Finite Group Databasebeta

← #13052658

Subgroups — #13052658

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
1C3×#6460C211yesmaximal, characteristic
2#10620921C211yesmaximal, characteristic
3C2×#66569C211yesmaximal, characteristic, complemented
4C2×#66557C211yesmaximal, characteristic
5#10620647C211yesmaximal, characteristic, complemented
6C7×#810C211yesmaximal, characteristic, complemented
7#10620506C211yesmaximal, characteristic, complemented
8#89588—31nomaximal, Hall
9C2×C7×Dic6C2211yescharacteristic
10C2×C3×C4×D7C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C4⋊Dic21C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×(C2×D7)⋊C4—21no—
17C3×C4⋊Dic7—21no—
18#66577—21no—
19(C2×D7)⋊Dic3—21no—
20#66569—22no—
21#66557—22no—
22#6509—31no—
23C2×#1144—31no—
24C2×#1131—31no—
25#6460D311yescharacteristic
26C7×#103—31no—
27#6390—31no—
28#6382—31no—
29#5052—71nomaximal, Hall
30C22×C3×D7D411yescharacteristic
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×C7×Dic3—21no—
35C7×Dic6—21no—
36C7×Dic6—21no—
37C2×C3×Dic7—21no—
38C2×C3×Dic7D411yescharacteristic
39C3×C4×D7—21no—
40C3×C4×D7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3⋊(C7⋊C8)D412yes—
44C7×C3⋊C8—21no—
45C2×C7×Q8—31no—
46C2×C4×D7C2×D311yescharacteristic
47C2×Dic14C2×D311yescharacteristic
48C2×C7×C8—31no—
49C7×C4⋊C4C2×D311yescharacteristic
50#1144—62no—
51(C2×D7)⋊C4—21no—
52(C2×D7)⋊C4—61no—
53C4⋊Dic7—21no—
54C4⋊Dic7—31no—
55Dic7⋊C4—61no—
56C2×C7⋊C8—31no—
57#1131—62no—
58C3×C22⋊Q8—71no—
59C2×#216—71no—
60#924—71no—
61C2×#208—71no—
62#835—71no—
63#810D711yescharacteristic, complemented
64#808—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×C3×D7—21no—
67C2×C3×D7—21no—
68C2×C3×D7—21no—
69C3×C4×C7—41no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—41no—
73C3×Dic7—21no—
74C3×Dic7—41no—
75C7×Dic3—41no—
76#470—211noSylow
77C22×D7#6711yescharacteristic
78C7×Q8—61no—
79C7×Q8—61no—
80C2×C4×C7—21no—
81C2×C4×C7—61no—
82C2×C4×C7C22×D311yescharacteristic
83C2×Dic7—21no—
84C2×Dic7—61no—
85C2×Dic7#6711yescharacteristic
86C4×D7—21no—
87C4×D7—21no—
88Dic14—21no—
89Dic14—21no—
90C7×C8—61no—
91C7⋊C8—32no—
92C2×C3×Q8—71no—
93C22×C3×C4—71no—
94C2×Dic6C2×D711yescharacteristic
95C3×C4⋊C4—141no—
96C3×C4⋊C4C2×D711yescharacteristic
97C3×C22⋊C4—141no—
98#217—141no—
99#216—142no—
100C4⋊Dic3—71no—
101Dic3⋊C4—141no—
102#208—142no—
103C2×C3⋊C8—71no—
104C2×C3⋊C8C2×D711yescharacteristic
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13711yescharacteristic
107C2×C3×C7#13711yescharacteristic
108C3×D7—41no—
109C2×Q16—211no—
110C2×M16—211no—
111C22⋊Q8—71no—
112C22⋊Q8—211no—
113#106—211no—
114#103—31no—
115#103—211no—
116C22×C7C2×#6711yescharacteristic
117C2×D7—21no—
118C2×D7—21no—
119C2×D7—21no—
120C4×C7—41no—
121C4×C7—121no—
122C4×C7D3×D411yescharacteristic
123C4×C7#23711yescharacteristic
124Dic7—21no—
125Dic7—41no—
126Dic7—121no—
127C23×C3—71no—
128C3×Q8—141no—
129C3×Q8—141no—
130C2×C3×C4—21no—
131C2×C3×C4—71no—
132C2×C3×C4—141no—
133C2×C3×C4—141no—
134C2×C3×C4—141no—
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—21no—
137C2×Dic3—141no—
138Dic6—21no—
139Dic6—21no—
140C3⋊C8—21no—
141C3⋊C8—72no—
142C3×C7#47011yescharacteristic, complemented, Hall
143C2×Q8—31no—
144C2×Q8—71no—
145C22×C4—71no—
146Q16—422no—
147M16—422no—
148C2×C8—31no—
149C2×C8—211no—
150C4⋊C4—141no—
151C4⋊C4—211no—
152C4⋊C4—421no—
153C4⋊C4D3×D711yescharacteristic
154C22⋊C4—141no—
155C22⋊C4—421no—
156C2×C7#93911yescharacteristic
157C2×C7#94211yescharacteristic
158C2×C7#95111yescharacteristic
159D7—41no—
160C22×C3—141no—
161C22×C3—141no—
162C22×C3—141no—
163C22×C3D4×D711yescharacteristic
164C3×C4—41no—
165C3×C4—141no—
166C3×C4—281no—
167C3×C4#115711yescharacteristic
168C3×C4D4×D711yescharacteristic
169Dic3—41no—
170Dic3—281no—
171C23—71no—
172Q8—61no—
173Q8—61no—
174Q8—141no—
175Q8—141no—
176C2×C4—21no—
177C2×C4—61no—
178C2×C4—71no—
179C2×C4—141no—
180C2×C4—141no—
181C2×C4—141no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—212no—
186C7#505211yescharacteristic, complemented, Sylow
187C2×C3—281no—
188C2×C3#644711yescharacteristic
189C2×C3#647811yescharacteristic
190C2×C3#648111yescharacteristic
191C22—141no—
192C22—141no—
193C22—141no—
194C22D7×#6711yescharacteristic
195C4—41no—
196C4—121no—
197C4—141no—
198C4—281no—
199C4—841no—
200C4#6666911yescharacteristic
201C4#6668111yescharacteristic
202C3#8958811yescharacteristic, complemented, Sylow
203C2—281no—
204C2#1062097011yescharacteristic
205C2#1062104611yescharacteristic
206C2#1062107711yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC22×C3×D7D4
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×#66569C2semidirect
5. #10620647C2semidirect
6. C7×#810C2semidirect
7. #10620506C2semidirect
63. #810D7semidirect
142. C3×C7#470semidirect
186. C7#5052semidirect
202. C3#89588semidirect

Maximal subgroups

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

subgroupindex
C3×#64602
#106209212
C2×#665692
C2×#665572
#106206472
C7×#8102
#106205062
#895883
#50527