Finite Group Databasebeta

← #11735481

Subgroups — #11735481

Order 896.

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

normal subgroups only

This group has 760 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×#6459C212yesmaximal, complemented
2C2×#6456C211yesmaximal, characteristic, complemented
3C2×#6448C211yesmaximal, characteristic, complemented
4C2×#6447C212yesmaximal, complemented
5C7×#399C211yesmaximal, characteristic, complemented
6#89764C211yesmaximal, characteristic, complemented
7#89760C212yesmaximal, complemented
8#89732C211yesmaximal, characteristic, complemented
9#89729C212yesmaximal, complemented
10#89443C211yesmaximal, characteristic, complemented
11#89421C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×D28C2211yescharacteristic
15C22×D28C2212yes—
16C22×C4×D7C2212yes—
17C22×Dic14C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2212yes—
20C2×C7×C22⋊C4C2212yes—
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×(C2×D7)⋊C4C2212yes—
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2212yes—
31C2×C4×Dic7C2212yes—
32C2×C4×Dic7C2211yescharacteristic
33#6459—28no—
34#6456—24no—
35#6448—24no—
36#6447—28no—
37C7×#95C2211yescharacteristic
38C7⋊#95C2211yescharacteristic
39#2540—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C23×D7C2311yescharacteristic
42C23×D7C2312yes—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2312yes—
47C2×C22⋊D7—24no—
48C2×C22⋊D7—24no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7—22no—
51C2×C22⋊D7—42no—
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C2×D28—24no—
57C2×D28—22no—
58C2×D28—24no—
59C2×D28—24no—
60C2×D28—42no—
61C2×C4×D7—24no—
62C2×C4×D7—24no—
63C2×C4×D7—24no—
64C2×Dic14—22no—
65C2×Dic14—24no—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—24no—
68C7×C22⋊C4—24no—
69C7×C22⋊C4—22no—
70C22⋊Dic7—22no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—24no—
73(C2×D7)⋊C4—24no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—24no—
76(C2×D7)⋊C4—24no—
77Dic7⋊C4—24no—
78Dic7⋊C4D418yescomplemented
79C4×Dic7—42no—
80C4×Dic7D414yescomplemented
81C2×#124—71no—
82C2×C22⋊Q8—72no—
83C2×#121—72no—
84C2×C4×D4—71no—
85#399—71no—
86#399D711yescharacteristic, complemented
87#392—71no—
88#392—71no—
89#390—72no—
90#390—72no—
91C4×C4⋊C4—71no—
92C23×C7—21no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—21no—
96C23×C7—21no—
97C23×C7C2411yescharacteristic
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—22no—
102C22×D7—22no—
103C22×D7—22no—
104C22×D7—21no—
105C22×D7—22no—
106C22×D7—22no—
107C22×D7—21no—
108C22×D7—21no—
109C22×D7#4114yes—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—21no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7C2×D412yes—
119C2×C4×C7C2×D414yes—
120C22⋊D7—48no—
121C22⋊D7—44no—
122C2×Dic7—22no—
123C2×Dic7—21no—
124C2×Dic7—22no—
125C2×Dic7—21no—
126C2×Dic7—22no—
127C2×Dic7—21no—
128C2×Dic7C2×D412yes—
129C2×Dic7C2×D414yes—
130C2×Dic7C2×D412yes—
131C2×Dic7C2×D414yes—
132D28—28no—
133D28—48no—
134C4×D7—48no—
135Dic14—44no—
136C22×Q8—71no—
137C22×D4—71no—
138C22×D4—72no—
139C22×D4—71no—
140C22×D4—72no—
141C23×C4—72no—
142#124—144no—
143C22⋊Q8—148no—
144#121—148no—
145C4×D4—144no—
146C2×C4⋊C4—72no—
147C2×C4⋊C4—72no—
148C2×C4⋊C4C2×D711yescharacteristic
149C2×C4⋊C4C2×D712yes—
150C2×C22⋊C4—71no—
151C2×C22⋊C4—72no—
152C2×C22⋊C4—72no—
153C2×C22⋊C4—72no—
154C2×C22⋊C4—72no—
155C2×C22⋊C4—71no—
156C2×C22⋊C4—71no—
157C2×C22⋊C4C2×D712yes—
158C2×C22⋊C4C2×D711yescharacteristic
159C2×C42—71no—
160C2×C42—72no—
161#95—71no—
162#95C2×D711yescharacteristic
163C22×C7—41no—
164C22×C7—41no—
165C22×C7—41no—
166C22×C7—42no—
167C22×C7—42no—
168C22×C7C22×D411yescharacteristic
169C22×C7C22×D412yes—
170C22×C7C2×#4112yes—
171C22×C7H(5,𝔽2)11yescharacteristic
172C22×C72−1+411yescharacteristic
173C2×D7—24no—
174C2×D7—24no—
175C2×D7—24no—
176C2×D7—42no—
177C2×D7—42no—
178C2×D7—42no—
179C2×D7—42no—
180C2×D7—41no—
181C2×D7—41no—
182C2×D7—41no—
183C2×D7—42no—
184C2×D7—42no—
185C4×C7—22no—
186C4×C7—24no—
187C4×C7—41no—
188C4×C7—42no—
189Dic7—22no—
190Dic7—24no—
191Dic7—41no—
192Dic7#12714yes—
193C24—72no—
194C24—71no—
195C24C22×D711yescharacteristic
196C2×Q8—144no—
197C2×Q8—142no—
198C2×D4—144no—
199C2×D4—144no—
200C2×D4—144no—
201C2×D4—144no—
202C2×D4—142no—
203C2×D4—144no—
204C2×D4—142no—
205C2×D4—144no—
206C2×D4—282no—
207C2×D4—282no—
208C22×C4—71no—
209C22×C4—72no—
210C22×C4—71no—
211C22×C4—71no—
212C22×C4—144no—
213C22×C4—144no—
214C22×C4—144no—
215C22×C4C22×D711yescharacteristic
216C22×C4C22×D712yes—
217C22×C4C22×D711yescharacteristic
218C22×C4C22×D712yes—
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—78no—
222C4⋊C4—144no—
223C22⋊C4—24no—
224C22⋊C4—22no—
225C22⋊C4—144no—
226C22⋊C4—144no—
227C22⋊C4—144no—
228C22⋊C4—142no—
229C22⋊C4—142no—
230C22⋊C4—144no—
231C22⋊C4—142no—
232C42—74no—
233C42—282no—
234C2×C7—81no—
235C2×C7#52511yescharacteristic
236C2×C7C2×#12711yescharacteristic
237C2×C7#53711yescharacteristic
238C2×C7#54612yes—
239C2×C7#54812yes—
240D7—44no—
241D7—81no—
242C23—21no—
243C23—22no—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—74no—
248C23—142no—
249C23—142no—
250C23—142no—
251C23—142no—
252C23—142no—
253C23—142no—
254C23—141no—
255C23—142no—
256C23—141no—
257C23—142no—
258C23—141no—
259C23C23×D711yescharacteristic
260Q8—284no—
261D4—148no—
262D4—288no—
263D4—284no—
264D4—288no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—72no—
274C2×C4—74no—
275C2×C4—72no—
276C2×C4—74no—
277C2×C4—141no—
278C2×C4—142no—
279C2×C4—141no—
280C2×C4—142no—
281C2×C4—141no—
282C2×C4—142no—
283C2×C4—288no—
284C2×C4D4×D712yes—
285C2×C4D4×D714yes—
286C7#254011yescharacteristic, complemented, Sylow
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—144no—
293C22—144no—
294C22—144no—
295C22—282no—
296C22—282no—
297C22—282no—
298C22—282no—
299C22—281no—
300C22—281no—
301C22—282no—
302C22—282no—
303C22—281no—
304C22C2×D4×D711yescharacteristic
305C22C2×D4×D712yes—
306C22#655111yescharacteristic
307C22D7×#4112yes—
308C22#655311yescharacteristic
309C4—24no—
310C4—22no—
311C4—42no—
312C4—41no—
313C4—74no—
314C4—142no—
315C4—144no—
316C4—281no—
317C2—81no—
318C2—284no—
319C2—561no—
320C2#9032011yescharacteristic
321C2#9032412yes—
322C2#9034512yes—
323C2#9034711yescharacteristic
324C2D7×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
hypercenter#399D7
hypocenterC7#2540
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6459C2semidirect
2. C2×#6456C2semidirect
3. C2×#6448C2semidirect
4. C2×#6447C2semidirect
5. C7×#399C2semidirect
6. #89764C2semidirect
7. #89760C2semidirect
8. #89732C2semidirect
9. #89729C2semidirect
10. #89443C2semidirect
11. #89421C2semidirect
78. Dic7⋊C4D4semidirect
80. C4×Dic7D4semidirect
86. #399D7semidirect
286. C7#2540semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64562
C2×#64482
C2×#64472
C7×#3992
#897642
#897602
#897322
#897292
#894432
#894212
#25407