Finite Group Databasebeta

← #13056383

Subgroups — #13056383

Order 1344.

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

normal subgroups only

This group has 304 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
2#10621419C211yesmaximal, characteristic, complemented
3#10621392C212yesmaximal, complemented
4C3×C7×#100C211yesmaximal, characteristic, complemented
5#10620664C212yesmaximal, complemented
6#89518—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×#4268—22no—
9#66730—22no—
10D4×D21—22no—
11#66728C2212yes—
12C2×Dic42C2211yescharacteristic, complemented
13C3×C7×M16C2212yescomplemented
14#66636—22no—
15C4×Dic21C2211yescharacteristic, complemented
16Dic84—22no—
17#66623—22no—
18#6548—31no—
19#6499—31no—
20#6472—32no—
21C7×#100D311yescharacteristic, complemented
22#6399—32no—
23#4746—71nomaximal, Hall
24C23×C3×C7D412yes—
25C22×D21—22no—
26C3×C7×D4—22no—
27C2×C3×C4×C7C2311yescharacteristic
28#4268—24no—
29#4268—22no—
30C2×Dic21—22no—
31C2×Dic21—21no—
32C2×Dic21—21no—
33D84D412yes—
34C4×D21—22no—
35Dic42—22no—
36Dic42D412yes—
37C3×C7×C8—22no—
38C2×C7×D4C2×D311yescharacteristic
39C2×C22⋊D7—62no—
40#1159—62no—
41D4×D7—62no—
42#1157—32no—
43C2×Dic14—31no—
44C7×M16C2×D312yescomplemented
45C22⋊Dic7—62no—
46C4×Dic7—31no—
47Dic28—62no—
48#1132—62no—
49#1004—71no—
50#936—71no—
51#909—72no—
52C3×#100D711yescharacteristic, complemented
53#825—72no—
54C22×C3×C7—22no—
55C22×C3×C7—42no—
56C22×C3×C7C2×D411yescharacteristic
57C2×D21—22no—
58C2×D21—82no—
59C3×C4×C7C2×D412yes—
60Dic21—22no—
61Dic21—41no—
62Dic21—41no—
63#455—211noSylow
64C23×C7D1212yes—
65C22×D7—62no—
66C7×D4—22no—
67C2×C4×C7C22×D311yescharacteristic
68C22⋊D7—64no—
69C22⋊D7—62no—
70C2×Dic7—62no—
71C2×Dic7—61no—
72C2×Dic7—61no—
73D28—32no—
74C4×D7—62no—
75Dic14—32no—
76Dic14—62no—
77C7×C8—22no—
78C2×C3×D4C2×D711yescharacteristic
79C2×#67—142no—
80#237—142no—
81D3×D4—142no—
82#235—72no—
83C2×Dic6—71no—
84C3×M16C2×D712yescomplemented
85#217—142no—
86C4×Dic3—71no—
87Dic12—142no—
88#204—142no—
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7#12011yescharacteristic
92D21—42no—
93H(5,𝔽2)—211no—
94#137—212no—
95#124—211no—
96#104—212no—
97#100D2111yescharacteristic, complemented
98C22×C7—22no—
99C22×C7—42no—
100C22×C7C2×D1211yescharacteristic
101C2×D7—62no—
102C2×D7—242no—
103C4×C7D3×D412yes—
104Dic7—62no—
105Dic7—121no—
106Dic7—121no—
107C23×C3D2812yes—
108C22×D3—142no—
109C3×D4—22no—
110C2×C3×C4C22×D711yescharacteristic
111#67—144no—
112#67—142no—
113C2×Dic3—142no—
114C2×Dic3—141no—
115C2×Dic3—141no—
116D12—72no—
117C4×D3—142no—
118Dic6—72no—
119Dic6—142no—
120C3×C8—22no—
121C3×C7#45511yescharacteristic, complemented, Hall
122#41—212no—
123#41—422no—
124C2×Q8—211no—
125C2×D4—422no—
126C2×D4—422no—
127C2×D4C2×D2111yescharacteristic
128Q16—422no—
129SD16—422no—
130M16C2×D2112yescomplemented
131C22⋊C4—422no—
132C42—211no—
133C2×C7—21no—
134C2×C7—42no—
135C2×C7#88211yescharacteristic
136D7—122no—
137C22×C3—22no—
138C22×C3—42no—
139C22×C3C2×D2811yescharacteristic
140C2×D3—142no—
141C2×D3—562no—
142C3×C4D4×D712yes—
143Dic3—142no—
144Dic3—281no—
145Dic3—281no—
146C23—422no—
147C23D8412yes—
148Q8—212no—
149Q8—422no—
150D4—22no—
151D4—212no—
152D4—424no—
153D4—422no—
154C2×C4—422no—
155C2×C4—422no—
156C2×C4—421no—
157C2×C4—421no—
158C2×C4C22×D2111yescharacteristic
159C8—22no—
160C7#474611yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3#644511yescharacteristic
164D3—282no—
165C22—22no—
166C22—42no—
167C22—422no—
168C22—1682no—
169C22C2×D8411yescharacteristic
170C4—422no—
171C4—841no—
172C4—841no—
173C4D4×D2112yes—
174C3#8951811yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—842no—
178C2#1062136511yescharacteristic
subgroupG/H
centerC2#10621365
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#100C2
socleC2×C3×C7#120
prime radical#10621657C2
hypercenter#100D21
hypocenterC3×C7#455
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. #10621419C2semidirect
3. #10621392C2semidirect
4. C3×C7×#100C2semidirect
5. #10620664C2semidirect
12. C2×Dic42C22semidirect
13. C3×C7×M16C22semidirect
15. C4×Dic21C22semidirect
21. C7×#100D3semidirect
44. C7×M16C2×D3semidirect
52. C3×#100D7semidirect
84. C3×M16C2×D7semidirect
97. #100D21semidirect
121. C3×C7#455semidirect
130. M16C2×D21semidirect
160. C7#4746semidirect
174. C3#89518semidirect

Maximal subgroups

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

subgroupindex
#106216572
#106214192
#106213922
C3×C7×#1002
#106206642
#895183
#47467