Finite Group Databasebeta

← #11736687

Subgroups — #11736687

Order 896.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#90212C211yesmaximal, characteristic
2#90191C211yesmaximal, characteristic, complemented
3C7×#428C211yesmaximal, characteristic, complemented
4#89903C211yesmaximal, characteristic
5#89891C211yesmaximal, characteristic, complemented
6#89770C211yesmaximal, characteristic, complemented
7C2×#6383C211yesmaximal, characteristic
8#89663C212yesmaximal, complemented
9#89662C212yesmaximal, complemented
10#89643C212yesmaximal, complemented
11#89642C212yesmaximal, complemented
12C22×D28C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C7×#117C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16#6489C2211yescharacteristic
17#6489C2211yescharacteristic
18#6487—22no—
19C22×C7⋊C8C2211yescharacteristic
20#6456C2212yes—
21#6456C2212yes—
22#6453—21no—
23#6453C2212yescomplemented
24#6451—21no—
25#6451C2212yescomplemented
26D7×C22⋊C4—22no—
27C4×D28—21no—
28C4×D28—21no—
29C7×#107C2212yescomplemented
30C7×#106C2212yescomplemented
31#6395C2212yes—
32#6395C2212yes—
33Dic7⋊C8C2212yescomplemented
34Dic7⋊C8C2212yescomplemented
35#6383C2212yes—
36#6383C2212yes—
37#6382C2211yescharacteristic, complemented
38#6382C2211yescharacteristic, complemented
39#6381C2211yescharacteristic, complemented
40#6381C2211yescharacteristic, complemented
41#3081—71nomaximal, Sylow
42C23×D7—21no—
43C22×C4×C7—21no—
44C22×C4×C7C2311yescharacteristic
45C2×D28—21no—
46C2×D28—22no—
47C2×D28Q814yescomplemented
48C2×D28C2312yes—
49C2×C4×D7—22no—
50C2×C4×D7—22no—
51C2×Dic14—22no—
52C7×M16—22no—
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2311yescharacteristic
56C7×C4⋊C4C2312yes—
57C7×C4⋊C4C2311yescharacteristic
58C7×C22⋊C4—21no—
59C42×C7—21no—
60C22⋊Dic7—21no—
61C22⋊Dic7D412yes—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64C4⋊Dic7—22no—
65C4⋊Dic7D411yescharacteristic
66C4⋊Dic7D411yescharacteristic
67C4⋊Dic7C2311yescharacteristic
68C4⋊Dic7C2311yescharacteristic
69Dic7⋊C4—22no—
70Dic7⋊C4—22no—
71C4×Dic7—21no—
72C4×Dic7C2312yes—
73C2×C7⋊C8—22no—
74C2×C7⋊C8C2311yescharacteristic
75C2×C7⋊C8C2311yescharacteristic
76#544—71no—
77#518—71no—
78#478—72no—
79#478—72no—
80#476—72no—
81#474—72no—
82#428D711yescharacteristic, complemented
83#427—71no—
84#424—71no—
85#418—71no—
86C2×#102—71no—
87C23×C7C2×D411yescharacteristic
88C22×D7—22no—
89C22×D7—24no—
90C22×D7—81no—
91C22×D7—81no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—81no—
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2×Q812yes—
98C2×C4×C7#4112yes—
99C2×C4×C7C2411yescharacteristic
100C2×Dic7—21no—
101C2×Dic7—21no—
102C2×Dic7C2×D412yes—
103C2×Dic7C2×D412yes—
104D28—24no—
105D28—22no—
106D28C2×Q814yes—
107C4×D7—82no—
108C4×D7—82no—
109Dic14—82no—
110C7×C8—22no—
111C7⋊C8—21no—
112C7⋊C8—21no—
113C22×D4—71no—
114C2×M16C2×D711yescharacteristic
115C22×C8—71no—
116C4⋊Q8—72no—
117C4⋊Q8—141no—
118#125—72no—
119#125—141no—
120C22⋊Q8—142no—
121C4×D4—72no—
122C4×D4—72no—
123C4×D4—141no—
124C4×D4—141no—
125#117—71no—
126#117—71no—
127#117C2×D711yescharacteristic
128C2×C4⋊C4C2×D711yescharacteristic
129C2×C22⋊C4—142no—
130#107—71no—
131#107—71no—
132#107C2×D712yescomplemented
133#106—71no—
134#106—71no—
135#106C2×D712yescomplemented
136C4⋊C8—72no—
137C4⋊C8—72no—
138#102—72no—
139#102—72no—
140#102—72no—
141#102—72no—
142C22×C7—21no—
143C22×C7—21no—
144C22×C7C22⋊Q812yes—
145C22×C7C22×D411yescharacteristic
146C2×D7—24no—
147C2×D7—42no—
148C2×D7—42no—
149C2×D7—84no—
150C4×C7—42no—
151C4×C7—41no—
152C4×C7—41no—
153C4×C7C22⋊Q812yes—
154C4×C7C22×Q811yescharacteristic
155C4×C7C2×#4111yescharacteristic
156Dic7—22no—
157Dic7—22no—
158Dic7—41no—
159Dic7—41no—
160C24—141no—
161C2×Q8—142no—
162C2×D4—74no—
163C2×D4—72no—
164C2×D4—142no—
165C2×D4—141no—
166C22×C4—21no—
167C22×C4—142no—
168C22×C4—142no—
169C22×C4C22×D711yescharacteristic
170M16—22no—
171C2×C8—71no—
172C2×C8—71no—
173C2×C8—142no—
174C2×C8C22×D712yes—
175C4⋊C4—21no—
176C4⋊C4—71no—
177C4⋊C4—71no—
178C4⋊C4—71no—
179C4⋊C4—71no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—142no—
183C4⋊C4C22×D712yes—
184C4⋊C4C22×D711yescharacteristic
185C4⋊C4C22×D711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—72no—
188C22⋊C4—142no—
189C22⋊C4—142no—
190C22⋊C4—141no—
191C42—21no—
192C42—72no—
193C42—141no—
194C2×C7—22no—
195C2×C7C2×C22⋊Q811yescharacteristic
196C2×C7#57611yescharacteristic
197C2×C7#57711yescharacteristic
198D7—44no—
199C23—142no—
200C23—144no—
201C23—561no—
202C23—561no—
203C23D4×D711yescharacteristic
204Q8—562no—
205D4—74no—
206D4—144no—
207D4—142no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—72no—
212C2×C4—72no—
213C2×C4—81no—
214C2×C4—141no—
215C2×C4—141no—
216C2×C4—562no—
217C2×C4—562no—
218C2×C4#115712yes—
219C2×C4D4×D711yescharacteristic
220C2×C4Q8×D712yes—
221C2×C4C23×D711yescharacteristic
222C8—22no—
223C8—141no—
224C8—141no—
225C7#308111yescharacteristic, complemented, Sylow
226C22—21no—
227C22—21no—
228C22—144no—
229C22—282no—
230C22—282no—
231C22—564no—
232C22#645912yes—
233C22C2×D4×D711yescharacteristic
234C4—42no—
235C4—41no—
236C4—41no—
237C4—142no—
238C4—142no—
239C4—281no—
240C4—281no—
241C4#645912yes—
242C4C2×#115711yescharacteristic
243C4C2×Q8×D711yescharacteristic
244C2—22no—
245C2—284no—
246C2C2×#645911yescharacteristic
247C2#9046511yescharacteristic
248C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radicalC22×D28C22
hypercenter#428D7
hypocenterC7#3081
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #90191C2semidirect
3. C7×#428C2semidirect
5. #89891C2semidirect
6. #89770C2semidirect
8. #89663C2semidirect
9. #89662C2semidirect
10. #89643C2semidirect
11. #89642C2semidirect
23. #6453C22semidirect
25. #6451C22semidirect
29. C7×#107C22semidirect
30. C7×#106C22semidirect
33. Dic7⋊C8C22semidirect
34. Dic7⋊C8C22semidirect
37. #6382C22semidirect
38. #6382C22semidirect
39. #6381C22semidirect
40. #6381C22semidirect
47. C2×D28Q8semidirect
82. #428D7semidirect
132. #107C2×D7semidirect
135. #106C2×D7semidirect
225. C7#3081semidirect

Maximal subgroups

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

subgroupindex
#902122
#901912
C7×#4282
#899032
#898912
#897702
C2×#63832
#896632
#896622
#896432
#896422
#30817