Finite Group Databasebeta

← #11729508

Subgroups — #11729508

Order 896.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C7×#529C211yesmaximal, characteristic
2#89704C211yesmaximal, characteristic, complemented
3#89697C211yesmaximal, characteristic, complemented
4#89333C212yesmaximal, complemented
5#89332C212yesmaximal
6C7×#126—21no—
7C7×#125C411yescharacteristic, complemented
8C7×#124C411yescharacteristic, complemented
9C7×#123—21no—
10C7×C22⋊Q8C2211yescharacteristic
11C7×#121C2211yescharacteristic
12C4×C7×Q8C411yescharacteristic, complemented
13C4×C7×D4—21no—
14C4×C7×D4C411yescharacteristic, complemented
15C7×#117—21no—
16C2×C42×C7C2211yescharacteristic
17C2×C4×Dic7—21no—
18C7⋊#95—21no—
19C7⋊#95C2212yes—
20C22⋊(C7⋊C8)C2212yes—
21C4⋊(C7⋊C8)—21no—
22C4×C7⋊C8—21no—
23#1500—71nomaximal, Sylow
24C2×C7×Q8C2×C411yescharacteristic
25C2×C7×D4—21no—
26C2×C7×D4C2×C411yescharacteristic
27C22×C4×C7—21no—
28C22×C4×C7D411yescharacteristic
29C22×C4×C7D411yescharacteristic
30C22×C4×C7C2311yescharacteristic
31C22×Dic7—22no—
32C7×C4⋊C4—21no—
33C7×C4⋊C4—21no—
34C7×C4⋊C4—21no—
35C7×C4⋊C4—21no—
36C7×C4⋊C4C2×C411yescharacteristic
37C7×C4⋊C4C2×C411yescharacteristic
38C7×C22⋊C4—21no—
39C7×C22⋊C4—21no—
40C7×C22⋊C4—21no—
41C7×C22⋊C4—21no—
42C7×C22⋊C4—21no—
43C42×C7—21no—
44C42×C7C2×C411yescharacteristic
45C42×C7C2×C411yescharacteristic
46C42×C7D411yescharacteristic
47C42×C7D411yescharacteristic
48C4×Dic7—24no—
49C2×C7⋊C8—22no—
50#529D711yescharacteristic
51#431—71no—
52#376—71no—
53#328—72no—
54#327—72no—
55C23×C7—21no—
56C23×C7C2×D411yescharacteristic
57C7×Q8—41no—
58C7×D4—41no—
59C7×D4—42no—
60C2×C4×C7—22no—
61C2×C4×C7—22no—
62C2×C4×C7—21no—
63C2×C4×C7—21no—
64C2×C4×C7—21no—
65C2×C4×C7—41no—
66C2×C4×C7—42no—
67C2×C4×C7C22⋊C411yescharacteristic
68C2×C4×C7C22⋊C411yescharacteristic
69C2×C4×C7C22⋊C411yescharacteristic
70C2×C4×C7C22⋊C411yescharacteristic
71C2×C4×C7C22×C411yescharacteristic
72C2×C4×C7C2×D411yescharacteristic
73C2×Dic7—24no—
74C2×Dic7—42no—
75C2×Dic7—42no—
76C7⋊C8—42no—
77#126—21no—
78#125Dic711yescharacteristic, complemented
79#124Dic711yescharacteristic, complemented
80#123—21no—
81C22⋊Q8C2×D711yescharacteristic
82#121C2×D711yescharacteristic
83C4×Q8Dic711yescharacteristic, complemented
84C4×D4—21no—
85C4×D4Dic711yescharacteristic, complemented
86#117—21no—
87C2×C42—141no—
88C2×C42C2×D711yescharacteristic
89C4⋊C8—141no—
90C22⋊C8—72no—
91C4×C8—141no—
92#95—72no—
93#95—141no—
94C22×C7—21no—
95C22×C7—21no—
96C22×C7—41no—
97C22×C7—41no—
98C22×C7—41no—
99C22×C7C2×C22⋊C411yescharacteristic
100C22×C7#13512yes—
101C4×C7—21no—
102C4×C7—21no—
103C4×C7—21no—
104C4×C7—21no—
105C4×C7—21no—
106C4×C7—41no—
107C4×C7—41no—
108C4×C7—41no—
109C4×C7#10412yes—
110Dic7—44no—
111C2×Q8C2×Dic711yescharacteristic
112C2×D4—21no—
113C2×D4C2×Dic711yescharacteristic
114C22×C4—21no—
115C22×C4—142no—
116C22×C4C22⋊D711yescharacteristic
117C22×C4C22⋊D711yescharacteristic
118C22×C4C22×D711yescharacteristic
119C2×C8—142no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4C2×Dic711yescharacteristic
125C4⋊C4C2×Dic711yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—21no—
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—21no—
131C42—21no—
132C42—144no—
133C42C2×Dic711yescharacteristic
134C42C2×Dic711yescharacteristic
135C42C22⋊D711yescharacteristic
136C42C22⋊D711yescharacteristic
137C2×C7—22no—
138C2×C7—81no—
139C2×C7#41011yescharacteristic
140C2×C7#41611yescharacteristic
141C2×C7C2×#10411yescharacteristic
142C23—21no—
143C23C2×C22⋊D711yescharacteristic
144Q8—41no—
145D4—41no—
146D4—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—144no—
155C2×C4—282no—
156C2×C4—282no—
157C2×C4C22⋊Dic711yescharacteristic
158C2×C4C22⋊Dic711yescharacteristic
159C2×C4C22⋊Dic711yescharacteristic
160C2×C4C22⋊Dic711yescharacteristic
161C2×C4C22×Dic711yescharacteristic
162C2×C4C2×C22⋊D711yescharacteristic
163C8—282no—
164C7#150011yescharacteristic, complemented, Sylow
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22#651312yes—
171C22C2×C22⋊Dic711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4—284no—
181C4#641112yes—
182C2—22no—
183C2—81no—
184C2#9000511yescharacteristic
185C2C2×#641111yescharacteristic
186C2#9001211yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic7
commutatorC2×C4×C7C22×C4
FrattiniC22×C4C22×D7
FittingC7×#529C2
socleC22×C7C2×C22⋊C4
prime radicalC7×#121C22
cosocleC22×C4×C7C23
hypercenter#529D7
hypocenterC7#1500
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
2. #89704C2semidirect
3. #89697C2semidirect
4. #89333C2semidirect
7. C7×#125C4semidirect
8. C7×#124C4semidirect
12. C4×C7×Q8C4semidirect
14. C4×C7×D4C4semidirect
78. #125Dic7semidirect
79. #124Dic7semidirect
83. C4×Q8Dic7semidirect
85. C4×D4Dic7semidirect
164. C7#1500semidirect

Maximal subgroups

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

subgroupindex
C7×#5292
#897042
#896972
#893332
#893322
#15007