Finite Group Databasebeta

← #13056520

Subgroups — #13056520

Order 1344.

268 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×D168C211yesmaximal, characteristic, complemented
2C2×C8×D21C211yesmaximal, characteristic, complemented
3#10621378C212yesmaximal, complemented
4C3×C7×#107C211yesmaximal, characteristic, complemented
5#10620648C212yesmaximal, complemented
6#89655—31nomaximal, Hall
7C2×D84—22no—
8C2×D84C2212yes—
9C2×C4×D21C2211yescharacteristic
10C2×C3×C7×C8C2211yescharacteristic
11C3×C7×C4⋊C4C2212yes—
12#66631—22no—
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14D168—22no—
15C8×D21—22no—
16C2×D56—31no—
17C2×C8×D7—31no—
18#6458—32no—
19C7×#107D311yescharacteristic, complemented
20#6383—32no—
21#4883—71nomaximal, Hall
22C22×D21—22no—
23C22×D21D411yescharacteristic
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7C2311yescharacteristic
26C2×Dic21D411yescharacteristic
27D84—22no—
28D84—22no—
29D84—44no—
30C4×D21—21no—
31C4×D21—21no—
32C3×C7×C8D412yescomplemented
33C3⋊(C7⋊C8)—21no—
34C2×D28—32no—
35C2×D28—62no—
36C2×C4×D7—31no—
37C2×C7×C8C2×D311yescharacteristic
38C7×C4⋊C4C2×D312yes—
39(C2×D7)⋊C4—62no—
40C2×C7⋊C8—31no—
41D56—62no—
42C8×D7—62no—
43C2×D24—71no—
44C2×D3×C8—71no—
45#895—72no—
46C3×#107D711yescharacteristic, complemented
47#809—72no—
48C22×C3×C7C2×D411yescharacteristic
49C2×D21—21no—
50C2×D21—21no—
51C2×D21—42no—
52C2×D21—42no—
53C2×D21—42no—
54C2×D21D812yescomplemented
55C3×C4×C7—42no—
56C3×C4×C7C2×D411yescharacteristic
57C3×C4×C7#4111yescharacteristic
58Dic21—21no—
59#466—211noSylow
60C22×D7—31no—
61C22×D7—62no—
62C2×C4×C7—22no—
63C2×C4×C7C22×D311yescharacteristic
64C2×Dic7—31no—
65D28—62no—
66D28—62no—
67D28—124no—
68C4×D7—61no—
69C4×D7—61no—
70C7×C8D1212yescomplemented
71C7⋊C8—61no—
72C2×D12—72no—
73C2×D12—142no—
74C2×C4×D3—71no—
75C2×C3×C8C2×D711yescharacteristic
76C3×C4⋊C4C2×D712yes—
77(C2×D3)⋊C4—142no—
78C2×C3⋊C8—71no—
79D24—142no—
80D3×C8—142no—
81C2×C3×C7#12111yescharacteristic
82C2×C3×C7C2×D811yescharacteristic
83C2×C3×C7#13511yescharacteristic
84D21—22no—
85D21—82no—
86C2×D8—211no—
87C22×C8—211no—
88#121—212no—
89#107D2111yescharacteristic, complemented
90#102—212no—
91C22×C7D3×D411yescharacteristic
92C2×D7—32no—
93C2×D7—61no—
94C2×D7—61no—
95C2×D7—122no—
96C2×D7—122no—
97C2×D7—122no—
98C4×C7—42no—
99C4×C7C2×D1211yescharacteristic
100C4×C7#23911yescharacteristic
101Dic7—61no—
102C22×D3—71no—
103C22×D3—142no—
104C2×C3×C4—22no—
105C2×C3×C4C22×D711yescharacteristic
106C2×Dic3—71no—
107D12—142no—
108D12—142no—
109D12—284no—
110C4×D3—141no—
111C4×D3—141no—
112C3×C8D2812yescomplemented
113C3⋊C8—141no—
114C3×C7#46611yescharacteristic, complemented, Hall
115C2×D4—212no—
116C2×D4—422no—
117C22×C4—211no—
118D8—422no—
119C2×C8—211no—
120C2×C8—422no—
121C2×C8C2×D2111yescharacteristic
122C4⋊C4C2×D2112yes—
123C22⋊C4—422no—
124C2×C7#89511yescharacteristic
125C2×C7D3×D811yescharacteristic
126C2×C7#91911yescharacteristic
127D7—62no—
128D7—242no—
129C22×C3D4×D711yescharacteristic
130C2×D3—72no—
131C2×D3—141no—
132C2×D3—141no—
133C2×D3—282no—
134C2×D3—282no—
135C2×D3—282no—
136C3×C4—42no—
137C3×C4C2×D2811yescharacteristic
138C3×C4#116111yescharacteristic
139Dic3—141no—
140C23—211no—
141C23—422no—
142D4—422no—
143D4—422no—
144D4—844no—
145C2×C4—22no—
146C2×C4—211no—
147C2×C4—421no—
148C2×C4—421no—
149C2×C4C22×D2111yescharacteristic
150C8—421no—
151C8D8412yescomplemented
152C7#488311yescharacteristic, complemented, Sylow
153C2×C3#645811yescharacteristic
154C2×C3D7×D811yescharacteristic
155C2×C3#648211yescharacteristic
156D3—142no—
157D3—562no—
158C22—212no—
159C22—421no—
160C22—421no—
161C22—842no—
162C22—842no—
163C22—842no—
164C22D4×D2111yescharacteristic
165C4—42no—
166C4—421no—
167C4C2×D8411yescharacteristic
168C4#6673211yescharacteristic
169C3#8965511yescharacteristic, complemented, Sylow
170C2—422no—
171C2—1682no—
172C2#1062137811yescharacteristic
173C2D8×D2111yescharacteristic
174C2#1062140211yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#107C2
socleC22×C3×C7C2×D4
hypercenter#107D21
hypocenterC3×C7#466
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
1. C2×D168C2semidirect
2. C2×C8×D21C2semidirect
3. #10621378C2semidirect
4. C3×C7×#107C2semidirect
5. #10620648C2semidirect
19. C7×#107D3semidirect
32. C3×C7×C8D4semidirect
46. C3×#107D7semidirect
54. C2×D21D8semidirect
70. C7×C8D12semidirect
89. #107D21semidirect
112. C3×C8D28semidirect
114. C3×C7#466semidirect
151. C8D84semidirect
152. C7#4883semidirect
169. C3#89655semidirect

Maximal subgroups

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

subgroupindex
C2×D1682
C2×C8×D212
#106213782
C3×C7×#1072
#106206482
#896553
#48837