Finite Group Databasebeta

← #11741131

Subgroups — #11741131

Order 896.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C7×#557C211yesmaximal, characteristic, complemented
2#90275C211yesmaximal, characteristic
3#89864C211yesmaximal, characteristic, complemented
4#89862C212yesmaximal, complemented
5#89858C212yesmaximal
6#89855C211yesmaximal, characteristic
7#89803C211yesmaximal, characteristic, complemented
8#89802C212yesmaximal, complemented
9#89800C212yesmaximal
10C4×#1143C211yesmaximal, characteristic
11#89796C211yesmaximal, characteristic, complemented
12C7×C4⋊Q8C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C7×#125—21no—
15C7×#125—22no—
16C4×C7×Q8—21no—
17C4×C7×Q8C2211yescharacteristic
18C4×C7×Q8C2212yes—
19C4×C7×Q8C2211yescharacteristic
20#6509—21no—
21C2×#1144C2212yes—
22C2×#1144C2212yes—
23C2×#1144C2212yes—
24C2×#1143C2211yescharacteristic
25C2×#1143C2212yes—
26#6461—22no—
27#6455—21no—
28#6438C2212yes—
29C4×D28C2211yescharacteristic
30#6432C2211yescharacteristic
31C4×Dic14C2212yes—
32#6409C2211yescharacteristic
33#6409C2212yes—
34#6384C2212yes—
35#6383C2211yescharacteristic
36#6382C2211yescharacteristic
37C4⋊(C7⋊C8)C2212yes—
38C4⋊(C7⋊C8)C2211yescharacteristic
39#6377C2212yes—
40C4×C7⋊C8C2211yescharacteristic
41#3398—71nomaximal, Sylow
42C2×C7×Q8D411yescharacteristic
43C2×C7×Q8C2311yescharacteristic
44C2×C7×Q8C2312yes—
45C2×D28C2311yescharacteristic
46C2×C4×D7—21no—
47C2×Dic14C2312yes—
48C7×C4⋊C4—22no—
49C7×C4⋊C4—21no—
50C7×C4⋊C4—22no—
51C7×C4⋊C4—21no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—21no—
54C7×C4⋊C4D411yescharacteristic
55C7×C4⋊C4D412yes—
56C7×C4⋊C4C2311yescharacteristic
57C42×C7—22no—
58C42×C7—21no—
59C42×C7C2312yes—
60C42×C7C2311yescharacteristic
61#1144—24no—
62#1144—42no—
63#1144—42no—
64#1143—22no—
65#1143—42no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—22no—
68(C2×D7)⋊C4—22no—
69C4⋊Dic7—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7C2311yescharacteristic
72Dic7⋊C4—22no—
73C4×Dic7—22no—
74C2×C7⋊C8C2312yes—
75C2×C7⋊C8C2311yescharacteristic
76C2×C7⋊C8C2311yescharacteristic
77#557D711yescharacteristic, complemented
78#556—71no—
79#497—72no—
80#494—71no—
81#464—72no—
82#462—72no—
83#462—71no—
84#461—71no—
85#446—71no—
86#441—72no—
87C4×SD16—71no—
88C22×D7—21no—
89C7×Q8—22no—
90C7×Q8—21no—
91C7×Q8—22no—
92C7×Q8—22no—
93C7×Q8—21no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7C2×D412yes—
98C2×C4×C7C2×D412yes—
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2411yescharacteristic
102C2×Dic7—21no—
103C2×Dic7—22no—
104D28—21no—
105D28—21no—
106C4×D7—42no—
107Dic14—22no—
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⋊Q8C2×D711yescharacteristic
119C4⋊Q8C2×D712yes—
120#126—142no—
121#125—21no—
122#125—22no—
123#124—72no—
124C22⋊Q8—141no—
125C4×Q8—21no—
126C4×Q8—72no—
127C4×Q8C2×D711yescharacteristic
128C4×Q8C2×D712yes—
129C4×Q8C2×D711yescharacteristic
130C4×D4—71no—
131#117—141no—
132#106—71no—
133C4⋊C8—71no—
134C4⋊C8—72no—
135#103—71no—
136#103—72no—
137#103—72no—
138#102—71no—
139#97—72no—
140C4×C8—71no—
141C22×C7C22×D411yescharacteristic
142C2×D7—41no—
143C2×D7—41no—
144C2×D7—41no—
145C4×C7—22no—
146C4×C7—22no—
147C4×C7—21no—
148C4×C7—41no—
149C4×C7—42no—
150C4×C7—41no—
151C4×C7#13712yes—
152C4×C7C2×#4111yescharacteristic
153C4×C7H(5,𝔽2)11yescharacteristic
154Dic7—42no—
155Dic7—41no—
156C2×Q8—72no—
157C2×Q8C22⋊D711yescharacteristic
158C2×Q8C22×D711yescharacteristic
159C2×Q8C22×D712yes—
160C2×D4—71no—
161C22×C4—141no—
162Q16—144no—
163Q16—282no—
164Q16—282no—
165SD16—142no—
166SD16—282no—
167C2×C8—71no—
168C2×C8—71no—
169C2×C8—72no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—71no—
177C4⋊C4—142no—
178C4⋊C4—141no—
179C4⋊C4—141no—
180C4⋊C4C22⋊D711yescharacteristic
181C4⋊C4C22⋊D712yes—
182C4⋊C4C22×D711yescharacteristic
183C22⋊C4—142no—
184C22⋊C4—141no—
185C22⋊C4—142no—
186C42—22no—
187C42—21no—
188C42—142no—
189C42C22×D711yescharacteristic
190C42C22×D712yes—
191C2×C7#55011yescharacteristic
192C2×C7C2×#13711yescharacteristic
193C2×C7#57811yescharacteristic
194D7—81no—
195C23—141no—
196Q8—22no—
197Q8—21no—
198Q8—22no—
199Q8—22no—
200Q8—21no—
201Q8—142no—
202Q8—142no—
203D4—141no—
204D4—141no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—141no—
209C2×C4—142no—
210C2×C4—282no—
211C2×C4C2×C22⋊D712yes—
212C2×C4C2×C22⋊D712yes—
213C2×C4C2×C22⋊D711yescharacteristic
214C2×C4C2×C22⋊D711yescharacteristic
215C2×C4C23×D711yescharacteristic
216C8—72no—
217C8—142no—
218C8—141no—
219C7#339811yescharacteristic, complemented, Sylow
220C22—281no—
221C22—281no—
222C22—281no—
223C22C22×C22⋊D711yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—41no—
228C4—42no—
229C4—41no—
230C4—281no—
231C4—282no—
232C4#650512yes—
233C4D7×#4111yescharacteristic
234C4#655311yescharacteristic
235C2—561no—
236C2C2×#650511yescharacteristic
237C2#9052511yescharacteristic
238C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#557C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#557D7
hypocenterC7#3398
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C7×#557C2semidirect
3. #89864C2semidirect
4. #89862C2semidirect
7. #89803C2semidirect
8. #89802C2semidirect
11. #89796C2semidirect
77. #557D7semidirect
219. C7#3398semidirect

Maximal subgroups

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

subgroupindex
C7×#5572
#902752
#898642
#898622
#898582
#898552
#898032
#898022
#898002
C4×#11432
#897962
#33987