Finite Group Databasebeta

← #13056392

Subgroups — #13056392

Order 1344.

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

normal subgroups only

This group has 208 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
1#10621425C211yesmaximal, characteristic, complemented
2C2×#66634C211yesmaximal, characteristic, complemented
3#10621391C211yesmaximal, characteristic
4#10621390C211yesmaximal, characteristic, complemented
5C3×C7×#101C211yesmaximal, characteristic, complemented
6#10620662C211yesmaximal, characteristic, complemented
7#10620661C211yesmaximal, characteristic, complemented
8#89526—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×D84C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13C3×C7×M16C2211yescharacteristic
14#66635—21no—
15#66634—21no—
16#66634—21no—
17#66634—21no—
18#66627C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20D168—21no—
21#66623—21no—
22#66622—21no—
23C8×D21—21no—
24#6505—31no—
25C2×#1143—31no—
26#6471—31no—
27#6470—31no—
28C7×#101D311yescharacteristic, complemented
29#6397—31no—
30#6396—31no—
31#4754—71nomaximal, Hall
32C22×D21—21no—
33C3×C7×Q8—21no—
34C3×C7×Q8—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—21no—
38D84—21no—
39D84—21no—
40D84D411yescharacteristic
41C4×D21—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×D28—31no—
51C7×M16C2×D311yescharacteristic
52C7×M16C2×D311yescharacteristic
53#1144—61no—
54#1143—61no—
55#1143—61no—
56#1143—61no—
57#1136—31no—
58C2×C7⋊C8—31no—
59D56—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63#942—71no—
64C2×#215—71no—
65#908—71no—
66#907—71no—
67C3×#101D711yescharacteristic, complemented
68#823—71no—
69#822—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C2×D21—41no—
73C2×D21—81no—
74C3×C4×C7—41no—
75C3×C4×C7C2×D411yescharacteristic
76C3×C4×C7C2×D411yescharacteristic
77Dic21—21no—
78#471—211noSylow
79C22×D7—61no—
80C7×Q8—21no—
81C7×Q8—21no—
82C2×C4×C7—21no—
83C2×C4×C7C22×D311yescharacteristic
84C22⋊D7—61no—
85D28—31no—
86D28—61no—
87D28—61no—
88C4×D7—61no—
89Dic14—31no—
90C7×C8—21no—
91C7×C8—21no—
92C7⋊C8—31no—
93C7⋊C8—31no—
94C7⋊C8—61no—
95C2×C3×Q8C2×D711yescharacteristic
96#235—71no—
97C2×D12—71no—
98C3×M16C2×D711yescharacteristic
99C3×M16C2×D711yescharacteristic
100#216—141no—
101#215—141no—
102#215—141no—
103#215—141no—
104#208—71no—
105C2×C3⋊C8—71no—
106D24—141no—
107#204—141no—
108ΓL(1,𝔽25)—141no—
109D3×C8—141no—
110C2×C3×C7—21no—
111C2×C3×C7#12111yescharacteristic
112D21—41no—
113D21—81no—
114#137—211no—
115AGL(1,ℤ/8)—211no—
116C2×SD16—211no—
117#131—211no—
118#108—211no—
119#101D2111yescharacteristic, complemented
120#100—211no—
121C22×C7#23511yescharacteristic
122C2×D7—61no—
123C2×D7—121no—
124C2×D7—241no—
125C4×C7—41no—
126C4×C7D3×D411yescharacteristic
127C4×C7D3×D411yescharacteristic
128Dic7—61no—
129C22×D3—141no—
130C3×Q8—21no—
131C3×Q8—21no—
132C2×C3×C4—21no—
133C2×C3×C4C22×D711yescharacteristic
134#67—141no—
135D12—71no—
136D12—141no—
137D12—141no—
138C4×D3—141no—
139Dic6—71no—
140C3×C8—21no—
141C3×C8—21no—
142C3⋊C8—71no—
143C3⋊C8—71no—
144C3⋊C8—141no—
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—211no—
147C2×Q8C2×D2111yescharacteristic
148C2×D4—211no—
149Q16—421no—
150SD16—421no—
151SD16—421no—
152SD16—421no—
153SD16—421no—
154D8—421no—
155M16—211no—
156M16—421no—
157M16C2×D2111yescharacteristic
158M16C2×D2111yescharacteristic
159C2×C8—211no—
160C2×C8—421no—
161C2×C7—21no—
162C2×C7#88411yescharacteristic
163D7—121no—
164D7—241no—
165C22×C3#115711yescharacteristic
166C2×D3—141no—
167C2×D3—281no—
168C2×D3—561no—
169C3×C4—41no—
170C3×C4D4×D711yescharacteristic
171C3×C4D4×D711yescharacteristic
172Dic3—141no—
173C23—421no—
174Q8—21no—
175Q8—21no—
176Q8—211no—
177D4—211no—
178D4—421no—
179D4—421no—
180D4—421no—
181C2×C4—21no—
182C2×C4—421no—
183C2×C4C22×D2111yescharacteristic
184C8—21no—
185C8—21no—
186C8—211no—
187C8—211no—
188C8—421no—
189C7#475411yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3#644711yescharacteristic
192D3—281no—
193D3—561no—
194C22—421no—
195C22—841no—
196C22—1681no—
197C22#6672811yescharacteristic
198C4—41no—
199C4—421no—
200C4D4×D2111yescharacteristic
201C4D4×D2111yescharacteristic
202C3#8952611yescharacteristic, complemented, Sylow
203C2—21no—
204C2—841no—
205C2—1681no—
206C2#1062136711yescharacteristic
subgroupG/H
centerC2#10621367
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#101C2
socleC2×C3×C7#121
prime radical#10621391C2
hypercenter#101D21
hypocenterC3×C7#471
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. #10621425C2semidirect
2. C2×#66634C2semidirect
4. #10621390C2semidirect
5. C3×C7×#101C2semidirect
6. #10620662C2semidirect
7. #10620661C2semidirect
28. C7×#101D3semidirect
67. C3×#101D7semidirect
119. #101D21semidirect
145. C3×C7#471semidirect
189. C7#4754semidirect
202. C3#89526semidirect

Maximal subgroups

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

subgroupindex
#106214252
C2×#666342
#106213912
#106213902
C3×C7×#1012
#106206622
#106206612
#895263
#47547