Finite Group Databasebeta

← #11737940

Subgroups — #11737940

Order 896.

374 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 115 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#90418C211yesmaximal, characteristic
2#90275C211yesmaximal, characteristic
3C7×#462C211yesmaximal, characteristic, complemented
4#89961C212yesmaximal, complemented
5#89958C212yesmaximal, complemented
6#89853C211yesmaximal, characteristic, complemented
7C4×#1144C211yesmaximal, characteristic, complemented
8#89623C211yesmaximal, characteristic, complemented
9#89610C211yesmaximal, characteristic
10#89587C212yesmaximal, complemented
11#89574C212yesmaximal, complemented
12C2×Q8×D7C2212yes—
13C2×C7×Q16C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×Q8C2211yescharacteristic
16#6509C2212yes—
17#6509C2212yes—
18C2×#1144C2211yescharacteristic
19C2×#1131C2212yes—
20#6461—22no—
21#6459—22no—
22#6456—21no—
23#6455C2212yes—
24#6451—21no—
25#6438—21no—
26C4×D28C2211yescharacteristic
27#6432C2211yescharacteristic
28C4×Dic14—21no—
29C4×Dic14C2211yescharacteristic
30C7×C4⋊C8C2211yescharacteristic
31C7×#103C2212yes—
32#6409C2212yes—
33#6409C2211yescharacteristic
34#6409C2212yes—
35(C2×D7)⋊C8C2212yes—
36#6391C2212yes—
37#6390C2212yes—
38#6384C2211yescharacteristic
39#6381C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3328—71nomaximal, Sylow
42C2×C7×Q8C2312yes—
43C2×C7×Q8C2311yescharacteristic
44Q8×D7—24no—
45Q8×D7—42no—
46C2×D28D411yescharacteristic
47C2×C4×D7—22no—
48C2×C4×D7C2312yes—
49C2×Dic14—21no—
50C2×Dic14—22no—
51C2×Dic14C2311yescharacteristic
52C7×Q16—42no—
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2311yescharacteristic
58C42×C7—21no—
59C42×C7C2311yescharacteristic
60#1144—22no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4D412yes—
65C4⋊Dic7—21no—
66C4⋊Dic7D411yescharacteristic
67C4⋊Dic7C2312yes—
68C4⋊Dic7C2311yescharacteristic
69Dic7⋊C4—21no—
70Dic7⋊C4—22no—
71Dic7⋊C4—22no—
72C4×Dic7—21no—
73C4×Dic7—22no—
74C2×C7⋊C8C2311yescharacteristic
75C2×C7⋊C8C2311yescharacteristic
76#1131—42no—
77#556—71no—
78D4×Q8—71no—
79#487—71no—
80#485—72no—
81#475—71no—
82#470—72no—
83#462D711yescharacteristic, complemented
84#451—72no—
85C4×Q16—71no—
86#436—71no—
87#419—72no—
88C22×D7C2×D412yes—
89C7×Q8—21no—
90C7×Q8—22no—
91C7×Q8—21no—
92C7×Q8—22no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2411yescharacteristic
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7C2×D412yes—
103D28—22no—
104C4×D7—22no—
105C4×D7—22no—
106C4×D7—44no—
107Dic14—21no—
108Dic14—44no—
109Dic14—42no—
110Dic14#4112yes—
111C7×C8—22no—
112C7⋊C8—21no—
113C7⋊C8—21no—
114C22×Q8—72no—
115C2×Q16—71no—
116C2×Q16C2×D712yes—
117C2×M16—72no—
118C4⋊Q8—71no—
119C4⋊Q8—141no—
120C4⋊Q8C2×D711yescharacteristic
121#126—142no—
122#124—141no—
123C22⋊Q8—72no—
124C22⋊Q8—72no—
125C22⋊Q8—142no—
126C4×Q8—71no—
127C4×Q8—141no—
128C4×Q8C2×D711yescharacteristic
129C4×D4—71no—
130C4×D4—141no—
131#117—72no—
132#107—71no—
133#106—72no—
134C4⋊C8C2×D711yescharacteristic
135#103—72no—
136#103—71no—
137#103—72no—
138#103—72no—
139#103—71no—
140#103C2×D712yes—
141C22⋊C8—72no—
142C4×C8—71no—
143C22×C7C22×D411yescharacteristic
144C2×D7—22no—
145C2×D7—22no—
146C2×D7—22no—
147C4×C7—21no—
148C4×C7—42no—
149C4×C7—41no—
150C4×C7—41no—
151C4×C7#13712yes—
152C4×C7C2×#4111yescharacteristic
153C4×C7H(5,𝔽2)11yescharacteristic
154Dic7—22no—
155Dic7—22no—
156Dic7—42no—
157Dic7—41no—
158C2×Q8—71no—
159C2×Q8—142no—
160C2×Q8—141no—
161C2×Q8—144no—
162C2×Q8—282no—
163C2×Q8C22×D712yes—
164C2×Q8C22×D711yescharacteristic
165C2×D4—71no—
166C22×C4—72no—
167C22×C4—142no—
168Q16—42no—
169Q16—142no—
170M16—282no—
171C2×C8—71no—
172C2×C8—71no—
173C2×C8C22×D712yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—71no—
178C4⋊C4—72no—
179C4⋊C4—71no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—141no—
183C4⋊C4—141no—
184C4⋊C4C22×D711yescharacteristic
185C22⋊C4—72no—
186C22⋊C4—142no—
187C22⋊C4—142no—
188C22⋊C4—142no—
189C42—21no—
190C42—142no—
191C42—141no—
192C42C22×D711yescharacteristic
193C2×C7#54611yescharacteristic
194C2×C7C2×#13711yescharacteristic
195C2×C7#57811yescharacteristic
196D7—42no—
197C23—72no—
198Q8—21no—
199Q8—22no—
200Q8—21no—
201Q8—22no—
202Q8—72no—
203Q8—141no—
204Q8—284no—
205Q8—282no—
206D4—142no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—72no—
211C2×C4—141no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—141no—
215C2×C4—142no—
216C2×C4—284no—
217C2×C4D4×D711yescharacteristic
218C2×C4D4×D711yescharacteristic
219C2×C4C23×D711yescharacteristic
220C8—22no—
221C8—141no—
222C8—141no—
223C7#332811yescharacteristic, complemented, Sylow
224C22—142no—
225C22—142no—
226C22—142no—
227C22C2×D4×D711yescharacteristic
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—142no—
233C4—142no—
234C4—282no—
235C4—281no—
236C4#648112yes—
237C4#654811yescharacteristic
238C4D7×#4111yescharacteristic
239C2—282no—
240C2#9030311yescharacteristic
241C2C2×#648111yescharacteristic
242C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radicalC2×D28D4
hypercenter#462D7
hypocenterC7#3328
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. C7×#462C2semidirect
4. #89961C2semidirect
5. #89958C2semidirect
6. #89853C2semidirect
7. C4×#1144C2semidirect
8. #89623C2semidirect
10. #89587C2semidirect
11. #89574C2semidirect
83. #462D7semidirect
223. C7#3328semidirect

Maximal subgroups

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

subgroupindex
#904182
#902752
C7×#4622
#899612
#899582
#898532
C4×#11442
#896232
#896102
#895872
#895742
#33287