Finite Group Databasebeta

← #11731560

Subgroups — #11731560

Order 896.

396 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 133 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×#381C211yesmaximal, characteristic
2#89984C212yesmaximal
3#89721C211yesmaximal, characteristic
4#89720C212yesmaximal
5#89719C211yesmaximal, characteristic
6#89433C212yesmaximal, complemented
7#89432C212yesmaximal, complemented
8#89429C212yesmaximal, complemented
9#89428C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C2×C7×C22⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2211yescharacteristic
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C7×#95C2212yes—
24C7×#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2211yescharacteristic
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2212yes—
31#2841—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C22×C4×C7—41no—
34C22×C4×C7—41no—
35C22×C4×C7Q814yescomplemented
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C2×Dic14—41no—
45C2×Dic14—42no—
46C7×C22⋊C4—41no—
47C7×C22⋊C4—41no—
48C22⋊Dic7—42no—
49C22⋊Dic7—42no—
50C4⋊Dic7—42no—
51C4⋊Dic7—41no—
52C4⋊Dic7—42no—
53C4⋊Dic7—42no—
54Dic7⋊C4—42no—
55Dic7⋊C4—42no—
56Dic7⋊C4—42no—
57#400—72no—
58#399—71no—
59#398—72no—
60#398—72no—
61#396—72no—
62#395—72no—
63#393—71no—
64#381D711yescharacteristic
65#380—72no—
66C23×C7—22no—
67C23×C7—21no—
68C23×C7—21no—
69C23×C7—22no—
70C23×C7C2×Q812yes—
71C23×C7C2411yescharacteristic
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—21no—
78C2×C4×C7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—44no—
82C2×C4×C7C2×Q814yes—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95Dic14—44no—
96C22×Q8—71no—
97C23×C4C2×D711yescharacteristic
98C2×C4⋊C4—72no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—72no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4—71no—
105C2×C22⋊C4—72no—
106C2×C22⋊C4—72no—
107C2×C22⋊C4C2×D711yescharacteristic
108C2×C22⋊C4C2×D711yescharacteristic
109#95—72no—
110#95—72no—
111#95—71no—
112#95—72no—
113#95—72no—
114#95—71no—
115#95C2×D712yes—
116#95C2×D712yes—
117C22×C7—22no—
118C22×C7—24no—
119C22×C7—41no—
120C22×C7—42no—
121C22×C7—41no—
122C22×C7C22×Q811yescharacteristic
123C22×C7H(5,𝔽2)11yescharacteristic
124C22×C7H(5,𝔽2)11yescharacteristic
125C22×C7H(5,𝔽2)12yes—
126C22×C72−1+412yes—
127C4×C7—24no—
128C4×C7—41no—
129C4×C7—42no—
130C4×C7—41no—
131Dic7—42no—
132Dic7—42no—
133Dic7—42no—
134Dic7—42no—
135C24C22×D711yescharacteristic
136C2×Q8—281no—
137C2×Q8—282no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—72no—
141C22×C4—72no—
142C22×C4—72no—
143C22×C4—72no—
144C22×C4Dic1414yescomplemented
145C22×C4C22×D711yescharacteristic
146C22×C4C22×D712yes—
147C22×C4C22×D711yescharacteristic
148C22×C4C22×D712yes—
149C4⋊C4—282no—
150C4⋊C4—282no—
151C4⋊C4—282no—
152C4⋊C4—282no—
153C4⋊C4—282no—
154C4⋊C4—281no—
155C4⋊C4—282no—
156C22⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—282no—
159C22⋊C4—282no—
160C2×C7—42no—
161C2×C7#54311yescharacteristic
162C2×C7#54412yes—
163C2×C7#56011yescharacteristic
164C2×C7#56212yes—
165C2×C7#56311yescharacteristic
166C23—21no—
167C23—21no—
168C23—22no—
169C23—22no—
170C23C2×Dic1412yes—
171C23C23×D711yescharacteristic
172Q8—284no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—142no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4C2×Dic1414yes—
196C7#284111yescharacteristic, complemented, Sylow
197C22—22no—
198C22—24no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22C22×Dic1411yescharacteristic
203C22#654811yescharacteristic
204C22#654811yescharacteristic
205C22#655312yes—
206C22#655412yes—
207C4—24no—
208C4—41no—
209C4—41no—
210C4—42no—
211C4—282no—
212C4—282no—
213C4—282no—
214C4—282no—
215C2—42no—
216C2#9017411yescharacteristic
217C2#9021012yes—
218C2#9034211yescharacteristic
219C2#9035711yescharacteristic
220C2#9035812yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
prime radicalC24×C7C23
hypercenter#381D7
hypocenterC7#2841
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
6. #89433C2semidirect
7. #89432C2semidirect
8. #89429C2semidirect
9. #89428C2semidirect
35. C22×C4×C7Q8semidirect
144. C22×C4Dic14semidirect
196. C7#2841semidirect

Maximal subgroups

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

subgroupindex
C7×#3812
#899842
#897212
#897202
#897192
#894332
#894322
#894292
#894282
#28417