Finite Group Databasebeta

← #11739061

Subgroups — #11739061

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#90423C211yesmaximal, characteristic
2#90418C211yesmaximal, characteristic
3C7×#487C211yesmaximal, characteristic, complemented
4#89866C211yesmaximal, characteristic, complemented
5#89862C211yesmaximal, characteristic, complemented
6#89588C212yesmaximal, complemented
7#89586C212yesmaximal, complemented
8#89580C212yesmaximal, complemented
9#89574C212yesmaximal, complemented
10C4×Dic28C211yesmaximal, characteristic, complemented
11#89460C211yesmaximal, characteristic
12C2×Q8×D7C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C7×C4⋊Q8C2211yescharacteristic
15#6510—21no—
16#6509C2212yes—
17Q8×Dic7—21no—
18C2×#1144C2212yes—
19C2×Dic28C2211yescharacteristic
20C2×#1131C2212yes—
21#6461—22no—
22#6460C2212yes—
23#6459—22no—
24#6456—21no—
25#6455C2212yes—
26#6451—21no—
27C4×D28C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C4×Dic14C2211yescharacteristic
30C7×#103C2212yes—
31C7×#103C2212yes—
32C4×C7×C8C2211yescharacteristic
33#6409C2212yes—
34(C2×D7)⋊C8C2212yes—
35#6392C2211yescharacteristic
36#6390C2211yescharacteristic
37#6390C2211yescharacteristic
38#6384C2212yes—
39#6382C2212yes—
40C4⋊(C7⋊C8)C2211yescharacteristic
41#3328—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2312yes—
44Q8×D7—24no—
45Q8×D7—42no—
46C2×D28D411yescharacteristic
47C2×C4×D7—22no—
48C2×C4×D7C2312yes—
49C2×Dic14—22no—
50C2×Dic14C2311yescharacteristic
51C2×Dic14C2311yescharacteristic
52C2×C7×C8C2311yescharacteristic
53C2×C7×C8C2311yescharacteristic
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2312yes—
58C42×C7C2311yescharacteristic
59#1144—42no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4D412yes—
64C4⋊Dic7—22no—
65C4⋊Dic7D411yescharacteristic
66C4⋊Dic7C2311yescharacteristic
67C4⋊Dic7C2311yescharacteristic
68Dic7⋊C4—21no—
69Dic7⋊C4—21no—
70Dic7⋊C4—22no—
71C4×Dic7—21no—
72C4×Dic7—21no—
73C4×Dic7—22no—
74C2×C7⋊C8C2312yes—
75Dic28—22no—
76#1131—42no—
77#556—71no—
78D4×Q8—71no—
79#487D711yescharacteristic, complemented
80#485—72no—
81#475—71no—
82#470—72no—
83#462—71no—
84#451—72no—
85C4×Q16—71no—
86#436—71no—
87#419—72no—
88C22×D7C2×D412yes—
89C7×Q8—22no—
90C7×Q8—22no—
91C7×Q8—42no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2411yescharacteristic
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7C2×D412yes—
102D28—22no—
103C4×D7—22no—
104C4×D7—22no—
105C4×D7—44no—
106Dic14—21no—
107Dic14—21no—
108Dic14—21no—
109Dic14—44no—
110Dic14#4112yes—
111C7×C8—21no—
112C7×C8—21no—
113C7⋊C8—22no—
114C22×Q8—72no—
115C2×Q16—72no—
116C2×Q16—71no—
117C2×M16—72no—
118C4⋊Q8—141no—
119C4⋊Q8C2×D711yescharacteristic
120C4⋊Q8C2×D711yescharacteristic
121#126—142no—
122#124—141no—
123C22⋊Q8—72no—
124C22⋊Q8—72no—
125C22⋊Q8—142no—
126C4×Q8—71no—
127C4×Q8—71no—
128C4×Q8—141no—
129C4×D4—71no—
130C4×D4—141no—
131#117—72no—
132#107—71no—
133#106—72no—
134C4⋊C8—71no—
135#103—72no—
136#103—71no—
137#103—71no—
138#103—72no—
139#103C2×D712yes—
140#103C2×D712yes—
141C22⋊C8—72no—
142C4×C8C2×D711yescharacteristic
143C22×C7C22×D411yescharacteristic
144C2×D7—22no—
145C2×D7—22no—
146C2×D7—22no—
147C4×C7—21no—
148C4×C7—42no—
149C4×C7—42no—
150C4×C7#13712yes—
151C4×C7C2×#4111yescharacteristic
152C4×C7H(5,𝔽2)11yescharacteristic
153Dic7—22no—
154Dic7—22no—
155Dic7—41no—
156Dic7—41no—
157Dic7—41no—
158C2×Q8—21no—
159C2×Q8—71no—
160C2×Q8—71no—
161C2×Q8—142no—
162C2×Q8—144no—
163C2×Q8—282no—
164C2×Q8C22×D712yes—
165C2×D4—71no—
166C22×C4—72no—
167C22×C4—142no—
168Q16—142no—
169Q16—282no—
170M16—282no—
171C2×C8—72no—
172C2×C8C22×D711yescharacteristic
173C2×C8C22×D711yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—71no—
178C4⋊C4—71no—
179C4⋊C4—71no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—141no—
183C4⋊C4—141no—
184C4⋊C4C22×D712yes—
185C22⋊C4—72no—
186C22⋊C4—142no—
187C22⋊C4—142no—
188C22⋊C4—142no—
189C42—142no—
190C42—141no—
191C42—141no—
192C42C22×D711yescharacteristic
193C2×C7#54611yescharacteristic
194C2×C7C2×#13711yescharacteristic
195C2×C7#57811yescharacteristic
196D7—42no—
197C23—72no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201Q8—72no—
202Q8—141no—
203Q8—141no—
204Q8—141no—
205Q8—284no—
206D4—142no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—72no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—141no—
214C2×C4—141no—
215C2×C4—141no—
216C2×C4—284no—
217C2×C4D4×D711yescharacteristic
218C2×C4D4×D711yescharacteristic
219C2×C4C23×D711yescharacteristic
220C8—21no—
221C8—21no—
222C8—142no—
223C7#332811yescharacteristic, complemented, Sylow
224C22—142no—
225C22—142no—
226C22—142no—
227C22C2×D4×D711yescharacteristic
228C4—21no—
229C4—42no—
230C4—42no—
231C4—142no—
232C4—142no—
233C4—281no—
234C4—281no—
235C4—281no—
236C4#647812yes—
237C4D7×#4111yescharacteristic
238C4#655311yescharacteristic
239C2—282no—
240C2#9036711yescharacteristic
241C2C2×#647811yescharacteristic
242C2#9045711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
prime radicalC2×D28D4
hypercenter#487D7
hypocenterC7#3328
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. C7×#487C2semidirect
4. #89866C2semidirect
5. #89862C2semidirect
6. #89588C2semidirect
7. #89586C2semidirect
8. #89580C2semidirect
9. #89574C2semidirect
10. C4×Dic28C2semidirect
79. #487D7semidirect
223. C7#3328semidirect

Maximal subgroups

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

subgroupindex
#904232
#904182
C7×#4872
#898662
#898622
#895882
#895862
#895802
#895742
C4×Dic282
#894602
#33287