Finite Group Databasebeta

← #13052628

Subgroups — #13052628

Order 1344.

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

normal subgroups only

This group has 208 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×D7×C4⋊C4C211yesmaximal, characteristic
2#10620921C211yesmaximal, characteristic
3#10620646C211yesmaximal, characteristic, complemented
4C7×#810C211yesmaximal, characteristic, complemented
5#10620525C211yesmaximal, characteristic, complemented
6#10620507C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic
8#89581—31nomaximal, Hall
9C2×C7×Dic6C2211yescharacteristic
10C2×C3×C4×D7—21no—
11C2×C3×C4×D7C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C4⋊Dic21C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C4⋊Dic7C2211yescharacteristic
17C3×Dic7⋊C4—21no—
18#66577—21no—
19(C2×D7)⋊Dic3—21no—
20#6509—31no—
21D7×C4⋊C4D311yescharacteristic
22C7×#103—31no—
23#6409—31no—
24(C2×D7)⋊C8—31no—
25#6392—31no—
26#6381—31no—
27#4970—71nomaximal, Hall
28C22×C3×D7D411yescharacteristic
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7C2311yescharacteristic
31C2×Dic21—21no—
32C2×C7×Dic3—21no—
33C7×Dic6—21no—
34C7×Dic6—21no—
35C2×C3×Dic7—21no—
36C2×C3×Dic7D411yescharacteristic
37C3×C4×D7—21no—
38C3×C4×D7—21no—
39C3×C4×D7—42no—
40C3⋊(C7⋊C8)—21no—
41C7×C3⋊C8—21no—
42C2×C7×Q8—31no—
43C2×C4×D7—21no—
44C2×C4×D7C2×D311yescharacteristic
45C2×C7×C8—31no—
46C7×C4⋊C4C2×D311yescharacteristic
47(C2×D7)⋊C4—61no—
48C4⋊Dic7—31no—
49C4⋊Dic7C2×D311yescharacteristic
50Dic7⋊C4—21no—
51Dic7⋊C4—61no—
52C2×C7⋊C8—31no—
53C2×C3×C4⋊C4—71no—
54#924—71no—
55#830—71no—
56#810—71no—
57#810D711yescharacteristic, complemented
58#807—71no—
59#807—71no—
60C22×C3×C7C2×D411yescharacteristic
61C2×C3×D7—21no—
62C2×C3×D7—21no—
63C2×C3×D7Q1612yes—
64C3×C4×C7—41no—
65C3×C4×C7#4111yescharacteristic
66C3×C4×C7#4111yescharacteristic
67Dic21—41no—
68C3×Dic7—21no—
69C3×Dic7—41no—
70C7×Dic3—41no—
71#484—211noSylow
72C22×D7#6711yescharacteristic
73C7×Q8—61no—
74C7×Q8—61no—
75C2×C4×C7—21no—
76C2×C4×C7—61no—
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—21no—
79C2×Dic7—61no—
80C2×Dic7#6711yescharacteristic
81C4×D7—21no—
82C4×D7—21no—
83C4×D7—42no—
84C7×C8—61no—
85C7⋊C8—61no—
86C22×C3×C4—71no—
87C22×C3×C4—141no—
88C2×Dic6C2×D711yescharacteristic
89C3×C4⋊C4—71no—
90C3×C4⋊C4—141no—
91C3×C4⋊C4C2×D711yescharacteristic
92#217—141no—
93C4⋊Dic3—71no—
94Dic3⋊C4—141no—
95C2×C3⋊C8—71no—
96C2×C3⋊C8C2×D711yescharacteristic
97C2×C3×C7#12311yescharacteristic
98C2×C3×C7C2×Q1611yescharacteristic
99C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
100C3×D7—22no—
101C22⋊Q8—211no—
102C2×C4⋊C4—71no—
103#107—211no—
104#107—211no—
105#103—31no—
106#103—211no—
107C22⋊C8—211no—
108C22×C7C2×#6711yescharacteristic
109C2×D7—21no—
110C2×D7—21no—
111C2×D7#21612yes—
112C4×C7—41no—
113C4×C7—121no—
114C4×C7#23511yescharacteristic
115C4×C7#23511yescharacteristic
116Dic7—21no—
117Dic7—41no—
118Dic7—121no—
119C23×C3—71no—
120C2×C3×C4—21no—
121C2×C3×C4—71no—
122C2×C3×C4—141no—
123C2×C3×C4—141no—
124C2×C3×C4—141no—
125C2×C3×C4—282no—
126C2×C3×C4C22×D711yescharacteristic
127C2×Dic3—21no—
128C2×Dic3—141no—
129Dic6—21no—
130Dic6—21no—
131C3⋊C8—21no—
132C3⋊C8—141no—
133C3×C7#48411yescharacteristic, complemented, Hall
134C2×Q8—31no—
135C22×C4—71no—
136C22×C4—141no—
137C2×C8—31no—
138C2×C8—211no—
139C4⋊C4—71no—
140C4⋊C4—141no—
141C4⋊C4—211no—
142C4⋊C4—421no—
143C4⋊C4D3×D711yescharacteristic
144C22⋊C4—421no—
145C2×C7#92911yescharacteristic
146C2×C7#93211yescharacteristic
147C2×C7C2×#21611yescharacteristic
148D7—22no—
149C22×C3—72no—
150C22×C3—141no—
151C22×C3—141no—
152C22×C3D4×D711yescharacteristic
153C3×C4—41no—
154C3×C4—141no—
155C3×C4—281no—
156C3×C4#115711yescharacteristic
157C3×C4#115911yescharacteristic
158Dic3—41no—
159Dic3—281no—
160C23—71no—
161Q8—61no—
162Q8—61no—
163C2×C4—21no—
164C2×C4—61no—
165C2×C4—71no—
166C2×C4—141no—
167C2×C4—141no—
168C2×C4—141no—
169C2×C4—282no—
170C2×C4—421no—
171C2×C4C2×D3×D711yescharacteristic
172C8—61no—
173C8—421no—
174C7#497011yescharacteristic, complemented, Sylow
175C2×C3—142no—
176C2×C3#644611yescharacteristic
177C2×C3#647711yescharacteristic
178C2×C3D7×Q1611yescharacteristic
179C22—72no—
180C22—141no—
181C22—141no—
182C22D7×#6711yescharacteristic
183C4—41no—
184C4—121no—
185C4—141no—
186C4—281no—
187C4—841no—
188C4#6667511yescharacteristic
189C4#6667611yescharacteristic
190C3#8958111yescharacteristic, complemented, Sylow
191C2—142no—
192C2#1062100911yescharacteristic
193C2#1062105311yescharacteristic
194C2D7×#21611yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC22×C3×D7D4
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#484
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #10620646C2semidirect
4. C7×#810C2semidirect
5. #10620525C2semidirect
6. #10620507C2semidirect
57. #810D7semidirect
133. C3×C7#484semidirect
174. C7#4970semidirect
190. C3#89581semidirect

Maximal subgroups

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

subgroupindex
C3×D7×C4⋊C42
#106209212
#106206462
C7×#8102
#106205252
#106205072
(C2×D7)⋊(C3⋊C8)2
#895813
#49707