Finite Group Databasebeta

← #11739719

Subgroups — #11739719

Order 896.

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

normal subgroups only

This group has 560 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×C7×#117C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D7C211yesmaximal, characteristic
3C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic, complemented
5#89764C211yesmaximal, characteristic, complemented
6#89754C211yesmaximal, characteristic, complemented
7#89752C211yesmaximal, characteristic, complemented
8#89731C212yesmaximal
9#89724C212yesmaximal, complemented
10C4×(C2×D7)⋊C4C212yesmaximal, complemented
11C4×Dic7⋊C4C212yesmaximal, complemented
12C23×C4×C7C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C7×#117—24no—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C42×C7C2212yes—
20C2×C22⋊Dic7C2211yescharacteristic
21C2×C22⋊Dic7C2212yes—
22C4×C22⋊D7—24no—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×Dic7⋊C4C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30C2×C4×Dic7C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2211yescharacteristic
37#2395—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7—24no—
41C22×C4×C7D414yes—
42C22×C4×C7C2312yes—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—22no—
47C2×C22⋊D7—22no—
48C2×C22⋊D7—22no—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2312yes—
54C2×C4×D7—22no—
55C2×C4×D7—24no—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—22no—
58C7×C22⋊C4—24no—
59C42×C7—24no—
60C22⋊Dic7—22no—
61C22⋊Dic7—24no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—24no—
65Dic7⋊C4—22no—
66Dic7⋊C4—24no—
67Dic7⋊C4—24no—
68C4×Dic7—22no—
69C4×Dic7—24no—
70C2×C4×D4—71no—
71C2×#117D711yescharacteristic, complemented
72#400—71no—
73#399—71no—
74#398—71no—
75#397—71no—
76#397—72no—
77#394—72no—
78C4×C4⋊C4—72no—
79C4×C22⋊C4—71no—
80C4×C22⋊C4—72no—
81C23×C7—22no—
82C23×C7—21no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7C2×D412yes—
86C23×C7C2411yescharacteristic
87C22×D7—21no—
88C22×D7—22no—
89C22×D7—22no—
90C22×D7—21no—
91C22×D7#4112yes—
92C2×C4×C7—22no—
93C2×C4×C7—28no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7C2×D412yes—
101C2×C4×C7C2×D412yes—
102C2×C4×C7#4114yes—
103C22⋊D7—28no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—21no—
108C2×Dic7—21no—
109C2×Dic7—21no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7#4112yes—
116C2×Dic7#4114yes—
117C4×D7—44no—
118C22×D4—71no—
119C23×C4—71no—
120C23×C4C2×D711yescharacteristic
121C4×D4—144no—
122#117—24no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4C2×D711yescharacteristic
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C22⋊C4—71no—
129C2×C22⋊C4—72no—
130C2×C22⋊C4—71no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4C2×D712yes—
134C2×C42—71no—
135C2×C42—72no—
136C2×C42C2×D712yes—
137#95—72no—
138#95—71no—
139#95—72no—
140#95—71no—
141#95—72no—
142#95—72no—
143C22×C7—22no—
144C22×C7—22no—
145C22×C7—22no—
146C22×C7—42no—
147C22×C7—42no—
148C22×C7C22×D411yescharacteristic
149C22×C7C2×#4112yes—
150C22×C7C2×#4111yescharacteristic
151C22×C7C2×#4111yescharacteristic
152C22×C7C2×#4112yes—
153C2×D7—22no—
154C2×D7—24no—
155C2×D7—41no—
156C2×D7—42no—
157C2×D7—41no—
158C4×C7—22no—
159C4×C7—42no—
160C4×C7—42no—
161C4×C7#12314yes—
162Dic7—22no—
163Dic7—24no—
164Dic7—41no—
165Dic7—42no—
166Dic7—41no—
167C24—71no—
168C24C22×D711yescharacteristic
169C2×D4—142no—
170C2×D4—142no—
171C2×D4—142no—
172C22×C4—24no—
173C22×C4—72no—
174C22×C4—71no—
175C22×C4—72no—
176C22×C4—71no—
177C22×C4—71no—
178C22×C4—144no—
179C22×C4—142no—
180C22×C4C22⋊D714yes—
181C22×C4C22×D712yes—
182C22×C4C22×D712yes—
183C22×C4C22×D711yescharacteristic
184C22×C4C22×D711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—144no—
188C4⋊C4—144no—
189C4⋊C4—142no—
190C22⋊C4—24no—
191C22⋊C4—144no—
192C22⋊C4—144no—
193C22⋊C4—144no—
194C22⋊C4—142no—
195C22⋊C4—142no—
196C42—24no—
197C42—144no—
198C42—142no—
199C2×C7—42no—
200C2×C7C2×#12311yescharacteristic
201C2×C7#52511yescharacteristic
202C2×C7#52912yes—
203C2×C7#52912yes—
204C2×C7#53211yescharacteristic
205D7—42no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—72no—
211C23—142no—
212C23—142no—
213C23—141no—
214C23—141no—
215C23C2×C22⋊D712yes—
216C23C23×D711yescharacteristic
217D4—148no—
218C2×C4—22no—
219C2×C4—28no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—74no—
227C2×C4—72no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4—141no—
231C2×C4—141no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—141no—
238C2×C4—141no—
239C2×C4—284no—
240C2×C4#115714yes—
241C2×C4C2×C22⋊D712yes—
242C2×C4C2×C22⋊D712yes—
243C7#239511yescharacteristic, complemented, Sylow
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—144no—
250C22—142no—
251C22—282no—
252C22—281no—
253C22—281no—
254C22C2×#115712yes—
255C22D7×#4111yescharacteristic
256C22D7×#4111yescharacteristic
257C22D7×#4112yes—
258C22C22×C22⋊D711yescharacteristic
259C4—22no—
260C4—42no—
261C4—42no—
262C4—144no—
263C4—142no—
264C4—282no—
265C4—281no—
266C4—281no—
267C4#649214yes—
268C2—42no—
269C2—282no—
270C2#9021912yes—
271C2#9021912yes—
272C2C2×#649211yescharacteristic
273C2#9051911yescharacteristic
274C2#9052311yescharacteristic
subgroupG/H
centerC22×C4C22×D7
commutator, socle, cosocleC23×C7C24
FrattiniC23C23×D7
FittingC2×C7×#117C2
prime radicalC22×C22⋊D7C22
hypercenterC2×#117D7
hypocenterC7#2395
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C7×#117C2semidirect
3. C4×C22⋊Dic7C2semidirect
4. #89765C2semidirect
5. #89764C2semidirect
6. #89754C2semidirect
7. #89752C2semidirect
9. #89724C2semidirect
10. C4×(C2×D7)⋊C4C2semidirect
11. C4×Dic7⋊C4C2semidirect
71. C2×#117D7semidirect
243. C7#2395semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1172
C2×C4×C22⋊D72
C4×C22⋊Dic72
#897652
#897642
#897542
#897522
#897312
#897242
C4×(C2×D7)⋊C42
C4×Dic7⋊C42
#23957