Finite Group Databasebeta

← #11727615

Subgroups — #11727615

Order 896.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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#90170C211yesmaximal, characteristic
2C4×C7×M16C211yesmaximal, characteristic, complemented
3#89698C211yesmaximal, characteristic, complemented
4#89260C212yesmaximal, complemented
5#89259C212yesmaximal, complemented
6C2×C7×M16—21no—
7C2×C42×C7C2211yescharacteristic
8#6493C411yescharacteristic
9#6492—21no—
10C4×C22⋊D7—21no—
11#6487C411yescharacteristic
12#6439—21no—
13#6438C2211yescharacteristic
14C4×D28C411yescharacteristic
15#6435—21no—
16#6433C2211yescharacteristic
17C4×Dic14C411yescharacteristic
18C7×#97C2212yes—
19C4×C7×C8—21no—
20C22⋊(C7⋊C8)—21no—
21C4⋊(C7⋊C8)—21no—
22#6377C2212yes—
23#1529—71nomaximal, Sylow
24C22×C4×C7C2×C411yescharacteristic
25C22×C4×C7D411yescharacteristic
26C22×C4×C7D411yescharacteristic
27C2×C22⋊D7—21no—
28C2×D28C2×C411yescharacteristic
29C2×C4×D7—21no—
30C2×Dic14C2×C411yescharacteristic
31C7×M16—24no—
32C2×C7×C8—22no—
33C42×C7C2×C411yescharacteristic
34C42×C7D411yescharacteristic
35C42×C7D411yescharacteristic
36C42×C7C2311yescharacteristic
37C22⋊Dic7—21no—
38(C2×D7)⋊C4—21no—
39(C2×D7)⋊C4—21no—
40(C2×D7)⋊C4—21no—
41(C2×D7)⋊C4—21no—
42C4⋊Dic7C2×C411yescharacteristic
43C4⋊Dic7C2×C411yescharacteristic
44Dic7⋊C4—21no—
45Dic7⋊C4—21no—
46Dic7⋊C4—21no—
47Dic7⋊C4—21no—
48C4×Dic7—21no—
49C2×C7⋊C8—22no—
50#529—71no—
51#432—71no—
52C4×M16D711yescharacteristic, complemented
53#330—72no—
54#329—72no—
55C23×C7C22⋊C411yescharacteristic
56C22×D7—21no—
57C2×C4×C7—22no—
58C2×C4×C7—41no—
59C2×C4×C7—41no—
60C2×C4×C7C22⋊C411yescharacteristic
61C2×C4×C7C22⋊C411yescharacteristic
62C2×C4×C7C22⋊C411yescharacteristic
63C2×C4×C7C22×C411yescharacteristic
64C2×C4×C7C2×D411yescharacteristic
65C2×C4×C7C2×D411yescharacteristic
66C22⋊D7—42no—
67C2×Dic7—21no—
68C2×Dic7—21no—
69C2×Dic7—21no—
70D28—41no—
71C4×D7—42no—
72Dic14—41no—
73C7×C8—24no—
74C7⋊C8—42no—
75C2×M16—21no—
76#126—141no—
77#125—71no—
78#124—71no—
79#123—141no—
80C22⋊Q8—71no—
81#121—71no—
82C4×Q8—71no—
83C4×D4—71no—
84C4×D4—141no—
85#117—141no—
86C2×C42C2×D711yescharacteristic
87C4⋊C8—141no—
88C22⋊C8—141no—
89#97—72no—
90#97C2×D712yes—
91C4×C8—21no—
92C22×C7—21no—
93C22×C7—21no—
94C22×C7—21no—
95C22×C7C2×C22⋊C411yescharacteristic
96C2×D7—41no—
97C2×D7—41no—
98C2×D7—41no—
99C4×C7—21no—
100C4×C7—21no—
101C4×C7—21no—
102C4×C7—21no—
103C4×C7—21no—
104C4×C7#10412yes—
105Dic7—41no—
106Dic7—41no—
107Dic7—41no—
108C2×Q8—71no—
109C2×D4—71no—
110C2×D4—141no—
111C22×C4—141no—
112C22×C4C4×D711yescharacteristic
113C22×C4D2811yescharacteristic
114C22×C4C22⋊D711yescharacteristic
115M16—24no—
116C2×C8—22no—
117C2×C8—142no—
118C4⋊C4—71no—
119C4⋊C4—71no—
120C4⋊C4—141no—
121C4⋊C4—141no—
122C4⋊C4—141no—
123C4⋊C4—141no—
124C22⋊C4—141no—
125C22⋊C4—141no—
126C22⋊C4—141no—
127C22⋊C4—141no—
128C22⋊C4—141no—
129C42—141no—
130C42C4×D711yescharacteristic
131C42D2811yescharacteristic
132C42C22⋊D711yescharacteristic
133C42C22×D711yescharacteristic
134C2×C7—41no—
135C2×C7#41311yescharacteristic
136C2×C7C2×#10411yescharacteristic
137C2×C7#42111yescharacteristic
138D7—81no—
139C23—141no—
140C23(C2×D7)⋊C411yescharacteristic
141Q8—281no—
142D4—281no—
143D4—282no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—141no—
148C2×C4—141no—
149C2×C4—141no—
150C2×C4—282no—
151C2×C4(C2×D7)⋊C411yescharacteristic
152C2×C4(C2×D7)⋊C411yescharacteristic
153C2×C4(C2×D7)⋊C411yescharacteristic
154C2×C4C2×C4×D711yescharacteristic
155C2×C4C2×D2811yescharacteristic
156C2×C4C2×C22⋊D711yescharacteristic
157C8—24no—
158C8—282no—
159C7#152911yescharacteristic, complemented, Sylow
160C22—21no—
161C22—21no—
162C22—21no—
163C22—281no—
164C22—281no—
165C22—281no—
166C22C2×(C2×D7)⋊C411yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—281no—
173C4—281no—
174C4—281no—
175C4#637912yes—
176C2—41no—
177C2—561no—
178C2C2×#637911yescharacteristic
179C2#8990411yescharacteristic
180C2#8991111yescharacteristic
subgroupG/H
centerC22C2×(C2×D7)⋊C4
commutatorC2×C4×C7C22×C4
FrattiniC42C22×D7
FittingC4×C7×M16C2
socleC22×C7C2×C22⋊C4
prime radical#6493C4
cosocleC42×C7C23
hypercenterC4×M16D7
hypocenterC7#1529
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. C4×C7×M16C2semidirect
3. #89698C2semidirect
4. #89260C2semidirect
5. #89259C2semidirect
52. C4×M16D7semidirect
159. C7#1529semidirect

Maximal subgroups

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

subgroupindex
#901702
C4×C7×M162
#896982
#892602
#892592
#15297