Finite Group Databasebeta

← #13056391

Subgroups — #13056391

Order 1344.

200 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C2×#66635C211yesmaximal, characteristic, complemented
2#10621425C211yesmaximal, characteristic
3#10621392C211yesmaximal, characteristic
4#10621390C211yesmaximal, characteristic
5C3×C7×#101C211yesmaximal, characteristic, complemented
6#10620663C211yesmaximal, characteristic, complemented
7#10620661C211yesmaximal, characteristic, complemented
8#89527—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×Dic42C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13C3×C7×M16C2211yescharacteristic
14#66635—21no—
15#66635—21no—
16#66635—21no—
17#66634—21no—
18#66627C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20Dic84—21no—
21#66623—21no—
22#66622—21no—
23C8×D21—21no—
24C2×#1144—31no—
25#6505—31no—
26#6472—31no—
27#6470—31no—
28C7×#101D311yescharacteristic, complemented
29#6398—31no—
30#6396—31no—
31#4755—71nomaximal, Hall
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—21no—
37C2×Dic21—21no—
38D84D411yescharacteristic
39C4×D21—21no—
40Dic42—21no—
41Dic42—21no—
42Dic42D411yescharacteristic
43C3×C7×C8—21no—
44C3×C7×C8—21no—
45C3⋊(C7⋊C8)—21no—
46C3⋊(C7⋊C8)D411yescharacteristic
47C3⋊(C7⋊C8)D411yescharacteristic
48C2×C7×Q8C2×D311yescharacteristic
49#1157—31no—
50C2×Dic14—31no—
51C7×M16C2×D311yescharacteristic
52C7×M16C2×D311yescharacteristic
53#1144—61no—
54#1144—61no—
55#1144—61no—
56#1143—61no—
57#1136—31no—
58C2×C7⋊C8—31no—
59Dic28—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63C2×#216—71no—
64#942—71no—
65#909—71no—
66#907—71no—
67C3×#101D711yescharacteristic, complemented
68#824—71no—
69#822—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C3×C4×C7—41no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7C2×D411yescharacteristic
75Dic21—21no—
76Dic21—41no—
77#473—211noSylow
78C7×Q8—21no—
79C7×Q8—21no—
80C2×C4×C7—21no—
81C2×C4×C7C22×D311yescharacteristic
82C22⋊D7—61no—
83C2×Dic7—61no—
84D28—31no—
85C4×D7—61no—
86Dic14—31no—
87Dic14—61no—
88Dic14—61no—
89C7×C8—21no—
90C7×C8—21no—
91C7⋊C8—31no—
92C7⋊C8—31no—
93C7⋊C8—61no—
94C2×C3×Q8C2×D711yescharacteristic
95#235—71no—
96C2×Dic6—71no—
97C3×M16C2×D711yescharacteristic
98C3×M16C2×D711yescharacteristic
99#216—141no—
100#216—141no—
101#216—141no—
102#215—141no—
103#208—71no—
104C2×C3⋊C8—71no—
105Dic12—141no—
106#204—141no—
107ΓL(1,𝔽25)—141no—
108D3×C8—141no—
109C2×C3×C7—21no—
110C2×C3×C7#12111yescharacteristic
111D21—41no—
112#137—211no—
113#137—211no—
114C2×Q16—211no—
115#131—211no—
116#108—211no—
117#101—211no—
118#101D2111yescharacteristic, complemented
119C22×C7#23511yescharacteristic
120C2×D7—61no—
121C4×C7—41no—
122C4×C7D3×D411yescharacteristic
123C4×C7D3×D411yescharacteristic
124Dic7—61no—
125Dic7—121no—
126C3×Q8—21no—
127C3×Q8—21no—
128C2×C3×C4—21no—
129C2×C3×C4C22×D711yescharacteristic
130#67—141no—
131C2×Dic3—141no—
132D12—71no—
133C4×D3—141no—
134Dic6—71no—
135Dic6—141no—
136Dic6—141no—
137C3×C8—21no—
138C3×C8—21no—
139C3⋊C8—71no—
140C3⋊C8—71no—
141C3⋊C8—141no—
142C3×C7#47311yescharacteristic, complemented, Hall
143#41—211no—
144C2×Q8—211no—
145C2×Q8C2×D2111yescharacteristic
146Q16—421no—
147Q16—421no—
148Q16—421no—
149Q16—421no—
150SD16—421no—
151SD16—421no—
152M16—211no—
153M16—421no—
154M16C2×D2111yescharacteristic
155M16C2×D2111yescharacteristic
156C2×C8—211no—
157C2×C8—421no—
158C2×C7—21no—
159C2×C7#88411yescharacteristic
160D7—121no—
161C22×C3#115711yescharacteristic
162C2×D3—141no—
163C3×C4—41no—
164C3×C4D4×D711yescharacteristic
165C3×C4D4×D711yescharacteristic
166Dic3—141no—
167Dic3—281no—
168Q8—21no—
169Q8—21no—
170Q8—211no—
171Q8—421no—
172Q8—421no—
173D4—211no—
174D4—421no—
175C2×C4—21no—
176C2×C4—421no—
177C2×C4—421no—
178C2×C4C22×D2111yescharacteristic
179C8—21no—
180C8—21no—
181C8—211no—
182C8—211no—
183C8—421no—
184C7#475511yescharacteristic, complemented, Sylow
185C2×C3—21no—
186C2×C3#644711yescharacteristic
187D3—281no—
188C22—421no—
189C22#6672811yescharacteristic
190C4—41no—
191C4—421no—
192C4—841no—
193C4D4×D2111yescharacteristic
194C4D4×D2111yescharacteristic
195C3#8952711yescharacteristic, complemented, Sylow
196C2—21no—
197C2—841no—
198C2#1062136711yescharacteristic
subgroupG/H
centerC2#10621367
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#101C2
socleC2×C3×C7#121
prime radical#66728C22
hypercenter#101D21
hypocenterC3×C7#473
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
1. C2×#66635C2semidirect
5. C3×C7×#101C2semidirect
6. #10620663C2semidirect
7. #10620661C2semidirect
28. C7×#101D3semidirect
67. C3×#101D7semidirect
118. #101D21semidirect
142. C3×C7#473semidirect
184. C7#4755semidirect
195. C3#89527semidirect

Maximal subgroups

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

subgroupindex
C2×#666352
#106214252
#106213922
#106213902
C3×C7×#1012
#106206632
#106206612
#895273
#47557