Finite Group Databasebeta

← #13056399

Subgroups — #13056399

Order 1344.

248 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621419C211yesmaximal, characteristic, complemented
2C2×#66632C211yesmaximal, characteristic
3C2×#66623C211yesmaximal, characteristic, complemented
4#10621376C211yesmaximal, characteristic, complemented
5C3×C7×#102C211yesmaximal, characteristic, complemented
6#10620653C211yesmaximal, characteristic, complemented
7#10620649C211yesmaximal, characteristic, complemented
8#89559—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C4×D21—21no—
12C2×Dic42C2211yescharacteristic
13C2×C3×C7×C8C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15#66636—21no—
16#66636—21no—
17#66632—22no—
18#66631—21no—
19C4×Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21#66623—22no—
22#6499—31no—
23C2×#1141—31no—
24C2×#1132—31no—
25#6456—31no—
26C7×#102D311yescharacteristic, complemented
27Dic7⋊C8—31no—
28#6384—31no—
29#4787—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D21—21no—
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×Dic21D411yescharacteristic
38C2×Dic21D411yescharacteristic
39D84—21no—
40D84D412yes—
41C4×D21—42no—
42Dic42—21no—
43Dic42—21no—
44C3×C7×C8—21no—
45C3⋊(C7⋊C8)—21no—
46C2×C7×D4C2×D311yescharacteristic
47C2×D28—31no—
48C2×C4×D7—61no—
49C2×Dic14—31no—
50C2×C7×C8C2×D311yescharacteristic
51C7×C4⋊C4C2×D311yescharacteristic
52C22⋊Dic7—61no—
53C22⋊Dic7—61no—
54#1141—62no—
55(C2×D7)⋊C4—61no—
56C4×Dic7—31no—
57C2×C7⋊C8—31no—
58#1132—62no—
59#936—71no—
60C2×#213—71no—
61C2×#204—71no—
62#893—71no—
63C3×#102D711yescharacteristic, complemented
64Dic3⋊C8—71no—
65#810—71no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7C2×D411yescharacteristic
70C2×D21—22no—
71C2×D21—41no—
72C2×D21—41no—
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—21no—
77Dic21—21no—
78Dic21—41no—
79#463—211noSylow
80C23×C7—21no—
81C22×D7—61no—
82C7×D4—21no—
83C7×D4—21no—
84C2×C4×C7—21no—
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—31no—
87C2×Dic7—31no—
88C2×Dic7—61no—
89D28—32no—
90D28—61no—
91C4×D7—122no—
92Dic14—61no—
93Dic14—61no—
94C7×C8—21no—
95C7⋊C8—61no—
96C2×C3×D4C2×D711yescharacteristic
97C2×D12—71no—
98C2×C4×D3—141no—
99C2×Dic6—71no—
100C2×C3×C8C2×D711yescharacteristic
101C3×C4⋊C4C2×D711yescharacteristic
102#217—141no—
103#217—141no—
104#213—142no—
105(C2×D3)⋊C4—141no—
106C4×Dic3—71no—
107C2×C3⋊C8—71no—
108#204—142no—
109C2×C3×C7—81no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7#13511yescharacteristic
112C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
113D21—42no—
114C2×SD16—211no—
115C2×D8—211no—
116#124—211no—
117C4×D4—211no—
118C4⋊C8—211no—
119#103—211no—
120#102D2111yescharacteristic, complemented
121C22×C7—41no—
122C22×C7—41no—
123C22×C7—41no—
124C22×C7D3×D411yescharacteristic
125C2×D7—62no—
126C2×D7—121no—
127C2×D7—121no—
128C4×C7—41no—
129C4×C7#23511yescharacteristic
130C4×C7D3×D411yescharacteristic
131Dic7—61no—
132Dic7—61no—
133Dic7—121no—
134C23×C3—21no—
135C22×D3—141no—
136C3×D4—21no—
137C3×D4—21no—
138C2×C3×C4—21no—
139C2×C3×C4C22×D711yescharacteristic
140C2×Dic3—71no—
141C2×Dic3—71no—
142C2×Dic3—141no—
143D12—72no—
144D12—141no—
145C4×D3—282no—
146Dic6—141no—
147Dic6—141no—
148C3×C8—21no—
149C3⋊C8—141no—
150C3×C7#46311yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—211no—
153C2×D4C2×D2111yescharacteristic
154C22×C4—421no—
155SD16—422no—
156D8—422no—
157C2×C8—211no—
158C2×C8C2×D2111yescharacteristic
159C4⋊C4C2×D2111yescharacteristic
160C22⋊C4—421no—
161C22⋊C4—421no—
162C22⋊C4—421no—
163C42—211no—
164C2×C7—81no—
165C2×C7#88411yescharacteristic
166C2×C7#91111yescharacteristic
167C2×C7#91611yescharacteristic
168D7—122no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3D4×D711yescharacteristic
173C2×D3—142no—
174C2×D3—281no—
175C2×D3—281no—
176C3×C4—41no—
177C3×C4#115711yescharacteristic
178C3×C4D4×D711yescharacteristic
179Dic3—141no—
180Dic3—141no—
181Dic3—281no—
182C23—21no—
183C23—421no—
184Q8—421no—
185Q8—421no—
186D4—21no—
187D4—21no—
188D4—212no—
189D4—421no—
190C2×C4—21no—
191C2×C4—211no—
192C2×C4—211no—
193C2×C4—421no—
194C2×C4—842no—
195C2×C4C22×D2111yescharacteristic
196C8—21no—
197C8—421no—
198C7#478711yescharacteristic, complemented, Sylow
199C2×C3—81no—
200C2×C3#644711yescharacteristic
201C2×C3#647411yescharacteristic
202C2×C3#647911yescharacteristic
203D3—282no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—422no—
208C22—841no—
209C22—841no—
210C22D4×D2111yescharacteristic
211C4—41no—
212C4—421no—
213C4—421no—
214C4—841no—
215C4#6672811yescharacteristic
216C4D4×D2111yescharacteristic
217C3#8955911yescharacteristic, complemented, Sylow
218C2—81no—
219C2—842no—
220C2#1062136711yescharacteristic
221C2#1062139411yescharacteristic
222C2#1062139911yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radicalC2×#66632C2
hypercenter#102D21
hypocenterC3×C7#463
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. #10621419C2semidirect
3. C2×#66623C2semidirect
4. #10621376C2semidirect
5. C3×C7×#102C2semidirect
6. #10620653C2semidirect
7. #10620649C2semidirect
26. C7×#102D3semidirect
63. C3×#102D7semidirect
120. #102D21semidirect
150. C3×C7#463semidirect
198. C7#4787semidirect
217. C3#89559semidirect

Maximal subgroups

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

subgroupindex
#106214192
C2×#666322
C2×#666232
#106213762
C3×C7×#1022
#106206532
#106206492
#895593
#47877