Finite Group Databasebeta

← #13053106

Subgroups — #13053106

Order 1344.

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

normal subgroups only

This group has 204 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
1D12×Dic7C211yesmaximal, characteristic
2#10620950C211yesmaximal, characteristic, complemented
3#10620657C211yesmaximal, characteristic, complemented
4C7×#821C211yesmaximal, characteristic
5C3×Dic7⋊C8C211yesmaximal, characteristic, complemented
6#10620520C211yesmaximal, characteristic, complemented
7#10620502C211yesmaximal, characteristic
8#89538—31nomaximal, Hall
9C2×C7×D12C2211yescharacteristic
10C2×D3×Dic7—21no—
11C2×C3×C7×C8C2211yescharacteristic
12C4⋊Dic21C2211yescharacteristic
13C4⋊Dic21C2211yescharacteristic
14C7×C4⋊Dic3C2211yescharacteristic
15C3×C4×Dic7C2211yescharacteristic
16C2×C3×C7⋊C8C2211yescharacteristic
17Dic3⋊Dic7—21no—
18#66577—21no—
19(C2×D3)⋊Dic7—21no—
20D4×Dic7—31no—
21#6453—31no—
22C7×#102—31no—
23#6406—31no—
24#6392—31no—
25Dic7⋊C8D311yescharacteristic, complemented
26#6382—31no—
27#4847—71nomaximal, Hall
28C22×D3×C7—21no—
29C2×C3×C4×C7C2311yescharacteristic
30C2×Dic21—21no—
31C2×Dic21—21no—
32C2×C7×Dic3—21no—
33C7×D12—21no—
34C7×D12Q812yes—
35C2×C3×Dic7D411yescharacteristic
36C2×C3×Dic7D411yescharacteristic
37D3×Dic7—42no—
38C3×C7×C8—21no—
39C3×C7⋊C8—21no—
40C2×C7×D4—31no—
41C22×Dic7—61no—
42C2×C7×C8C2×D311yescharacteristic
43C7×C4⋊C4—31no—
44C22⋊Dic7—61no—
45C4⋊Dic7—31no—
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#821D711yescharacteristic
56#818—71no—
57#817—71no—
58C22×C3×C7C2×D411yescharacteristic
59C2×D3×C7—22no—
60C2×D3×C7—41no—
61C2×D3×C7—41no—
62C3×C4×C7C2×Q811yescharacteristic
63C3×C4×C7#4111yescharacteristic
64Dic21—41no—
65Dic21—41no—
66C3×Dic7—21no—
67C3×Dic7—21no—
68C7×Dic3—41no—
69#478—211noSylow
70C23×C7—61no—
71C7×D4—32no—
72C7×D4—61no—
73C2×C4×C7—61no—
74C2×C4×C7C22×D311yescharacteristic
75C2×Dic7—61no—
76C2×Dic7—61no—
77C2×Dic7—122no—
78C2×Dic7D1211yescharacteristic
79C2×Dic7D1211yescharacteristic
80C7×C8—21no—
81C7⋊C8—21no—
82C2×D12C2×D711yescharacteristic
83C2×C4×D3—141no—
84C2×C3×C8—71no—
85C2×C3×C8C2×D711yescharacteristic
86C3×C42—71no—
87(C2×D3)⋊C4—141no—
88C4⋊Dic3—71no—
89C4⋊Dic3—71no—
90C4⋊Dic3C2×D711yescharacteristic
91Dic3⋊C4—141no—
92Dic3⋊C4—141no—
93C2×C3×C7C22⋊Q811yescharacteristic
94C2×C3×C7#13511yescharacteristic
95C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
96D3×C7—42no—
97#125—211no—
98C4×D4—211no—
99#107—211no—
100#106—211no—
101C4⋊C8—71no—
102#102—31no—
103#102—211no—
104C22×C7—62no—
105C22×C7—121no—
106C22×C7—121no—
107C22×C7C2×D1211yescharacteristic
108C4×C7—121no—
109C4×C7#23711yescharacteristic
110C4×C7D3×Q811yescharacteristic
111Dic7—21no—
112Dic7—21no—
113Dic7—121no—
114Dic7—121no—
115C22×D3—21no—
116C2×C3×C4—71no—
117C2×C3×C4—71no—
118C2×C3×C4C22×D711yescharacteristic
119C2×Dic3—21no—
120C2×Dic3—141no—
121C2×Dic3—141no—
122D12—21no—
123D12Dic1412yes—
124C4×D3—282no—
125C3×C8—21no—
126C3×C8—141no—
127C3×C7#47811yescharacteristic, complemented, Hall
128C2×D4—31no—
129C22×C4—421no—
130C2×C8—71no—
131C2×C8D3×D711yescharacteristic
132C4⋊C4—31no—
133C4⋊C4—211no—
134C4⋊C4—211no—
135C4⋊C4—421no—
136C4⋊C4—421no—
137C22⋊C4—421no—
138C42—71no—
139C2×C7—122no—
140C2×C7#89711yescharacteristic
141C2×C7#90411yescharacteristic
142C2×C7#90811yescharacteristic
143C22×C3D4×D711yescharacteristic
144C2×D3—22no—
145C2×D3—41no—
146C2×D3—41no—
147C3×C4—141no—
148C3×C4—141no—
149C3×C4C2×Dic1411yescharacteristic
150C3×C4#115911yescharacteristic
151Dic3—41no—
152Dic3—281no—
153Dic3—281no—
154C23—61no—
155D4—32no—
156D4—61no—
157C2×C4—61no—
158C2×C4—71no—
159C2×C4—71no—
160C2×C4—421no—
161C2×C4—421no—
162C2×C4—842no—
163C2×C4C2×D3×D711yescharacteristic
164C8—21no—
165C8—141no—
166C7#484711yescharacteristic, complemented, Sylow
167C2×C3#644111yescharacteristic
168C2×C3#647511yescharacteristic
169C2×C3#647711yescharacteristic
170D3—42no—
171C22—62no—
172C22—121no—
173C22—121no—
174C22D7×D1211yescharacteristic
175C4—121no—
176C4—141no—
177C4—141no—
178C4—841no—
179C4—841no—
180C4D3×Dic1411yescharacteristic
181C4#6668411yescharacteristic
182C3#8953811yescharacteristic, complemented, Sylow
183C2—122no—
184C2#1062081111yescharacteristic
185C2#1062083211yescharacteristic
186C2#1062097911yescharacteristic
subgroupG/H
centerC22D7×D12
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×C8C22
socleC22×C3×C7C2×D4
prime radicalC2×C7×D12C22
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
3. #10620657C2semidirect
5. C3×Dic7⋊C8C2semidirect
6. #10620520C2semidirect
25. Dic7⋊C8D3semidirect
127. C3×C7#478semidirect
166. C7#4847semidirect
182. C3#89538semidirect

Maximal subgroups

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

subgroupindex
D12×Dic72
#106209502
#106206572
C7×#8212
C3×Dic7⋊C82
#106205202
#106205022
#895383
#48477