Finite Group Databasebeta

← #13054760

Subgroups — #13054760

Order 1344.

292 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#6551C211yesmaximal, characteristic, complemented
2#10621430C211yesmaximal, characteristic, complemented
3#10620863C212yesmaximal, complemented
4C7×#836C211yesmaximal, characteristic, complemented
5#10620511C212yesmaximal, complemented
6#89525—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C2×D84C2211yescharacteristic
9C3×#1161—22no—
10C3×Q8×D7—22no—
11C3×#1157—22no—
12C3×#1157C2212yes—
13#66631—22no—
14C4×Dic21C2211yescharacteristic
15C7×#208C2212yes—
16#66566—22no—
17#66561—22no—
18#6551D311yescharacteristic, complemented
19#6510—31no—
20#6471—32no—
21C7×#101—31no—
22#6399—32no—
23#5245—71nomaximal, Hall
24C22×D21—21no—
25C3×C7×Q8—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28C2×Dic21—21no—
29D84—22no—
30C3×C22⋊D7—22no—
31C3×D28—22no—
32C3×D28D412yes—
33C3×C4×D7—24no—
34C3×C4×D7—22no—
35C3×Dic14—22no—
36C3×Dic14D412yes—
37C7×C3⋊C8—22no—
38C2×C7×Q8C2×D311yescharacteristic
39#1161—22no—
40Q8×D7—22no—
41#1157—22no—
42#1157C2×D312yes—
43C2×D28—31no—
44C7×M16—32no—
45(C2×D7)⋊C4—62no—
46C4×Dic7—31no—
47D56—62no—
48#1132—62no—
49C3×2−1+4—71no—
50#949—72no—
51#947—71no—
52#837—72no—
53#836D711yescharacteristic, complemented
54C22×C3×C7C2×D411yescharacteristic
55C2×D21—42no—
56C2×D21—41no—
57C2×C3×D7—22no—
58C3×C4×C7—22no—
59C3×C4×C7C2×D412yes—
60Dic21—41no—
61C3×Dic7—22no—
62#454—211noSylow
63C22×D7—61no—
64C7×Q8—22no—
65C2×C4×C7#6712yes—
66C2×C4×C7C22×D311yescharacteristic
67C22⋊D7—22no—
68C2×Dic7—61no—
69D28—22no—
70D28—62no—
71D28#6712yes—
72C4×D7—24no—
73C4×D7—22no—
74Dic14—22no—
75Dic14#6712yes—
76C7×C8—62no—
77C3×#41—72no—
78C3×#41—142no—
79C3×#41—142no—
80C2×C3×Q8—142no—
81C2×C3×Q8C2×D711yescharacteristic
82C2×D12—71no—
83#215—142no—
84#213—142no—
85(C2×D3)⋊C4—142no—
86C4×Dic3—71no—
87#208C2×D712yes—
88C2×C3×C7—21no—
89C2×C3×C7#12011yescharacteristic
90D21—81no—
91C3×D7—42no—
922−1+4—71no—
93AGL(1,ℤ/8)—212no—
94#124—211no—
95#104—212no—
96#101—31no—
97C22×C7C2×#6711yescharacteristic
98C2×D7—22no—
99C2×D7—122no—
100C2×D7—121no—
101C4×C7—22no—
102C4×C7C2×#6712yes—
103Dic7—22no—
104Dic7—121no—
105C22×D3—141no—
106C3×Q8—22no—
107C3×Q8—72no—
108C3×Q8—142no—
109C3×D4—72no—
110C3×D4—142no—
111C3×D4—142no—
112C2×C3×C4—144no—
113C2×C3×C4—142no—
114C2×C3×C4D2812yes—
115C2×C3×C4C22×D711yescharacteristic
116C2×Dic3—141no—
117D12—142no—
118C3⋊C8—22no—
119C3×C7#45411yescharacteristic, complemented, Hall
120#41—72no—
121#41—142no—
122#41—142no—
123C2×Q8—142no—
124C2×Q8D3×D711yescharacteristic
125C2×D4—211no—
126SD16—422no—
127D8—422no—
128M16—32no—
129C22⋊C4—422no—
130C42—211no—
131C2×C7—21no—
132C2×C7#95311yescharacteristic
133D7—42no—
134D7—241no—
135C22×C3—142no—
136C22×C3C2×D2811yescharacteristic
137C2×D3—282no—
138C2×D3—281no—
139C3×C4—22no—
140C3×C4—142no—
141C3×C4D4×D712yes—
142Dic3—281no—
143C23—421no—
144Q8—22no—
145Q8—72no—
146Q8—142no—
147D4—72no—
148D4—142no—
149D4—142no—
150D4—422no—
151C2×C4—144no—
152C2×C4—142no—
153C2×C4—421no—
154C2×C4#424612yes—
155C2×C4C2×D3×D711yescharacteristic
156C8—62no—
157C7#524511yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3—282no—
160C2×C3#644511yescharacteristic
161D3—561no—
162C22—142no—
163C22—842no—
164C22—841no—
165C22C2×#424611yescharacteristic
166C4—22no—
167C4—142no—
168C4—841no—
169C4D7×#6712yes—
170C3#8952511yescharacteristic, complemented, Sylow
171C2—21no—
172C2—282no—
173C2—1681no—
174C2#1062112611yescharacteristic
subgroupG/H
centerC2#10621126
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
hypercenterC2×Q8D3×D7
hypocenterC3×C7#454
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#6551C2semidirect
2. #10621430C2semidirect
3. #10620863C2semidirect
4. C7×#836C2semidirect
5. #10620511C2semidirect
18. #6551D3semidirect
53. #836D7semidirect
119. C3×C7#454semidirect
157. C7#5245semidirect
170. C3#89525semidirect

Maximal subgroups

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

subgroupindex
C3×#65512
#106214302
#106208632
C7×#8362
#106205112
#895253
#52457