Finite Group Databasebeta

← #11736545

Subgroups — #11736545

Order 896.

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

normal subgroups only

This group has 458 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
1C2×#6472C211yesmaximal, characteristic
2C22×Dic28C211yesmaximal, characteristic, complemented
3#90217C211yesmaximal, characteristic
4#90207C211yesmaximal, characteristic, complemented
5C7×#424C211yesmaximal, characteristic, complemented
6#89892C211yesmaximal, characteristic, complemented
7#89885C211yesmaximal, characteristic
8#89621C214yesmaximal, complemented
9#89617C214yesmaximal, complemented
10C2×#1157C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C22×C7×C8C2211yescharacteristic
15C7×#117C2211yescharacteristic
16#6489C2211yescharacteristic
17#6472—24no—
18C2×Dic28—24no—
19C2×Dic28C2214yes—
20C2×Dic28C2212yes—
21C2×#1132C2212yes—
22#6460—22no—
23#6450—22no—
24#6449—22no—
25#6440—22no—
26#6438C2212yes—
27#6432C2212yes—
28C4×Dic14C2214yes—
29C7×C4⋊C8C2214yes—
30#6395C2212yes—
31#6390C2214yes—
32#6390C2212yes—
33#3044—71nomaximal, Sylow
34C22×C4×C7C2311yescharacteristic
35C2×C22⋊D7—21no—
36C22×Dic7—21no—
37C22×Dic7—21no—
38#1157—22no—
39#1157—22no—
40C2×D28C2311yescharacteristic
41C2×C4×D7—21no—
42C2×Dic14—21no—
43C2×Dic14—22no—
44C2×Dic14—22no—
45C2×Dic14—22no—
46C2×Dic14—21no—
47C2×Dic14D414yes—
48C2×Dic14C2311yescharacteristic
49C2×Dic14C2312yes—
50C2×Dic14C2312yes—
51C7×M16—22no—
52C2×C7×C8—22no—
53C2×C7×C8C2312yes—
54C2×C7×C8C2312yes—
55C7×C4⋊C4D412yes—
56C7×C22⋊C4D412yes—
57C42×C7C2312yes—
58C22⋊Dic7—21no—
59(C2×D7)⋊C4—24no—
60C4⋊Dic7—24no—
61C4⋊Dic7C2312yes—
62Dic7⋊C4—24no—
63C4×Dic7—24no—
64C4×Dic7—21no—
65Dic28—28no—
66Dic28—24no—
67#1132—24no—
68C2×#137—71no—
69C22×Q16—71no—
70#536—71no—
71#519—71no—
72#464—74no—
73#462—74no—
74#424D711yescharacteristic, complemented
75#419—71no—
76#416—71no—
77C23×C7C2×D411yescharacteristic
78C22×D7—21no—
79C2×C4×C7C2×D414yes—
80C2×C4×C7C2×D411yescharacteristic
81C2×C4×C7C2×D412yes—
82C2×C4×C7#4112yes—
83C2×C4×C7C2411yescharacteristic
84C22⋊D7—42no—
85C2×Dic7—24no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—21no—
89C2×Dic7—22no—
90C2×Dic7—42no—
91D28—21no—
92D28—21no—
93C4×D7—42no—
94Dic14—24no—
95Dic14—22no—
96Dic14—22no—
97Dic14—21no—
98Dic14—22no—
99Dic14—21no—
100Dic14—42no—
101Dic14C2×D414yes—
102C7×C8—22no—
103C7×C8—22no—
104C2×#41—71no—
105C22×Q8—71no—
106C22×Q8—71no—
107#137—144no—
108C2×Q16—74no—
109C2×Q16—72no—
110C2×Q16—144no—
111C2×SD16—72no—
112C2×M16C2×D711yescharacteristic
113C22×C8C2×D711yescharacteristic
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⋊C8C2×D714yes—
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#12112yes—
134C4×C7C22×D411yescharacteristic
135C4×C7C2×#4111yescharacteristic
136Dic7—24no—
137Dic7—42no—
138Dic7—41no—
139Dic7—42no—
140#41—142no—
141#41—142no—
142C2×Q8—74no—
143C2×Q8—72no—
144C2×Q8—72no—
145C2×Q8—71no—
146C2×Q8—141no—
147C2×Q8—141no—
148C2×Q8—142no—
149C2×Q8—142no—
150C2×Q8—142no—
151C2×D4—71no—
152C2×D4—141no—
153C22×C4—141no—
154C22×C4—141no—
155C22×C4—141no—
156C22×C4C22×D711yescharacteristic
157Q16—148no—
158Q16—144no—
159SD16—144no—
160M16—22no—
161C2×C8—22no—
162C2×C8C22×D712yes—
163C2×C8C22×D712yes—
164C4⋊C4—72no—
165C4⋊C4—144no—
166C4⋊C4—144no—
167C4⋊C4D2812yes—
168C22⋊C4—141no—
169C22⋊C4—144no—
170C22⋊C4D2812yes—
171C42—141no—
172C42—144no—
173C42C22×D712yes—
174C2×C7—22no—
175C2×C7C2×#12111yescharacteristic
176C2×C7#57812yes—
177D7—81no—
178C23—141no—
179C23C2×D2811yescharacteristic
180Q8—74no—
181Q8—144no—
182Q8—142no—
183Q8—142no—
184Q8—141no—
185Q8—142no—
186Q8—141no—
187Q8—282no—
188D4—141no—
189D4—141no—
190D4—282no—
191C2×C4—144no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—141no—
196C2×C4—282no—
197C2×C4—282no—
198C2×C4C2×D2814yes—
199C2×C4C2×D2811yescharacteristic
200C2×C4D4×D712yes—
201C2×C4#116112yes—
202C2×C4C23×D711yescharacteristic
203C8—22no—
204C8—22no—
205C7#304411yescharacteristic, complemented, Sylow
206C22—22no—
207C22—282no—
208C22—281no—
209C22#645812yes—
210C22C22×D2811yescharacteristic
211C4—24no—
212C4—144no—
213C4—282no—
214C4—281no—
215C4—282no—
216C4#645812yes—
217C4C2×D4×D711yescharacteristic
218C4C2×#116111yescharacteristic
219C2—22no—
220C2—561no—
221C2C2×#645811yescharacteristic
222C2#9044412yes—
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#424C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#424D7
hypocenterC7#3044
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic28C2semidirect
4. #90207C2semidirect
5. C7×#424C2semidirect
6. #89892C2semidirect
8. #89621C2semidirect
9. #89617C2semidirect
74. #424D7semidirect
205. C7#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#64722
C22×Dic282
#902172
#902072
C7×#4242
#898922
#898852
#896212
#896172
#30447