Finite Group Databasebeta

← #13054535

Subgroups — #13054535

Order 1344.

336 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C7×#1004C211yesmaximal, characteristic, complemented
2#10621423C211yesmaximal, characteristic, complemented
3#10620867C212yesmaximal, complemented
4C3×#6407C211yesmaximal, characteristic, complemented
5#10620513C212yesmaximal, complemented
6#90013—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×#4268—22no—
9C2×D84C2211yescharacteristic, complemented
10C2×C7×#67—22no—
11C7×#237—22no—
12D3×C7×D4—22no—
13C7×#235C2212yes—
14C4×Dic21C2211yescharacteristic, complemented
15C3×#1136C2212yescomplemented
16#66568—22no—
17#66562—22no—
18C7×H(5,𝔽2)—31no—
19#6512—32no—
20#6503—31no—
21#6411—32no—
22#6407D311yescharacteristic, complemented
23#4747—71nomaximal, Hall
24C23×C3×C7D412yes—
25C22×D21—21no—
26C22×D3×C7—22no—
27C3×C7×D4—22no—
28C2×C3×C4×C7C2311yescharacteristic
29#4268—44no—
30C2×Dic21—21no—
31D84—22no—
32C7×#67—24no—
33C7×#67—22no—
34C2×C7×Dic3—22no—
35C7×D12D412yescomplemented
36C4×D3×C7—22no—
37C7×Dic6D412yescomplemented
38C3×C7⋊C8—22no—
39C7×#41—32no—
40C7×#41—62no—
41C2×C7×D4—62no—
42C2×C7×D4—62no—
43C2×C7×D4C2×D311yescharacteristic
44C2×C22⋊D7—62no—
45C2×D28—31no—
46#1143—62no—
47#1141—62no—
48C4×Dic7—31no—
49#1136C2×D312yescomplemented
50#1004D711yescharacteristic, complemented
51#940—71no—
52#908—72no—
53C3×#100—71no—
54#825—72no—
55C22×C3×C7—22no—
56C22×C3×C7—42no—
57C22×C3×C7C2×D411yescharacteristic
58C2×D21—42no—
59C2×D21—41no—
60C2×D3×C7—22no—
61C2×D3×C7—82no—
62C3×C4×C7C2×D412yes—
63Dic21—22no—
64C7×Dic3—22no—
65#453—211noSylow
66C23×C7—62no—
67C23×C7D1212yes—
68C22×D7—61no—
69C7×Q8—32no—
70C7×D4—22no—
71C7×D4—32no—
72C7×D4—64no—
73C7×D4—62no—
74C2×C4×C7—62no—
75C2×C4×C7—62no—
76C2×C4×C7C22×D311yescharacteristic
77C22⋊D7—124no—
78C2×Dic7—61no—
79D28—62no—
80C7⋊C8—22no—
81C2×C3×D4C2×D711yescharacteristic
82C2×#67—22no—
83C2×#67—142no—
84#237—22no—
85D3×D4—22no—
86#235C2×D712yes—
87C2×D12—71no—
88C3×M16—72no—
89C4×Dic3—71no—
90D24—142no—
91#204—142no—
92C2×C3×C7—21no—
93C2×C3×C7—42no—
94C2×C3×C7#12011yescharacteristic
95D21—81no—
96D3×C7—42no—
97H(5,𝔽2)—31no—
98AGL(1,ℤ/8)—212no—
99#127—211no—
100#104—212no—
101#100—71no—
102C22×C7—22no—
103C22×C7—42no—
104C22×C7—62no—
105C22×C7—242no—
106C22×C7C2×D1211yescharacteristic
107C2×D7—122no—
108C2×D7—121no—
109C4×C7—62no—
110C4×C7D3×D412yes—
111Dic7—62no—
112C23×C3C22⋊D712yes—
113C22×D3—22no—
114C22×D3—141no—
115C3×D4—22no—
116C2×C3×C4C22×D711yescharacteristic
117#67—24no—
118#67—22no—
119#67—284no—
120C2×Dic3—22no—
121C2×Dic3—141no—
122D12—142no—
123D12C22⋊D712yescomplemented
124C4×D3—22no—
125Dic6C22⋊D712yescomplemented
126C3×C8—142no—
127C3×C7#45311yescharacteristic, complemented, Hall
128#41—32no—
129#41—62no—
130C2×D4—62no—
131C2×D4—62no—
132C2×D4—211no—
133C2×D4—422no—
134C2×D4D3×D711yescharacteristic
135SD16—422no—
136D8—422no—
137M16—72no—
138C42—211no—
139C2×C7—21no—
140C2×C7—42no—
141C2×C7—122no—
142C2×C7#88211yescharacteristic
143D7—241no—
144C22×C3—22no—
145C22×C3—42no—
146C22×C3C2×C22⋊D711yescharacteristic
147C2×D3—22no—
148C2×D3—82no—
149C2×D3—282no—
150C2×D3—281no—
151C3×C4C2×C22⋊D712yes—
152Dic3—22no—
153Dic3—142no—
154C23—62no—
155C23—421no—
156C23#424712yes—
157Q8—32no—
158D4—22no—
159D4—32no—
160D4—64no—
161D4—62no—
162D4—422no—
163D4—844no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—421no—
167C2×C4C2×D3×D711yescharacteristic
168C8—142no—
169C7#474711yescharacteristic, complemented, Sylow
170C2×C3—21no—
171C2×C3—42no—
172C2×C3#651611yescharacteristic
173D3—42no—
174D3—561no—
175C22—22no—
176C22—42no—
177C22—62no—
178C22—242no—
179C22—842no—
180C22—841no—
181C22C2×#424711yescharacteristic
182C4—62no—
183C4—422no—
184C4D3×C22⋊D712yes—
185C3#9001311yescharacteristic, complemented, Sylow
186C2—21no—
187C2—42no—
188C2—122no—
189C2—1681no—
190C2#1062113211yescharacteristic
subgroupG/H
centerC2#10621132
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
hypercenterC2×D4D3×D7
hypocenterC3×C7#453
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C7×#1004C2semidirect
2. #10621423C2semidirect
3. #10620867C2semidirect
4. C3×#6407C2semidirect
5. #10620513C2semidirect
9. C2×D84C22semidirect
14. C4×Dic21C22semidirect
15. C3×#1136C22semidirect
22. #6407D3semidirect
35. C7×D12D4semidirect
37. C7×Dic6D4semidirect
49. #1136C2×D3semidirect
50. #1004D7semidirect
123. D12C22⋊D7semidirect
125. Dic6C22⋊D7semidirect
127. C3×C7#453semidirect
169. C7#4747semidirect
185. C3#90013semidirect

Maximal subgroups

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

subgroupindex
C7×#10042
#106214232
#106208672
C3×#64072
#106205132
#900133
#47477