Finite Group Databasebeta

← #13056425

Subgroups — #13056425

Order 1344.

240 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 65 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#10621427C211yesmaximal, characteristic, complemented
2C2×#66634C211yesmaximal, characteristic
3C2×#66623C211yesmaximal, characteristic
4#10621376C211yesmaximal, characteristic
5C3×C7×#103C211yesmaximal, characteristic, complemented
6#10620653C211yesmaximal, characteristic, complemented
7#10620649C211yesmaximal, characteristic, complemented
8#89590—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C4×D21—21no—
12C2×Dic42C2211yescharacteristic
13C2×C3×C7×C8C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15#66634—22no—
16#66631—21no—
17#66629—21no—
18#66629—21no—
19C4×Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21#66623—22no—
22#6507—31no—
23C2×#1143—31no—
24C2×#1132—31no—
25#6456—31no—
26C7×#103D311yescharacteristic, complemented
27Dic7⋊C8—31no—
28#6384—31no—
29#4818—71nomaximal, Hall
30C22×D21—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×Dic21D411yescharacteristic
38C2×Dic21D411yescharacteristic
39D84—21no—
40D84D412yes—
41C4×D21—42no—
42Dic42—21no—
43Dic42—21no—
44C3×C7×C8—21no—
45C3⋊(C7⋊C8)—21no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×D28—31no—
48C2×C4×D7—61no—
49C2×Dic14—31no—
50C2×C7×C8C2×D311yescharacteristic
51C7×C4⋊C4C2×D311yescharacteristic
52#1143—62no—
53(C2×D7)⋊C4—61no—
54Dic7⋊C4—61no—
55Dic7⋊C4—61no—
56C4×Dic7—31no—
57C2×C7⋊C8—31no—
58#1132—62no—
59#944—71no—
60C2×#215—71no—
61C2×#204—71no—
62#893—71no—
63C3×#103D711yescharacteristic, complemented
64Dic3⋊C8—71no—
65#810—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—22no—
68C2×D21—41no—
69C2×D21—41no—
70C3×C4×C7—41no—
71C3×C4×C7—41no—
72C3×C4×C7C2×D411yescharacteristic
73C3×C4×C7#4111yescharacteristic
74Dic21—21no—
75Dic21—41no—
76Dic21SD1612yescomplemented
77#461—211noSylow
78C22×D7—61no—
79C7×Q8—21no—
80C7×Q8—21no—
81C2×C4×C7—21no—
82C2×C4×C7—21no—
83C2×C4×C7C22×D311yescharacteristic
84C2×Dic7—31no—
85C2×Dic7—31no—
86C2×Dic7—61no—
87D28—32no—
88D28—61no—
89C4×D7—122no—
90Dic14—61no—
91Dic14—61no—
92C7×C8—21no—
93C7⋊C8—61no—
94C2×C3×Q8C2×D711yescharacteristic
95C2×D12—71no—
96C2×C4×D3—141no—
97C2×Dic6—71no—
98C2×C3×C8C2×D711yescharacteristic
99C3×C4⋊C4C2×D711yescharacteristic
100#215—142no—
101(C2×D3)⋊C4—141no—
102Dic3⋊C4—141no—
103Dic3⋊C4—141no—
104C4×Dic3—71no—
105C2×C3⋊C8—71no—
106#204—142no—
107C2×C3×C7#12111yescharacteristic
108C2×C3×C7C2×SD1611yescharacteristic
109C2×C3×C7#13711yescharacteristic
110D21—42no—
111C2×SD16—211no—
112C2×SD16—211no—
113C4⋊Q8—211no—
114C4×D4—211no—
115C4⋊C8—211no—
116#103—211no—
117#103D2111yescharacteristic, complemented
118C22×C7D3×D411yescharacteristic
119C2×D7—62no—
120C2×D7—121no—
121C2×D7—121no—
122C4×C7—41no—
123C4×C7—41no—
124C4×C7#23511yescharacteristic
125C4×C7D3×D411yescharacteristic
126Dic7—32no—
127Dic7—61no—
128Dic7—121no—
129C22×D3—141no—
130C3×Q8—21no—
131C3×Q8—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4C22×D711yescharacteristic
135C2×Dic3—71no—
136C2×Dic3—71no—
137C2×Dic3—141no—
138D12—72no—
139D12—141no—
140C4×D3—282no—
141Dic6—141no—
142Dic6—141no—
143C3×C8—21no—
144C3⋊C8—141no—
145C3×C7#46111yescharacteristic, complemented, Hall
146C2×Q8—211no—
147C2×Q8C2×D2111yescharacteristic
148C2×D4—211no—
149C22×C4—421no—
150SD16—422no—
151SD16—422no—
152C2×C8—211no—
153C2×C8C2×D2111yescharacteristic
154C4⋊C4—421no—
155C4⋊C4—421no—
156C4⋊C4C2×D2111yescharacteristic
157C22⋊C4—421no—
158C42—211no—
159C2×C7#88411yescharacteristic
160C2×C7D3×SD1611yescharacteristic
161C2×C7#91811yescharacteristic
162D7—122no—
163C22×C3D4×D711yescharacteristic
164C2×D3—142no—
165C2×D3—281no—
166C2×D3—281no—
167C3×C4—41no—
168C3×C4—41no—
169C3×C4#115711yescharacteristic
170C3×C4D4×D711yescharacteristic
171Dic3—72no—
172Dic3—141no—
173Dic3—281no—
174C23—421no—
175Q8—21no—
176Q8—21no—
177Q8—421no—
178Q8—421no—
179D4—212no—
180D4—421no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—211no—
184C2×C4—211no—
185C2×C4—421no—
186C2×C4—842no—
187C2×C4C22×D2111yescharacteristic
188C8—21no—
189C8—421no—
190C7#481811yescharacteristic, complemented, Sylow
191C2×C3#644711yescharacteristic
192C2×C3D7×SD1611yescharacteristic
193C2×C3#648111yescharacteristic
194D3—282no—
195C22—422no—
196C22—841no—
197C22—841no—
198C22D4×D2111yescharacteristic
199C4—41no—
200C4—41no—
201C4—212no—
202C4—421no—
203C4—841no—
204C4#6672811yescharacteristic
205C4D4×D2111yescharacteristic
206C3#8959011yescharacteristic, complemented, Sylow
207C2—842no—
208C2#1062136711yescharacteristic
209C2SD16×D2111yescharacteristic
210C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenter#103D21
hypocenterC3×C7#461
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #10621427C2semidirect
5. C3×C7×#103C2semidirect
6. #10620653C2semidirect
7. #10620649C2semidirect
26. C7×#103D3semidirect
63. C3×#103D7semidirect
76. Dic21SD16semidirect
117. #103D21semidirect
145. C3×C7#461semidirect
190. C7#4818semidirect
206. C3#89590semidirect

Maximal subgroups

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

subgroupindex
#106214272
C2×#666342
C2×#666232
#106213762
C3×C7×#1032
#106206532
#106206492
#895903
#48187