Finite Group Databasebeta

← #11734607

Subgroups — #11734607

Order 896.

630 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6503C211yesmaximal, characteristic, complemented
2C2×#6461C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic, complemented
5#89763C211yesmaximal, characteristic, complemented
6#89733C212yesmaximal, complemented
7#89732C212yesmaximal, complemented
8#89732C212yesmaximal, complemented
9#89731C212yesmaximal, complemented
10#89443C211yesmaximal, characteristic, complemented
11#89423C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C22⋊D7C2212yes—
15C22×C22⋊D7C2212yes—
16C22×D28C2211yescharacteristic
17C22×D28C2212yes—
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2212yes—
20C2×C7×C22⋊C4C2212yes—
21C2×C22⋊Dic7C2212yes—
22#6503—24no—
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×(C2×D7)⋊C4C2212yes—
29C2×C4⋊Dic7C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×C4×Dic7C2211yescharacteristic
32#6461—24no—
33C7×#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2818—71nomaximal, Sylow
38C24×C7C2312yes—
39C23×D7C2311yescharacteristic
40C23×D7C2312yes—
41C2×C7×D4—22no—
42C2×C7×D4—42no—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—24no—
48C2×C22⋊D7—24no—
49C2×C22⋊D7—42no—
50C2×C22⋊D7—42no—
51C2×C22⋊D7—42no—
52C2×C22⋊D7—42no—
53C2×C22⋊D7—42no—
54C2×C22⋊D7—42no—
55C2×C22⋊D7—42no—
56C22×Dic7C2312yes—
57C22×Dic7C2311yescharacteristic
58C22×Dic7C2311yescharacteristic
59C22×Dic7C2311yescharacteristic
60C2×D28—22no—
61C2×D28—42no—
62C2×D28—42no—
63C2×D28—42no—
64C2×D28—42no—
65C7×C4⋊C4—22no—
66C7×C22⋊C4—42no—
67C7×C22⋊C4—42no—
68C22⋊Dic7—42no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—22no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—42no—
73(C2×D7)⋊C4—42no—
74(C2×D7)⋊C4—42no—
75C4⋊Dic7—22no—
76Dic7⋊C4—22no—
77C4×Dic7D414yescomplemented
78C2×#127—71no—
79C2×#126—71no—
80#394—71no—
81#394—72no—
82#394—72no—
83#394D711yescharacteristic, complemented
84#392—71no—
85#392—72no—
86#392—72no—
87#392—71no—
88#383—71no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7C2411yescharacteristic
97C22×D7—21no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—22no—
102C22×D7—22no—
103C22×D7—21no—
104C22×D7—22no—
105C22×D7—22no—
106C22×D7—22no—
107C22×D7—22no—
108C22×D7—21no—
109C7×D4—44no—
110C2×C4×C7—22no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7C2×D412yes—
120C22⋊D7—48no—
121C22⋊D7—48no—
122C22⋊D7—48no—
123C2×Dic7—21no—
124C2×Dic7—21no—
125C2×Dic7—22no—
126C2×Dic7—21no—
127C2×Dic7—22no—
128C2×Dic7—22no—
129C2×Dic7—22no—
130C2×Dic7—22no—
131C2×Dic7C2×D412yes—
132C2×Dic7C2×D412yes—
133D28—44no—
134D28—48no—
135C22×D4—72no—
136C22×D4—72no—
137C22×D4—72no—
138C22×D4—71no—
139C22×D4—72no—
140C22×D4C2×D711yescharacteristic
141#127—144no—
142#126—144no—
143C2×C4⋊C4—71no—
144C2×C4⋊C4—71no—
145C2×C4⋊C4C2×D711yescharacteristic
146C2×C22⋊C4—71no—
147C2×C22⋊C4—71no—
148C2×C22⋊C4—71no—
149C2×C22⋊C4—72no—
150C2×C22⋊C4—72no—
151C2×C22⋊C4—72no—
152C2×C22⋊C4—72no—
153C2×C22⋊C4C2×D712yes—
154C2×C22⋊C4C2×D712yes—
155C2×C42—71no—
156#95—72no—
157#95—72no—
158#95—71no—
159#95C2×D711yescharacteristic
160C22×C7—42no—
161C22×C7—42no—
162C22×C7—42no—
163C22×C7—42no—
164C22×C7—42no—
165C22×C7—42no—
166C22×C7—42no—
167C22×C7C22×D411yescharacteristic
168C22×C7H(5,𝔽2)12yes—
169C22×C7H(5,𝔽2)11yescharacteristic
170C22×C7H(5,𝔽2)11yescharacteristic
171C22×C7H(5,𝔽2)12yes—
172C2×D7—42no—
173C2×D7—42no—
174C2×D7—42no—
175C2×D7—41no—
176C2×D7—41no—
177C2×D7—41no—
178C2×D7—42no—
179C2×D7—42no—
180C2×D7—42no—
181C2×D7—42no—
182C2×D7—42no—
183C2×D7—42no—
184C4×C7—22no—
185C4×C7—41no—
186C4×C7—41no—
187C4×C7—42no—
188Dic7—22no—
189Dic7—22no—
190Dic7—41no—
191Dic7—42no—
192C24—71no—
193C24—72no—
194C24C22×D712yes—
195C2×D4—22no—
196C2×D4—42no—
197C2×D4—144no—
198C2×D4—144no—
199C2×D4—142no—
200C2×D4—282no—
201C2×D4—282no—
202C2×D4—282no—
203C2×D4—282no—
204C2×D4—282no—
205C2×D4—282no—
206C2×D4—282no—
207C2×D4—282no—
208C2×D4—282no—
209C2×D4—282no—
210C2×D4—282no—
211C22×C4—71no—
212C22×C4—71no—
213C22×C4—72no—
214C22×C4—71no—
215C22×C4C22×D712yes—
216C22×C4C22×D711yescharacteristic
217C22×C4C22×D711yescharacteristic
218C22×C4C22×D711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—142no—
221C4⋊C4—142no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—142no—
225C22⋊C4—142no—
226C22⋊C4—142no—
227C22⋊C4—282no—
228C22⋊C4—282no—
229C22⋊C4—282no—
230C22⋊C4—282no—
231C42—74no—
232C2×C7—82no—
233C2×C7#53511yescharacteristic
234C2×C7#53511yescharacteristic
235C2×C7#53511yescharacteristic
236C2×C7#56012yes—
237C2×C7#56012yes—
238D7—82no—
239D7—81no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—141no—
248C23—142no—
249C23—142no—
250C23—142no—
251C23—142no—
252C23—142no—
253C23—141no—
254C23—142no—
255C23—142no—
256C23—141no—
257C23—142no—
258C23—142no—
259C23C23×D711yescharacteristic
260D4—44no—
261D4—284no—
262D4—288no—
263D4—288no—
264D4—288no—
265D4—288no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—72no—
276C2×C4—72no—
277C2×C4—141no—
278C2×C4—141no—
279C2×C4—142no—
280C2×C4—141no—
281C2×C4—142no—
282C2×C4—142no—
283C2×C4—142no—
284C2×C4—142no—
285C2×C4D4×D712yes—
286C7#281811yescharacteristic, complemented, Sylow
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—282no—
295C22—281no—
296C22—282no—
297C22—281no—
298C22—282no—
299C22—282no—
300C22—282no—
301C22—282no—
302C22—282no—
303C22—282no—
304C22—282no—
305C22—281no—
306C22C2×D4×D711yescharacteristic
307C22#654812yes—
308C22#654811yescharacteristic
309C22#655311yescharacteristic
310C22#655312yes—
311C4—22no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—142no—
316C4—142no—
317C4—282no—
318C4—281no—
319C2—82no—
320C2—562no—
321C2—561no—
322C2#9028311yescharacteristic
323C2#9029811yescharacteristic
324C2#9031112yes—
325C2#9035712yes—
326C2#9036511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
hypercenter#394D7
hypocenterC7#2818
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#6503C2semidirect
2. C2×#6461C2semidirect
3. C7×#394C2semidirect
4. #89995C2semidirect
5. #89763C2semidirect
6. #89733C2semidirect
7. #89732C2semidirect
8. #89732C2semidirect
9. #89731C2semidirect
10. #89443C2semidirect
11. #89423C2semidirect
77. C4×Dic7D4semidirect
83. #394D7semidirect
286. C7#2818semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
C2×#64612
C7×#3942
#899952
#897632
#897332
#897322
#897312
#894432
#894232
#28187