Finite Group Databasebeta

← #13053108

Subgroups — #13053108

Order 1344.

204 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 62 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#10620980C211yesmaximal, characteristic
2#10620950C211yesmaximal, characteristic, complemented
3#10620660C211yesmaximal, characteristic
4C7×#817C211yesmaximal, characteristic, complemented
5C3×Dic7⋊C8C211yesmaximal, characteristic, complemented
6#10620523C211yesmaximal, characteristic, complemented
7#10620504C211yesmaximal, characteristic
8#89643—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×D21⋊C4—21no—
11C2×C3×C7×C8C2211yescharacteristic
12C4⋊Dic21C2211yescharacteristic
13C7×C4⋊Dic3C2211yescharacteristic
14C7×C4⋊Dic3C2211yescharacteristic
15C3×C4×Dic7C2211yescharacteristic
16C2×C3×C7⋊C8C2211yescharacteristic
17Dic3⋊Dic7—21no—
18#66577—21no—
19#66575—21no—
20#6456—31no—
21#6453—31no—
22C7×#106—31no—
23#6395—31no—
24Dic7⋊C8D311yescharacteristic, complemented
25#6383—31no—
26#6381—31no—
27#4847—71nomaximal, Hall
28C22×D21—21no—
29C2×C3×C4×C7C2311yescharacteristic
30C2×Dic21—21no—
31D84—21no—
32D84Q812yes—
33C2×C7×Dic3—21no—
34C2×C7×Dic3—21no—
35C2×C3×Dic7D411yescharacteristic
36C2×C3×Dic7D411yescharacteristic
37D21⋊C4—42no—
38C3×C7×C8—21no—
39C3×C7⋊C8—21no—
40C2×D28—31no—
41C2×C4×D7—61no—
42C2×C7×C8C2×D311yescharacteristic
43C7×C4⋊C4—31no—
44C7×C4⋊C4—31no—
45(C2×D7)⋊C4—61no—
46C4⋊Dic7—31no—
47C4⋊Dic7—61no—
48Dic7⋊C4—61no—
49C4×Dic7C2×D311yescharacteristic
50C2×C7⋊C8C2×D311yescharacteristic
51C4×D12—71no—
52#870—71no—
53C3×C4⋊C8—71no—
54#821—71no—
55#821—71no—
56#818—71no—
57#817D711yescharacteristic, complemented
58C22×C3×C7C2×D411yescharacteristic
59C2×D21—22no—
60C2×D21—41no—
61C2×D21—41no—
62C3×C4×C7C2×Q811yescharacteristic
63C3×C4×C7#4111yescharacteristic
64Dic21—41no—
65C3×Dic7—21no—
66C3×Dic7—21no—
67C7×Dic3—41no—
68C7×Dic3—41no—
69#478—211noSylow
70C22×D7—61no—
71C2×C4×C7—61no—
72C2×C4×C7—61no—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—61no—
75C2×Dic7D1211yescharacteristic
76C2×Dic7D1211yescharacteristic
77D28—32no—
78D28—61no—
79C4×D7—122no—
80C7×C8—21no—
81C7⋊C8—21no—
82C2×D12—71no—
83C2×C4×D3—141no—
84C2×C3×C8—71no—
85C2×C3×C8C2×D711yescharacteristic
86C3×C42—71no—
87(C2×D3)⋊C4—141no—
88C4⋊Dic3—71no—
89C4⋊Dic3C2×D711yescharacteristic
90C4⋊Dic3C2×D711yescharacteristic
91Dic3⋊C4—141no—
92Dic3⋊C4—141no—
93C2×C3×C7C22⋊Q811yescharacteristic
94C2×C3×C7#13511yescharacteristic
95C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
96D21—42no—
97#125—211no—
98C4×D4—211no—
99#107—211no—
100#106—31no—
101C4⋊C8—71no—
102#102—211no—
103#102—211no—
104C22×C7C2×D1211yescharacteristic
105C2×D7—62no—
106C2×D7—121no—
107C2×D7—121no—
108C4×C7—121no—
109C4×C7—121no—
110C4×C7#23711yescharacteristic
111C4×C7D3×Q811yescharacteristic
112Dic7—21no—
113Dic7—21no—
114Dic7—121no—
115C22×D3—141no—
116C2×C3×C4—71no—
117C2×C3×C4—71no—
118C2×C3×C4C22×D711yescharacteristic
119C2×Dic3—21no—
120C2×Dic3—21no—
121C2×Dic3—141no—
122D12—72no—
123D12—141no—
124C4×D3—282no—
125C3×C8—21no—
126C3×C8—141no—
127C3×C7#47811yescharacteristic, complemented, Hall
128C2×D4—211no—
129C22×C4—421no—
130C2×C8—71no—
131C2×C8D3×D711yescharacteristic
132C4⋊C4—31no—
133C4⋊C4—31no—
134C4⋊C4—211no—
135C4⋊C4—421no—
136C4⋊C4—421no—
137C22⋊C4—421no—
138C42—71no—
139C2×C7#89711yescharacteristic
140C2×C7#90411yescharacteristic
141C2×C7#90811yescharacteristic
142D7—122no—
143C22×C3D4×D711yescharacteristic
144C2×D3—142no—
145C2×D3—281no—
146C2×D3—281no—
147C3×C4—141no—
148C3×C4—141no—
149C3×C4#115711yescharacteristic
150C3×C4Q8×D711yescharacteristic
151Dic3—41no—
152Dic3—41no—
153Dic3—281no—
154C23—421no—
155D4—212no—
156D4—421no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—71no—
160C2×C4—71no—
161C2×C4—421no—
162C2×C4—842no—
163C2×C4C2×D3×D711yescharacteristic
164C8—21no—
165C8—141no—
166C7#484711yescharacteristic, complemented, Sylow
167C2×C3#645911yescharacteristic
168C2×C3#647711yescharacteristic
169C2×C3#647911yescharacteristic
170D3—282no—
171C22—422no—
172C22—841no—
173C22—841no—
174C22D7×D1211yescharacteristic
175C4—121no—
176C4—121no—
177C4—141no—
178C4—141no—
179C4—841no—
180C4D21⋊Q811yescharacteristic
181C4#6668311yescharacteristic
182C3#8964311yescharacteristic, complemented, Sylow
183C2—842no—
184C2#1062081111yescharacteristic
185C2#1062083111yescharacteristic
186C2#1062098111yescharacteristic
subgroupG/H
centerC22D7×D12
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×C8C22
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenterC2×C8D3×D7
hypocenterC3×C7#478
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #10620950C2semidirect
4. C7×#817C2semidirect
5. C3×Dic7⋊C8C2semidirect
6. #10620523C2semidirect
24. Dic7⋊C8D3semidirect
57. #817D7semidirect
127. C3×C7#478semidirect
166. C7#4847semidirect
182. C3#89643semidirect

Maximal subgroups

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

subgroupindex
#106209802
#106209502
#106206602
C7×#8172
C3×Dic7⋊C82
#106205232
#106205042
#896433
#48477