Finite Group Databasebeta

← #13056450

Subgroups — #13056450

Order 1344.

220 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621429C211yesmaximal, characteristic
2C2×#66635C211yesmaximal, characteristic, complemented
3C2×#66622C211yesmaximal, characteristic
4#10621380C211yesmaximal, characteristic
5C3×C7×#103C211yesmaximal, characteristic, complemented
6#10620655C211yesmaximal, characteristic, complemented
7#10620647C211yesmaximal, characteristic, complemented
8#89588—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C4×D21C2211yescharacteristic
11C2×Dic42C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66635—22no—
15#66631—21no—
16#66631—21no—
17C4⋊Dic21—21no—
18C4⋊Dic21C2211yescharacteristic
19#66629—21no—
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21#66622—22no—
22#6509—31no—
23C2×#1144—31no—
24C2×#1131—31no—
25#6460—31no—
26C7×#103D311yescharacteristic, complemented
27#6390—31no—
28#6382—31no—
29#4816—71nomaximal, Hall
30C22×D21D411yescharacteristic
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×Dic21—21no—
38C2×Dic21D411yescharacteristic
39C4×D21—21no—
40C4×D21—21no—
41Dic42—21no—
42Dic42—21no—
43C3×C7×C8—21no—
44C3⋊(C7⋊C8)D412yes—
45C2×C7×Q8C2×D311yescharacteristic
46C2×C4×D7—31no—
47C2×Dic14—31no—
48C2×C7×C8C2×D311yescharacteristic
49C7×C4⋊C4C2×D311yescharacteristic
50#1144—62no—
51(C2×D7)⋊C4—61no—
52(C2×D7)⋊C4—61no—
53C4⋊Dic7—31no—
54C4⋊Dic7—61no—
55Dic7⋊C4—61no—
56C2×C7⋊C8—31no—
57#1131—62no—
58#946—71no—
59C2×#216—71no—
60C2×ΓL(1,𝔽25)—71no—
61#897—71no—
62C3×#103D711yescharacteristic, complemented
63#816—71no—
64#808—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×D21—21no—
67C2×D21—21no—
68C2×D21—21no—
69C3×C4×C7—41no—
70C3×C4×C7—41no—
71C3×C4×C7C2×D411yescharacteristic
72C3×C4×C7#4111yescharacteristic
73Dic21—21no—
74Dic21—41no—
75Dic21—41no—
76#470—211noSylow
77C22×D7—31no—
78C7×Q8—21no—
79C7×Q8—21no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7C22×D311yescharacteristic
83C2×Dic7—31no—
84C2×Dic7—61no—
85C2×Dic7—61no—
86C4×D7—61no—
87C4×D7—61no—
88Dic14—61no—
89Dic14—61no—
90C7×C8—21no—
91C7⋊C8—32no—
92C2×C3×Q8C2×D711yescharacteristic
93C2×C4×D3—71no—
94C2×Dic6—71no—
95C2×C3×C8C2×D711yescharacteristic
96C3×C4⋊C4C2×D711yescharacteristic
97#216—142no—
98(C2×D3)⋊C4—141no—
99(C2×D3)⋊C4—141no—
100C4⋊Dic3—71no—
101C4⋊Dic3—141no—
102Dic3⋊C4—141no—
103C2×C3⋊C8—71no—
104ΓL(1,𝔽25)—142no—
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13711yescharacteristic
107C2×C3×C7#13711yescharacteristic
108D21—41no—
109C2×Q16—211no—
110C2×M16—211no—
111C22⋊Q8—211no—
112C22⋊Q8—211no—
113#106—211no—
114#103—211no—
115#103D2111yescharacteristic, complemented
116C22×C7D3×D411yescharacteristic
117C2×D7—61no—
118C2×D7—61no—
119C2×D7—61no—
120C4×C7—41no—
121C4×C7—41no—
122C4×C7#23511yescharacteristic
123C4×C7D3×D411yescharacteristic
124Dic7—61no—
125Dic7—121no—
126Dic7—121no—
127C22×D3—71no—
128C3×Q8—21no—
129C3×Q8—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4C22×D711yescharacteristic
133C2×Dic3—71no—
134C2×Dic3—141no—
135C2×Dic3—141no—
136C4×D3—141no—
137C4×D3—141no—
138Dic6—141no—
139Dic6—141no—
140C3×C8—21no—
141C3⋊C8—72no—
142C3×C7#47011yescharacteristic, complemented, Hall
143C2×Q8—211no—
144C2×Q8C2×D2111yescharacteristic
145C22×C4—211no—
146Q16—422no—
147M16—422no—
148C2×C8—211no—
149C2×C8C2×D2111yescharacteristic
150C4⋊C4—211no—
151C4⋊C4—421no—
152C4⋊C4—421no—
153C4⋊C4C2×D2111yescharacteristic
154C22⋊C4—421no—
155C22⋊C4—421no—
156C2×C7#88411yescharacteristic
157C2×C7#91511yescharacteristic
158C2×C7#91811yescharacteristic
159D7—121no—
160C22×C3D4×D711yescharacteristic
161C2×D3—141no—
162C2×D3—141no—
163C2×D3—141no—
164C3×C4—41no—
165C3×C4—41no—
166C3×C4#115711yescharacteristic
167C3×C4D4×D711yescharacteristic
168Dic3—141no—
169Dic3—281no—
170Dic3—281no—
171C23—211no—
172Q8—21no—
173Q8—21no—
174Q8—421no—
175Q8—421no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—211no—
179C2×C4—421no—
180C2×C4—421no—
181C2×C4—421no—
182C2×C4—421no—
183C2×C4C22×D2111yescharacteristic
184C8—21no—
185C8—212no—
186C7#481611yescharacteristic, complemented, Sylow
187C2×C3#644711yescharacteristic
188C2×C3#647811yescharacteristic
189C2×C3#648111yescharacteristic
190D3—281no—
191C22—421no—
192C22—421no—
193C22—421no—
194C22D4×D2111yescharacteristic
195C4—41no—
196C4—41no—
197C4—421no—
198C4—841no—
199C4—841no—
200C4#6672811yescharacteristic
201C4D4×D2111yescharacteristic
202C3#8958811yescharacteristic, complemented, Sylow
203C2—841no—
204C2#1062136711yescharacteristic
205C2#1062139811yescharacteristic
206C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
hypercenter#103D21
hypocenterC3×C7#470
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#66635C2semidirect
5. C3×C7×#103C2semidirect
6. #10620655C2semidirect
7. #10620647C2semidirect
26. C7×#103D3semidirect
62. C3×#103D7semidirect
115. #103D21semidirect
142. C3×C7#470semidirect
186. C7#4816semidirect
202. C3#89588semidirect

Maximal subgroups

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

subgroupindex
#106214292
C2×#666352
C2×#666222
#106213802
C3×C7×#1032
#106206552
#106206472
#895883
#48167