Finite Group Databasebeta

← #13056348

Subgroups — #13056348

Order 1344.

240 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#66623C211yesmaximal, characteristic
2C4×D84C211yesmaximal, characteristic
3C4×Dic42C211yesmaximal, characteristic, complemented
4C3×C7×#97C211yesmaximal, characteristic, complemented
5#10620660C211yesmaximal, characteristic
6#10620657C211yesmaximal, characteristic, complemented
7#10620655C211yesmaximal, characteristic, complemented
8#89482—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×C4×D21—21no—
11C2×Dic42C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C2×C3×C7×C8C2211yescharacteristic
14C3×C42×C7C2211yescharacteristic
15#66631—21no—
16C4⋊Dic21C2211yescharacteristic
17C4⋊Dic21C2211yescharacteristic
18#66629—21no—
19C4×Dic21—21no—
20#66623C414yescomplemented
21C2×#1132—31no—
22C4×D28—31no—
23C4×Dic14—31no—
24C7×#97D311yescharacteristic, complemented
25#6395—31no—
26#6392—31no—
27#6390—31no—
28#4710—71nomaximal, Hall
29C22×D21—21no—
30C2×C3×C4×C7D411yescharacteristic
31C2×C3×C4×C7D411yescharacteristic
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×Dic21—21no—
36D84—21no—
37D84C2×C412yes—
38C4×D21—42no—
39Dic42—21no—
40Dic42C2×C412yescomplemented
41C3×C7×C8—21no—
42C3×C7×C8C2×C412yescomplemented
43C2×D28—31no—
44C2×C4×D7—61no—
45C2×Dic14—31no—
46C2×C7×C8C2×D311yescharacteristic
47C2×C7×C8C2×D311yescharacteristic
48C42×C7C2×D311yescharacteristic
49(C2×D7)⋊C4—61no—
50C4⋊Dic7—31no—
51C4⋊Dic7—31no—
52Dic7⋊C4—61no—
53C4×Dic7—61no—
54#1132—34no—
55C2×#204—71no—
56C4×D12—71no—
57C4×Dic6—71no—
58C3×#97D711yescharacteristic, complemented
59#821—71no—
60#818—71no—
61#816—71no—
62C22×C3×C7C2×D411yescharacteristic
63C2×D21—22no—
64C2×D21—41no—
65C2×D21—41no—
66C3×C4×C7—21no—
67C3×C4×C7—21no—
68C3×C4×C7C22×C411yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—22no—
71Dic21—41no—
72Dic21—41no—
73#440—211noSylow
74C22×D7—61no—
75C2×C4×C7D1211yescharacteristic
76C2×C4×C7D1211yescharacteristic
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—61no—
79C2×Dic7—61no—
80C2×Dic7—61no—
81D28—32no—
82D28—61no—
83C4×D7—122no—
84Dic14—32no—
85Dic14—61no—
86C7×C8—21no—
87C7×C8C4×D312yescomplemented
88C2×D12—71no—
89C2×C4×D3—141no—
90C2×Dic6—71no—
91C2×C3×C8C2×D711yescharacteristic
92C2×C3×C8C2×D711yescharacteristic
93C3×C42C2×D711yescharacteristic
94(C2×D3)⋊C4—141no—
95C4⋊Dic3—71no—
96C4⋊Dic3—71no—
97Dic3⋊C4—141no—
98C4×Dic3—141no—
99#204—74no—
100C2×C3×C7C4×D411yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C2×C3×C7#13711yescharacteristic
103D21—42no—
104C2×SD16—211no—
105C4×Q8—211no—
106C4×D4—211no—
107#107—211no—
108#103—211no—
109#102—211no—
110#97D2111yescharacteristic, complemented
111C22×C7C2×D1211yescharacteristic
112C2×D7—62no—
113C2×D7—121no—
114C2×D7—121no—
115C4×C7—21no—
116C4×C7—21no—
117C4×C7C2×C4×D311yescharacteristic
118C4×C7#23511yescharacteristic
119Dic7—62no—
120Dic7—121no—
121Dic7—121no—
122C22×D3—141no—
123C2×C3×C4D2811yescharacteristic
124C2×C3×C4D2811yescharacteristic
125C2×C3×C4C22×D711yescharacteristic
126C2×Dic3—141no—
127C2×Dic3—141no—
128C2×Dic3—141no—
129D12—72no—
130D12—141no—
131C4×D3—282no—
132Dic6—72no—
133Dic6—141no—
134C3×C8—21no—
135C3×C8C4×D712yescomplemented
136C3×C7#44011yescharacteristic, complemented, Hall
137C2×Q8—211no—
138C2×D4—211no—
139C22×C4—421no—
140SD16—214no—
141C2×C8C2×D2111yescharacteristic
142C2×C8C2×D2111yescharacteristic
143C4⋊C4—211no—
144C4⋊C4—211no—
145C4⋊C4—421no—
146C22⋊C4—421no—
147C42—421no—
148C42C2×D2111yescharacteristic
149C2×C7C4×D1211yescharacteristic
150C2×C7#90811yescharacteristic
151C2×C7#90911yescharacteristic
152D7—122no—
153C22×C3C2×D2811yescharacteristic
154C2×D3—142no—
155C2×D3—281no—
156C2×D3—281no—
157C3×C4—21no—
158C3×C4—21no—
159C3×C4C2×C4×D711yescharacteristic
160C3×C4#115711yescharacteristic
161Dic3—142no—
162Dic3—281no—
163Dic3—281no—
164C23—421no—
165Q8—212no—
166Q8—421no—
167D4—212no—
168D4—421no—
169C2×C4—421no—
170C2×C4—421no—
171C2×C4—421no—
172C2×C4—842no—
173C2×C4D8411yescharacteristic
174C2×C4D8411yescharacteristic
175C2×C4C22×D2111yescharacteristic
176C8—21no—
177C8C4×D2112yescomplemented
178C7#471011yescharacteristic, complemented, Sylow
179C2×C3C4×D2811yescharacteristic
180C2×C3#647111yescharacteristic
181C2×C3#647211yescharacteristic
182D3—282no—
183C22—422no—
184C22—841no—
185C22—841no—
186C22C2×D8411yescharacteristic
187C4—21no—
188C4—21no—
189C4—422no—
190C4—841no—
191C4—841no—
192C4C2×C4×D2111yescharacteristic
193C4#6672811yescharacteristic
194C3#8948211yescharacteristic, complemented, Sylow
195C2—842no—
196C2C4×D8411yescharacteristic
197C2#1062139111yescharacteristic
198C2#1062139211yescharacteristic
subgroupG/H
centerC22C2×D84
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C22×D21
FittingC3×C7×#97C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
cosocleC2×C3×C4×C7C23
hypercenter#97D21
hypocenterC3×C7#440
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
3. C4×Dic42C2semidirect
4. C3×C7×#97C2semidirect
6. #10620657C2semidirect
7. #10620655C2semidirect
20. #66623C4semidirect
24. C7×#97D3semidirect
40. Dic42C2×C4semidirect
42. C3×C7×C8C2×C4semidirect
58. C3×#97D7semidirect
87. C7×C8C4×D3semidirect
110. #97D21semidirect
135. C3×C8C4×D7semidirect
136. C3×C7#440semidirect
177. C8C4×D21semidirect
178. C7#4710semidirect
194. C3#89482semidirect

Maximal subgroups

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

subgroupindex
C2×#666232
C4×D842
C4×Dic422
C3×C7×#972
#106206602
#106206572
#106206552
#894823
#47107