Finite Group Databasebeta

← #11739732

Subgroups — #11739732

Order 896.

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

normal subgroups only

This group has 612 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×C7×#117C211yesmaximal, characteristic, complemented
2C2×#6493C211yesmaximal, characteristic
3C2×#6487C211yesmaximal, characteristic, complemented
4C2×#6438C211yesmaximal, characteristic, complemented
5#89766C214yesmaximal, complemented
6#89733C214yesmaximal, complemented
7#89711C211yesmaximal, characteristic, complemented
8#89707C211yesmaximal, characteristic, complemented
9#89701C211yesmaximal, characteristic, complemented
10C23×C4×C7C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C22×D28C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C7×#117—24no—
15C2×C7×C4⋊C4C2212yes—
16C2×C7×C22⋊C4C2212yes—
17C2×C42×C7C2211yescharacteristic
18C2×C42×C7C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20#6493—24no—
21C2×(C2×D7)⋊C4C2214yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2214yes—
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2214yes—
27#6487—24no—
28C2×Dic7⋊C4C2212yes—
29#6438—24no—
30C7⋊#95C2214yes—
31#2620—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C23×D7C2312yes—
34C22×C4×C7—22no—
35C22×C4×C7—22no—
36C22×C4×C7D414yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C2×C22⋊D7—24no—
42C2×C22⋊D7—44no—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C2×D28—22no—
47C2×D28—22no—
48C2×D28—22no—
49C2×Dic14—22no—
50C2×Dic14—22no—
51C2×Dic14—22no—
52C7×C4⋊C4—24no—
53C7×C22⋊C4—24no—
54C42×C7—22no—
55C42×C7D414yescomplemented
56C22⋊Dic7—24no—
57(C2×D7)⋊C4—28no—
58(C2×D7)⋊C4—24no—
59(C2×D7)⋊C4—44no—
60C4⋊Dic7—22no—
61C4⋊Dic7—22no—
62C4⋊Dic7—44no—
63Dic7⋊C4—24no—
64C2×#124—71no—
65C2×C22⋊Q8—71no—
66C2×#121—71no—
67C2×#117D711yescharacteristic, complemented
68#396—74no—
69#394—74no—
70#391—71no—
71#390—71no—
72#384—71no—
73C23×C7—21no—
74C23×C7—21no—
75C23×C7—22no—
76C23×C7—22no—
77C23×C7C2×D412yes—
78C23×C7C2411yescharacteristic
79C22×D7—22no—
80C22×D7—22no—
81C22×D7—22no—
82C22×D7—24no—
83C22×D7—24no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—28no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7C2×D412yes—
93C2×C4×C7C2×D412yes—
94C2×C4×C7C2×D414yes—
95C2×C4×C7C2×D412yes—
96C2×C4×C7#4112yes—
97C22⋊D7—48no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—24no—
104C2×Dic7—24no—
105D28—44no—
106Dic14—44no—
107C22×Q8—71no—
108C22×D4—72no—
109C22×D4—71no—
110C23×C4C2×D711yescharacteristic
111#124—144no—
112C22⋊Q8—144no—
113#121—144no—
114#117—24no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—74no—
119C2×C4⋊C4C2×D712yes—
120C2×C22⋊C4—72no—
121C2×C22⋊C4—72no—
122C2×C22⋊C4—74no—
123C2×C22⋊C4—74no—
124C2×C22⋊C4C2×D712yes—
125C2×C42C2×D711yescharacteristic
126C2×C42C2×D711yescharacteristic
127#95—74no—
128C22×C7—22no—
129C22×C7—22no—
130C22×C7—22no—
131C22×C7—42no—
132C22×C7—42no—
133C22×C7C22×D411yescharacteristic
134C22×C7C22×D411yescharacteristic
135C22×C7C2×#4111yescharacteristic
136C22×C7H(5,𝔽2)12yes—
137C22×C72−1+412yes—
138C2×D7—42no—
139C2×D7—42no—
140C2×D7—42no—
141C2×D7—44no—
142C2×D7—44no—
143C4×C7—22no—
144C4×C7—24no—
145C4×C7—42no—
146C4×C7#12114yes—
147Dic7—42no—
148Dic7—42no—
149Dic7—42no—
150C24—72no—
151C24C22×D711yescharacteristic
152C2×Q8—142no—
153C2×Q8—142no—
154C2×Q8—142no—
155C2×D4—144no—
156C2×D4—142no—
157C2×D4—142no—
158C2×D4—142no—
159C2×D4—284no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—72no—
163C22×C4—72no—
164C22×C4—72no—
165C22×C4D2814yes—
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C22×C4C22×D712yes—
169C22×C4C22×D711yescharacteristic
170C4⋊C4—24no—
171C4⋊C4—144no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—284no—
175C22⋊C4—24no—
176C22⋊C4—144no—
177C22⋊C4—144no—
178C22⋊C4—148no—
179C22⋊C4—284no—
180C42—22no—
181C42C22⋊D714yescomplemented
182C2×C7—42no—
183C2×C7C2×#12111yescharacteristic
184C2×C7#53511yescharacteristic
185C2×C7#53611yescharacteristic
186C2×C7#53712yes—
187C2×C7#54212yes—
188D7—82no—
189C23—21no—
190C23—22no—
191C23—22no—
192C23—21no—
193C23—142no—
194C23—142no—
195C23—144no—
196C23—144no—
197C23—142no—
198C23C2×D2812yes—
199C23C23×D711yescharacteristic
200Q8—284no—
201D4—288no—
202D4—284no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—28no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—144no—
216C2×C4—144no—
217C2×C4—142no—
218C2×C4C2×D2812yes—
219C2×C4C2×D2812yes—
220C2×C4#115712yes—
221C2×C4C2×C22⋊D714yes—
222C2×C4C2×C22⋊D712yes—
223C7#262011yescharacteristic, complemented, Sylow
224C22—22no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22—42no—
229C22—282no—
230C22—282no—
231C22—284no—
232C22—282no—
233C22—284no—
234C22C22×D2811yescharacteristic
235C22C2×#115711yescharacteristic
236C22#655312yes—
237C22#655412yes—
238C22C22×C22⋊D711yescharacteristic
239C4—22no—
240C4—24no—
241C4—42no—
242C4—282no—
243C4—282no—
244C4—282no—
245C4#649314yes—
246C2—42no—
247C2—562no—
248C2#9021611yescharacteristic
249C2#9021711yescharacteristic
250C2#9022512yes—
251C2C2×#649311yescharacteristic
252C2#9052412yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#117C2
prime radicalC2×#6493C2
hypercenterC2×#117D7
hypocenterC7#2620
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C7×#117C2semidirect
3. C2×#6487C2semidirect
4. C2×#6438C2semidirect
5. #89766C2semidirect
6. #89733C2semidirect
7. #89711C2semidirect
8. #89707C2semidirect
9. #89701C2semidirect
55. C42×C7D4semidirect
67. C2×#117D7semidirect
181. C42C22⋊D7semidirect
223. C7#2620semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1172
C2×#64932
C2×#64872
C2×#64382
#897662
#897332
#897112
#897072
#897012
#26207