Finite Group Databasebeta

← #13056519

Subgroups — #13056519

Order 1344.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621378C211yesmaximal, characteristic
2C4⋊C4×D21C211yesmaximal, characteristic, complemented
3C3×C7×#107C211yesmaximal, characteristic, complemented
4#10620660C211yesmaximal, characteristic, complemented
5#10620659C211yesmaximal, characteristic, complemented
6#10620648C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89654—31nomaximal, Hall
9C2×D84—21no—
10C2×D84C2211yescharacteristic
11C2×C4×D21—21no—
12C2×C4×D21C2211yescharacteristic
13C2×C3×C7×C8C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15C3×C7×C4⋊C4C2211yescharacteristic
16#66631—21no—
17C4⋊Dic21C2211yescharacteristic
18#66629—21no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6458—31no—
21D7×C4⋊C4—31no—
22C7×#107D311yescharacteristic, complemented
23#6395—31no—
24(C2×D7)⋊C8—31no—
25#6383—31no—
26#6381—31no—
27#4882—71nomaximal, Hall
28C22×D21—21no—
29C22×D21D411yescharacteristic
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21D411yescharacteristic
35D84—21no—
36D84—21no—
37D84—42no—
38C4×D21—21no—
39C4×D21—21no—
40C4×D21—42no—
41C3×C7×C8—21no—
42C3⋊(C7⋊C8)—21no—
43C2×D28—31no—
44C2×D28—61no—
45C2×C4×D7—31no—
46C2×C4×D7—61no—
47C2×C7×C8C2×D311yescharacteristic
48C7×C4⋊C4C2×D311yescharacteristic
49C7×C4⋊C4C2×D311yescharacteristic
50(C2×D7)⋊C4—61no—
51C4⋊Dic7—31no—
52Dic7⋊C4—61no—
53C2×C7⋊C8—31no—
54#895—71no—
55D3×C4⋊C4—71no—
56C3×#107D711yescharacteristic, complemented
57#821—71no—
58(C2×D3)⋊C8—71no—
59#809—71no—
60#807—71no—
61C22×C3×C7C2×D411yescharacteristic
62C2×D21—21no—
63C2×D21—21no—
64C2×D21—41no—
65C2×D21—41no—
66C2×D21—41no—
67C2×D21D812yes—
68C3×C4×C7—41no—
69C3×C4×C7—41no—
70C3×C4×C7#4111yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—21no—
73Dic21—41no—
74#480—211noSylow
75C22×D7—31no—
76C22×D7—61no—
77C2×C4×C7—21no—
78C2×C4×C7—21no—
79C2×C4×C7C22×D311yescharacteristic
80C2×Dic7—31no—
81C2×Dic7—61no—
82D28—61no—
83D28—61no—
84D28—122no—
85C4×D7—61no—
86C4×D7—61no—
87C4×D7—122no—
88C7×C8—21no—
89C7⋊C8—61no—
90C2×D12—71no—
91C2×D12—141no—
92C2×C4×D3—71no—
93C2×C4×D3—141no—
94C2×C3×C8C2×D711yescharacteristic
95C3×C4⋊C4C2×D711yescharacteristic
96C3×C4⋊C4C2×D711yescharacteristic
97(C2×D3)⋊C4—141no—
98C4⋊Dic3—71no—
99Dic3⋊C4—141no—
100C2×C3⋊C8—71no—
101C2×C3×C7#12311yescharacteristic
102C2×C3×C7C2×D811yescharacteristic
103C2×C3×C7#13711yescharacteristic
104D21—22no—
105D21—81no—
106#121—211no—
107C2×C4⋊C4—211no—
108#107—211no—
109#107D2111yescharacteristic, complemented
110#102—211no—
111#102—211no—
112C22⋊C8—211no—
113C22×C7D3×D411yescharacteristic
114C2×D7—32no—
115C2×D7—61no—
116C2×D7—61no—
117C2×D7—121no—
118C2×D7—121no—
119C2×D7—121no—
120C4×C7—41no—
121C4×C7—41no—
122C4×C7#23511yescharacteristic
123C4×C7#23911yescharacteristic
124Dic7—61no—
125Dic7—121no—
126C22×D3—71no—
127C22×D3—141no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4C22×D711yescharacteristic
131C2×Dic3—71no—
132C2×Dic3—141no—
133D12—141no—
134D12—141no—
135D12—282no—
136C4×D3—141no—
137C4×D3—141no—
138C4×D3—282no—
139C3×C8—21no—
140C3⋊C8—141no—
141C3×C7#48011yescharacteristic, complemented, Hall
142C2×D4—211no—
143C2×D4—421no—
144C22×C4—211no—
145C22×C4—421no—
146C2×C8—211no—
147C2×C8C2×D2111yescharacteristic
148C4⋊C4—211no—
149C4⋊C4—421no—
150C4⋊C4C2×D2111yescharacteristic
151C4⋊C4C2×D2111yescharacteristic
152C22⋊C4—421no—
153C2×C7#89411yescharacteristic
154C2×C7D3×D811yescharacteristic
155C2×C7#91811yescharacteristic
156D7—62no—
157D7—241no—
158C22×C3D4×D711yescharacteristic
159C2×D3—72no—
160C2×D3—141no—
161C2×D3—141no—
162C2×D3—281no—
163C2×D3—281no—
164C2×D3—281no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4#115711yescharacteristic
168C3×C4#116111yescharacteristic
169Dic3—141no—
170Dic3—281no—
171C23—211no—
172C23—421no—
173D4—421no—
174D4—421no—
175D4—842no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—211no—
179C2×C4—421no—
180C2×C4—421no—
181C2×C4—421no—
182C2×C4—842no—
183C2×C4C22×D2111yescharacteristic
184C8—21no—
185C8—421no—
186C7#488211yescharacteristic, complemented, Sylow
187C2×C3#645711yescharacteristic
188C2×C3D7×D811yescharacteristic
189C2×C3#648111yescharacteristic
190D3—142no—
191D3—561no—
192C22—212no—
193C22—421no—
194C22—421no—
195C22—841no—
196C22—841no—
197C22—841no—
198C22D4×D2111yescharacteristic
199C4—41no—
200C4—41no—
201C4—421no—
202C4—841no—
203C4#6672811yescharacteristic
204C4#6673211yescharacteristic
205C3#8965411yescharacteristic, complemented, Sylow
206C2—422no—
207C2—1681no—
208C2#1062137711yescharacteristic
209C2D8×D2111yescharacteristic
210C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#107C2
socleC22×C3×C7C2×D4
prime radical#10621378C2
hypercenter#107D21
hypocenterC3×C7#480
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×D21C2semidirect
3. C3×C7×#107C2semidirect
4. #10620660C2semidirect
5. #10620659C2semidirect
6. #10620648C2semidirect
7. #10620646C2semidirect
22. C7×#107D3semidirect
56. C3×#107D7semidirect
109. #107D21semidirect
141. C3×C7#480semidirect
186. C7#4882semidirect
205. C3#89654semidirect

Maximal subgroups

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

subgroupindex
#106213782
C4⋊C4×D212
C3×C7×#1072
#106206602
#106206592
#106206482
#106206462
#896543
#48827