Finite Group Databasebeta

← #11740436

Subgroups — #11740436

Order 896.

592 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 137 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×#124C211yesmaximal, characteristic, complemented
2C2×#6510C211yesmaximal, characteristic, complemented
3#90002C211yesmaximal, characteristic, complemented
4#89995C212yesmaximal, complemented
5#89733C212yesmaximal, complemented
6#89732C212yesmaximal, complemented
7#89729C212yesmaximal, complemented
8#89716C212yesmaximal, complemented
9#89712C212yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C7×D4C2211yescharacteristic
12C22×C22⋊D7C2212yes—
13C22×C22⋊D7C2212yes—
14C22×D28C2211yescharacteristic
15C7×#124—24no—
16C2×C7×C22⋊C4C2212yes—
17C2×C7×C22⋊C4C2212yes—
18C2×C42×C7C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21#6510—24no—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×C4⋊Dic7C2211yescharacteristic
27C2×Dic7⋊C4C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33#2733—71nomaximal, Sylow
34C24×C7C2312yes—
35C23×D7C2312yes—
36C2×C7×Q8—41no—
37C2×C7×Q8—41no—
38C2×C7×Q8D414yescomplemented
39C2×C7×D4—22no—
40C2×C7×D4—41no—
41C2×C7×D4—41no—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—24no—
47C2×C22⋊D7—42no—
48C2×C22⋊D7—42no—
49C2×C22⋊D7—42no—
50C2×C22⋊D7—42no—
51C2×C22⋊D7—42no—
52C22×Dic7C2312yes—
53C22×Dic7C2312yes—
54C22×Dic7C2312yes—
55C2×D28—22no—
56C2×D28—41no—
57C2×D28—41no—
58C7×C22⋊C4—24no—
59C7×C22⋊C4—24no—
60C42×C7—22no—
61C22⋊Dic7—24no—
62C22⋊Dic7—42no—
63(C2×D7)⋊C4—24no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—42no—
67C4⋊Dic7—41no—
68Dic7⋊C4—42no—
69C4×Dic7—22no—
70C4×Dic7—41no—
71C2×#124—71no—
72C2×#124D711yescharacteristic, complemented
73#394—72no—
74#392—72no—
75#392—72no—
76#391—71no—
77#390—72no—
78#388—72no—
79#388—72no—
80C23×C7—22no—
81C23×C7—22no—
82C23×C7—22no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7C2411yescharacteristic
88C22×D7—22no—
89C22×D7—22no—
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C7×Q8—44no—
96C7×D4—44no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7C2×D412yes—
107C2×C4×C7C2×D412yes—
108C2×C4×C7C2×D412yes—
109C22⋊D7—48no—
110C22⋊D7—48no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7#4114yes—
120D28—44no—
121C22×Q8C2×D711yescharacteristic
122C22×D4—72no—
123C22×D4—72no—
124C22×D4—71no—
125C22×D4C2×D711yescharacteristic
126#124—24no—
127#124—144no—
128C2×C4⋊C4—72no—
129C2×C4⋊C4—71no—
130C2×C22⋊C4—72no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4—72no—
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4C2×D712yes—
137C2×C22⋊C4C2×D712yes—
138C2×C42—71no—
139C2×C42—71no—
140C2×C42C2×D711yescharacteristic
141#95—72no—
142#95—72no—
143#95—72no—
144C22×C7—42no—
145C22×C7—42no—
146C22×C7—42no—
147C22×C7—42no—
148C22×C7—42no—
149C22×C7—42no—
150C22×C7—42no—
151C22×C7C22×D411yescharacteristic
152C22×C7C2×#4112yes—
153C22×C7H(5,𝔽2)11yescharacteristic
154C22×C7H(5,𝔽2)11yescharacteristic
155C22×C7H(5,𝔽2)12yes—
156C2×D7—42no—
157C2×D7—42no—
158C2×D7—42no—
159C2×D7—42no—
160C2×D7—42no—
161C2×D7—42no—
162C2×D7—42no—
163C4×C7—22no—
164C4×C7—22no—
165C4×C7—22no—
166C4×C7—42no—
167Dic7—24no—
168Dic7—42no—
169Dic7—42no—
170C24—72no—
171C24C22×D712yes—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8C22⋊D714yescomplemented
175C2×D4—22no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—144no—
179C2×D4—142no—
180C2×D4—281no—
181C2×D4—281no—
182C2×D4—282no—
183C2×D4—282no—
184C2×D4—282no—
185C2×D4—282no—
186C2×D4—282no—
187C22×C4—72no—
188C22×C4—72no—
189C22×C4—72no—
190C22×C4C22×D712yes—
191C22×C4C22×D711yescharacteristic
192C22×C4C22×D711yescharacteristic
193C22×C4C22×D711yescharacteristic
194C4⋊C4—282no—
195C4⋊C4—281no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—144no—
199C22⋊C4—144no—
200C22⋊C4—144no—
201C22⋊C4—144no—
202C22⋊C4—282no—
203C22⋊C4—282no—
204C42—22no—
205C42—142no—
206C42—281no—
207C2×C7—82no—
208C2×C7#53411yescharacteristic
209C2×C7#53812yes—
210C2×C7#55012yes—
211C2×C7#55511yescharacteristic
212C2×C7H(3,𝔽4)11yescharacteristic
213D7—82no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—142no—
222C23—142no—
223C23—142no—
224C23—142no—
225C23—142no—
226C23—142no—
227C23—142no—
228C23C23×D711yescharacteristic
229Q8—44no—
230D4—44no—
231D4—284no—
232D4—288no—
233D4—288no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—74no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4C2×C22⋊D712yes—
253C2×C4C2×C22⋊D712yes—
254C2×C4C2×C22⋊D712yes—
255C7#273311yescharacteristic, complemented, Sylow
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—282no—
264C22—282no—
265C22—282no—
266C22—282no—
267C22—282no—
268C22—282no—
269C22—282no—
270C22#654811yescharacteristic
271C22#654811yescharacteristic
272C22D7×#4112yes—
273C22#655312yes—
274C22C22×C22⋊D711yescharacteristic
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—144no—
280C4—282no—
281C4—282no—
282C2—82no—
283C2—562no—
284C2#9036111yescharacteristic
285C2#9036912yes—
286C2#9037411yescharacteristic
287C2#9049511yescharacteristic
288C2#9052512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
hypercenterC2×#124D7
hypocenterC7#2733
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C7×#124C2semidirect
2. C2×#6510C2semidirect
3. #90002C2semidirect
4. #89995C2semidirect
5. #89733C2semidirect
6. #89732C2semidirect
7. #89729C2semidirect
8. #89716C2semidirect
9. #89712C2semidirect
38. C2×C7×Q8D4semidirect
72. C2×#124D7semidirect
174. C2×Q8C22⋊D7semidirect
255. C7#2733semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×#65102
#900022
#899952
#897332
#897322
#897292
#897162
#897122
#27337