Finite Group Databasebeta

← #11739820

Subgroups — #11739820

Order 896.

604 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 165 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
2C2×#6487C212yesmaximal, complemented
3C2×C4×Dic14C211yesmaximal, characteristic, complemented
4#89994C212yesmaximal
5#89984C212yesmaximal, complemented
6#89752C212yesmaximal, complemented
7#89716C214yesmaximal, complemented
8#89708C211yesmaximal, characteristic, complemented
9C22×C7×D4C2211yescharacteristic
10C23×C4×C7C2212yes—
11C22×Dic14C2211yescharacteristic
12C4×C7×D4—24no—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2212yes—
15C2×C42×C7C2211yescharacteristic
16C2×C22⋊Dic7C2214yes—
17C2×C22⋊Dic7C2214yes—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2212yes—
20#6487—28no—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2214yes—
23C2×C4×Dic7C2212yes—
24C4×Dic14—24no—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2214yes—
28#2536—71nomaximal, Sylow
29C24×C7C2312yes—
30C2×C7×D4—24no—
31C2×C7×D4—22no—
32C22×C4×C7—24no—
33C22×C4×C7—24no—
34C22×C4×C7—24no—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2314yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C2×Dic14—22no—
43C2×Dic14—22no—
44C2×Dic14D414yescomplemented
45C7×C4⋊C4—22no—
46C7×C22⋊C4—24no—
47C42×C7D414yes—
48C22⋊Dic7—28no—
49C22⋊Dic7—28no—
50C4⋊Dic7—22no—
51C4⋊Dic7—24no—
52Dic7⋊C4—24no—
53Dic7⋊C4—28no—
54C4×Dic7—24no—
55C2×C22⋊Q8—72no—
56C2×C4×Q8—71no—
57C2×C4×D4D711yescharacteristic
58#398—72no—
59#394—72no—
60#388—74no—
61#382—71no—
62#380—72no—
63C23×C7—22no—
64C23×C7—22no—
65C23×C7—22no—
66C23×C7—22no—
67C23×C7—22no—
68C23×C7—22no—
69C23×C7#4114yes—
70C23×C7C2411yescharacteristic
71C7×D4—28no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—48no—
78C2×C4×C7C2×D412yes—
79C2×C4×C7C2×D412yes—
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×D412yes—
82C2×C4×C7#4112yes—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—24no—
87C2×Dic7—24no—
88C2×Dic7—24no—
89C2×Dic7—24no—
90C2×Dic7C2×D414yes—
91Dic14—28no—
92C22×Q8—71no—
93C22×D4C2×D711yescharacteristic
94C23×C4C2×D712yes—
95C22⋊Q8—148no—
96C4×Q8—144no—
97C4×D4—24no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4—71no—
100C2×C4⋊C4—74no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4C2×D711yescharacteristic
103C2×C22⋊C4—74no—
104C2×C22⋊C4—74no—
105C2×C22⋊C4C2×D712yes—
106C2×C42—72no—
107C2×C42C2×D711yescharacteristic
108#95—74no—
109#95—72no—
110#95—72no—
111C22×C7—24no—
112C22×C7—24no—
113C22×C7—24no—
114C22×C7—42no—
115C22×C7—42no—
116C22×C7—42no—
117C22×C7—42no—
118C22×C7C22×D411yescharacteristic
119C22×C7C22×D411yescharacteristic
120C22×C7C2×#4112yes—
121C22×C7C2×#4111yescharacteristic
122C22×C72−1+412yes—
123C4×C7—22no—
124C4×C7—22no—
125C4×C7—42no—
126C4×C7#12114yes—
127Dic7—24no—
128Dic7—42no—
129Dic7—44no—
130C24C22×D712yes—
131C2×Q8—74no—
132C2×Q8—142no—
133C2×Q8—142no—
134C2×D4—24no—
135C2×D4—22no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—72no—
140C22×C4—74no—
141C22×C4—72no—
142C22×C4C22×D712yes—
143C22×C4C22×D711yescharacteristic
144C22×C4C22×D711yescharacteristic
145C22×C4C22×D711yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—148no—
148C4⋊C4—144no—
149C4⋊C4—144no—
150C4⋊C4—142no—
151C22⋊C4—24no—
152C22⋊C4—148no—
153C22⋊C4—148no—
154C42—144no—
155C42C22⋊D714yes—
156C2×C7—44no—
157C2×C7C2×#12111yescharacteristic
158C2×C7#52511yescharacteristic
159C2×C7#53611yescharacteristic
160C2×C7#54812yes—
161C2×C7#55212yes—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23#115714yes—
169C23C23×D711yescharacteristic
170Q8—148no—
171D4—28no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—48no—
178C2×C4—74no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4—144no—
182C2×C4—144no—
183C2×C4—144no—
184C2×C4—144no—
185C2×C4—142no—
186C2×C4D4×D712yes—
187C2×C4#115912yes—
188C2×C4C2×C22⋊D712yes—
189C2×C4C2×C22⋊D712yes—
190C2×C4C2×C22⋊D712yes—
191C7#253611yescharacteristic, complemented, Sylow
192C22—24no—
193C22—24no—
194C22—24no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22C2×#115712yes—
200C22C2×D4×D711yescharacteristic
201C22C2×#115911yescharacteristic
202C22#655412yes—
203C22C22×C22⋊D711yescharacteristic
204C4—22no—
205C4—22no—
206C4—42no—
207C4—144no—
208C4—282no—
209C4—284no—
210C4#650114yes—
211C2—44no—
212C2#9023212yes—
213C2#9024312yes—
214C2#9048211yescharacteristic
215C2C2×#650111yescharacteristic
216C2#9051611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C4×C7×D4C2
prime radicalC22×C7×D4C22
hypercenterC2×C4×D4D7
hypocenterC7#2536
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#6487C2semidirect
3. C2×C4×Dic14C2semidirect
5. #89984C2semidirect
6. #89752C2semidirect
7. #89716C2semidirect
8. #89708C2semidirect
44. C2×Dic14D4semidirect
191. C7#2536semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D42
C2×#64872
C2×C4×Dic142
#899942
#899842
#897522
#897162
#897082
#25367