Finite Group Databasebeta

← #11739046

Subgroups — #11739046

Order 896.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#90416C211yesmaximal, characteristic
2D7×C4⋊Q8C211yesmaximal, characteristic, complemented
3C7×#487C211yesmaximal, characteristic, complemented
4#89860C211yesmaximal, characteristic, complemented
5#89853C211yesmaximal, characteristic, complemented
6#89575C214yesmaximal, complemented
7#89567C214yesmaximal, complemented
8#89466C211yesmaximal, characteristic, complemented
9C4×#1131C211yesmaximal, characteristic, complemented
10C2×#1161C2212yes—
11C2×Q8×D7—21no—
12C7×C4⋊Q8C2211yescharacteristic
13C7×C4⋊Q8C2211yescharacteristic
14#6510C2212yes—
15#6507—21no—
16C2×#1131C2212yes—
17#6458—21no—
18#6458—21no—
19#6456—21no—
20#6456—21no—
21D7×C4⋊C4—21no—
22D7×C4⋊C4C2212yes—
23#6451—21no—
24#6451C2212yes—
25#6437C2211yescharacteristic
26C42×D7C2211yescharacteristic
27#6432C2211yescharacteristic
28C7×#103C2214yes—
29C4×C7×C8C2211yescharacteristic
30#6409C2214yes—
31#6395C2214yes—
32#6389C2212yes—
33#6383C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3141—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8C2312yes—
38#1161—28no—
39Q8×D7—44no—
40C2×D28—22no—
41C2×D28—21no—
42C2×D28—21no—
43C2×D28C2312yes—
44C2×C4×D7—22no—
45C2×C4×D7—22no—
46C2×C4×D7D411yescharacteristic
47C2×C4×D7D411yescharacteristic
48C2×C4×D7C2311yescharacteristic
49C2×Dic14—21no—
50C2×Dic14—22no—
51C2×C7×C8C2312yes—
52C7×C4⋊C4—21no—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2312yes—
56C42×C7C2311yescharacteristic
57(C2×D7)⋊C4—22no—
58(C2×D7)⋊C4—22no—
59C4⋊Dic7—21no—
60C4⋊Dic7C2312yes—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63C4×Dic7D411yescharacteristic
64C4×Dic7D411yescharacteristic
65C4×Dic7C2311yescharacteristic
66C2×C7⋊C8C2312yes—
67#1131—24no—
68#532—71no—
69C2×C4⋊Q8—71no—
70#489—74no—
71#487—71no—
72#487D711yescharacteristic, complemented
73#486—71no—
74#486—71no—
75#417—74no—
76C4×M16—71no—
77C22×D7—22no—
78C22×D7C2×D411yescharacteristic
79C7×Q8—24no—
80C7×Q8—42no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2×D411yescharacteristic
85C2×C4×C7C2411yescharacteristic
86C2×Dic7—22no—
87C2×Dic7C2×D411yescharacteristic
88C2×Dic7C2×D411yescharacteristic
89C2×Dic7C2×D411yescharacteristic
90D28—24no—
91D28—44no—
92D28—42no—
93D28—42no—
94C4×D7—22no—
95C4×D7—22no—
96C4×D7—44no—
97C4×D7—44no—
98C4×D7#4114yes—
99Dic14—44no—
100Dic14—42no—
101C7×C8—22no—
102C7⋊C8—22no—
103C2×#41—72no—
104C22×Q8—141no—
105C2×M16—72no—
106C4⋊Q8—72no—
107C4⋊Q8—71no—
108C4⋊Q8—141no—
109C4⋊Q8—141no—
110C4⋊Q8C2×D711yescharacteristic
111C4⋊Q8C2×D711yescharacteristic
112#127—71no—
113#124—72no—
114#121—141no—
115#121—141no—
116C4×D4—141no—
117C4×D4—141no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—141no—
120C2×C42—71no—
121#103—74no—
122#103C2×D714yes—
123#102—74no—
124#102—74no—
125#97—72no—
126C4×C8—71no—
127C4×C8C2×D711yescharacteristic
128C22×C7C22×D411yescharacteristic
129C2×D7—21no—
130C2×D7—21no—
131C2×D7—42no—
132C2×D7—42no—
133C2×D7—42no—
134C2×D7#12412yes—
135C4×C7—42no—
136C4×C7—42no—
137C4×C7AGL(1,ℤ/8)12yes—
138C4×C7#13712yes—
139C4×C7C2×#4112yes—
140Dic7—21no—
141Dic7—21no—
142Dic7—42no—
143Dic7#12412yes—
144#41—148no—
145C2×Q8—21no—
146C2×Q8—142no—
147C2×Q8—141no—
148C2×Q8—284no—
149C2×Q8C22×D712yes—
150C2×D4—72no—
151C2×D4—142no—
152C2×D4—141no—
153C2×D4—141no—
154C22×C4—71no—
155C22×C4—71no—
156C22×C4—71no—
157C22×C4—142no—
158C22×C4—142no—
159M16—144no—
160C2×C8—72no—
161C2×C8C22×D712yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—72no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4—141no—
169C4⋊C4C22×D712yes—
170C22⋊C4—142no—
171C22⋊C4—142no—
172C42—71no—
173C42—71no—
174C42—71no—
175C42C22×D711yescharacteristic
176C2×C7C2×#12411yescharacteristic
177C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C7C2×#13711yescharacteristic
179D7—22no—
180D7—82no—
181C23—71no—
182C23—142no—
183Q8—24no—
184Q8—42no—
185Q8—284no—
186Q8—282no—
187D4—144no—
188D4—284no—
189D4—282no—
190D4—282no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—71no—
194C2×C4—74no—
195C2×C4—71no—
196C2×C4—71no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—284no—
201C2×C4—284no—
202C2×C4D4×D711yescharacteristic
203C2×C4D4×D711yescharacteristic
204C2×C4C23×D711yescharacteristic
205C8—22no—
206C8—142no—
207C7#314111yescharacteristic, complemented, Sylow
208C22—72no—
209C22—141no—
210C22—141no—
211C22—282no—
212C22—282no—
213C22—282no—
214C22C2×D4×D711yescharacteristic
215C4—42no—
216C4—42no—
217C4—72no—
218C4—141no—
219C4—141no—
220C4—282no—
221C4#647712yes—
222C4#648112yes—
223C4D7×#4112yes—
224C2—142no—
225C2—562no—
226C2D7×#12411yescharacteristic
227C2C2×#647711yescharacteristic
228C2C2×#648111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#487D7
hypocenterC7#3141
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
2. D7×C4⋊Q8C2semidirect
3. C7×#487C2semidirect
4. #89860C2semidirect
5. #89853C2semidirect
6. #89575C2semidirect
7. #89567C2semidirect
8. #89466C2semidirect
9. C4×#1131C2semidirect
72. #487D7semidirect
207. C7#3141semidirect

Maximal subgroups

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

subgroupindex
#904162
D7×C4⋊Q82
C7×#4872
#898602
#898532
#895752
#895672
#894662
C4×#11312
#31417