Finite Group Databasebeta

← #13056390

Subgroups — #13056390

Order 1344.

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

normal subgroups only

This group has 292 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#10621660C211yesmaximal, characteristic, complemented
2#10621430C211yesmaximal, characteristic, complemented
3#10621391C212yesmaximal, complemented
4C3×C7×#101C211yesmaximal, characteristic, complemented
5#10620664C212yesmaximal, complemented
6#89525—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8#66732—22no—
9Q8×D21—22no—
10#66728—22no—
11#66728C2212yes—
12C2×D84C2211yescharacteristic
13C3×C7×M16C2212yes—
14#66631—22no—
15C4×Dic21C2211yescharacteristic
16D168—22no—
17#66623—22no—
18#6551—31no—
19#6510—31no—
20#6471—32no—
21C7×#101D311yescharacteristic, complemented
22#6399—32no—
23#4753—71nomaximal, Hall
24C22×D21—21no—
25C3×C7×Q8—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28#4268—22no—
29C2×Dic21—21no—
30D84—22no—
31D84—22no—
32D84D412yes—
33C4×D21—24no—
34C4×D21—22no—
35Dic42—22no—
36Dic42D412yes—
37C3×C7×C8—22no—
38C2×C7×Q8C2×D311yescharacteristic
39#1161—62no—
40Q8×D7—62no—
41#1157—32no—
42#1157—62no—
43C2×D28—31no—
44C7×M16C2×D312yes—
45(C2×D7)⋊C4—62no—
46C4×Dic7—31no—
47D56—62no—
48#1132—62no—
49#1007—71no—
50#947—71no—
51#908—72no—
52C3×#101D711yescharacteristic, complemented
53#825—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D21—22no—
56C2×D21—42no—
57C2×D21—41no—
58C3×C4×C7—22no—
59C3×C4×C7C2×D412yes—
60Dic21—22no—
61Dic21—41no—
62#454—211noSylow
63C22×D7—61no—
64C7×Q8—22no—
65C2×C4×C7D1212yes—
66C2×C4×C7C22×D311yescharacteristic
67C22⋊D7—62no—
68C2×Dic7—61no—
69D28—32no—
70D28—62no—
71D28—62no—
72C4×D7—64no—
73C4×D7—62no—
74Dic14—32no—
75Dic14—62no—
76C7×C8—22no—
77C2×C3×Q8C2×D711yescharacteristic
78#239—142no—
79D3×Q8—142no—
80#235—72no—
81#235—142no—
82C2×D12—71no—
83C3×M16C2×D712yes—
84(C2×D3)⋊C4—142no—
85C4×Dic3—71no—
86D24—142no—
87#204—142no—
88C2×C3×C7—21no—
89C2×C3×C7#12011yescharacteristic
90D21—42no—
91D21—81no—
922−1+4—211no—
93AGL(1,ℤ/8)—212no—
94#124—211no—
95#104—212no—
96#101D2111yescharacteristic, complemented
97C22×C7C2×D1211yescharacteristic
98C2×D7—62no—
99C2×D7—122no—
100C2×D7—121no—
101C4×C7—22no—
102C4×C7D3×D412yes—
103Dic7—62no—
104Dic7—121no—
105C22×D3—141no—
106C3×Q8—22no—
107C2×C3×C4D2812yes—
108C2×C3×C4C22×D711yescharacteristic
109#67—142no—
110C2×Dic3—141no—
111D12—72no—
112D12—142no—
113D12—142no—
114C4×D3—144no—
115C4×D3—142no—
116Dic6—72no—
117Dic6—142no—
118C3×C8—22no—
119C3×C7#45411yescharacteristic, complemented, Hall
120#41—212no—
121#41—422no—
122#41—422no—
123C2×Q8—422no—
124C2×Q8C2×D2111yescharacteristic
125C2×D4—211no—
126SD16—422no—
127D8—422no—
128M16C2×D2112yes—
129C22⋊C4—422no—
130C42—211no—
131C2×C7—21no—
132C2×C7#88211yescharacteristic
133D7—122no—
134D7—241no—
135C22×C3C2×D2811yescharacteristic
136C2×D3—142no—
137C2×D3—282no—
138C2×D3—281no—
139C3×C4—22no—
140C3×C4D4×D712yes—
141Dic3—142no—
142Dic3—281no—
143C23—421no—
144Q8—22no—
145Q8—212no—
146Q8—422no—
147D4—212no—
148D4—422no—
149D4—422no—
150D4—422no—
151C2×C4—424no—
152C2×C4—422no—
153C2×C4—421no—
154C2×C4D8412yes—
155C2×C4C22×D2111yescharacteristic
156C8—22no—
157C7#475311yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3#644511yescharacteristic
160D3—282no—
161D3—561no—
162C22—422no—
163C22—842no—
164C22—841no—
165C22C2×D8411yescharacteristic
166C4—22no—
167C4—422no—
168C4—841no—
169C4D4×D2112yes—
170C3#8952511yescharacteristic, complemented, Sylow
171C2—21no—
172C2—842no—
173C2—1681no—
174C2#1062136511yescharacteristic
subgroupG/H
centerC2#10621365
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#101C2
socleC2×C3×C7#120
hypercenter#101D21
hypocenterC3×C7#454
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #10621660C2semidirect
2. #10621430C2semidirect
3. #10621391C2semidirect
4. C3×C7×#101C2semidirect
5. #10620664C2semidirect
21. C7×#101D3semidirect
52. C3×#101D7semidirect
96. #101D21semidirect
119. C3×C7#454semidirect
157. C7#4753semidirect
170. C3#89525semidirect

Maximal subgroups

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

subgroupindex
#106216602
#106214302
#106213912
C3×C7×#1012
#106206642
#895253
#47537