Finite Group Databasebeta

← #13056672

Subgroups — #13056672

Order 1344.

212 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 67 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×C7×#121C211yesmaximal, characteristic
2#10621409C211yesmaximal, characteristic, complemented
3#10620671C211yesmaximal, characteristic, complemented
4#10620671C211yesmaximal, characteristic, complemented
5#10620669C211yesmaximal, characteristic, complemented
6#10620647C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89807—31nomaximal, Hall
9C2×C3×C7×D4—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C3×C7×C22⋊C4—21no—
14#66636—21no—
15C4⋊Dic21C2211yescharacteristic
16C4⋊Dic21C2211yescharacteristic
17C4×Dic21—21no—
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×#121D311yescharacteristic
21#6489—31no—
22#6406—31no—
23#6406—31no—
24C22⋊(C7⋊C8)—31no—
25#6382—31no—
26#6381—31no—
27#5035—71nomaximal, Hall
28C23×C3×C7—21no—
29C23×C3×C7D411yescharacteristic
30C3×C7×D4—21no—
31C3×C7×D4—21no—
32C3×C7×D4—42no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7D411yescharacteristic
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21—21no—
39C2×Dic21—21no—
40C3⋊(C7⋊C8)—21no—
41C3⋊(C7⋊C8)—21no—
42C2×C7×D4—21no—
43C2×C7×D4C2×D311yescharacteristic
44C22×C4×C7C2×D311yescharacteristic
45C7×C4⋊C4C2×D311yescharacteristic
46C7×C22⋊C4—21no—
47C22⋊Dic7—61no—
48C4⋊Dic7—31no—
49C4⋊Dic7—31no—
50C4×Dic7—61no—
51C2×C7⋊C8—31no—
52C2×C7⋊C8—31no—
53C3×#121D711yescharacteristic
54#926—71no—
55#832—71no—
56#832—71no—
57#830—71no—
58#808—71no—
59#807—71no—
60C22×C3×C7—21no—
61C22×C3×C7—21no—
62C22×C3×C7—21no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7—41no—
66C22×C3×C7C2×D411yescharacteristic
67C3×C4×C7—21no—
68C3×C4×C7—41no—
69C3×C4×C7#4111yescharacteristic
70C3×C4×C7#4111yescharacteristic
71Dic21—41no—
72Dic21—41no—
73#482—211noSylow
74C23×C7—21no—
75C23×C7#6711yescharacteristic
76C7×D4—21no—
77C7×D4—21no—
78C7×D4—42no—
79C2×C4×C7—21no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7#6711yescharacteristic
83C2×C4×C7C22×D311yescharacteristic
84C2×Dic7—61no—
85C2×Dic7—61no—
86C7⋊C8—61no—
87C7⋊C8—61no—
88C2×C3×D4—21no—
89C2×C3×D4C2×D711yescharacteristic
90C22×C3×C4C2×D711yescharacteristic
91C3×C4⋊C4C2×D711yescharacteristic
92C3×C22⋊C4—21no—
93#217—141no—
94C4⋊Dic3—71no—
95C4⋊Dic3—71no—
96C4×Dic3—141no—
97C2×C3⋊C8—71no—
98C2×C3⋊C8—71no—
99C2×C3×C7—41no—
100C2×C3×C7—81no—
101C2×C3×C7#12311yescharacteristic
102C2×C3×C7#13511yescharacteristic
103C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
104#121D2111yescharacteristic
105#117—211no—
106#107—211no—
107#106—211no—
108#102—211no—
109#102—211no—
110C22⋊C8—211no—
111C22×C7—21no—
112C22×C7—21no—
113C22×C7—21no—
114C22×C7—41no—
115C22×C7—41no—
116C22×C7—41no—
117C22×C7C2×#6711yescharacteristic
118C4×C7—21no—
119C4×C7—41no—
120C4×C7#23711yescharacteristic
121C4×C7#23711yescharacteristic
122Dic7—121no—
123Dic7—121no—
124C23×C3—21no—
125C23×C3C22⋊D711yescharacteristic
126C3×D4—21no—
127C3×D4—21no—
128C3×D4—42no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4C22⋊D711yescharacteristic
133C2×C3×C4C22×D711yescharacteristic
134C2×Dic3—141no—
135C2×Dic3—141no—
136C3⋊C8—141no—
137C3⋊C8—141no—
138C3×C7#48211yescharacteristic, complemented, Hall
139C2×D4—21no—
140C2×D4C2×D2111yescharacteristic
141C22×C4C2×D2111yescharacteristic
142C2×C8—211no—
143C2×C8—211no—
144C4⋊C4—211no—
145C4⋊C4—211no—
146C4⋊C4C2×D2111yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—421no—
149C42—421no—
150C2×C7—41no—
151C2×C7—81no—
152C2×C7#93211yescharacteristic
153C2×C7#93511yescharacteristic
154C2×C7#95011yescharacteristic
155C22×C3—21no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3C2×C22⋊D711yescharacteristic
162C3×C4—21no—
163C3×C4—41no—
164C3×C4#115911yescharacteristic
165C3×C4#115911yescharacteristic
166Dic3—281no—
167Dic3—281no—
168C23—21no—
169C23#426811yescharacteristic
170D4—21no—
171D4—21no—
172D4—42no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—421no—
177C2×C4—421no—
178C2×C4#426811yescharacteristic
179C2×C4C22×D2111yescharacteristic
180C8—421no—
181C8—421no—
182C7#503511yescharacteristic, complemented, Sylow
183C2×C3—41no—
184C2×C3—81no—
185C2×C3#649511yescharacteristic
186C2×C3#649811yescharacteristic
187C2×C3#651311yescharacteristic
188C22—21no—
189C22—21no—
190C22—21no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22C2×#426811yescharacteristic
195C4—21no—
196C4—41no—
197C4—841no—
198C4—841no—
199C4#6673011yescharacteristic
200C4#6673011yescharacteristic
201C3#8980711yescharacteristic, complemented, Sylow
202C2—41no—
203C2—81no—
204C2#1062141511yescharacteristic
205C2#1062141811yescharacteristic
206C2#1062143311yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
Fitting, prime radicalC3×C7×#121C2
socleC22×C3×C7C2×D4
hypercenter#121D21
hypocenterC3×C7#482
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #10621409C2semidirect
3. #10620671C2semidirect
4. #10620671C2semidirect
5. #10620669C2semidirect
6. #10620647C2semidirect
7. #10620646C2semidirect
138. C3×C7#482semidirect
182. C7#5035semidirect
201. C3#89807semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
#106214092
#106206712
#106206692
#106206472
#106206462
#898073
#50357