Finite Group Databasebeta

← #13056386

Subgroups — #13056386

Order 1344.

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

normal subgroups only

This group has 216 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#10621415C211yesmaximal, characteristic, complemented
2C2×#66632C211yesmaximal, characteristic, complemented
3#10621391C211yesmaximal, characteristic, complemented
4#10621390C211yesmaximal, characteristic, complemented
5C3×C7×#100C211yesmaximal, characteristic, complemented
6#10620662C211yesmaximal, characteristic, complemented
7#10620661C211yesmaximal, characteristic, complemented
8#89521—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×D84C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13C3×C7×M16C2211yescharacteristic
14#66633—21no—
15#66632—21no—
16#66632—21no—
17#66632—21no—
18#66627C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20D168—21no—
21#66623—21no—
22#66622—21no—
23C8×D21—21no—
24#6495—31no—
25C2×#1141—31no—
26#6471—31no—
27#6470—31no—
28C7×#100D311yescharacteristic, complemented
29#6397—31no—
30#6396—31no—
31#4749—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21—21no—
34C3×C7×D4—21no—
35C3×C7×D4—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×D4C2×D311yescharacteristic
49#1157—31no—
50C2×D28—31no—
51C7×M16C2×D311yescharacteristic
52C7×M16C2×D311yescharacteristic
53#1142—61no—
54#1141—61no—
55#1141—61no—
56#1141—61no—
57#1136—31no—
58C2×C7⋊C8—31no—
59D56—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63#932—71no—
64C2×#213—71no—
65#908—71no—
66#907—71no—
67C3×#100D711yescharacteristic, complemented
68#823—71no—
69#822—71no—
70C22×C3×C7—41no—
71C22×C3×C7—81no—
72C22×C3×C7#4111yescharacteristic
73C2×D21—21no—
74C2×D21—41no—
75C2×D21—81no—
76C3×C4×C7C2×D411yescharacteristic
77C3×C4×C7C2×D411yescharacteristic
78Dic21—21no—
79#472—211noSylow
80C23×C7—21no—
81C22×D7—61no—
82C7×D4—21no—
83C7×D4—21no—
84C2×C4×C7C22×D311yescharacteristic
85C22⋊D7—61no—
86D28—31no—
87D28—61no—
88D28—61no—
89C4×D7—61no—
90Dic14—31no—
91C7×C8—21no—
92C7×C8—21no—
93C7⋊C8—31no—
94C7⋊C8—31no—
95C7⋊C8—61no—
96C2×C3×D4C2×D711yescharacteristic
97#235—71no—
98C2×D12—71no—
99C3×M16C2×D711yescharacteristic
100C3×M16C2×D711yescharacteristic
101#214—141no—
102#213—141no—
103#213—141no—
104#213—141no—
105#208—71no—
106C2×C3⋊C8—71no—
107D24—141no—
108#204—141no—
109ΓL(1,𝔽25)—141no—
110D3×C8—141no—
111C2×C3×C7—21no—
112C2×C3×C7—81no—
113C2×C3×C7#12111yescharacteristic
114D21—41no—
115D21—81no—
116AGL(1,ℤ/8)—211no—
117AGL(1,ℤ/8)—211no—
118C2×D8—211no—
119#131—211no—
120#108—211no—
121#100—211no—
122#100D2111yescharacteristic, complemented
123C22×C7—41no—
124C22×C7—81no—
125C22×C7#23511yescharacteristic
126C2×D7—61no—
127C2×D7—121no—
128C2×D7—241no—
129C4×C7D3×D411yescharacteristic
130C4×C7D3×D411yescharacteristic
131Dic7—61no—
132C23×C3—21no—
133C22×D3—141no—
134C3×D4—21no—
135C3×D4—21no—
136C2×C3×C4C22×D711yescharacteristic
137#67—141no—
138D12—71no—
139D12—141no—
140D12—141no—
141C4×D3—141no—
142Dic6—71no—
143C3×C8—21no—
144C3×C8—21no—
145C3⋊C8—71no—
146C3⋊C8—71no—
147C3⋊C8—141no—
148C3×C7#47211yescharacteristic, complemented, Hall
149#41—211no—
150C2×D4—211no—
151C2×D4C2×D2111yescharacteristic
152SD16—421no—
153SD16—421no—
154D8—421no—
155D8—421no—
156D8—421no—
157D8—421no—
158M16—211no—
159M16—421no—
160M16C2×D2111yescharacteristic
161M16C2×D2111yescharacteristic
162C2×C8—211no—
163C2×C8—421no—
164C2×C7—21no—
165C2×C7—81no—
166C2×C7#88411yescharacteristic
167D7—121no—
168D7—241no—
169C22×C3—41no—
170C22×C3—81no—
171C22×C3#115711yescharacteristic
172C2×D3—141no—
173C2×D3—281no—
174C2×D3—561no—
175C3×C4D4×D711yescharacteristic
176C3×C4D4×D711yescharacteristic
177Dic3—141no—
178C23—21no—
179C23—421no—
180Q8—211no—
181D4—21no—
182D4—21no—
183D4—211no—
184D4—421no—
185D4—421no—
186D4—421no—
187C2×C4—421no—
188C2×C4C22×D2111yescharacteristic
189C8—21no—
190C8—21no—
191C8—211no—
192C8—211no—
193C8—421no—
194C7#474911yescharacteristic, complemented, Sylow
195C2×C3—21no—
196C2×C3—81no—
197C2×C3#644711yescharacteristic
198D3—281no—
199D3—561no—
200C22—41no—
201C22—81no—
202C22—421no—
203C22—841no—
204C22—1681no—
205C22#6672811yescharacteristic
206C4—421no—
207C4D4×D2111yescharacteristic
208C4D4×D2111yescharacteristic
209C3#8952111yescharacteristic, complemented, Sylow
210C2—21no—
211C2—81no—
212C2—841no—
213C2—1681no—
214C2#1062136711yescharacteristic
subgroupG/H
centerC2#10621367
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#100C2
socleC2×C3×C7#121
hypercenter#100D21
hypocenterC3×C7#472
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
1. #10621415C2semidirect
2. C2×#66632C2semidirect
3. #10621391C2semidirect
4. #10621390C2semidirect
5. C3×C7×#100C2semidirect
6. #10620662C2semidirect
7. #10620661C2semidirect
28. C7×#100D3semidirect
67. C3×#100D7semidirect
122. #100D21semidirect
148. C3×C7#472semidirect
194. C7#4749semidirect
209. C3#89521semidirect

Maximal subgroups

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

subgroupindex
#106214152
C2×#666322
#106213912
#106213902
C3×C7×#1002
#106206622
#106206612
#895213
#47497