Finite Group Databasebeta

← #11726113

Subgroups — #11726113

Order 896.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#90363C211yesmaximal, characteristic
2#89827C211yesmaximal, characteristic
3#89475C211yesmaximal, characteristic, complemented
4#89474C211yesmaximal, characteristic, complemented
5C7×#329C211yesmaximal, characteristic
6#89260C211yesmaximal, characteristic, complemented
7#89260C211yesmaximal, characteristic, complemented
8C7×#124C2211yescharacteristic
9#6507—21no—
10#6498—21no—
11#6442—21no—
12#6442—21no—
13#6441—21no—
14#6441—21no—
15#6433C2211yescharacteristic
16#6433C2211yescharacteristic
17#6432C2211yescharacteristic
18C7×#97C2211yescharacteristic
19C7×#97C2211yescharacteristic
20#6409—21no—
21#6406—21no—
22#6392—21no—
23#6391—21no—
24#6390—21no—
25#6390—21no—
26#6377C2211yescharacteristic
27#1664—71nomaximal, Sylow
28C2×C7×Q8D411yescharacteristic
29C2×C7×D4D411yescharacteristic
30C22×Dic7—21no—
31C2×Dic14—21no—
32C2×Dic14—21no—
33C2×C7×C8—21no—
34C2×C7×C8—21no—
35C7×C22⋊C4—21no—
36C7×C22⋊C4—21no—
37C42×C7C2311yescharacteristic
38C22⋊Dic7—21no—
39C22⋊Dic7—21no—
40C22⋊Dic7—21no—
41C4⋊Dic7—21no—
42C4⋊Dic7—21no—
43C4⋊Dic7D411yescharacteristic
44C4⋊Dic7D411yescharacteristic
45C4⋊Dic7D411yescharacteristic
46C4⋊Dic7D411yescharacteristic
47Dic7⋊C4—21no—
48Dic7⋊C4—21no—
49Dic7⋊C4—21no—
50Dic7⋊C4—21no—
51Dic7⋊C4—21no—
52Dic7⋊C4—21no—
53C4×Dic7—21no—
54C2×C7⋊C8—21no—
55#563—71no—
56#501—71no—
57#491—71no—
58#489—71no—
59#330—71no—
60#330—71no—
61#329D711yescharacteristic
62C23×C7—21no—
63C7×Q8—41no—
64C7×D4—41no—
65C2×C4×C7—21no—
66C2×C4×C7C2×D411yescharacteristic
67C2×C4×C7C2×D411yescharacteristic
68C2×C4×C7C2×D411yescharacteristic
69C2×Dic7—21no—
70C2×Dic7—21no—
71C2×Dic7—21no—
72C2×Dic7—21no—
73C2×Dic7—41no—
74C2×Dic7—81no—
75Dic14—42no—
76Dic14—81no—
77C7×C8—22no—
78C7×C8—41no—
79C7⋊C8—41no—
80C4⋊Q8—71no—
81C4⋊Q8—141no—
82#126—141no—
83#126—141no—
84#125—71no—
85#125—71no—
86#124C2×D711yescharacteristic
87#123—141no—
88C22⋊Q8—141no—
89C22⋊Q8—141no—
90#107—141no—
91#106—141no—
92#103—141no—
93#103—141no—
94#103—141no—
95#102—141no—
96#97—71no—
97#97C2×D711yescharacteristic
98#97C2×D711yescharacteristic
99C22×C7—41no—
100C22×C7—41no—
101C22×C7—41no—
102C22×C7#12011yescharacteristic
103C4×C7—21no—
104C4×C7—21no—
105C4×C7—21no—
106C4×C7—41no—
107Dic7—41no—
108Dic7—41no—
109Dic7—41no—
110Dic7—41no—
111Dic7—81no—
112C2×Q8—141no—
113C2×Q8—141no—
114C2×Q8D2811yescharacteristic
115C2×D4D2811yescharacteristic
116C22×C4—141no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—141no—
120C4⋊C4—71no—
121C4⋊C4—71no—
122C4⋊C4—71no—
123C4⋊C4—71no—
124C4⋊C4—141no—
125C4⋊C4—141no—
126C4⋊C4—141no—
127C4⋊C4—141no—
128C4⋊C4—141no—
129C4⋊C4—141no—
130C4⋊C4—141no—
131C4⋊C4—141no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—141no—
135C22⋊C4—141no—
136C22⋊C4—141no—
137C42—141no—
138C42C22×D711yescharacteristic
139C2×C7—81no—
140C2×C7#45411yescharacteristic
141C2×C7#45511yescharacteristic
142C2×C7#45611yescharacteristic
143C23—21no—
144Q8—41no—
145Q8—282no—
146Q8—561no—
147D4—41no—
148C2×C4—21no—
149C2×C4—141no—
150C2×C4—141no—
151C2×C4—141no—
152C2×C4—141no—
153C2×C4—281no—
154C2×C4—561no—
155C2×C4C2×D2811yescharacteristic
156C2×C4D4×D711yescharacteristic
157C2×C4D4×D711yescharacteristic
158C8—22no—
159C8—41no—
160C8—281no—
161C7#166411yescharacteristic, complemented, Sylow
162C22—41no—
163C22—41no—
164C22—41no—
165C22#644511yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—281no—
171C4—281no—
172C4—281no—
173C4—281no—
174C4—561no—
175C2—81no—
176C2#8951811yescharacteristic
177C2#8959811yescharacteristic
178C2#8959911yescharacteristic
subgroupG/H
centerC22#6445
commutator, cosocleC42×C7C23
FrattiniC42C22×D7
FittingC7×#329C2
socleC22×C7#120
prime radicalC2×C7×D4D4
hypercenter#329D7
hypocenterC7#1664
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
3. #89475C2semidirect
4. #89474C2semidirect
6. #89260C2semidirect
7. #89260C2semidirect
161. C7#1664semidirect

Maximal subgroups

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

subgroupindex
#903632
#898272
#894752
#894742
C7×#3292
#892602
#16647