Finite Group Databasebeta

← #11730960

Subgroups — #11730960

Order 896.

568 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 157 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×#6493C211yesmaximal, characteristic
2C2×#6489C211yesmaximal, characteristic, complemented
3C4×C7×C22⋊C4C211yesmaximal, characteristic, complemented
4#89733C214yesmaximal, complemented
5#89711C211yesmaximal, characteristic, complemented
6#89711C211yesmaximal, characteristic, complemented
7#89708C211yesmaximal, characteristic, complemented
8#89707C211yesmaximal, characteristic, complemented
9#89442C214yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C22×D28C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C42×C7C2211yescharacteristic
15C2×C42×C7C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17#6493—24no—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2214yes—
20C2×(C2×D7)⋊C4C2214yes—
21#6489—24no—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2214yes—
26C2×C4×Dic7C2212yes—
27C7×#95C2212yes—
28C7⋊#95C2214yes—
29#2500—71nomaximal, Sylow
30C24×C7C2311yescharacteristic
31C23×D7C2312yes—
32C22×C4×C7—22no—
33C22×C4×C7—22no—
34C22×C4×C7D414yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C2×C22⋊D7—24no—
40C2×C22⋊D7—44no—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C2×D28—22no—
45C2×D28—22no—
46C2×D28—22no—
47C7×C22⋊C4—24no—
48C42×C7—22no—
49C42×C7—22no—
50C22⋊Dic7—24no—
51(C2×D7)⋊C4—24no—
52(C2×D7)⋊C4—28no—
53(C2×D7)⋊C4—44no—
54C4⋊Dic7—22no—
55C4⋊Dic7—22no—
56C4⋊Dic7—22no—
57C4⋊Dic7—44no—
58C4×Dic7—24no—
59C2×#121—71no—
60C2×#117—71no—
61#394—74no—
62#390—71no—
63#390—71no—
64#388—74no—
65#384—71no—
66#382—71no—
67C4×C22⋊C4D711yescharacteristic, complemented
68C23×C7—22no—
69C23×C7—21no—
70C23×C7—21no—
71C23×C7—22no—
72C23×C7C2×D412yes—
73C23×C7C2411yescharacteristic
74C22×D7—22no—
75C22×D7—24no—
76C22×D7—22no—
77C22×D7—22no—
78C22×D7—24no—
79C2×C4×C7—22no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7—22no—
83C2×C4×C7—28no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7C2×D412yes—
88C2×C4×C7C2×D412yes—
89C2×C4×C7#4114yes—
90C2×C4×C7#4112yes—
91C2×C4×C7#4112yes—
92C22⋊D7—48no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—24no—
97C2×Dic7—24no—
98C2×Dic7—24no—
99D28—44no—
100C22×D4—72no—
101C22×D4—71no—
102C23×C4C2×D711yescharacteristic
103#121—144no—
104#117—144no—
105C2×C4⋊C4—71no—
106C2×C4⋊C4—71no—
107C2×C4⋊C4—71no—
108C2×C4⋊C4—74no—
109C2×C22⋊C4—72no—
110C2×C22⋊C4—72no—
111C2×C22⋊C4—74no—
112C2×C22⋊C4—74no—
113C2×C22⋊C4C2×D712yes—
114C2×C42—72no—
115C2×C42C2×D711yescharacteristic
116C2×C42C2×D711yescharacteristic
117#95—74no—
118#95C2×D712yes—
119C22×C7—22no—
120C22×C7—22no—
121C22×C7—22no—
122C22×C7—42no—
123C22×C7—42no—
124C22×C7C22×D411yescharacteristic
125C22×C7C2×#4112yes—
126C22×C7C2×#4111yescharacteristic
127C22×C7C2×#4111yescharacteristic
128C22×C7H(5,𝔽2)12yes—
129C2×D7—42no—
130C2×D7—42no—
131C2×D7—42no—
132C2×D7—44no—
133C2×D7—44no—
134C4×C7—22no—
135C4×C7—24no—
136C4×C7—42no—
137C4×C7#12314yes—
138Dic7—42no—
139Dic7—42no—
140Dic7—42no—
141C24—72no—
142C24C22×D711yescharacteristic
143C2×D4—144no—
144C2×D4—142no—
145C2×D4—142no—
146C2×D4—142no—
147C2×D4—284no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—72no—
151C22×C4—72no—
152C22×C4—72no—
153C22×C4D2814yes—
154C22×C4C22×D712yes—
155C22×C4C22×D711yescharacteristic
156C22×C4C22×D712yes—
157C22×C4C22×D711yescharacteristic
158C4⋊C4—142no—
159C4⋊C4—142no—
160C4⋊C4—142no—
161C4⋊C4—284no—
162C22⋊C4—24no—
163C22⋊C4—144no—
164C22⋊C4—144no—
165C22⋊C4—148no—
166C22⋊C4—284no—
167C42—22no—
168C42—22no—
169C42—144no—
170C2×C7—42no—
171C2×C7C2×#12311yescharacteristic
172C2×C7#53211yescharacteristic
173C2×C7#53511yescharacteristic
174C2×C7#55312yes—
175C2×C7#55512yes—
176D7—82no—
177C23—22no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23—142no—
182C23—144no—
183C23—142no—
184C23—142no—
185C23—144no—
186C23C2×D2812yes—
187C23C23×D711yescharacteristic
188D4—288no—
189D4—284no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—28no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—144no—
202C2×C4—144no—
203C2×C4—144no—
204C2×C4C2×D2812yes—
205C2×C4C2×D2812yes—
206C2×C4#115714yes—
207C2×C4#115912yes—
208C2×C4#115912yes—
209C7#250011yescharacteristic, complemented, Sylow
210C22—22no—
211C22—22no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—282no—
216C22—282no—
217C22—282no—
218C22—284no—
219C22—284no—
220C22C22×D2811yescharacteristic
221C22C2×#115712yes—
222C22C2×#115911yescharacteristic
223C22C2×#115911yescharacteristic
224C22#655312yes—
225C4—22no—
226C4—24no—
227C4—42no—
228C4—282no—
229C4—282no—
230C4—282no—
231C4#644914yes—
232C2—42no—
233C2—562no—
234C2#9016811yescharacteristic
235C2C2×#644911yescharacteristic
236C2#9021611yescharacteristic
237C2#9025412yes—
238C2#9025612yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC4×C7×C22⋊C4C2
prime radicalC2×#6493C2
hypercenterC4×C22⋊C4D7
hypocenterC7#2500
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#6489C2semidirect
3. C4×C7×C22⋊C4C2semidirect
4. #89733C2semidirect
5. #89711C2semidirect
6. #89711C2semidirect
7. #89708C2semidirect
8. #89707C2semidirect
9. #89442C2semidirect
67. C4×C22⋊C4D7semidirect
209. C7#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#64932
C2×#64892
C4×C7×C22⋊C42
#897332
#897112
#897082
#897072
#894422
#25007