Finite Group Databasebeta

← #11740301

Subgroups — #11740301

Order 896.

624 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#123C211yesmaximal, characteristic
2C2×#6498C211yesmaximal, characteristic
3C2×D4×Dic7C211yesmaximal, characteristic
4C22×Dic7⋊C4C211yesmaximal, characteristic, complemented
5#89983C211yesmaximal, characteristic, complemented
6#89756C212yesmaximal, complemented
7#89744C212yesmaximal, complemented
8#89724C212yesmaximal, complemented
9#89715C211yesmaximal, characteristic, complemented
10#89714C212yesmaximal, complemented
11#89714C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C23×Dic7C2211yescharacteristic
15C23×Dic7C2211yescharacteristic
16C23×Dic7C2211yescharacteristic
17C7×#123—24no—
18C2×C7×C4⋊C4C2212yes—
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2212yes—
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25#6498—24no—
26D4×Dic7—24no—
27C2×C4⋊Dic7C2211yescharacteristic
28C2×Dic7⋊C4—24no—
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2212yes—
31C2×Dic7⋊C4C2212yes—
32C2×Dic7⋊C4C2212yes—
33C2×C4×Dic7C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2212yes—
38C7⋊#95C2212yes—
39#2486—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C24×C7C2311yescharacteristic
42C2×C7×D4—22no—
43C2×C7×D4—22no—
44C2×C7×D4D414yes—
45C22×C4×C7—22no—
46C22×C4×C7—22no—
47C22×C4×C7—22no—
48C22×C4×C7C2312yes—
49C22×C4×C7C2312yes—
50C22×C4×C7C2311yescharacteristic
51C22×Dic7—24no—
52C22×Dic7—22no—
53C22×Dic7—22no—
54C22×Dic7—22no—
55C22×Dic7—22no—
56C22×Dic7—24no—
57C22×Dic7D414yes—
58C22×Dic7C2311yescharacteristic
59C22×Dic7C2311yescharacteristic
60C22×Dic7C2312yes—
61C22×Dic7C2312yes—
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C7×C4⋊C4—24no—
65C7×C22⋊C4—22no—
66C7×C22⋊C4—24no—
67C22⋊Dic7—22no—
68C22⋊Dic7—22no—
69C22⋊Dic7—24no—
70C22⋊Dic7—22no—
71C4⋊Dic7—22no—
72Dic7⋊C4—24no—
73Dic7⋊C4—24no—
74Dic7⋊C4—24no—
75Dic7⋊C4—24no—
76C4×Dic7—22no—
77C2×#123—71no—
78C2×#123D711yescharacteristic
79C2×C4×D4—71no—
80C22×C4⋊C4—71no—
81#398—72no—
82#397—72no—
83#389—72no—
84#381—71no—
85#381—71no—
86#380—72no—
87#380—71no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—21no—
91C23×C7—22no—
92C23×C7—21no—
93C23×C7—22no—
94C23×C7C2×D412yes—
95C23×C7C2×D412yes—
96C23×C7C2×D412yes—
97C23×C7#4112yes—
98C23×C7C2411yescharacteristic
99C7×D4—28no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—44no—
109C2×C4×C7C2×D412yes—
110C2×Dic7—22no—
111C2×Dic7—28no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—21no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—44no—
123C2×Dic7—44no—
124C2×Dic7C2×D412yes—
125C2×Dic7C2×D412yes—
126C2×Dic7#4114yes—
127C22×D4C2×D711yescharacteristic
128C23×C4—71no—
129C23×C4—71no—
130C23×C4—71no—
131C23×C4C2×D711yescharacteristic
132#123—24no—
133#123—144no—
134C4×D4—144no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—72no—
137C2×C4⋊C4—71no—
138C2×C4⋊C4—72no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4—144no—
141C2×C4⋊C4C2×D712yes—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—71no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4C2×D711yescharacteristic
147C2×C22⋊C4C2×D712yes—
148C2×C42—71no—
149#95—72no—
150#95—72no—
151#95—72no—
152#95—72no—
153#95—72no—
154C22×C7—22no—
155C22×C7—22no—
156C22×C7—24no—
157C22×C7—24no—
158C22×C7—22no—
159C22×C7—24no—
160C22×C7—41no—
161C22×C7—41no—
162C22×C7—42no—
163C22×C7#12114yes—
164C22×C7C22×D411yescharacteristic
165C22×C7C22×D411yescharacteristic
166C22×C7C2×#4111yescharacteristic
167C22×C7C2×#4112yes—
168C22×C72−1+412yes—
169C4×C7—22no—
170C4×C7—42no—
171C4×C7—42no—
172Dic7—22no—
173Dic7—42no—
174Dic7—41no—
175Dic7—41no—
176Dic7—42no—
177Dic7#12314yescomplemented
178C24C22×D711yescharacteristic
179C24C22×D711yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4C22⋊D714yes—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—71no—
187C22×C4—74no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—71no—
191C22×C4—72no—
192C22×C4—72no—
193C22×C4—144no—
194C22×C4—142no—
195C22×C4—144no—
196C22×C4—142no—
197C22×C4—142no—
198C22×C4—142no—
199C22×C4C22×D712yes—
200C22×C4C22×D712yes—
201C22×C4C22×D711yescharacteristic
202C4⋊C4—24no—
203C4⋊C4—144no—
204C4⋊C4—144no—
205C4⋊C4—144no—
206C4⋊C4—142no—
207C4⋊C4—144no—
208C22⋊C4—22no—
209C22⋊C4—24no—
210C22⋊C4—142no—
211C22⋊C4—142no—
212C22⋊C4—142no—
213C22⋊C4—144no—
214C42—142no—
215C2×C7—24no—
216C2×C7—42no—
217C2×C7C2×#12111yescharacteristic
218C2×C7C2×#12311yescharacteristic
219C2×C7#53611yescharacteristic
220C2×C7#54712yes—
221C2×C7#55212yes—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23D4×D712yes—
229C23#115912yes—
230C23C2×C22⋊D712yes—
231C23C2×C22⋊D712yes—
232C23C23×D711yescharacteristic
233D4—28no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—44no—
243C2×C4—72no—
244C2×C4—74no—
245C2×C4—72no—
246C2×C4—148no—
247C2×C4—142no—
248C2×C4—141no—
249C2×C4—141no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4—284no—
259C2×C4—284no—
260C2×C4C2×C22⋊D712yes—
261C7#248611yescharacteristic, complemented, Sylow
262C22—24no—
263C22—22no—
264C22—24no—
265C22—22no—
266C22—22no—
267C22—24no—
268C22—41no—
269C22—42no—
270C22—41no—
271C22#650214yes—
272C22C2×D4×D711yescharacteristic
273C22C2×#115911yescharacteristic
274C22D7×#4112yes—
275C22#655412yes—
276C22C22×C22⋊D711yescharacteristic
277C4—22no—
278C4—42no—
279C4—42no—
280C4—74no—
281C4—142no—
282C4—282no—
283C4—282no—
284C4—281no—
285C4—281no—
286C2—24no—
287C2—42no—
288C2#9033911yescharacteristic
289C2#9034112yes—
290C2D7×#12311yescharacteristic
291C2C2×#650211yescharacteristic
292C2#9051512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#123C2
prime radicalC22×C7×D4C22
hypercenterC2×#123D7
hypocenterC7#2486
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
4. C22×Dic7⋊C4C2semidirect
5. #89983C2semidirect
6. #89756C2semidirect
7. #89744C2semidirect
8. #89724C2semidirect
9. #89715C2semidirect
10. #89714C2semidirect
11. #89714C2semidirect
177. Dic7#123semidirect
261. C7#2486semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1232
C2×#64982
C2×D4×Dic72
C22×Dic7⋊C42
#899832
#897562
#897442
#897242
#897152
#897142
#24867