Finite Group Databasebeta

← #13053571

Subgroups — #13053571

Order 1344.

208 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 52 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#10621390C211yesmaximal, characteristic, complemented
2#10620883C211yesmaximal, characteristic, complemented
3#10620874C211yesmaximal, characteristic
4C2×#66563C211yesmaximal, characteristic, complemented
5C7×#824C211yesmaximal, characteristic, complemented
6C3×#6397C211yesmaximal, characteristic, complemented
7#10620527C211yesmaximal, characteristic, complemented
8#89526—31nomaximal, Hall
9#66728C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14#66622—21no—
15C8×D21—21no—
16C7×#208C2211yescharacteristic
17C3×#1136C2211yescharacteristic
18#66571—21no—
19#66568—21no—
20#66565—21no—
21#66563—21no—
22#66563—21no—
23#66561—21no—
24#6505—31no—
25C2×#1143—31no—
26#6471—31no—
27#6470—31no—
28C7×#101—31no—
29#6397D311yescharacteristic, complemented
30#6396—31no—
31#4748—71nomaximal, Hall
32C22×C3×D7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34#4268—21no—
35D84D411yescharacteristic
36C4×D21—21no—
37Dic42D411yescharacteristic
38C2×C7×Dic3—21no—
39C7×Dic6—21no—
40C7×Dic6—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—
48C2×C7×Q8—31no—
49#1157—31no—
50C2×D28C2×D311yescharacteristic
51C7×M16—31no—
52C7×M16C2×D311yescharacteristic
53#1144—61no—
54#1143—61no—
55#1143—61no—
56#1143—61no—
57#1136C2×D311yescharacteristic
58C2×C7⋊C8—31no—
59D56—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63C2×#214—71no—
64#932—71no—
65#909—71no—
66#907—71no—
67C3×#100—71no—
68#824D711yescharacteristic, complemented
69#822—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C2×C3×D7—41no—
73C2×C3×D7—81no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76Dic21—21no—
77C7×Dic3—41no—
78#471—211noSylow
79C22×D7—21no—
80C7×Q8—61no—
81C7×Q8—61no—
82C2×C4×C7—61no—
83C2×C4×C7C22×D311yescharacteristic
84C22⋊D7—61no—
85D28—21no—
86D28—21no—
87D28—31no—
88C4×D7—61no—
89Dic14—31no—
90C7×C8—21no—
91C7×C8—61no—
92C7⋊C8—21no—
93C7⋊C8—31no—
94C7⋊C8—31no—
95C2×C3×D4—71no—
96#235—71no—
97C2×Dic6C2×D711yescharacteristic
98C3×M16—71no—
99C3×M16C2×D711yescharacteristic
100#214—141no—
101#214—141no—
102#214—141no—
103#213—141no—
104#208C2×D711yescharacteristic
105C2×C3⋊C8—71no—
106Dic12—141no—
107#204—141no—
108ΓL(1,𝔽25)—141no—
109D3×C8—141no—
110C2×C3×C7—21no—
111C2×C3×C7#12111yescharacteristic
112D21—41no—
113C3×D7—81no—
114#137—211no—
115AGL(1,ℤ/8)—211no—
116C2×SD16—211no—
117#131—211no—
118#108—211no—
119#101—31no—
120#100—71no—
121C22×C7#23511yescharacteristic
122C2×D7—41no—
123C2×D7—61no—
124C2×D7—81no—
125C4×C7—121no—
126C4×C7D3×D411yescharacteristic
127C4×C7D3×D411yescharacteristic
128Dic7—61no—
129C23×C3—141no—
130C3×D4—141no—
131C3×D4—141no—
132C2×C3×C4C22×D711yescharacteristic
133#67—141no—
134C2×Dic3—21no—
135D12—71no—
136C4×D3—141no—
137Dic6—21no—
138Dic6—21no—
139Dic6—71no—
140C3×C8—21no—
141C3×C8—141no—
142C3⋊C8—21no—
143C3⋊C8—71no—
144C3⋊C8—71no—
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—211no—
147C2×Q8—31no—
148C2×D4—71no—
149Q16—421no—
150SD16—421no—
151SD16—421no—
152SD16—421no—
153SD16—421no—
154D8—421no—
155M16—31no—
156M16—71no—
157M16—421no—
158M16D3×D711yescharacteristic
159C2×C8—211no—
160C2×C8—421no—
161C2×C7—21no—
162C2×C7#88411yescharacteristic
163D7—81no—
164D7—121no—
165C22×C3—281no—
166C22×C3—561no—
167C22×C3#115711yescharacteristic
168C2×D3—141no—
169C3×C4D4×D711yescharacteristic
170C3×C4D4×D711yescharacteristic
171Dic3—41no—
172Dic3—141no—
173C23—141no—
174Q8—61no—
175Q8—61no—
176Q8—211no—
177D4—141no—
178D4—141no—
179D4—211no—
180D4—421no—
181C2×C4—61no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—21no—
185C8—61no—
186C8—141no—
187C8—211no—
188C8—211no—
189C7#474811yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—561no—
192C2×C3#644711yescharacteristic
193D3—281no—
194C22—281no—
195C22—421no—
196C22—561no—
197C22#6667511yescharacteristic
198C4—121no—
199C4—421no—
200C4#6668111yescharacteristic
201C4#6669411yescharacteristic
202C3#8952611yescharacteristic, complemented, Sylow
203C2—21no—
204C2—561no—
205C2—841no—
206C2#1062102311yescharacteristic
subgroupG/H
centerC2#10621023
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#121
prime radical#10620874C2
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. #10621390C2semidirect
2. #10620883C2semidirect
4. C2×#66563C2semidirect
5. C7×#824C2semidirect
6. C3×#6397C2semidirect
7. #10620527C2semidirect
29. #6397D3semidirect
68. #824D7semidirect
145. C3×C7#471semidirect
189. C7#4748semidirect
202. C3#89526semidirect

Maximal subgroups

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

subgroupindex
#106213902
#106208832
#106208742
C2×#665632
C7×#8242
C3×#63972
#106205272
#895263
#47487