Finite Group Databasebeta

← #11739189

Subgroups — #11739189

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#90415C211yesmaximal, characteristic
2#90388C211yesmaximal, characteristic, complemented
3C7×#491C211yesmaximal, characteristic, complemented
4#89864C211yesmaximal, characteristic, complemented
5#89840C211yesmaximal, characteristic, complemented
6#89582C212yesmaximal
7#89580C212yesmaximal, complemented
8#89570C212yesmaximal, complemented
9#89565C212yesmaximal, complemented
10#89487C211yesmaximal, characteristic, complemented
11#89481C211yesmaximal, characteristic, complemented
12C2×#1161C2211yescharacteristic
13C2×Q8×D7C2211yescharacteristic
14C7×C4⋊Q8C2211yescharacteristic
15C7×#125C2211yescharacteristic
16#6510C2211yescharacteristic
17#6507C2211yescharacteristic
18C2×#1144C2212yes—
19C2×#1144C2212yes—
20C2×Dic28C2212yes—
21C2×#1132C2212yes—
22#6461—22no—
23#6460C2212yes—
24#6459—22no—
25#6457—22no—
26#6453—22no—
27#6450C2212yes—
28#6438C2211yescharacteristic
29#6435C2211yescharacteristic
30#6432C2211yescharacteristic
31C7×#103C2212yes—
32C7×#103C2212yes—
33C7×#97C2211yescharacteristic
34(C2×D7)⋊C8C2212yes—
35Dic7⋊C8C2212yes—
36#6384C2212yes—
37#6383C2212yes—
38C4×C7⋊C8C2211yescharacteristic
39#3223—71nomaximal, Sylow
40C2×C7×Q8C2311yescharacteristic
41C2×C7×Q8C2311yescharacteristic
42#1161—42no—
43Q8×D7—42no—
44C2×D28—21no—
45C2×D28C2311yescharacteristic
46C2×C4×D7—21no—
47C2×C4×D7—21no—
48C2×C4×D7C2311yescharacteristic
49C2×Dic14—21no—
50C2×Dic14C2312yes—
51C2×Dic14C2311yescharacteristic
52C2×C7×C8C2312yes—
53C7×C4⋊C4—22no—
54C7×C4⋊C4—22no—
55C7×C4⋊C4C2312yes—
56C42×C7C2311yescharacteristic
57#1144—42no—
58#1144—42no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4D412yes—
63C4⋊Dic7—22no—
64C4⋊Dic7—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4D412yes—
68C4×Dic7—22no—
69C4×Dic7C2311yescharacteristic
70C2×C7⋊C8C2312yes—
71Dic28—42no—
72#1132—42no—
73#541—71no—
74#536—71no—
75#497—71no—
76#494—71no—
77#491D711yescharacteristic, complemented
78#488—71no—
79#464—72no—
80#462—72no—
81#452—72no—
82#451—72no—
83#433—71no—
84C22×D7—21no—
85C22×D7C2×D411yescharacteristic
86C7×Q8—22no—
87C7×Q8—22no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7C2×D412yes—
92C2×C4×C7C2411yescharacteristic
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7C2×D412yes—
96C2×Dic7C2×D411yescharacteristic
97D28—22no—
98D28—42no—
99C4×D7—22no—
100C4×D7—42no—
101C4×D7—42no—
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⋊Q8C2×D711yescharacteristic
117#126—142no—
118#125—142no—
119#125C2×D711yescharacteristic
120#124—71no—
121#124—71no—
122#123—142no—
123C22⋊Q8—72no—
124C22⋊Q8—142no—
125C4×Q8—72no—
126#117—71no—
127C4⋊C8—72no—
128#103—72no—
129#103C2×D712yes—
130#103C2×D712yes—
131#102—72no—
132C22⋊C8—72no—
133#97C2×D711yescharacteristic
134C4×C8—71no—
135C22×C7C22×D411yescharacteristic
136C2×D7—21no—
137C2×D7—22no—
138C2×D7—41no—
139C2×D7—42no—
140C4×C7—22no—
141C4×C7—41no—
142C4×C7—41no—
143C4×C7—42no—
144C4×C7H(5,𝔽2)12yes—
145Dic7—21no—
146Dic7—22no—
147Dic7—41no—
148Dic7—42no—
149#41—282no—
150C2×Q8—71no—
151C2×Q8—72no—
152C2×Q8—141no—
153C2×Q8—282no—
154C2×Q8C22×D711yescharacteristic
155C2×Q8C22×D711yescharacteristic
156C2×D4—71no—
157C2×D4—141no—
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—22no—
169C4⋊C4—72no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4C22×D712yes—
175C22⋊C4—72no—
176C22⋊C4—142no—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C42—71no—
180C42—142no—
181C42C22×D711yescharacteristic
182C2×C7#53511yescharacteristic
183C2×C7#57812yes—
184D7—41no—
185D7—81no—
186C23—71no—
187C23—141no—
188Q8—22no—
189Q8—22no—
190Q8—142no—
191Q8—142no—
192Q8—142no—
193Q8—282no—
194D4—142no—
195D4—282no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—72no—
200C2×C4—71no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—141no—
204C2×C4—282no—
205C2×C4—282no—
206C2×C4D4×D712yes—
207C2×C4C23×D711yescharacteristic
208C8—22no—
209C8—142no—
210C7#322311yescharacteristic, complemented, Sylow
211C22—142no—
212C22—141no—
213C22—282no—
214C22—281no—
215C22C2×D4×D711yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—142no—
221C4—141no—
222C4—281no—
223C4—282no—
224C4#655312yes—
225C2—281no—
226C2—561no—
227C2#9036511yescharacteristic
228C2#9047012yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#491C2
socleC22×C7C22×D4
prime radicalC2×#1161C22
hypercenter#491D7
hypocenterC7#3223
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #90388C2semidirect
3. C7×#491C2semidirect
4. #89864C2semidirect
5. #89840C2semidirect
7. #89580C2semidirect
8. #89570C2semidirect
9. #89565C2semidirect
10. #89487C2semidirect
11. #89481C2semidirect
77. #491D7semidirect
210. C7#3223semidirect

Maximal subgroups

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

subgroupindex
#904152
#903882
C7×#4912
#898642
#898402
#895822
#895802
#895702
#895652
#894872
#894812
#32237