Finite Group Databasebeta

← #11739071

Subgroups — #11739071

Order 896.

338 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 115 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#90423C211yesmaximal, characteristic
2#90413C211yesmaximal, characteristic, complemented
3C7×#487C211yesmaximal, characteristic, complemented
4#89862C211yesmaximal, characteristic, complemented
5#89856C211yesmaximal, characteristic
6#89570C212yesmaximal
7#89567C212yesmaximal
8#89563C212yesmaximal, complemented
9#89561C212yesmaximal, complemented
10C4×#1132C211yesmaximal, characteristic
11#89451C211yesmaximal, characteristic, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C7×C4⋊Q8C2211yescharacteristic
14#6510C2212yes—
15Q8×Dic7C2212yes—
16#6507—22no—
17C2×#1144C2212yes—
18C2×#1132C2211yescharacteristic
19#6461—22no—
20#6460—21no—
21#6455—21no—
22#6451—21no—
23#6451C2212yes—
24#6450—21no—
25#6450C2212yes—
26C4×D28C2211yescharacteristic
27C4×Dic14C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C7×#103C2212yes—
30C7×#103C2212yes—
31C4×C7×C8C2211yescharacteristic
32#6409C2212yes—
33#6395C2211yescharacteristic
34#6391C2211yescharacteristic
35#6390C2211yescharacteristic
36#6389C2212yes—
37Dic7⋊C8C2212yes—
38#6383C2212yes—
39#6382C2212yes—
40C4⋊(C7⋊C8)C2211yescharacteristic
41#3369—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2312yes—
44C2×D28C2311yescharacteristic
45C2×C4×D7—21no—
46C2×Dic14—22no—
47C2×Dic14D411yescharacteristic
48C2×Dic14C2311yescharacteristic
49C2×C7×C8C2311yescharacteristic
50C2×C7×C8C2311yescharacteristic
51C7×C4⋊C4—21no—
52C7×C4⋊C4—21no—
53C7×C4⋊C4—21no—
54C7×C4⋊C4C2312yes—
55C42×C7C2311yescharacteristic
56#1144—24no—
57(C2×D7)⋊C4—21no—
58(C2×D7)⋊C4—22no—
59(C2×D7)⋊C4—22no—
60C4⋊Dic7—22no—
61C4⋊Dic7D411yescharacteristic
62C4⋊Dic7C2311yescharacteristic
63C4⋊Dic7C2311yescharacteristic
64Dic7⋊C4—21no—
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4D412yes—
68C4×Dic7—21no—
69C4×Dic7—22no—
70C4×Dic7—22no—
71C4×Dic7C2312yes—
72C2×C7⋊C8C2312yes—
73#1132—22no—
74Q82—71no—
75#556—71no—
76#489—72no—
77#487D711yescharacteristic, complemented
78#476—71no—
79#475—72no—
80#464—72no—
81#462—71no—
82#446—71no—
83#441—72no—
84C4×SD16—71no—
85C22×D7—21no—
86C7×Q8—22no—
87C7×Q8—22no—
88C7×Q8—81no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2411yescharacteristic
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7C2×D412yes—
98C2×Dic7C2×D412yes—
99D28—21no—
100D28—21no—
101C4×D7—42no—
102Dic14—22no—
103Dic14—21no—
104Dic14—82no—
105Dic14#4112yes—
106C7×C8—21no—
107C7×C8—21no—
108C7⋊C8—22no—
109C2×Q16—72no—
110C2×SD16—71no—
111C4⋊Q8—72no—
112C4⋊Q8—141no—
113C4⋊Q8—142no—
114C4⋊Q8C2×D711yescharacteristic
115C4⋊Q8C2×D711yescharacteristic
116#126—142no—
117#124—72no—
118C22⋊Q8—141no—
119C4×Q8—71no—
120C4×Q8—72no—
121C4×Q8—72no—
122C4×Q8—71no—
123C4×Q8—141no—
124C4×D4—71no—
125#117—141no—
126#106—71no—
127#106—72no—
128C4⋊C8—72no—
129C4⋊C8—71no—
130#103—72no—
131#103—71no—
132#103C2×D712yes—
133#103C2×D712yes—
134#102—71no—
135#102—72no—
136#97—72no—
137C4×C8C2×D711yescharacteristic
138C22×C7C22×D411yescharacteristic
139C2×D7—41no—
140C2×D7—41no—
141C2×D7—41no—
142C4×C7—21no—
143C4×C7—42no—
144C4×C7—42no—
145C4×C7#13712yes—
146C4×C7C2×#4111yescharacteristic
147C4×C72−1+411yescharacteristic
148Dic7—22no—
149Dic7—22no—
150Dic7—22no—
151Dic7—41no—
152Dic7—41no—
153C2×Q8—21no—
154C2×Q8—71no—
155C2×Q8—71no—
156C2×Q8—142no—
157C2×Q8C22×D712yes—
158C2×D4—71no—
159C22×C4—141no—
160Q16—144no—
161SD16—142no—
162C2×C8—72no—
163C2×C8C22×D711yescharacteristic
164C2×C8C22×D711yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—72no—
169C4⋊C4—71no—
170C4⋊C4—71no—
171C4⋊C4—71no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—141no—
176C4⋊C4C22×D712yes—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C22⋊C4—141no—
180C42—72no—
181C42—142no—
182C42—142no—
183C42—141no—
184C42C22×D711yescharacteristic
185C2×C7#54811yescharacteristic
186C2×C7C2×#13711yescharacteristic
187C2×C7#57711yescharacteristic
188D7—81no—
189C23—141no—
190Q8—22no—
191Q8—22no—
192Q8—72no—
193Q8—81no—
194Q8—142no—
195Q8—141no—
196Q8—562no—
197D4—141no—
198D4—141no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—72no—
202C2×C4—72no—
203C2×C4—141no—
204C2×C4—141no—
205C2×C4—141no—
206C2×C4—282no—
207C2×C4D4×D711yescharacteristic
208C2×C4D4×D711yescharacteristic
209C2×C4C23×D711yescharacteristic
210C8—21no—
211C8—21no—
212C8—142no—
213C7#336911yescharacteristic, complemented, Sylow
214C22—281no—
215C22—281no—
216C22—281no—
217C22C2×D4×D711yescharacteristic
218C4—21no—
219C4—42no—
220C4—42no—
221C4—142no—
222C4—142no—
223C4—142no—
224C4—281no—
225C4—281no—
226C4#648112yes—
227C4D7×#4111yescharacteristic
228C4#655411yescharacteristic
229C2—561no—
230C2#9036811yescharacteristic
231C2#9045311yescharacteristic
232C2C2×#648111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#487D7
hypocenterC7#3369
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #90413C2semidirect
3. C7×#487C2semidirect
4. #89862C2semidirect
8. #89563C2semidirect
9. #89561C2semidirect
11. #89451C2semidirect
77. #487D7semidirect
213. C7#3369semidirect

Maximal subgroups

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

subgroupindex
#904232
#904132
C7×#4872
#898622
#898562
#895702
#895672
#895632
#895612
C4×#11322
#894512
#33697