41—142no—
142C2×Q8—21no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—74no—
146C2×Q8—72no—
147C2×Q8—71no—
148C2×Q8—142no—
149C2×Q8—141no—
150C2×Q8C22×D712yes—
151C2×D4—71no—
152C2×D4—141no—
153C22×C4—21no—
154C22×C4—141no—
155C22×C4—141no—
156C22×C4C22×D711yescharacteristic
157Q16—144no—
158Q16—148no—
159SD16—144no—
160M16—22no—
161C2×C8—72no—
162C2×C8—142no—
163C2×C8C22×D712yes—
164C4⋊C4—72no—
165C4⋊C4—144no—
166C4⋊C4—144no—
167C4⋊C4C22×D712yes—
168C22⋊C4—21no—
169C22⋊C4—72no—
170C22⋊C4—144no—
171C42—21no—
172C42—72no—
173C42—144no—
174C2×C7—22no—
175C2×C7C2×#12111yescharacteristic
176C2×C7#57812yes—
177D7—81no—
178C23—141no—
179C23D4×D711yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—42no—
183Q8—74no—
184Q8—144no—
185Q8—142no—
186Q8—141no—
187Q8—141no—
188D4—141no—
189D4—141no—
190D4—282no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—74no—
195C2×C4—144no—
196C2×C4—142no—
197C2×C4—141no—
198C2×C4—282no—
199C2×C4#115712yes—
200C2×C4D4×D711yescharacteristic
201C2×C4D4×D712yes—
202C2×C4C23×D711yescharacteristic
203C8—22no—
204C8—142no—
205C7#304411yescharacteristic, complemented, Sylow
206C22—22no—
207C22—281no—
208C22—282no—
209C22#644712yes—
210C22C2×D4×D711yescharacteristic
211C4—42no—
212C4—42no—
213C4—144no—
214C4—144no—
215C4—281no—
216C4#644712yes—
217C4C2×#115711yescharacteristic
218C4C2×D4×D711yescharacteristic
219C2—22no—
220C2—561no—
221C2C2×#644711yescharacteristic
222C2#9047012yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#419C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#419D7
hypocenterC7#3044
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#1144C2semidirect
4. #90207C2semidirect
5. C7×#419C2semidirect
6. #89891C2semidirect
8. #89570C2semidirect
9. #89565C2semidirect
75. #419D7semidirect
205. C7#3044semidirect

Maximal subgroups

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

subgroupindex
#905072
C22×#11442
C2×#64722
#902072
C7×#4192
#898912
#897742
#895702
#895652
#30447