Finite Group Databasebeta

← #11741322

Subgroups — #11741322

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
1C22×C7×Q16C211yesmaximal, characteristic, complemented
2#90507C211yesmaximal, characteristic
3#90503C211yesmaximal, characteristic, complemented
4C2×#6505C211yesmaximal, characteristic
5#89997C211yesmaximal, characteristic, complemented
6#89957C214yesmaximal, complemented
7#89954C214yesmaximal, complemented
8#89885C211yesmaximal, characteristic
9#89874C211yesmaximal, characteristic, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C7×Q8C2211yescharacteristic
12C2×#1157C2211yescharacteristic
13C2×C7×Q16—24no—
14C2×C7×Q16C2214yes—
15C22×C7×C8C2211yescharacteristic
16#6510C2212yes—
17#6509—22no—
18Q8×Dic7—22no—
19Q8×Dic7C2214yes—
20#6507C2212yes—
21C2×#1144C2212yes—
22#6505—24no—
23C2×#1143C2212yes—
24#6492—22no—
25#6489—22no—
26#6489C2211yescharacteristic
27#6489C2211yescharacteristic
28C2×#1136C2211yescharacteristic
29#6409C2214yes—
30#6395C2212yes—
31#6390C2212yes—
32Dic7⋊C8C2214yes—
33#3044—71nomaximal, Sylow
34C2×C7×Q8—21no—
35C2×C7×Q8—21no—
36C2×C7×Q8—22no—
37C2×C7×Q8—22no—
38C2×C7×Q8—22no—
39C2×C7×Q8D414yes—
40C2×C7×Q8C2312yes—
41C2×C7×Q8C2312yes—
42C22×C4×C7—21no—
43C22×C4×C7—21no—
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—21no—
46#1157—22no—
47#1157—22no—
48C2×D28C2311yescharacteristic
49C2×C4×D7—21no—
50C2×Dic14C2311yescharacteristic
51C7×Q16—28no—
52C2×C7×C8—22no—
53C2×C7×C8C2312yes—
54C22⋊Dic7—21no—
55C22⋊Dic7D412yes—
56#1144—24no—
57#1143—24no—
58(C2×D7)⋊C4—24no—
59C4⋊Dic7—24no—
60C4⋊Dic7D412yes—
61C4⋊Dic7C2312yes—
62Dic7⋊C4—24no—
63C4×Dic7—21no—
64C4×Dic7—24no—
65C4×Dic7C2312yes—
66#1136—22no—
67C2×C7⋊C8C2312yes—
68C2×#137—71no—
69C22×Q16D711yescharacteristic, complemented
70#536—71no—
71#519—71no—
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×Q8—22no—
82C7×Q8—22no—
83C7×Q8—42no—
84C7×Q8C2×D414yes—
85C2×C4×C7—24no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—42no—
89C2×C4×C7C2×D411yescharacteristic
90C2×C4×C7C2×D412yes—
91C2×C4×C7#4112yes—
92C2×C4×C7C2411yescharacteristic
93C22⋊D7—42no—
94C2×Dic7—22no—
95C2×Dic7—21no—
96C2×Dic7C2×D414yes—
97D28—21no—
98D28—21no—
99C4×D7—42no—
100Dic14—21no—
101Dic14—21no—
102C7×C8—22no—
103C7⋊C8—22no—
104C2×#41—71no—
105C22×Q8C2×D711yescharacteristic
106C22×Q8C2×D711yescharacteristic
107#137—144no—
108C2×Q16—24no—
109C2×Q16—72no—
110C2×Q16C2×D714yes—
111C2×SD16—72no—
112C2×M16—71no—
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—71no—
122#117—142no—
123C4⋊C8—74no—
124#103—74no—
125#103—72no—
126#102—72no—
127C22×C7—22no—
128C22×C7#12112yes—
129C22×C7C22×D411yescharacteristic
130C2×D7—41no—
131C2×D7—42no—
132C4×C7—24no—
133C4×C7—42no—
134C4×C7#12112yes—
135C4×C7C22×D411yescharacteristic
136C4×C7C2×#4111yescharacteristic
137Dic7—24no—
138Dic7—41no—
139Dic7—42no—
140#41—142no—
141#41—142no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—71no—
148C2×Q8C22⋊D714yes—
149C2×Q8C22×D712yes—
150C2×Q8C22×D712yes—
151C2×D4—71no—
152C2×D4—141no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—141no—
156C22×C4C22×D711yescharacteristic
157Q16—28no—
158Q16—144no—
159SD16—144no—
160M16—142no—
161C2×C8—22no—
162C2×C8—72no—
163C2×C8C22×D712yes—
164C4⋊C4—72no—
165C4⋊C4—72no—
166C4⋊C4—144no—
167C4⋊C4—144no—
168C22⋊C4—72no—
169C22⋊C4—144no—
170C22⋊C4—141no—
171C42—72no—
172C42—144no—
173C42—141no—
174C2×C7—22no—
175C2×C7C2×#12111yescharacteristic
176C2×C7#57812yes—
177D7—81no—
178C23—141no—
179C23D4×D711yescharacteristic
180Q8—24no—
181Q8—22no—
182Q8—22no—
183Q8—22no—
184Q8—42no—
185Q8—141no—
186Q8—141no—
187Q8C2×C22⋊D714yes—
188D4—141no—
189D4—141no—
190D4—282no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—74no—
196C2×C4—142no—
197C2×C4—141no—
198C2×C4—282no—
199C2×C4D4×D711yescharacteristic
200C2×C4#115912yes—
201C2×C4C2×C22⋊D712yes—
202C2×C4C23×D711yescharacteristic
203C8—22no—
204C8—142no—
205C7#304411yescharacteristic, complemented, Sylow
206C22—22no—
207C22—282no—
208C22—281no—
209C22#650212yes—
210C22C2×D4×D711yescharacteristic
211C4—24no—
212C4—42no—
213C4—144no—
214C4—282no—
215C4—281no—
216C4#650212yes—
217C4C2×#115911yescharacteristic
218C4C22×C22⋊D711yescharacteristic
219C2—22no—
220C2—561no—
221C2#9045712yes—
222C2C2×#650211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC22×C7×Q16C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenterC22×Q16D7
hypocenterC7#3044
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C22×C7×Q16C2semidirect
3. #90503C2semidirect
5. #89997C2semidirect
6. #89957C2semidirect
7. #89954C2semidirect
9. #89874C2semidirect
69. C22×Q16D7semidirect
205. C7#3044semidirect

Maximal subgroups

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

subgroupindex
C22×C7×Q162
#905072
#905032
C2×#65052
#899972
#899572
#899542
#898852
#898742
#30447