Finite Group Databasebeta

← #13052851

Subgroups — #13052851

Order 1344.

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

normal subgroups only

This group has 220 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#10621380C211yesmaximal, characteristic
2#10620952C211yesmaximal, characteristic
3C2×#66570C211yesmaximal, characteristic, complemented
4C2×#66559C211yesmaximal, characteristic
5C7×#808C211yesmaximal, characteristic, complemented
6C3×#6384C211yesmaximal, characteristic, complemented
7#10620510C211yesmaximal, characteristic, complemented
8#89640—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13#66631—21no—
14C4⋊Dic21—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18Dic3⋊Dic7—21no—
19#66575—21no—
20#66570—22no—
21#66559—22no—
22C2×Dic28—31no—
23C2×#1131—31no—
24#6460—31no—
25#6460—31no—
26C7×#106—31no—
27#6384—31no—
28#6384D311yescharacteristic, complemented
29#4816—71nomaximal, Hall
30C22×D21D411yescharacteristic
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21D411yescharacteristic
35C4×D21—21no—
36C4×D21—21no—
37Dic42—21no—
38Dic42—21no—
39C2×C7×Dic3—21no—
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3×C7⋊C8—21no—
44C7×C3⋊C8D412yes—
45C2×C4×D7—31no—
46C2×Dic14—31no—
47C2×Dic14C2×D311yescharacteristic
48C2×C7×C8—31no—
49C7×C4⋊C4—31no—
50C7×C4⋊C4C2×D311yescharacteristic
51(C2×D7)⋊C4—61no—
52(C2×D7)⋊C4—61no—
53C4⋊Dic7—61no—
54C4⋊Dic7—61no—
55C2×C7⋊C8C2×D311yescharacteristic
56Dic28—62no—
57#1131—62no—
58#946—71no—
59C2×#216—71no—
60C2×ΓL(1,𝔽25)—71no—
61#897—71no—
62C3×#103—71no—
63#816—71no—
64#808D711yescharacteristic, complemented
65C22×C3×C7C2×D411yescharacteristic
66C2×D21—21no—
67C2×D21—21no—
68C2×D21—21no—
69C3×C4×C7—41no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—21no—
73Dic21—41no—
74C3×Dic7—41no—
75C7×Dic3—41no—
76#470—211noSylow
77C22×D7—31no—
78C2×C4×C7—21no—
79C2×C4×C7—61no—
80C2×C4×C7C22×D311yescharacteristic
81C2×Dic7—21no—
82C2×Dic7—31no—
83C2×Dic7—61no—
84C4×D7—61no—
85C4×D7—61no—
86Dic14—21no—
87Dic14—21no—
88Dic14—61no—
89Dic14—61no—
90C7×C8—32no—
91C7⋊C8—21no—
92C2×C3×Q8—71no—
93C2×C4×D3—71no—
94C2×Dic6—71no—
95C2×C3×C8—71no—
96C3×C4⋊C4C2×D711yescharacteristic
97#216—142no—
98(C2×D3)⋊C4—141no—
99(C2×D3)⋊C4—141no—
100C4⋊Dic3—141no—
101C4⋊Dic3C2×D711yescharacteristic
102Dic3⋊C4—141no—
103C2×C3⋊C8C2×D711yescharacteristic
104ΓL(1,𝔽25)—142no—
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13711yescharacteristic
107C2×C3×C7#13711yescharacteristic
108D21—41no—
109C2×Q16—211no—
110C2×M16—211no—
111C22⋊Q8—211no—
112C22⋊Q8—211no—
113#106—31no—
114#103—71no—
115#103—211no—
116C22×C7D3×D411yescharacteristic
117C2×D7—61no—
118C2×D7—61no—
119C2×D7—61no—
120C4×C7—41no—
121C4×C7—121no—
122C4×C7#23511yescharacteristic
123C4×C7D3×D411yescharacteristic
124Dic7—41no—
125Dic7—61no—
126Dic7—121no—
127C22×D3—71no—
128C3×Q8—141no—
129C3×Q8—141no—
130C2×C3×C4—21no—
131C2×C3×C4—141no—
132C2×C3×C4C22×D711yescharacteristic
133C2×Dic3—21no—
134C2×Dic3—71no—
135C2×Dic3—141no—
136C4×D3—141no—
137C4×D3—141no—
138Dic6—141no—
139Dic6—141no—
140C3×C8—141no—
141C3⋊C8D2812yes—
142C3×C7#47011yescharacteristic, complemented, Hall
143C2×Q8—71no—
144C2×Q8—211no—
145C22×C4—211no—
146Q16—422no—
147M16—422no—
148C2×C8—31no—
149C2×C8—71no—
150C4⋊C4—31no—
151C4⋊C4—421no—
152C4⋊C4—421no—
153C4⋊C4D3×D711yescharacteristic
154C22⋊C4—421no—
155C22⋊C4—421no—
156C2×C7#88411yescharacteristic
157C2×C7#91511yescharacteristic
158C2×C7#91811yescharacteristic
159D7—121no—
160C22×C3D4×D711yescharacteristic
161C2×D3—141no—
162C2×D3—141no—
163C2×D3—141no—
164C3×C4—41no—
165C3×C4—281no—
166C3×C4C2×D2811yescharacteristic
167C3×C4#116111yescharacteristic
168Dic3—41no—
169Dic3—141no—
170Dic3—281no—
171C23—211no—
172Q8—141no—
173Q8—141no—
174Q8—421no—
175Q8—421no—
176C2×C4—21no—
177C2×C4—61no—
178C2×C4—141no—
179C2×C4—211no—
180C2×C4—421no—
181C2×C4—421no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—32no—
185C8—141no—
186C7#481611yescharacteristic, complemented, Sylow
187C2×C3#645811yescharacteristic
188C2×C3#647811yescharacteristic
189C2×C3#647811yescharacteristic
190D3—281no—
191C22—421no—
192C22—421no—
193C22—421no—
194C22#6669411yescharacteristic
195C4—41no—
196C4—121no—
197C4—281no—
198C4—421no—
199C4—841no—
200C4#6667711yescharacteristic
201C4D3×D2811yescharacteristic
202C3#8964011yescharacteristic, complemented, Sylow
203C2—841no—
204C2#1062102111yescharacteristic
205C2#1062105011yescharacteristic
206C2#1062108611yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#470
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#66570C2semidirect
5. C7×#808C2semidirect
6. C3×#6384C2semidirect
7. #10620510C2semidirect
28. #6384D3semidirect
64. #808D7semidirect
142. C3×C7#470semidirect
186. C7#4816semidirect
202. C3#89640semidirect

Maximal subgroups

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

subgroupindex
#106213802
#106209522
C2×#665702
C2×#665592
C7×#8082
C3×#63842
#106205102
#896403
#48167