Finite Group Databasebeta

← #13052529

Subgroups — #13052529

Order 1344.

212 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 60 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×#6458C211yesmaximal, characteristic
2#10620920C211yesmaximal, characteristic, complemented
3#10620647C211yesmaximal, characteristic, complemented
4C7×#807C211yesmaximal, characteristic, complemented
5#10620500C211yesmaximal, characteristic, complemented
6#10620499C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89657—31nomaximal, Hall
9C2×C3×D28—21no—
10C2×C3×D28C2211yescharacteristic
11C2×C3×C4×D7C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C4⋊Dic21C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×(C2×D7)⋊C4—21no—
18(C2×D7)⋊Dic3—21no—
19Dic3×Dic7—21no—
20#6458D311yescharacteristic
21#6455—31no—
22C7×#107—31no—
23#6395—31no—
24(C2×D7)⋊C8—31no—
25#6383—31no—
26#6382—31no—
27#5035—71nomaximal, Hall
28C22×C3×D7—21no—
29C22×C3×D7D411yescharacteristic
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×C7×Dic3—21no—
34C2×C3×Dic7D411yescharacteristic
35C3×D28—21no—
36C3×D28—21no—
37C3×D28—42no—
38C3×C4×D7—21no—
39C3×C4×D7—21no—
40C3⋊(C7⋊C8)—21no—
41C7×C3⋊C8—21no—
42C2×D28—21no—
43C2×D28C2×D311yescharacteristic
44C2×C4×D7C2×D311yescharacteristic
45C2×C7×C8—31no—
46C7×C4⋊C4—31no—
47C7×C4⋊C4C2×D311yescharacteristic
48(C2×D7)⋊C4—21no—
49(C2×D7)⋊C4—61no—
50C4⋊Dic7—31no—
51C4×Dic7—61no—
52C2×C7⋊C8—31no—
53C3×#121—71no—
54#926—71no—
55#832—71no—
56#832—71no—
57#830—71no—
58#808—71no—
59#807D711yescharacteristic, complemented
60C22×C3×C7C2×D411yescharacteristic
61C2×C3×D7—21no—
62C2×C3×D7—21no—
63C2×C3×D7—21no—
64C2×C3×D7—41no—
65C2×C3×D7—41no—
66C2×C3×D7—41no—
67C3×C4×C7—41no—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—41no—
71C3×Dic7—21no—
72C7×Dic3—41no—
73#482—211noSylow
74C22×D7—21no—
75C22×D7#6711yescharacteristic
76C2×C4×C7—21no—
77C2×C4×C7—61no—
78C2×C4×C7C22×D311yescharacteristic
79C2×Dic7—61no—
80C2×Dic7#6711yescharacteristic
81D28—21no—
82D28—21no—
83D28—42no—
84C4×D7—21no—
85C4×D7—21no—
86C7×C8—61no—
87C7⋊C8—61no—
88C2×C3×D4—71no—
89C2×C3×D4—141no—
90C22×C3×C4—71no—
91C3×C4⋊C4C2×D711yescharacteristic
92C3×C22⋊C4—141no—
93#217—141no—
94C4⋊Dic3—71no—
95C4⋊Dic3C2×D711yescharacteristic
96C4×Dic3—141no—
97C2×C3⋊C8—71no—
98C2×C3⋊C8C2×D711yescharacteristic
99C2×C3×C7#12311yescharacteristic
100C2×C3×C7#13511yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C3×D7—41no—
103C3×D7—81no—
104#121—71no—
105#117—211no—
106#107—31no—
107#106—211no—
108#102—211no—
109#102—211no—
110C22⋊C8—211no—
111C22×C7C2×#6711yescharacteristic
112C2×D7—21no—
113C2×D7—21no—
114C2×D7—21no—
115C2×D7—41no—
116C2×D7—41no—
117C2×D7—41no—
118C4×C7—41no—
119C4×C7—121no—
120C4×C7#23711yescharacteristic
121C4×C7#23711yescharacteristic
122Dic7—21no—
123Dic7—121no—
124C23×C3—71no—
125C23×C3—141no—
126C3×D4—141no—
127C3×D4—141no—
128C3×D4—282no—
129C2×C3×C4—21no—
130C2×C3×C4—71no—
131C2×C3×C4—141no—
132C2×C3×C4—141no—
133C2×C3×C4C22×D711yescharacteristic
134C2×Dic3—21no—
135C2×Dic3—141no—
136C3⋊C8—21no—
137C3⋊C8—141no—
138C3×C7#48211yescharacteristic, complemented, Hall
139C2×D4—71no—
140C2×D4—141no—
141C22×C4—71no—
142C2×C8—31no—
143C2×C8—211no—
144C4⋊C4—31no—
145C4⋊C4—211no—
146C4⋊C4D3×D711yescharacteristic
147C22⋊C4—141no—
148C22⋊C4—421no—
149C42—421no—
150C2×C7#93211yescharacteristic
151C2×C7#93511yescharacteristic
152C2×C7#95011yescharacteristic
153D7—41no—
154D7—81no—
155C22×C3—141no—
156C22×C3—141no—
157C22×C3—141no—
158C22×C3—281no—
159C22×C3—281no—
160C22×C3—281no—
161C22×C3D4×D711yescharacteristic
162C3×C4—41no—
163C3×C4—141no—
164C3×C4#115711yescharacteristic
165C3×C4#116111yescharacteristic
166Dic3—41no—
167Dic3—281no—
168C23—71no—
169C23—141no—
170D4—141no—
171D4—141no—
172D4—282no—
173C2×C4—21no—
174C2×C4—61no—
175C2×C4—71no—
176C2×C4—141no—
177C2×C4—141no—
178C2×C4—421no—
179C2×C4C2×D3×D711yescharacteristic
180C8—61no—
181C8—421no—
182C7#503511yescharacteristic, complemented, Sylow
183C2×C3—281no—
184C2×C3—561no—
185C2×C3#645711yescharacteristic
186C2×C3#647411yescharacteristic
187C2×C3#648211yescharacteristic
188C22—141no—
189C22—141no—
190C22—141no—
191C22—281no—
192C22—281no—
193C22—281no—
194C22D7×#6711yescharacteristic
195C4—41no—
196C4—121no—
197C4—141no—
198C4—841no—
199C4#6666911yescharacteristic
200C4#6667011yescharacteristic
201C3#8965711yescharacteristic, complemented, Sylow
202C2—281no—
203C2—561no—
204C2#1062096911yescharacteristic
205C2#1062105111yescharacteristic
206C2#1062107811yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6458C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#482
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #10620920C2semidirect
3. #10620647C2semidirect
4. C7×#807C2semidirect
5. #10620500C2semidirect
6. #10620499C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
59. #807D7semidirect
138. C3×C7#482semidirect
182. C7#5035semidirect
201. C3#89657semidirect

Maximal subgroups

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

subgroupindex
C3×#64582
#106209202
#106206472
C7×#8072
#106205002
#106204992
(C2×D7)⋊(C3⋊C8)2
#896573
#50357