Finite Group Databasebeta

← #13054488

Subgroups — #13054488

Order 1344.

240 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 62 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#10621421C211yesmaximal, characteristic
2C2×C3×#1142C211yesmaximal, characteristic, complemented
3#10620952C211yesmaximal, characteristic, complemented
4C2×#66559C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620523C211yesmaximal, characteristic, complemented
7#10620506C211yesmaximal, characteristic, complemented
8#89552—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×C4×D21C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13#66636—21no—
14C4⋊Dic21C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×#1142—22no—
18C2×C3×C7⋊C8C2211yescharacteristic
19Dic3⋊Dic7—21no—
20#66575—21no—
21#66559—22no—
22#6501—31no—
23C2×#1142D311yescharacteristic, complemented
24C2×#1131—31no—
25#6460—31no—
26C7×#102—31no—
27#6390—31no—
28#6381—31no—
29#5174—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D21D411yescharacteristic
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C2×C3×C4×C7C2311yescharacteristic
35#4268—42no—
36C2×Dic21—21no—
37C2×Dic21D411yescharacteristic
38C4×D21—21no—
39C4×D21—21no—
40C2×C7×Dic3—21no—
41C2×C3×Dic7—21no—
42C3×Dic14—21no—
43C3×Dic14—21no—
44C3×C7⋊C8D412yescomplemented
45C7×C3⋊C8—21no—
46C2×C7×D4C2×D311yescharacteristic
47C2×C22⋊D7—61no—
48C2×C4×D7—31no—
49C2×Dic14C2×D311yescharacteristic
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52C22⋊Dic7—61no—
53#1142—22no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7—31no—
56C4⋊Dic7—61no—
57C2×C7⋊C8C2×D311yescharacteristic
58#1131—62no—
59C2×C3×SD16—71no—
60#946—71no—
61#938—71no—
62C2×ΓL(1,𝔽25)—71no—
63#835—71no—
64#832D711yescharacteristic, complemented
65#818—71no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7C2×D411yescharacteristic
70C2×D21—21no—
71C2×D21—21no—
72C2×D21—21no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7#4111yescharacteristic
75Dic21—21no—
76Dic21—41no—
77C3×Dic7—41no—
78C7×Dic3—41no—
79#468—211noSylow
80C23×C7—21no—
81C22×D7—31no—
82C7×D4—21no—
83C7×D4—21no—
84C2×C4×C7—61no—
85C2×C4×C7C22×D311yescharacteristic
86C22⋊D7—122no—
87C2×Dic7—21no—
88C2×Dic7—31no—
89C2×Dic7—61no—
90C4×D7—61no—
91C4×D7—61no—
92Dic14—21no—
93Dic14—21no—
94C7×C8—61no—
95C7⋊C8#6712yescomplemented
96C2×C3×Q8—71no—
97C2×C3×D4C2×D711yescharacteristic
98C2×#67—141no—
99C2×C4×D3—71no—
100C3×SD16—142no—
101C2×C3×C8—71no—
102#217—141no—
103(C2×D3)⋊C4—141no—
104C4⋊Dic3—71no—
105C4⋊Dic3C2×D711yescharacteristic
106Dic3⋊C4—141no—
107C2×C3⋊C8C2×D711yescharacteristic
108ΓL(1,𝔽25)—142no—
109C2×C3×C7—81no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
112C2×C3×C7#13711yescharacteristic
113D21—41no—
114C2×SD16—71no—
115C2×M16—211no—
116C22⋊Q8—211no—
117#121—211no—
118#107—211no—
119#103—211no—
120#102—31no—
121C22×C7—41no—
122C22×C7—41no—
123C22×C7—41no—
124C22×C7D3×D411yescharacteristic
125C2×D7—61no—
126C2×D7—61no—
127C2×D7—61no—
128C4×C7—121no—
129C4×C7#23711yescharacteristic
130C4×C7C2×#6711yescharacteristic
131Dic7—41no—
132Dic7—61no—
133Dic7—121no—
134C23×C3—21no—
135C22×D3—71no—
136C3×Q8—141no—
137C3×Q8—141no—
138C3×D4—21no—
139C3×D4—21no—
140C2×C3×C4—141no—
141C2×C3×C4C22×D711yescharacteristic
142#67—282no—
143C2×Dic3—21no—
144C2×Dic3—71no—
145C2×Dic3—141no—
146C4×D3—141no—
147C4×D3—141no—
148C3×C8—72no—
149C3⋊C8—21no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8—71no—
152C2×D4—421no—
153C2×D4D3×D711yescharacteristic
154C22×C4—211no—
155SD16—142no—
156M16—422no—
157C2×C8—31no—
158C2×C8—71no—
159C4⋊C4—31no—
160C4⋊C4—211no—
161C4⋊C4—421no—
162C22⋊C4—421no—
163C22⋊C4—421no—
164C2×C7—81no—
165C2×C7#91411yescharacteristic
166C2×C7#91511yescharacteristic
167C2×C7#93811yescharacteristic
168D7—121no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3D4×D711yescharacteristic
173C2×D3—141no—
174C2×D3—141no—
175C2×D3—141no—
176C3×C4—281no—
177C3×C4#115711yescharacteristic
178C3×C4D4×D711yescharacteristic
179Dic3—41no—
180Dic3—141no—
181Dic3—281no—
182C23—21no—
183C23—211no—
184Q8—141no—
185Q8—141no—
186D4—21no—
187D4—21no—
188D4—842no—
189C2×C4—61no—
190C2×C4—141no—
191C2×C4—211no—
192C2×C4—421no—
193C2×C4—421no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—72no—
198C7#517411yescharacteristic, complemented, Sylow
199C2×C3—81no—
200C2×C3#644711yescharacteristic
201C2×C3#647411yescharacteristic
202C2×C3#647811yescharacteristic
203D3—281no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—421no—
208C22—421no—
209C22—421no—
210C22#6669411yescharacteristic
211C4—121no—
212C4—281no—
213C4—421no—
214C4—841no—
215C4#6668311yescharacteristic
216C4D7×#6711yescharacteristic
217C3#8955211yescharacteristic, complemented, Sylow
218C2—81no—
219C2—841no—
220C2#1062105011yescharacteristic
221C2#1062106211yescharacteristic
222C2#1062109511yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radical#10621421C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#468
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1142C2semidirect
3. #10620952C2semidirect
4. C2×#66559C2semidirect
5. C7×#832C2semidirect
6. #10620523C2semidirect
7. #10620506C2semidirect
23. C2×#1142D3semidirect
44. C3×C7⋊C8D4semidirect
64. #832D7semidirect
95. C7⋊C8#67semidirect
150. C3×C7#468semidirect
198. C7#5174semidirect
217. C3#89552semidirect

Maximal subgroups

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

subgroupindex
#106214212
C2×C3×#11422
#106209522
C2×#665592
C7×#8322
#106205232
#106205062
#895523
#51747