Finite Group Databasebeta

← #11739188

Subgroups — #11739188

Order 896.

388 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 109 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#90415C211yesmaximal, characteristic, complemented
2#90389C211yesmaximal, characteristic, complemented
3C7×#491C211yesmaximal, characteristic, complemented
4#89857C211yesmaximal, characteristic, complemented
5#89840C211yesmaximal, characteristic, complemented
6#89591C212yesmaximal, complemented
7#89590C212yesmaximal, complemented
8#89583C212yesmaximal, complemented
9#89578C212yesmaximal, complemented
10#89484C211yesmaximal, characteristic, complemented
11#89481C211yesmaximal, characteristic, complemented
12C2×#1161C2211yescharacteristic
13C2×Q8×D7C2211yescharacteristic
14C7×C4⋊Q8C2211yescharacteristic
15C7×#125C2211yescharacteristic
16#6510C2211yescharacteristic
17#6507C2211yescharacteristic
18C2×#1143C2212yes—
19C2×#1143C2212yes—
20C2×D56C2212yes—
21C2×#1132C2212yes—
22#6459—22no—
23#6458—22no—
24#6458C2212yes—
25#6457—22no—
26#6457—22no—
27#6456C2212yes—
28#6438C2211yescharacteristic
29#6437C2211yescharacteristic
30#6435C2211yescharacteristic
31C7×#103C2212yes—
32C7×#103C2212yes—
33C7×#97C2211yescharacteristic
34(C2×D7)⋊C8C2212yes—
35Dic7⋊C8C2212yes—
36#6384C2212yes—
37#6383C2212yes—
38C4×C7⋊C8C2211yescharacteristic
39#3221—71nomaximal, Sylow
40C2×C7×Q8C2311yescharacteristic
41C2×C7×Q8C2311yescharacteristic
42#1161—42no—
43Q8×D7—42no—
44C2×D28—21no—
45C2×D28—22no—
46C2×D28—22no—
47C2×D28C2312yes—
48C2×D28C2311yescharacteristic
49C2×C4×D7—21no—
50C2×C4×D7—21no—
51C2×C4×D7—22no—
52C2×C4×D7C2311yescharacteristic
53C2×Dic14—21no—
54C2×Dic14C2311yescharacteristic
55C2×C7×C8C2312yes—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—22no—
58C7×C4⋊C4C2312yes—
59C42×C7C2311yescharacteristic
60#1143—42no—
61#1143—42no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4D412yes—
67Dic7⋊C4—22no—
68Dic7⋊C4D412yes—
69C4×Dic7C2311yescharacteristic
70C2×C7⋊C8C2312yes—
71D56—42no—
72#1132—42no—
73#540—71no—
74#536—71no—
75#496—71no—
76#492—71no—
77#491D711yescharacteristic, complemented
78#488—71no—
79#463—72no—
80#461—72no—
81#449—72no—
82#448—72no—
83#433—71no—
84C22×D7—21no—
85C22×D7—22no—
86C22×D7C2×D411yescharacteristic
87C7×Q8—22no—
88C7×Q8—22no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7C2×D412yes—
93C2×C4×C7C2411yescharacteristic
94C2×Dic7—21no—
95C2×Dic7C2×D412yes—
96C2×Dic7C2×D411yescharacteristic
97D28—22no—
98D28—22no—
99D28—22no—
100D28—42no—
101D28—82no—
102D28—82no—
103C4×D7—22no—
104C4×D7—42no—
105C4×D7—42no—
106C4×D7—82no—
107Dic14—22no—
108Dic14—42no—
109C7×C8—22no—
110C7⋊C8—22no—
111C2×#41—71no—
112C22×Q8—71no—
113C2×SD16—72no—
114C2×SD16—72no—
115C2×SD16—72no—
116C2×D8—72no—
117C4⋊Q8—71no—
118C4⋊Q8C2×D711yescharacteristic
119#127—71no—
120#125C2×D711yescharacteristic
121#124—71no—
122#124—71no—
123#123—142no—
124#123—142no—
125C22⋊Q8—142no—
126#121—72no—
127#121—142no—
128C4×D4—72no—
129#117—71no—
130C4⋊C8—72no—
131#103—72no—
132#103C2×D712yes—
133#103C2×D712yes—
134#102—72no—
135C22⋊C8—72no—
136#97C2×D711yescharacteristic
137C4×C8—71no—
138C22×C7C22×D411yescharacteristic
139C2×D7—21no—
140C2×D7—22no—
141C2×D7—41no—
142C2×D7—42no—
143C2×D7—42no—
144C2×D7—42no—
145C2×D7—42no—
146C4×C7—22no—
147C4×C7—41no—
148C4×C7—41no—
149C4×C7—42no—
150C4×C7H(5,𝔽2)12yes—
151Dic7—21no—
152Dic7—22no—
153Dic7—41no—
154#41—282no—
155C2×Q8—71no—
156C2×Q8—141no—
157C2×Q8—282no—
158C2×Q8C22×D711yescharacteristic
159C2×Q8C22×D711yescharacteristic
160C2×D4—71no—
161C2×D4—72no—
162C2×D4—141no—
163C2×D4—142no—
164C2×D4—142no—
165C22×C4—71no—
166C22×C4—141no—
167C22×C4—142no—
168C22×C4—141no—
169SD16—282no—
170SD16—282no—
171SD16—282no—
172D8—282no—
173C2×C8—72no—
174C2×C8C22×D712yes—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—72no—
178C4⋊C4—142no—
179C4⋊C4C22×D712yes—
180C22⋊C4—72no—
181C22⋊C4—142no—
182C22⋊C4—142no—
183C22⋊C4—142no—
184C22⋊C4—142no—
185C42—71no—
186C42C22×D711yescharacteristic
187C2×C7#53511yescharacteristic
188C2×C7#57712yes—
189D7—41no—
190D7—81no—
191D7—82no—
192C23—71no—
193C23—141no—
194C23—142no—
195Q8—22no—
196Q8—22no—
197Q8—142no—
198Q8—282no—
199D4—142no—
200D4—142no—
201D4—142no—
202D4—282no—
203D4—562no—
204D4—562no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—72no—
209C2×C4—71no—
210C2×C4—142no—
211C2×C4—141no—
212C2×C4—282no—
213C2×C4—282no—
214C2×C4—562no—
215C2×C4D4×D712yes—
216C2×C4C23×D711yescharacteristic
217C8—22no—
218C8—142no—
219C7#322111yescharacteristic, complemented, Sylow
220C22—142no—
221C22—141no—
222C22—282no—
223C22—282no—
224C22—282no—
225C22—281no—
226C22—282no—
227C22C2×D4×D711yescharacteristic
228C4—22no—
229C4—42no—
230C4—41no—
231C4—41no—
232C4—142no—
233C4—141no—
234C4—281no—
235C4#655312yes—
236C2—281no—
237C2—562no—
238C2—561no—
239C2#9036511yescharacteristic
240C2#9046912yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#491C2
socleC22×C7C22×D4
hypercenter#491D7
hypocenterC7#3221
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #90415C2semidirect
2. #90389C2semidirect
3. C7×#491C2semidirect
4. #89857C2semidirect
5. #89840C2semidirect
6. #89591C2semidirect
7. #89590C2semidirect
8. #89583C2semidirect
9. #89578C2semidirect
10. #89484C2semidirect
11. #89481C2semidirect
77. #491D7semidirect
219. C7#3221semidirect

Maximal subgroups

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

subgroupindex
#904152
#903892
C7×#4912
#898572
#898402
#895912
#895902
#895832
#895782
#894842
#894812
#32217