Finite Group Databasebeta

← #13053858

Subgroups — #13053858

Order 1344.

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

normal subgroups only

This group has 212 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#10621413C211yesmaximal, characteristic
2C7×#926C211yesmaximal, characteristic, complemented
3C3×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4#10620523C211yesmaximal, characteristic, complemented
5#10620520C211yesmaximal, characteristic, complemented
6#10620504C211yesmaximal, characteristic, complemented
7#10620504C211yesmaximal, characteristic, complemented
8#89775—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×#4268—21no—
11C2×D84C2211yescharacteristic
12#66631—21no—
13C4⋊Dic21C2211yescharacteristic
14C7×#217—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C4×C7×Dic3—21no—
18C2×C3×C7⋊C8C2211yescharacteristic
19C2×C3×C7⋊C8C2211yescharacteristic
20C7×#117—31no—
21#6493—31no—
22C22⋊(C7⋊C8)D311yescharacteristic, complemented
23#6383—31no—
24#6383—31no—
25#6382—31no—
26#6381—31no—
27#4737—71nomaximal, Hall
28C23×C3×C7D411yescharacteristic
29C22×D21—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7D411yescharacteristic
33C2×C3×C4×C7C2311yescharacteristic
34#4268—42no—
35C2×Dic21—21no—
36D84—21no—
37D84—21no—
38C2×C7×Dic3—21no—
39C2×C7×Dic3—21no—
40C3×C7⋊C8—21no—
41C3×C7⋊C8—21no—
42C22×C4×C7C2×D311yescharacteristic
43C2×C22⋊D7—61no—
44C2×D28—31no—
45C7×C4⋊C4—31no—
46C7×C4⋊C4—31no—
47C7×C22⋊C4—61no—
48C42×C7—61no—
49(C2×D7)⋊C4—61no—
50C4⋊Dic7—31no—
51C2×C7⋊C8C2×D311yescharacteristic
52C2×C7⋊C8C2×D311yescharacteristic
53#930—71no—
54#926D711yescharacteristic, complemented
55C3×C22⋊C8—71no—
56#821—71no—
57#821—71no—
58#818—71no—
59#817—71no—
60C22×C3×C7—21no—
61C22×C3×C7—21no—
62C22×C3×C7—21no—
63C22×C3×C7C2×D411yescharacteristic
64C2×D21—41no—
65C2×D21—41no—
66C2×D21—41no—
67C3×C4×C7—21no—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—41no—
71C7×Dic3—41no—
72C7×Dic3—41no—
73#482—211noSylow
74C23×C7D1211yescharacteristic
75C22×D7—61no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—61no—
79C2×C4×C7—61no—
80C2×C4×C7D1211yescharacteristic
81C2×C4×C7C22×D311yescharacteristic
82C22⋊D7—122no—
83C2×Dic7—61no—
84D28—61no—
85D28—61no—
86C7⋊C8—21no—
87C7⋊C8—21no—
88C22×C3×C4C2×D711yescharacteristic
89C2×#67—141no—
90C2×D12—71no—
91C2×C3×C8—71no—
92C2×C3×C8—71no—
93#217—21no—
94(C2×D3)⋊C4—141no—
95C4⋊Dic3—71no—
96C4⋊Dic3C2×D711yescharacteristic
97C4⋊Dic3C2×D711yescharacteristic
98C4×Dic3—21no—
99C2×C3×C7—41no—
100C2×C3×C7#12311yescharacteristic
101C2×C3×C7#13511yescharacteristic
102C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
103D21—81no—
104#121—211no—
105#117—31no—
106#107—211no—
107#106—211no—
108#102—211no—
109#102—211no—
110C22⋊C8—71no—
111C22×C7—21no—
112C22×C7—21no—
113C22×C7—21no—
114C22×C7C2×D1211yescharacteristic
115C2×D7—121no—
116C2×D7—121no—
117C2×D7—121no—
118C4×C7—21no—
119C4×C7—121no—
120C4×C7—121no—
121C4×C7#23711yescharacteristic
122C4×C7#23711yescharacteristic
123Dic7—121no—
124C23×C3C22⋊D711yescharacteristic
125C22×D3—141no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4C22⋊D711yescharacteristic
129C2×C3×C4C22×D711yescharacteristic
130#67—282no—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—141no—
134D12—141no—
135D12—141no—
136C3×C8—141no—
137C3×C8—141no—
138C3×C7#48211yescharacteristic, complemented, Hall
139C2×D4—211no—
140C2×D4—421no—
141C22×C4D3×D711yescharacteristic
142C2×C8—71no—
143C2×C8—71no—
144C4⋊C4—31no—
145C4⋊C4—31no—
146C4⋊C4—211no—
147C22⋊C4—61no—
148C22⋊C4—421no—
149C42—61no—
150C2×C7—41no—
151C2×C7#88611yescharacteristic
152C2×C7#90411yescharacteristic
153C2×C7#90811yescharacteristic
154D7—241no—
155C22×C3—21no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3C2×C22⋊D711yescharacteristic
159C2×D3—281no—
160C2×D3—281no—
161C2×D3—281no—
162C3×C4—21no—
163C3×C4#115711yescharacteristic
164C3×C4#115711yescharacteristic
165Dic3—41no—
166Dic3—41no—
167Dic3—281no—
168C23—421no—
169C23#424711yescharacteristic
170D4—421no—
171D4—421no—
172D4—842no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—421no—
178C2×C4#424711yescharacteristic
179C2×C4C2×D3×D711yescharacteristic
180C8—141no—
181C8—141no—
182C7#473711yescharacteristic, complemented, Sylow
183C2×C3—41no—
184C2×C3#649211yescharacteristic
185C2×C3#651211yescharacteristic
186C2×C3#651311yescharacteristic
187D3—561no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—841no—
192C22—841no—
193C22—841no—
194C22C2×#424711yescharacteristic
195C4—21no—
196C4—121no—
197C4—121no—
198C4—841no—
199C4#6668311yescharacteristic
200C4#6668311yescharacteristic
201C3#8977511yescharacteristic, complemented, Sylow
202C2—41no—
203C2—1681no—
204C2#1062086511yescharacteristic
205C2#1062086711yescharacteristic
206C2#1062110411yescharacteristic
subgroupG/H
centerC22C2×#4247
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radical#10621413C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#482
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C7×#926C2semidirect
3. C3×C22⋊(C7⋊C8)C2semidirect
4. #10620523C2semidirect
5. #10620520C2semidirect
6. #10620504C2semidirect
7. #10620504C2semidirect
22. C22⋊(C7⋊C8)D3semidirect
54. #926D7semidirect
138. C3×C7#482semidirect
182. C7#4737semidirect
201. C3#89775semidirect

Maximal subgroups

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

subgroupindex
#106214132
C7×#9262
C3×C22⋊(C7⋊C8)2
#106205232
#106205202
#106205042
#897753
#47377