Finite Group Databasebeta

← #13053768

Subgroups — #13053768

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
1C7×#926C211yesmaximal, characteristic, complemented
2C3×#6493C211yesmaximal, characteristic
3#10620669C211yesmaximal, characteristic, complemented
4#10620524C211yesmaximal, characteristic, complemented
5#10620521C211yesmaximal, characteristic, complemented
6#10620499C211yesmaximal, characteristic, complemented
7#10620499C211yesmaximal, characteristic, complemented
8#89775—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C2×C3×D28C2211yescharacteristic
12C2×C3⋊(C7⋊C8)C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×#217—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C4×C7×Dic3—21no—
18C3×(C2×D7)⋊C4—21no—
19C3×C4⋊Dic7C2211yescharacteristic
20C7×#117—31no—
21#6493D311yescharacteristic
22C22⋊(C7⋊C8)—31no—
23#6383—31no—
24#6383—31no—
25#6382—31no—
26#6381—31no—
27#5035—71nomaximal, Hall
28C23×C3×C7D411yescharacteristic
29C22×C3×D7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7D411yescharacteristic
33C2×C3×C4×C7C2311yescharacteristic
34C2×C7×Dic3—21no—
35C2×C7×Dic3—21no—
36C3×C22⋊D7—42no—
37C2×C3×Dic7—21no—
38C3×D28—21no—
39C3×D28—21no—
40C3⋊(C7⋊C8)—21no—
41C3⋊(C7⋊C8)—21no—
42C22×C4×C7C2×D311yescharacteristic
43C2×C22⋊D7—21no—
44C2×D28C2×D311yescharacteristic
45C7×C4⋊C4—31no—
46C7×C4⋊C4—31no—
47C7×C22⋊C4—61no—
48C42×C7—61no—
49(C2×D7)⋊C4—21no—
50C4⋊Dic7C2×D311yescharacteristic
51C2×C7⋊C8—31no—
52C2×C7⋊C8—31no—
53C3×#121—71no—
54#926D711yescharacteristic, complemented
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×C7C2×D411yescharacteristic
64C2×C3×D7—41no—
65C2×C3×D7—41no—
66C2×C3×D7—41no—
67C3×C4×C7—21no—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70C3×Dic7—41no—
71C7×Dic3—41no—
72C7×Dic3—41no—
73#482—211noSylow
74C23×C7#6711yescharacteristic
75C22×D7—21no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—61no—
79C2×C4×C7—61no—
80C2×C4×C7#6711yescharacteristic
81C2×C4×C7C22×D311yescharacteristic
82C22⋊D7—42no—
83C2×Dic7—21no—
84D28—21no—
85D28—21no—
86C7⋊C8—61no—
87C7⋊C8—61no—
88C2×C3×D4—71no—
89C2×C3×D4—141no—
90C22×C3×C4C2×D711yescharacteristic
91C3×C4⋊C4—71no—
92C3×C22⋊C4—141no—
93#217—21no—
94C4⋊Dic3C2×D711yescharacteristic
95C4⋊Dic3C2×D711yescharacteristic
96C4×Dic3—21no—
97C2×C3⋊C8—71no—
98C2×C3⋊C8—71no—
99C2×C3×C7—41no—
100C2×C3×C7#12311yescharacteristic
101C2×C3×C7#13511yescharacteristic
102C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
103C3×D7—81no—
104#121—71no—
105#117—31no—
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×C7C2×#6711yescharacteristic
115C2×D7—41no—
116C2×D7—41no—
117C2×D7—41no—
118C4×C7—21no—
119C4×C7—121no—
120C4×C7—121no—
121C4×C7#23711yescharacteristic
122C4×C7#23711yescharacteristic
123Dic7—41no—
124C23×C3—141no—
125C23×C3C22⋊D711yescharacteristic
126C3×D4—141no—
127C3×D4—141no—
128C3×D4—282no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—141no—
132C2×C3×C4C22⋊D711yescharacteristic
133C2×C3×C4C22×D711yescharacteristic
134C2×Dic3—21no—
135C2×Dic3—21no—
136C3⋊C8—141no—
137C3⋊C8—141no—
138C3×C7#48211yescharacteristic, complemented, Hall
139C2×D4—71no—
140C2×D4—141no—
141C22×C4D3×D711yescharacteristic
142C2×C8—211no—
143C2×C8—211no—
144C4⋊C4—31no—
145C4⋊C4—31no—
146C4⋊C4—71no—
147C22⋊C4—61no—
148C22⋊C4—141no—
149C42—61no—
150C2×C7—41no—
151C2×C7#93211yescharacteristic
152C2×C7#93511yescharacteristic
153C2×C7#95011yescharacteristic
154D7—81no—
155C22×C3—21no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—281no—
159C22×C3—281no—
160C22×C3—281no—
161C22×C3C2×C22⋊D711yescharacteristic
162C3×C4—21no—
163C3×C4—281no—
164C3×C4#115711yescharacteristic
165C3×C4#115711yescharacteristic
166Dic3—41no—
167Dic3—41no—
168C23—141no—
169C23#424511yescharacteristic
170D4—141no—
171D4—141no—
172D4—282no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—141no—
178C2×C4#424511yescharacteristic
179C2×C4C2×D3×D711yescharacteristic
180C8—421no—
181C8—421no—
182C7#503511yescharacteristic, complemented, Sylow
183C2×C3—41no—
184C2×C3—561no—
185C2×C3#649211yescharacteristic
186C2×C3#651211yescharacteristic
187C2×C3#651311yescharacteristic
188C22—21no—
189C22—21no—
190C22—21no—
191C22—281no—
192C22—281no—
193C22—281no—
194C22C2×#424511yescharacteristic
195C4—21no—
196C4—121no—
197C4—121no—
198C4—281no—
199C4#6668311yescharacteristic
200C4#6668311yescharacteristic
201C3#8977511yescharacteristic, complemented, Sylow
202C2—41no—
203C2—561no—
204C2#1062085911yescharacteristic
205C2#1062086011yescharacteristic
206C2#1062109811yescharacteristic
subgroupG/H
centerC22C2×#4245
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC3×#6493C2
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
1. C7×#926C2semidirect
3. #10620669C2semidirect
4. #10620524C2semidirect
5. #10620521C2semidirect
6. #10620499C2semidirect
7. #10620499C2semidirect
54. #926D7semidirect
138. C3×C7#482semidirect
182. C7#5035semidirect
201. C3#89775semidirect

Maximal subgroups

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

subgroupindex
C7×#9262
C3×#64932
#106206692
#106205242
#106205212
#106204992
#897753
#50357