Finite Group Databasebeta

← #13052986

Subgroups — #13052986

Order 1344.

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

normal subgroups only

This group has 228 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
1#10621022C211yesmaximal, characteristic
2D3×C4⋊Dic7C211yesmaximal, characteristic, complemented
3#10620660C211yesmaximal, characteristic, complemented
4C7×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
5C3×#6392C211yesmaximal, characteristic, complemented
6#10620525C211yesmaximal, characteristic, complemented
7#10620503C211yesmaximal, characteristic, complemented
8#89508—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×C4×D3×C7C2211yescharacteristic
11C2×#4247—21no—
12C2×D3×Dic7—21no—
13C2×C3×C7×C8C2211yescharacteristic
14C4⋊Dic21C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C4⋊Dic7C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18Dic3⋊Dic7—21no—
19#66575—21no—
20#6493—31no—
21C2×C4⋊Dic7—31no—
22C7×C22⋊C8—31no—
23#6395—31no—
24#6395—31no—
25#6392—31no—
26#6392D311yescharacteristic, complemented
27#4882—71nomaximal, Hall
28C22×D21—21no—
29C22×D3×C7D411yescharacteristic
30C2×C3×C4×C7C2311yescharacteristic
31C2×Dic21—21no—
32D84—21no—
33D84—21no—
34C2×C7×Dic3D411yescharacteristic
35C4×D3×C7—21no—
36C4×D3×C7—21no—
37C2×C3×Dic7—21no—
38C2×C3×Dic7—21no—
39#4247—42no—
40D3×Dic7—42no—
41C3×C7×C8—21no—
42C7×C3⋊C8—21no—
43C22×C4×C7—31no—
44C2×C22⋊D7—61no—
45C22×Dic7—61no—
46C2×D28—31no—
47C2×C7×C8—31no—
48C2×C7×C8C2×D311yescharacteristic
49(C2×D7)⋊C4—61no—
50C4⋊Dic7—31no—
51C4⋊Dic7—61no—
52C4⋊Dic7C2×D311yescharacteristic
53C4⋊Dic7C2×D311yescharacteristic
54#895—71no—
55D3×C4⋊C4—71no—
56C3×#107—71no—
57#821—71no—
58(C2×D3)⋊C8D711yescharacteristic, complemented
59#809—71no—
60#807—71no—
61C22×C3×C7C2×D411yescharacteristic
62C2×D21—41no—
63C2×D21—41no—
64C2×D21—41no—
65C2×D3×C7—21no—
66C2×D3×C7—21no—
67C2×D3×C7D812yes—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—41no—
71C3×Dic7—41no—
72C3×Dic7—41no—
73C7×Dic3—21no—
74#480—211noSylow
75C23×C7—31no—
76C22×D7—61no—
77C2×C4×C7—31no—
78C2×C4×C7—61no—
79C2×C4×C7—61no—
80C2×C4×C7C22×D311yescharacteristic
81C22⋊D7—122no—
82C2×Dic7—21no—
83C2×Dic7—21no—
84C2×Dic7—61no—
85C2×Dic7—122no—
86D28—61no—
87D28—61no—
88C7×C8—21no—
89C7×C8—61no—
90C2×D12—71no—
91C2×D12—141no—
92C2×C4×D3—141no—
93C2×C4×D3C2×D711yescharacteristic
94C2×C3×C8C2×D711yescharacteristic
95C3×C4⋊C4—71no—
96C3×C4⋊C4—71no—
97(C2×D3)⋊C4—141no—
98C4⋊Dic3—71no—
99Dic3⋊C4—141no—
100C2×C3⋊C8C2×D711yescharacteristic
101C2×C3×C7#12311yescharacteristic
102C2×C3×C7C2×D811yescharacteristic
103C2×C3×C7#13711yescharacteristic
104D21—81no—
105D3×C7—22no—
106#121—211no—
107C2×C4⋊C4—211no—
108#107—71no—
109#107—211no—
110#102—211no—
111#102—211no—
112C22⋊C8—31no—
113C22×C7—32no—
114C22×C7—61no—
115C22×C7—61no—
116C22×C7D3×D411yescharacteristic
117C2×D7—121no—
118C2×D7—121no—
119C2×D7—121no—
120C4×C7—61no—
121C4×C7#23511yescharacteristic
122C4×C7#23911yescharacteristic
123Dic7—41no—
124Dic7—41no—
125Dic7—121no—
126C22×D3—141no—
127C22×D3D2811yescharacteristic
128C2×C3×C4—141no—
129C2×C3×C4—141no—
130C2×C3×C4C22×D711yescharacteristic
131C2×Dic3—141no—
132C2×Dic3D2811yescharacteristic
133D12—141no—
134D12—141no—
135D12—282no—
136C4×D3—21no—
137C4×D3—21no—
138C4×D3—282no—
139C3×C8—21no—
140C3⋊C8—21no—
141C3×C7#48011yescharacteristic, complemented, Hall
142C2×D4—211no—
143C2×D4—421no—
144C22×C4—31no—
145C22×C4—421no—
146C2×C8—31no—
147C2×C8D3×D711yescharacteristic
148C4⋊C4—71no—
149C4⋊C4—71no—
150C4⋊C4—211no—
151C4⋊C4—421no—
152C22⋊C4—421no—
153C2×C7—62no—
154C2×C7#89411yescharacteristic
155C2×C7D3×D811yescharacteristic
156C2×C7#91811yescharacteristic
157D7—241no—
158C22×C3C2×D2811yescharacteristic
159C2×D3—21no—
160C2×D3—21no—
161C2×D3—281no—
162C2×D3—281no—
163C2×D3—281no—
164C2×D3D5612yes—
165C3×C4—281no—
166C3×C4—281no—
167C3×C4#115911yescharacteristic
168C3×C4#115911yescharacteristic
169Dic3—21no—
170Dic3—281no—
171C23—31no—
172C23—421no—
173D4—421no—
174D4—421no—
175D4—842no—
176C2×C4—31no—
177C2×C4—61no—
178C2×C4—61no—
179C2×C4—141no—
180C2×C4—141no—
181C2×C4—421no—
182C2×C4—842no—
183C2×C4C2×D3×D711yescharacteristic
184C8—21no—
185C8—61no—
186C7#488211yescharacteristic, complemented, Sylow
187C2×C3#644911yescharacteristic
188C2×C3C2×D5611yescharacteristic
189C2×C3#647211yescharacteristic
190D3—22no—
191D3—561no—
192C22—32no—
193C22—61no—
194C22—61no—
195C22—841no—
196C22—841no—
197C22—841no—
198C22D3×D2811yescharacteristic
199C4—61no—
200C4—281no—
201C4—281no—
202C4—841no—
203C4#6667211yescharacteristic
204C4#6668611yescharacteristic
205C3#8950811yescharacteristic, complemented, Sylow
206C2—62no—
207C2—1681no—
208C2D3×D5611yescharacteristic
209C2#1062082511yescharacteristic
210C2#1062098911yescharacteristic
subgroupG/H
centerC22D3×D28
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×C8C22
socleC22×C3×C7C2×D4
prime radical#10621022C2
hypercenterC2×C8D3×D7
hypocenterC3×C7#480
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. D3×C4⋊Dic7C2semidirect
3. #10620660C2semidirect
4. C7×(C2×D3)⋊C8C2semidirect
5. C3×#6392C2semidirect
6. #10620525C2semidirect
7. #10620503C2semidirect
26. #6392D3semidirect
58. (C2×D3)⋊C8D7semidirect
141. C3×C7#480semidirect
186. C7#4882semidirect
205. C3#89508semidirect

Maximal subgroups

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

subgroupindex
#106210222
D3×C4⋊Dic72
#106206602
C7×(C2×D3)⋊C82
C3×#63922
#106205252
#106205032
#895083
#48827