Finite Group Databasebeta

← #13054821

Subgroups — #13054821

Order 1344.

216 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 58 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#10621431C211yesmaximal, characteristic, complemented
2C7×#949C211yesmaximal, characteristic, complemented
3C3×#6512C211yesmaximal, characteristic, complemented
4C2×#66560C211yesmaximal, characteristic, complemented
5#10620527C211yesmaximal, characteristic, complemented
6#10620494C211yesmaximal, characteristic, complemented
7#10620493C211yesmaximal, characteristic, complemented
8#89971—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12#66627—21no—
13#66627C2211yescharacteristic
14C2×C3⋊(C7⋊C8)—21no—
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C7×#215—21no—
17C7×#213—21no—
18C7×#208C2211yescharacteristic
19C3×#1143—21no—
20C3×#1141—21no—
21C3×#1136C2211yescharacteristic
22#66560—21no—
23#66560—21no—
24C7×AGL(1,ℤ/8)—31no—
25#6512D311yescharacteristic, complemented
26#6511—31no—
27C2×#1141—31no—
28#6407—31no—
29#6397—31no—
30#6396—31no—
31#5199—71nomaximal, Hall
32C22×D3×C7—21no—
33C22×C3×D7—21no—
34C3×C7×Q8D411yescharacteristic
35C3×C7×D4—21no—
36C3×C7×D4D411yescharacteristic
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39C7×D12—21no—
40C7×D12—21no—
41C3×D28—21no—
42C3×D28—21no—
43C3⋊(C7⋊C8)—21no—
44C3⋊(C7⋊C8)D411yescharacteristic
45C3⋊(C7⋊C8)D411yescharacteristic
46C3×C7⋊C8—21no—
47C7×C3⋊C8—21no—
48C7×#41C2×D311yescharacteristic
49C2×C7×D4—31no—
50C2×D28C2×D311yescharacteristic
51C7×SD16—61no—
52C7×D8—61no—
53C7×M16—31no—
54#1143—21no—
55#1141—21no—
56#1141—61no—
57#1141—61no—
58#1136—31no—
59#1136—61no—
60#1136C2×D311yescharacteristic
61C2×C7⋊C8—31no—
62C2×C7⋊C8—61no—
63C3×AGL(1,ℤ/8)—71no—
64#949D711yescharacteristic, complemented
65#948—71no—
66C2×#213—71no—
67#833—71no—
68#823—71no—
69#822—71no—
70C22×C3×C7—21no—
71C22×C3×C7#4111yescharacteristic
72C2×D3×C7—41no—
73C2×D3×C7—81no—
74C2×C3×D7—41no—
75C2×C3×D7—81no—
76C3×C4×C7—21no—
77C3×C4×C7C2×D411yescharacteristic
78C3×C4×C7C2×D411yescharacteristic
79#472—211noSylow
80C23×C7—61no—
81C22×D7—21no—
82C7×Q8#6711yescharacteristic
83C7×D4—21no—
84C7×D4—61no—
85C7×D4—61no—
86C7×D4#6711yescharacteristic
87C2×C4×C7—21no—
88C2×C4×C7C22×D311yescharacteristic
89D28—21no—
90D28—21no—
91C7×C8—61no—
92C7⋊C8—21no—
93C7⋊C8—31no—
94C7⋊C8—31no—
95C7⋊C8—61no—
96C3×#41C2×D711yescharacteristic
97C2×C3×D4—71no—
98C2×D12C2×D711yescharacteristic
99C3×SD16—141no—
100C3×D8—141no—
101C3×M16—71no—
102#215—21no—
103#213—21no—
104#213—141no—
105#213—141no—
106#208—71no—
107#208—141no—
108#208C2×D711yescharacteristic
109C2×C3⋊C8—71no—
110C2×C3⋊C8—141no—
111C2×C3×C7—21no—
112C2×C3×C7—41no—
113C2×C3×C7#12111yescharacteristic
114C3×D7—81no—
115D3×C7—81no—
116AGL(1,ℤ/8)—31no—
117AGL(1,ℤ/8)—71no—
118C2×D8—211no—
119#131—211no—
120#108—211no—
121#100—211no—
122#100—211no—
123C22×C7—21no—
124C22×C7—121no—
125C22×C7—241no—
126C22×C7#23711yescharacteristic
127C2×D7—41no—
128C2×D7—81no—
129C4×C7—21no—
130C4×C7D3×D411yescharacteristic
131C4×C7C2×#6711yescharacteristic
132C23×C3—141no—
133C22×D3—21no—
134C3×Q8C22⋊D711yescharacteristic
135C3×D4—21no—
136C3×D4—141no—
137C3×D4—141no—
138C3×D4C22⋊D711yescharacteristic
139C2×C3×C4—21no—
140C2×C3×C4C22×D711yescharacteristic
141D12—21no—
142D12—21no—
143C3×C8—141no—
144C3⋊C8—21no—
145C3⋊C8—71no—
146C3⋊C8—71no—
147C3⋊C8—141no—
148C3×C7#47211yescharacteristic, complemented, Hall
149#41D3×D711yescharacteristic
150C2×D4—31no—
151C2×D4—71no—
152SD16—61no—
153SD16—141no—
154D8—61no—
155D8—141no—
156D8—421no—
157D8—421no—
158M16—31no—
159M16—71no—
160M16—211no—
161M16—421no—
162C2×C8—211no—
163C2×C8—421no—
164C2×C7—21no—
165C2×C7—41no—
166C2×C7—241no—
167C2×C7#93911yescharacteristic
168D7—81no—
169C22×C3—21no—
170C22×C3—281no—
171C22×C3—561no—
172C22×C3#115911yescharacteristic
173C2×D3—41no—
174C2×D3—81no—
175C3×C4—21no—
176C3×C4D4×D711yescharacteristic
177C3×C4C2×C22⋊D711yescharacteristic
178C23—61no—
179C23—141no—
180Q8#424511yescharacteristic
181D4—21no—
182D4—61no—
183D4—61no—
184D4—141no—
185D4—141no—
186D4#424511yescharacteristic
187C2×C4—21no—
188C2×C4C2×D3×D711yescharacteristic
189C8—61no—
190C8—141no—
191C8—211no—
192C8—211no—
193C8—421no—
194C7#519911yescharacteristic, complemented, Sylow
195C2×C3—21no—
196C2×C3—41no—
197C2×C3—561no—
198C2×C3#650211yescharacteristic
199D3—81no—
200C22—21no—
201C22—121no—
202C22—241no—
203C22—281no—
204C22—561no—
205C22#6668411yescharacteristic
206C4—21no—
207C4C2×#424511yescharacteristic
208C4#6669411yescharacteristic
209C3#8997111yescharacteristic, complemented, Sylow
210C2—21no—
211C2—41no—
212C2—241no—
213C2—561no—
214C2#1062113311yescharacteristic
subgroupG/H
centerC2#10621133
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×#41C22
socleC2×C3×C7#121
hypercenter#41D3×D7
hypocenterC3×C7#472
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #10621431C2semidirect
2. C7×#949C2semidirect
3. C3×#6512C2semidirect
4. C2×#66560C2semidirect
5. #10620527C2semidirect
6. #10620494C2semidirect
7. #10620493C2semidirect
25. #6512D3semidirect
64. #949D7semidirect
148. C3×C7#472semidirect
194. C7#5199semidirect
209. C3#89971semidirect

Maximal subgroups

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

subgroupindex
#106214312
C7×#9492
C3×#65122
C2×#665602
#106205272
#106204942
#106204932
#899713
#51997