Finite Group Databasebeta

← #11739092

Subgroups — #11739092

Order 896.

364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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#90388C211yesmaximal, characteristic, complemented
2#90366C211yesmaximal, characteristic
3C7×#488C211yesmaximal, characteristic, complemented
4#89841C211yesmaximal, characteristic, complemented
5#89832C211yesmaximal, characteristic, complemented
6#89580C212yesmaximal, complemented
7#89565C212yesmaximal, complemented
8#89555C212yesmaximal
9#89539C212yesmaximal, complemented
10#89471C211yesmaximal, characteristic, complemented
11#89462C211yesmaximal, characteristic, complemented
12C2×Q8×D7C2211yescharacteristic
13C2×#1159C2211yescharacteristic
14C7×#125C2211yescharacteristic
15C7×#124C2211yescharacteristic
16#6507C2211yescharacteristic
17C2×#1144C2212yes—
18#6499C2211yescharacteristic
19C2×#1142C2212yes—
20C2×Dic28C2212yes—
21C2×Dic28C2212yes—
22#6461—22no—
23#6460C2212yes—
24#6453—22no—
25#6450C2212yes—
26#6446—22no—
27#6441—22no—
28#6435C2211yescharacteristic
29#6432C2211yescharacteristic
30#6432C2211yescharacteristic
31C7×#103C2212yes—
32C7×#102C2212yes—
33C4×C7×C8C2211yescharacteristic
34(C2×D7)⋊C8C2212yes—
35Dic7⋊C8C2212yes—
36#6384C2212yes—
37#6384C2212yes—
38#6377C2211yescharacteristic
39#3223—71nomaximal, Sylow
40C2×C7×Q8C2311yescharacteristic
41C2×C7×D4C2311yescharacteristic
42C2×C22⋊D7—21no—
43C22×Dic7—21no—
44Q8×D7—42no—
45#1159—42no—
46C2×C4×D7—21no—
47C2×C4×D7C2311yescharacteristic
48C2×Dic14—21no—
49C2×Dic14C2312yes—
50C2×Dic14C2311yescharacteristic
51C2×Dic14C2311yescharacteristic
52C2×C7×C8C2312yes—
53C7×C4⋊C4—22no—
54C7×C4⋊C4C2312yes—
55C7×C22⋊C4—22no—
56C42×C7C2311yescharacteristic
57C22⋊Dic7—22no—
58#1144—42no—
59#1142—42no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4D412yes—
62C4⋊Dic7—22no—
63C4⋊Dic7—22no—
64C4⋊Dic7—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4D412yes—
68C4×Dic7—22no—
69C4×Dic7C2311yescharacteristic
70C2×C7⋊C8C2312yes—
71Dic28—42no—
72Dic28—42no—
73#541—71no—
74#536—71no—
75#497—71no—
76#494—71no—
77#491—71no—
78#488D711yescharacteristic, complemented
79#464—72no—
80#462—72no—
81#452—72no—
82#451—72no—
83#433—71no—
84C23×C7—21no—
85C22×D7C2×D411yescharacteristic
86C7×Q8—22no—
87C7×D4—22no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7C2×D412yes—
91C2×C4×C7C2411yescharacteristic
92C22⋊D7—42no—
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—42no—
97C2×Dic7C2×D412yes—
98C2×Dic7C2×D411yescharacteristic
99C4×D7—22no—
100C4×D7—42no—
101Dic14—22no—
102Dic14—22no—
103Dic14—22no—
104Dic14—22no—
105Dic14—42no—
106C7×C8—22no—
107C7⋊C8—22no—
108C2×#41—71no—
109C22×Q8—71no—
110C2×Q16—72no—
111C2×Q16—72no—
112C2×Q16—72no—
113C2×SD16—72no—
114C4⋊Q8—71no—
115C4⋊Q8—71no—
116C4⋊Q8—71no—
117#126—142no—
118#125—142no—
119#125C2×D711yescharacteristic
120#124—71no—
121#124C2×D711yescharacteristic
122#123—142no—
123C22⋊Q8—72no—
124C22⋊Q8—142no—
125C4×Q8—72no—
126#117—71no—
127C4⋊C8—72no—
128#103—72no—
129#103—72no—
130#103C2×D712yes—
131#102C2×D712yes—
132C22⋊C8—72no—
133#97—71no—
134C4×C8C2×D711yescharacteristic
135C22×C7—41no—
136C22×C7—42no—
137C22×C7C22×D411yescharacteristic
138C2×D7—21no—
139C2×D7—22no—
140C4×C7—22no—
141C4×C7—41no—
142C4×C7—42no—
143C4×C7H(5,𝔽2)12yes—
144Dic7—21no—
145Dic7—22no—
146Dic7—41no—
147Dic7—42no—
148Dic7—41no—
149#41—282no—
150C2×Q8—71no—
151C2×Q8—71no—
152C2×Q8—72no—
153C2×Q8—141no—
154C2×Q8—282no—
155C2×Q8C22×D711yescharacteristic
156C2×D4—141no—
157C2×D4C22×D711yescharacteristic
158C22×C4—71no—
159C22×C4—141no—
160C22×C4—141no—
161Q16—282no—
162Q16—282no—
163Q16—282no—
164SD16—282no—
165C2×C8—72no—
166C2×C8C22×D712yes—
167C4⋊C4—22no—
168C4⋊C4—72no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4C22×D712yes—
175C22⋊C4—22no—
176C22⋊C4—72no—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C42—71no—
180C42—142no—
181C42C22×D711yescharacteristic
182C2×C7—81no—
183C2×C7#53511yescharacteristic
184C2×C7#57812yes—
185D7—41no—
186C23—21no—
187C23—71no—
188Q8—22no—
189Q8—142no—
190Q8—142no—
191Q8—142no—
192Q8—142no—
193Q8—282no—
194D4—22no—
195D4—282no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—72no—
199C2×C4—71no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—141no—
203C2×C4—141no—
204C2×C4—282no—
205C2×C4—282no—
206C2×C4D4×D712yes—
207C2×C4C23×D711yescharacteristic
208C8—22no—
209C8—142no—
210C7#322311yescharacteristic, complemented, Sylow
211C22—42no—
212C22—41no—
213C22—142no—
214C22—141no—
215C22C2×D4×D711yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—142no—
220C4—141no—
221C4—281no—
222C4—281no—
223C4—282no—
224C4#655312yes—
225C2—81no—
226C2—281no—
227C2#9036511yescharacteristic
228C2#9046212yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#488C2
socleC22×C7C22×D4
prime radicalC2×#1159C22
hypercenter#488D7
hypocenterC7#3223
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #90388C2semidirect
3. C7×#488C2semidirect
4. #89841C2semidirect
5. #89832C2semidirect
6. #89580C2semidirect
7. #89565C2semidirect
9. #89539C2semidirect
10. #89471C2semidirect
11. #89462C2semidirect
78. #488D7semidirect
210. C7#3223semidirect

Maximal subgroups

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

subgroupindex
#903882
#903662
C7×#4882
#898412
#898322
#895802
#895652
#895552
#895392
#894712
#894622
#32237