Finite Group Databasebeta

← #13053591

Subgroups — #13053591

Order 1344.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C3×#6472C211yesmaximal, characteristic, complemented
2C2×#66566C211yesmaximal, characteristic, complemented
3#10620863C211yesmaximal, characteristic
4#10620845C211yesmaximal, characteristic, complemented
5#10620662C211yesmaximal, characteristic, complemented
6C7×#822C211yesmaximal, characteristic, complemented
7#10620496C211yesmaximal, characteristic, complemented
8#89670—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C3×#1157C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13#66627C2211yescharacteristic
14C7×#208C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×Dic28—21no—
17C3×#1132—21no—
18#66566—21no—
19#66566—21no—
20#66566—21no—
21#66561—21no—
22#66557—21no—
23D7×C3⋊C8—21no—
24#6472D311yescharacteristic, complemented
25#6471—31no—
26C2×#1132—31no—
27#6464—31no—
28C7×#108—31no—
29#6398—31no—
30#6397—31no—
31#5203—71nomaximal, Hall
32C22×D21—21no—
33C2×C3×C4×C7C2311yescharacteristic
34D84—21no—
35D84—21no—
36C3×C22⋊D7—21no—
37C2×C3×Dic7—21no—
38C3×D28D411yescharacteristic
39C3×C4×D7—21no—
40C3×Dic14—21no—
41C3×Dic14—21no—
42C3×Dic14D411yescharacteristic
43C3×C7×C8—21no—
44C3⋊(C7⋊C8)—21no—
45C7×C3⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48#1157C2×D311yescharacteristic
49C2×D28—31no—
50C2×Dic14C2×D311yescharacteristic
51C7×M16—31no—
52C7×M16C2×D311yescharacteristic
53C2×C7×C8—31no—
54#1136—31no—
55Dic28—21no—
56D56—61no—
57#1132—21no—
58#1132—61no—
59#1132—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63C3×#137—71no—
64#949—71no—
65#948—71no—
66C2×#215—71no—
67#836—71no—
68#823—71no—
69#822D711yescharacteristic, complemented
70C22×C3×C7#4111yescharacteristic
71C2×D21—41no—
72C2×D21—81no—
73C2×C3×D7—21no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76C3×Dic7—21no—
77C3×Dic7—41no—
78#471—211noSylow
79C22×D7—61no—
80C2×C4×C7C22×D311yescharacteristic
81C22⋊D7—21no—
82C2×Dic7—21no—
83D28—61no—
84D28—61no—
85D28#6711yescharacteristic
86C4×D7—21no—
87Dic14—21no—
88Dic14—21no—
89Dic14#6711yescharacteristic
90C7×C8—21no—
91C7×C8—31no—
92C7×C8—31no—
93C7×C8—61no—
94C7⋊C8—61no—
95C3×#41—71no—
96C2×C3×Q8—71no—
97C2×D12—71no—
98C3×Q16—141no—
99C3×SD16—141no—
100C3×M16C2×D711yescharacteristic
101#215—141no—
102#215—141no—
103#215—141no—
104#213—141no—
105#208—71no—
106#208—141no—
107#208C2×D711yescharacteristic
108C2×C3⋊C8—141no—
109C2×C3⋊C8C2×D711yescharacteristic
110C2×C3×C7—21no—
111C2×C3×C7#12111yescharacteristic
112D21—81no—
113C3×D7—41no—
114#137—71no—
115AGL(1,ℤ/8)—211no—
116C2×SD16—211no—
117#131—211no—
118#108—31no—
119#101—211no—
120#100—211no—
121C22×C7#23711yescharacteristic
122C2×D7—21no—
123C2×D7—121no—
124C2×D7—241no—
125C4×C7D3×D411yescharacteristic
126C4×C7C2×#6711yescharacteristic
127Dic7—21no—
128Dic7—41no—
129C22×D3—141no—
130C3×Q8—71no—
131C3×Q8—141no—
132C3×Q8—141no—
133C3×D4—71no—
134C3×D4—141no—
135C2×C3×C4—141no—
136C2×C3×C4—141no—
137C2×C3×C4C22×D711yescharacteristic
138D12—141no—
139D12—141no—
140C3×C8—21no—
141C3⋊C8—21no—
142C3⋊C8—141no—
143C3⋊C8D2811yescharacteristic
144C3⋊C8D2811yescharacteristic
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—71no—
147C2×Q8—71no—
148C2×D4—211no—
149Q16—141no—
150SD16—141no—
151SD16—421no—
152SD16—421no—
153SD16—421no—
154D8—421no—
155M16—31no—
156M16—211no—
157M16—421no—
158M16D3×D711yescharacteristic
159C2×C8—31no—
160C2×C8—421no—
161C2×C7—21no—
162C2×C7#93911yescharacteristic
163D7—41no—
164D7—241no—
165C22×C3—141no—
166C22×C3#116111yescharacteristic
167C2×D3—281no—
168C2×D3—561no—
169C3×C4—141no—
170C3×C4—281no—
171C3×C4C2×D2811yescharacteristic
172C3×C4D4×D711yescharacteristic
173C23—421no—
174Q8—71no—
175Q8—141no—
176Q8—141no—
177D4—71no—
178D4—141no—
179D4—421no—
180D4—421no—
181C2×C4—141no—
182C2×C4—141no—
183C2×C4C2×D3×D711yescharacteristic
184C8—21no—
185C8—31no—
186C8—31no—
187C8—61no—
188C8—421no—
189C7#520311yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—281no—
192C2×C3#645811yescharacteristic
193D3—561no—
194C22—141no—
195C22—841no—
196C22—1681no—
197C22#6667011yescharacteristic
198C4—141no—
199C4—281no—
200C4D3×D2811yescharacteristic
201C4D7×#6711yescharacteristic
202C3#8967011yescharacteristic, complemented, Sylow
203C2—21no—
204C2—281no—
205C2—1681no—
206C2#1062097111yescharacteristic
subgroupG/H
centerC2#10620971
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#121
prime radical#10620863C2
hypercenterM16D3×D7
hypocenterC3×C7#471
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. C3×#6472C2semidirect
2. C2×#66566C2semidirect
4. #10620845C2semidirect
5. #10620662C2semidirect
6. C7×#822C2semidirect
7. #10620496C2semidirect
24. #6472D3semidirect
69. #822D7semidirect
145. C3×C7#471semidirect
189. C7#5203semidirect
202. C3#89670semidirect

Maximal subgroups

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

subgroupindex
C3×#64722
C2×#665662
#106208632
#106208452
#106206622
C7×#8222
#106204962
#896703
#52037