Finite Group Databasebeta

← #11737396

Subgroups — #11737396

Order 896.

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

normal subgroups only

This group has 350 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#90275C211yesmaximal, characteristic
2#90261C211yesmaximal, characteristic, complemented
3C7×#446C211yesmaximal, characteristic, complemented
4#89621C211yesmaximal, characteristic, complemented
5#89621C212yesmaximal, complemented
6#89617C212yesmaximal
7#89613C211yesmaximal, characteristic
8#89488C212yesmaximal, complemented
9#89487C212yesmaximal
10#89471C211yesmaximal, characteristic, complemented
11C4×#1132C211yesmaximal, characteristic
12C4×C7×Q8C2211yescharacteristic
13#6509—21no—
14Q8×Dic7—21no—
15C2×Dic28C2212yes—
16C2×Dic28C2212yes—
17C2×Dic28C2212yes—
18C2×#1132C2211yescharacteristic
19C2×#1132C2212yes—
20#6461—22no—
21#6455—21no—
22#6453—21no—
23#6453—22no—
24#6438C2212yes—
25C4×D28C2211yescharacteristic
26#6432C2211yescharacteristic
27#6432C2211yescharacteristic
28#6432C2212yes—
29C4×Dic14C2212yes—
30C4×Dic14C2211yescharacteristic
31C4×Dic14C2212yes—
32C7×C4⋊C8C2211yescharacteristic
33C7×C4⋊C8C2212yes—
34C7×#97C2212yes—
35C4×C7×C8C2211yescharacteristic
36#6395C2211yescharacteristic
37#6391C2211yescharacteristic
38#6390C2212yes—
39#6390C2212yes—
40#6390C2211yescharacteristic
41#3398—71nomaximal, Sylow
42C2×C7×Q8D411yescharacteristic
43C2×D28C2311yescharacteristic
44C2×C4×D7—21no—
45C2×Dic14C2311yescharacteristic
46C2×Dic14C2312yes—
47C2×Dic14C2312yes—
48C2×C7×C8C2311yescharacteristic
49C2×C7×C8C2312yes—
50C2×C7×C8C2311yescharacteristic
51C7×C4⋊C4D411yescharacteristic
52C7×C4⋊C4D412yes—
53C42×C7C2311yescharacteristic
54C42×C7C2312yes—
55(C2×D7)⋊C4—21no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58C4⋊Dic7—22no—
59C4⋊Dic7—21no—
60C4⋊Dic7—22no—
61C4⋊Dic7—21no—
62C4⋊Dic7—21no—
63C4⋊Dic7—21no—
64C4⋊Dic7C2311yescharacteristic
65C4⋊Dic7C2311yescharacteristic
66Dic7⋊C4—22no—
67Dic7⋊C4—21no—
68Dic7⋊C4—22no—
69C4×Dic7—22no—
70C4×Dic7—21no—
71C4×Dic7—22no—
72Dic28—24no—
73Dic28—42no—
74Dic28—42no—
75#1132—22no—
76#1132—42no—
77#557—71no—
78#556—71no—
79#497—72no—
80#494—71no—
81#464—72no—
82#462—72no—
83#462—71no—
84#461—71no—
85#446D711yescharacteristic, complemented
86#441—72no—
87C4×SD16—71no—
88C22×D7—21no—
89C7×Q8—22no—
90C2×C4×C7C2×D412yes—
91C2×C4×C7C2×D412yes—
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2411yescharacteristic
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100D28—21no—
101D28—21no—
102C4×D7—42no—
103Dic14—21no—
104Dic14—22no—
105Dic14—22no—
106Dic14—22no—
107Dic14—21no—
108Dic14—22no—
109C7×C8—22no—
110C7×C8—21no—
111C7×C8#4112yes—
112C2×Q16—72no—
113C2×Q16—72no—
114C2×Q16—72no—
115C2×SD16—71no—
116C2×SD16—72no—
117C4⋊Q8—71no—
118C4⋊Q8—71no—
119C4⋊Q8—72no—
120#126—142no—
121#125—141no—
122#125—142no—
123#124—72no—
124C22⋊Q8—141no—
125C4×Q8—72no—
126C4×Q8—71no—
127C4×Q8—72no—
128C4×Q8—141no—
129C4×Q8C2×D711yescharacteristic
130C4×D4—71no—
131#117—141no—
132#106—71no—
133C4⋊C8C2×D711yescharacteristic
134C4⋊C8C2×D712yes—
135#103—71no—
136#103—72no—
137#103—72no—
138#102—71no—
139#97C2×D712yes—
140C4×C8C2×D711yescharacteristic
141C22×C7C22×D411yescharacteristic
142C2×D7—41no—
143C2×D7—41no—
144C2×D7—41no—
145C4×C7—22no—
146C4×C7—22no—
147C4×C7—21no—
148C4×C7#13712yes—
149C4×C7C2×#4111yescharacteristic
150C4×C7H(5,𝔽2)11yescharacteristic
151Dic7—41no—
152Dic7—42no—
153Dic7—42no—
154Dic7—41no—
155Dic7—41no—
156C2×Q8—72no—
157C2×Q8—71no—
158C2×Q8—72no—
159C2×Q8D2811yescharacteristic
160C2×D4—71no—
161C22×C4—141no—
162Q16—144no—
163Q16—282no—
164Q16—282no—
165SD16—142no—
166SD16—282no—
167C2×C8C22×D711yescharacteristic
168C2×C8C22×D712yes—
169C2×C8C22×D711yescharacteristic
170C4⋊C4—71no—
171C4⋊C4—71no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—141no—
175C4⋊C4—141no—
176C4⋊C4—142no—
177C4⋊C4—141no—
178C4⋊C4—142no—
179C4⋊C4—141no—
180C4⋊C4—141no—
181C4⋊C4D2811yescharacteristic
182C4⋊C4D2812yes—
183C22⋊C4—142no—
184C22⋊C4—141no—
185C22⋊C4—142no—
186C42—142no—
187C42—142no—
188C42—141no—
189C42C22×D711yescharacteristic
190C42C22×D712yes—
191C2×C7#55011yescharacteristic
192C2×C7C2×#13711yescharacteristic
193C2×C7#57811yescharacteristic
194D7—81no—
195C23—141no—
196Q8—22no—
197Q8—141no—
198Q8—142no—
199Q8—142no—
200Q8—142no—
201Q8—142no—
202Q8—141no—
203D4—141no—
204D4—141no—
205C2×C4—141no—
206C2×C4—141no—
207C2×C4—142no—
208C2×C4—141no—
209C2×C4—142no—
210C2×C4—282no—
211C2×C4C2×D2812yes—
212C2×C4C2×D2812yes—
213C2×C4C2×D2811yescharacteristic
214C2×C4C2×D2811yescharacteristic
215C2×C4C23×D711yescharacteristic
216C8—22no—
217C8—21no—
218C8#116112yes—
219C7#339811yescharacteristic, complemented, Sylow
220C22—281no—
221C22—281no—
222C22—281no—
223C22C22×D2811yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—281no—
228C4—281no—
229C4—282no—
230C4—281no—
231C4—282no—
232C4#647212yes—
233C4C2×#116111yescharacteristic
234C4#655311yescharacteristic
235C2—561no—
236C2#9026811yescharacteristic
237C2C2×#647211yescharacteristic
238C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#446C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#446D7
hypocenterC7#3398
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
2. #90261C2semidirect
3. C7×#446C2semidirect
4. #89621C2semidirect
5. #89621C2semidirect
8. #89488C2semidirect
10. #89471C2semidirect
85. #446D7semidirect
219. C7#3398semidirect

Maximal subgroups

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

subgroupindex
#902752
#902612
C7×#4462
#896212
#896172
#896132
#894882
#894872
#894712
C4×#11322
#33987