Finite Group Databasebeta

← #13056714

Subgroups — #13056714

Order 1344.

336 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621657C211yesmaximal, characteristic, complemented
2C3×C7×#127C211yesmaximal, characteristic, complemented
3#10621415C212yesmaximal, complemented
4#10620672C211yesmaximal, characteristic, complemented
5#10620644C212yesmaximal, complemented
6#89849—31nomaximal, Hall
7C2×C3×C7×D4—22no—
8C2×C3×C7×D4C2211yescharacteristic
9C2×C3×C7×D4C2211yescharacteristic, complemented
10C2×#4268—22no—
11#66730—22no—
12D4×D21—22no—
13#66728C2212yes—
14C3×C42×C7C2211yescharacteristic, complemented
15#66633—22no—
16#66632—22no—
17#66627C2212yescomplemented
18#6548—31no—
19C7×#127D311yescharacteristic, complemented
20#6495—32no—
21#6407—31no—
22#6379—32no—
23#5077—71nomaximal, Hall
24C23×C3×C7—21no—
25C23×C3×C7D412yes—
26C22×D21—22no—
27C3×C7×D4—22no—
28C3×C7×D4—22no—
29C3×C7×D4—44no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32#4268—24no—
33#4268—22no—
34C2×Dic21—22no—
35D84D412yescomplemented
36C4×D21—22no—
37Dic42D412yescomplemented
38C3⋊(C7⋊C8)—22no—
39C2×C7×D4—22no—
40C2×C7×D4C2×D311yescharacteristic
41C2×C7×D4C2×D311yescharacteristic, complemented
42C2×C22⋊D7—62no—
43#1159—62no—
44D4×D7—62no—
45#1157—32no—
46C42×C7C2×D311yescharacteristic, complemented
47#1142—62no—
48#1141—62no—
49#1136—32no—
50#1004—71no—
51C3×#127D711yescharacteristic, complemented
52#932—72no—
53#833—71no—
54#805—72no—
55C22×C3×C7—22no—
56C22×C3×C7—42no—
57C22×C3×C7—42no—
58C22×C3×C7—41no—
59C22×C3×C7C2×D411yescharacteristic
60C2×D21—22no—
61C2×D21—82no—
62C3×C4×C7—22no—
63C3×C4×C7C2×D412yes—
64Dic21—22no—
65#453—211noSylow
66C23×C7—21no—
67C23×C7#6712yes—
68C22×D7—62no—
69C7×D4—22no—
70C7×D4—22no—
71C7×D4—44no—
72C2×C4×C7—21no—
73C2×C4×C7C22×D311yescharacteristic
74C22⋊D7—64no—
75C22⋊D7—62no—
76C2×Dic7—62no—
77D28—32no—
78C4×D7—62no—
79Dic14—32no—
80C7⋊C8—62no—
81C2×C3×D4—22no—
82C2×C3×D4C2×D711yescharacteristic
83C2×C3×D4C2×D711yescharacteristic, complemented
84C2×#67—142no—
85#237—142no—
86D3×D4—142no—
87#235—72no—
88C3×C42C2×D711yescharacteristic, complemented
89#214—142no—
90#213—142no—
91#208—72no—
92C2×C3×C7—21no—
93C2×C3×C7—42no—
94C2×C3×C7—81no—
95C2×C3×C7#12011yescharacteristic
96D21—42no—
97H(5,𝔽2)—211no—
98AGL(1,ℤ/8)—212no—
99#127D2111yescharacteristic, complemented
100#104—212no—
101#100—211no—
102C22×C7—22no—
103C22×C7—42no—
104C22×C7—42no—
105C22×C7—41no—
106C22×C7C2×#6711yescharacteristic
107C2×D7—62no—
108C2×D7—242no—
109C4×C7—22no—
110C4×C7D3×D412yes—
111Dic7—62no—
112C23×C3—21no—
113C23×C3C22⋊D712yes—
114C22×D3—142no—
115C3×D4—22no—
116C3×D4—22no—
117C3×D4—44no—
118C2×C3×C4—21no—
119C2×C3×C4C22×D711yescharacteristic
120#67—144no—
121#67—142no—
122C2×Dic3—142no—
123D12—72no—
124C4×D3—142no—
125Dic6—72no—
126C3⋊C8—142no—
127C3×C7#45311yescharacteristic, complemented, Hall
128#41—212no—
129#41—422no—
130C2×D4—22no—
131C2×D4—422no—
132C2×D4—422no—
133C2×D4C2×D2111yescharacteristic
134C2×D4C2×D2111yescharacteristic, complemented
135SD16—422no—
136D8—422no—
137M16—212no—
138C42C2×D2111yescharacteristic, complemented
139C2×C7—21no—
140C2×C7—42no—
141C2×C7—81no—
142C2×C7#93711yescharacteristic
143D7—122no—
144C22×C3—22no—
145C22×C3—42no—
146C22×C3—42no—
147C22×C3—41no—
148C22×C3C2×C22⋊D711yescharacteristic
149C2×D3—142no—
150C2×D3—562no—
151C3×C4—22no—
152C3×C4D4×D712yes—
153Dic3—142no—
154C23—21no—
155C23—422no—
156C23#426812yes—
157Q8—212no—
158D4—22no—
159D4—22no—
160D4—44no—
161D4—212no—
162D4—424no—
163D4—422no—
164C2×C4—21no—
165C2×C4—422no—
166C2×C4—422no—
167C2×C4C22×D2111yescharacteristic
168C8—422no—
169C7#507711yescharacteristic, complemented, Sylow
170C2×C3—21no—
171C2×C3—42no—
172C2×C3—81no—
173C2×C3#650011yescharacteristic
174D3—282no—
175C22—22no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—422no—
180C22—1682no—
181C22C2×#426811yescharacteristic
182C4—22no—
183C4—422no—
184C4D4×D2112yes—
185C3#8984911yescharacteristic, complemented, Sylow
186C2—21no—
187C2—42no—
188C2—81no—
189C2—842no—
190C2#1062142011yescharacteristic
subgroupG/H
centerC2#10621420
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#127C2
socleC2×C3×C7#120
hypercenter#127D21
hypocenterC3×C7#453
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #10621657C2semidirect
2. C3×C7×#127C2semidirect
3. #10621415C2semidirect
4. #10620672C2semidirect
5. #10620644C2semidirect
9. C2×C3×C7×D4C22semidirect
14. C3×C42×C7C22semidirect
17. #66627C22semidirect
19. C7×#127D3semidirect
35. D84D4semidirect
37. Dic42D4semidirect
41. C2×C7×D4C2×D3semidirect
46. C42×C7C2×D3semidirect
51. C3×#127D7semidirect
83. C2×C3×D4C2×D7semidirect
88. C3×C42C2×D7semidirect
99. #127D21semidirect
127. C3×C7#453semidirect
134. C2×D4C2×D21semidirect
138. C42C2×D21semidirect
169. C7#5077semidirect
185. C3#89849semidirect

Maximal subgroups

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

subgroupindex
#106216572
C3×C7×#1272
#106214152
#106206722
#106206442
#898493
#50777