Finite Group Databasebeta

← #11739819

Subgroups — #11739819

Order 896.

584 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C2×C4×C7×D4C211yesmaximal, characteristic
2C22×C4⋊Dic7C211yesmaximal, characteristic, complemented
3#89992C212yesmaximal
4#89984C212yesmaximal, complemented
5#89984C211yesmaximal, characteristic, complemented
6#89758C212yesmaximal, complemented
7#89721C212yesmaximal, complemented
8#89718C212yesmaximal, complemented
9#89700C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C23×C4×C7C2211yescharacteristic
12C23×C4×C7C2211yescharacteristic
13C23×Dic7C2212yes—
14C4×C7×D4—24no—
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C42×C7C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2214yes—
21C2×C4⋊Dic7—24no—
22C2×C4⋊Dic7C2212yes—
23C2×C4⋊Dic7C2212yes—
24C2×C4⋊Dic7C2212yes—
25C2×Dic7⋊C4C2214yes—
26C7⋊#95C2214yes—
27C7⋊#95C2214yes—
28C7⋊#95C2212yes—
29#2487—71nomaximal, Sylow
30C24×C7C2311yescharacteristic
31C24×C7C2311yescharacteristic
32C2×C7×D4—22no—
33C2×C7×D4—22no—
34C2×C7×D4Q814yes—
35C22×C4×C7—24no—
36C22×C4×C7—22no—
37C22×C4×C7—24no—
38C22×C4×C7D414yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×Dic7—24no—
45C22×Dic7—44no—
46C22×Dic7C2314yes—
47C22×Dic7C2314yes—
48C7×C4⋊C4—22no—
49C7×C22⋊C4—22no—
50C7×C22⋊C4—22no—
51C42×C7—22no—
52C22⋊Dic7—24no—
53C22⋊Dic7—44no—
54C4⋊Dic7—24no—
55C4⋊Dic7—24no—
56C4⋊Dic7—24no—
57Dic7⋊C4—44no—
58C2×C4×D4D711yescharacteristic
59C22×C4⋊C4—71no—
60#400—72no—
61#399—72no—
62#389—72no—
63#386—72no—
64#385—72no—
65#380—72no—
66#380—71no—
67C23×C7—22no—
68C23×C7—22no—
69C23×C7—21no—
70C23×C7—22no—
71C23×C7—21no—
72C23×C7—22no—
73C23×C7C2×D412yes—
74C23×C7C2×Q812yes—
75C23×C7C2×Q812yes—
76C23×C7#4112yes—
77C23×C7C2411yescharacteristic
78C7×D4—28no—
79C2×C4×C7—21no—
80C2×C4×C7—22no—
81C2×C4×C7—21no—
82C2×C4×C7—28no—
83C2×C4×C7—21no—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—44no—
88C2×C4×C7C2×D412yes—
89C2×C4×C7C2×D412yes—
90C2×C4×C7C2×Q812yes—
91C2×C4×C7#4114yes—
92C2×Dic7—24no—
93C2×Dic7—24no—
94C2×Dic7—24no—
95C2×Dic7—24no—
96C2×Dic7—24no—
97C2×Dic7—24no—
98C2×Dic7—48no—
99C22×D4C2×D711yescharacteristic
100C23×C4—72no—
101C23×C4C2×D711yescharacteristic
102C23×C4C2×D711yescharacteristic
103C4×D4—24no—
104C2×C4⋊C4—72no—
105C2×C4⋊C4—72no—
106C2×C4⋊C4—72no—
107C2×C4⋊C4—74no—
108C2×C4⋊C4—144no—
109C2×C4⋊C4C2×D711yescharacteristic
110C2×C22⋊C4—72no—
111C2×C22⋊C4—74no—
112C2×C22⋊C4C2×D711yescharacteristic
113C2×C22⋊C4C2×D711yescharacteristic
114C2×C42C2×D711yescharacteristic
115#95—74no—
116#95—72no—
117#95—74no—
118C22×C7—22no—
119C22×C7—22no—
120C22×C7—24no—
121C22×C7—24no—
122C22×C7—22no—
123C22×C7—24no—
124C22×C7—41no—
125C22×C7—41no—
126C22×C7—42no—
127C22×C7C22⋊Q814yes—
128C22×C7C22×D411yescharacteristic
129C22×C7C22×Q811yescharacteristic
130C22×C7C2×#4111yescharacteristic
131C22×C7C2×#4112yes—
132C22×C7H(5,𝔽2)12yes—
133C4×C7—22no—
134C4×C7—22no—
135C4×C7—41no—
136C4×C7—41no—
137C4×C7#12314yes—
138Dic7—44no—
139Dic7—44no—
140C24C22×D711yescharacteristic
141C24C22×D711yescharacteristic
142C2×D4—22no—
143C2×D4—22no—
144C2×D4Dic1414yes—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—74no—
149C22×C4—74no—
150C22×C4—144no—
151C22×C4—284no—
152C22×C4C22⋊D714yes—
153C22×C4C22×D711yescharacteristic
154C22×C4C22×D711yescharacteristic
155C22×C4C22×D711yescharacteristic
156C22×C4C22×D711yescharacteristic
157C22×C4C22×D711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—144no—
160C4⋊C4—144no—
161C4⋊C4—144no—
162C4⋊C4—284no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—144no—
166C22⋊C4—284no—
167C42—22no—
168C2×C7—24no—
169C2×C7—42no—
170C2×C7C2×C22⋊Q811yescharacteristic
171C2×C7C2×#12311yescharacteristic
172C2×C7#53511yescharacteristic
173C2×C7#55312yes—
174C2×C7#55412yes—
175C23—22no—
176C23—22no—
177C23—21no—
178C23—22no—
179C23—22no—
180C23—21no—
181C23C2×Dic1412yes—
182C23C2×Dic1412yes—
183C23#115712yes—
184C23C2×C22⋊D712yes—
185C23C23×D711yescharacteristic
186D4—28no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—28no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—44no—
196C2×C4—144no—
197C2×C4—144no—
198C2×C4—144no—
199C2×C4—144no—
200C2×C4—144no—
201C2×C4—144no—
202C2×C4—288no—
203C2×C4C2×Dic1412yes—
204C2×C4#115914yes—
205C2×C4C2×C22⋊D712yes—
206C2×C4C2×C22⋊D712yes—
207C7#248711yescharacteristic, complemented, Sylow
208C22—22no—
209C22—24no—
210C22—24no—
211C22—22no—
212C22—22no—
213C22—24no—
214C22—41no—
215C22—42no—
216C22—41no—
217C22#648714yes—
218C22C22×Dic1411yescharacteristic
219C22C2×#115711yescharacteristic
220C22C2×#115912yes—
221C22#655312yes—
222C22C22×C22⋊D711yescharacteristic
223C4—22no—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—284no—
228C4—284no—
229C4#649814yes—
230C2—24no—
231C2—42no—
232C2#9023412yes—
233C2#9025612yes—
234C2C2×#648711yescharacteristic
235C2C2×#649811yescharacteristic
236C2#9052111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C4×C7×D4C2
prime radicalC22×C7×D4C22
hypercenterC2×C4×D4D7
hypocenterC7#2487
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic7C2semidirect
4. #89984C2semidirect
5. #89984C2semidirect
6. #89758C2semidirect
7. #89721C2semidirect
8. #89718C2semidirect
9. #89700C2semidirect
207. C7#2487semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D42
C22×C4⋊Dic72
#899922
#899842
#897582
#897212
#897182
#897002
#24877