Finite Group Databasebeta

← #11736397

Subgroups — #11736397

Order 896.

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

normal subgroups only

This group has 372 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
1#90507C211yesmaximal, characteristic
2#90210C211yesmaximal, characteristic, complemented
3C7×#419C211yesmaximal, characteristic, complemented
4C2×#6409C211yesmaximal, characteristic, complemented
5#89905C211yesmaximal, characteristic
6#89894C211yesmaximal, characteristic, complemented
7#89774C211yesmaximal, characteristic
8#89572C212yesmaximal, complemented
9#89569C212yesmaximal, complemented
10#89567C212yesmaximal, complemented
11#89564C212yesmaximal, complemented
12C22×C7×Q8C2211yescharacteristic
13C2×#1157C2211yescharacteristic
14C2×C7×M16C2211yescharacteristic
15C7×#117C2211yescharacteristic
16#6510C2212yes—
17#6509—21no—
18#6509—21no—
19#6507C2212yes—
20#6492—22no—
21#6489C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23C22×C7⋊C8C2211yescharacteristic
24#6453C2212yes—
25#6451C2212yes—
26#6441—22no—
27#6433—21no—
28#6432—21no—
29C7×#103C2212yes—
30C7×#103C2212yes—
31#6409C2212yes—
32#6409C2212yes—
33#6395C2212yes—
34#6390C2212yes—
35#6389C2212yes—
36C8×Dic7C2212yes—
37#6384C2211yescharacteristic
38#6384C2211yescharacteristic
39#6383C2211yescharacteristic
40#6383C2211yescharacteristic
41#3137—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8—22no—
44C2×C7×Q8—22no—
45C2×C7×Q8C2312yes—
46C22×C4×C7—21no—
47C22×C4×C7C2311yescharacteristic
48C2×C22⋊D7—21no—
49C22×Dic7—21no—
50#1157—22no—
51#1157—22no—
52C2×D28C2311yescharacteristic
53C2×C4×D7—21no—
54C2×Dic14—22no—
55C2×Dic14C2311yescharacteristic
56C7×M16—22no—
57C2×C7×C8C2312yes—
58C7×C4⋊C4C2311yescharacteristic
59C7×C4⋊C4C2311yescharacteristic
60C7×C22⋊C4—21no—
61C42×C7—21no—
62C22⋊Dic7D412yes—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—22no—
65C4⋊Dic7—21no—
66C4⋊Dic7—22no—
67C4⋊Dic7D411yescharacteristic
68C4⋊Dic7D411yescharacteristic
69C4⋊Dic7C2312yes—
70Dic7⋊C4—22no—
71Dic7⋊C4—22no—
72Dic7⋊C4—22no—
73Dic7⋊C4—22no—
74C4×Dic7C2312yes—
75C2×C7⋊C8—22no—
76C2×C7⋊C8C2311yescharacteristic
77C2×C7⋊C8C2311yescharacteristic
78#544—71no—
79#536—71no—
80#491—72no—
81#489—72no—
82#488—72no—
83#487—72no—
84#419D711yescharacteristic, complemented
85#417—71no—
86#416—71no—
87C2×#103—71no—
88#405—71no—
89C23×C7C2×D411yescharacteristic
90C22×D7—21no—
91C7×Q8—22no—
92C7×Q8—22no—
93C7×Q8—42no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—42no—
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7#4112yes—
100C2×C4×C7#4112yes—
101C2×C4×C7C2411yescharacteristic
102C22⋊D7—42no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—81no—
106C2×Dic7C2×D412yes—
107C2×Dic7C2×D412yes—
108D28—21no—
109D28—21no—
110C4×D7—42no—
111Dic14—21no—
112Dic14—21no—
113Dic14—82no—
114C7×C8—22no—
115C7⋊C8—21no—
116C7⋊C8—21no—
117C2×#41—71no—
118C22×Q8C2×D711yescharacteristic
119C2×M16C2×D711yescharacteristic
120C22×C8—71no—
121C4⋊Q8—72no—
122C4⋊Q8—72no—
123C4⋊Q8—141no—
124#125—72no—
125#125—141no—
126#124—72no—
127#123—142no—
128C22⋊Q8—142no—
129C22⋊Q8—141no—
130C22⋊Q8—141no—
131#117—71no—
132#117C2×D711yescharacteristic
133C2×C4⋊C4—71no—
134#103—72no—
135#103—72no—
136#103—72no—
137#103—71no—
138#103—71no—
139#103C2×D712yes—
140#103C2×D712yes—
141#102—72no—
142#102—71no—
143#102—71no—
144#97—72no—
145C4×C8—72no—
146C22×C7—21no—
147C22×C7—21no—
148C22×C7#12412yes—
149C22×C7C22×D411yescharacteristic
150C2×D7—41no—
151C2×D7—41no—
152C2×D7—41no—
153C4×C7—42no—
154C4×C7—41no—
155C4×C7—41no—
156C4×C7#12412yes—
157C4×C7C2×#4111yescharacteristic
158C4×C7C2×#4111yescharacteristic
159Dic7—22no—
160Dic7—22no—
161Dic7—41no—
162Dic7—42no—
163#41—142no—
164#41—142no—
165C2×Q8—21no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—71no—
169C2×Q8—142no—
170C2×Q8C22×D712yes—
171C2×D4—71no—
172C2×D4—141no—
173C22×C4—21no—
174C22×C4—141no—
175C22×C4—141no—
176C22×C4C22×D711yescharacteristic
177M16—22no—
178C2×C8—71no—
179C2×C8—71no—
180C2×C8—142no—
181C2×C8C22×D712yes—
182C4⋊C4—71no—
183C4⋊C4—71no—
184C4⋊C4—72no—
185C4⋊C4—142no—
186C4⋊C4—142no—
187C4⋊C4—142no—
188C4⋊C4—142no—
189C4⋊C4—142no—
190C4⋊C4—141no—
191C4⋊C4C22×D711yescharacteristic
192C4⋊C4C22×D711yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—72no—
195C22⋊C4—142no—
196C22⋊C4—142no—
197C42—21no—
198C42—72no—
199C2×C7—22no—
200C2×C7C2×#12411yescharacteristic
201C2×C7#57711yescharacteristic
202C2×C7#57811yescharacteristic
203D7—81no—
204C23—141no—
205C23D4×D711yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—42no—
209Q8—141no—
210Q8—141no—
211Q8—562no—
212D4—141no—
213D4—141no—
214D4—282no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—72no—
220C2×C4—72no—
221C2×C4—142no—
222C2×C4—141no—
223C2×C4—282no—
224C2×C4—561no—
225C2×C4#115712yes—
226C2×C4D4×D711yescharacteristic
227C2×C4#115912yes—
228C2×C4C23×D711yescharacteristic
229C8—22no—
230C8—141no—
231C8—141no—
232C7#313711yescharacteristic, complemented, Sylow
233C22—21no—
234C22—21no—
235C22—281no—
236C22—281no—
237C22—281no—
238C22#644812yes—
239C22C2×D4×D711yescharacteristic
240C4—42no—
241C4—41no—
242C4—41no—
243C4—142no—
244C4—142no—
245C4—281no—
246C4—282no—
247C4#644812yes—
248C4C2×#115711yescharacteristic
249C4C2×#115911yescharacteristic
250C2—22no—
251C2—561no—
252C2C2×#644811yescharacteristic
253C2#9046911yescharacteristic
254C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#419C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#419D7
hypocenterC7#3137
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. #90210C2semidirect
3. C7×#419C2semidirect
4. C2×#6409C2semidirect
6. #89894C2semidirect
8. #89572C2semidirect
9. #89569C2semidirect
10. #89567C2semidirect
11. #89564C2semidirect
84. #419D7semidirect
232. C7#3137semidirect

Maximal subgroups

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

subgroupindex
#905072
#902102
C7×#4192
C2×#64092
#899052
#898942
#897742
#895722
#895692
#895672
#895642
#31377