Finite Group Databasebeta

← #13056489

Subgroups — #13056489

Order 1344.

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

normal subgroups only

This group has 216 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#10621376C211yesmaximal, characteristic
2#10621371C211yesmaximal, characteristic, complemented
3C3×C7×#106C211yesmaximal, characteristic, complemented
4#10620660C211yesmaximal, characteristic
5#10620653C211yesmaximal, characteristic, complemented
6#10620648C211yesmaximal, characteristic
7#10620647C211yesmaximal, characteristic, complemented
8#89642—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×C4×D21—21no—
11C2×Dic42—21no—
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15#66631—21no—
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6456—31no—
21#6451—31no—
22C7×#106D311yescharacteristic, complemented
23#6395—31no—
24Dic7⋊C8—31no—
25#6383—31no—
26#6382—31no—
27#4870—71nomaximal, Hall
28C22×D21—21no—
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C2×Dic21D411yescharacteristic
35D84—21no—
36D84Q812yescomplemented
37C4×D21—42no—
38Dic42—42no—
39C3×C7×C8—21no—
40C3⋊(C7⋊C8)—21no—
41C2×D28—31no—
42C2×C4×D7—61no—
43C2×Dic14—61no—
44C2×C7×C8C2×D311yescharacteristic
45C7×C4⋊C4C2×D311yescharacteristic
46C7×C4⋊C4C2×D311yescharacteristic
47(C2×D7)⋊C4—61no—
48C4⋊Dic7—31no—
49Dic7⋊C4—61no—
50C4×Dic7—31no—
51C2×C7⋊C8—31no—
52#893—71no—
53#888—71no—
54C3×#106D711yescharacteristic, complemented
55#821—71no—
56Dic3⋊C8—71no—
57#809—71no—
58#808—71no—
59C22×C3×C7C2×D411yescharacteristic
60C2×D21—22no—
61C2×D21—41no—
62C2×D21—41no—
63C3×C4×C7—41no—
64C3×C4×C7—41no—
65C3×C4×C7C2×Q811yescharacteristic
66C3×C4×C7#4111yescharacteristic
67Dic21—21no—
68Dic21—41no—
69Dic21SD1612yes—
70#476—211noSylow
71C22×D7—61no—
72C2×C4×C7—21no—
73C2×C4×C7—21no—
74C2×C4×C7C22×D311yescharacteristic
75C2×Dic7—31no—
76C2×Dic7—31no—
77C2×Dic7—61no—
78D28—32no—
79D28—61no—
80C4×D7—122no—
81Dic14—122no—
82C7×C8—21no—
83C7⋊C8—61no—
84C2×D12—71no—
85C2×C4×D3—141no—
86C2×Dic6—141no—
87C2×C3×C8C2×D711yescharacteristic
88C3×C4⋊C4C2×D711yescharacteristic
89C3×C4⋊C4C2×D711yescharacteristic
90(C2×D3)⋊C4—141no—
91C4⋊Dic3—71no—
92Dic3⋊C4—141no—
93C4×Dic3—71no—
94C2×C3⋊C8—71no—
95C2×C3×C7C22⋊Q811yescharacteristic
96C2×C3×C7C2×SD1611yescharacteristic
97C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
98D21—42no—
99C4⋊Q8—211no—
100C4×D4—211no—
101#106—211no—
102#106D2111yescharacteristic, complemented
103C4⋊C8—211no—
104#102—211no—
105#102—211no—
106C22×C7D3×D411yescharacteristic
107C2×D7—62no—
108C2×D7—121no—
109C2×D7—121no—
110C4×C7—41no—
111C4×C7—41no—
112C4×C7#23511yescharacteristic
113C4×C7D3×Q811yescharacteristic
114Dic7—32no—
115Dic7—61no—
116Dic7—121no—
117C22×D3—141no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4C22×D711yescharacteristic
121C2×Dic3—71no—
122C2×Dic3—71no—
123C2×Dic3—141no—
124D12—72no—
125D12—141no—
126C4×D3—282no—
127Dic6—282no—
128C3×C8—21no—
129C3⋊C8—141no—
130C3×C7#47611yescharacteristic, complemented, Hall
131C2×Q8—421no—
132C2×D4—211no—
133C22×C4—421no—
134C2×C8—211no—
135C2×C8C2×D2111yescharacteristic
136C4⋊C4—211no—
137C4⋊C4—421no—
138C4⋊C4C2×D2111yescharacteristic
139C4⋊C4C2×D2111yescharacteristic
140C22⋊C4—421no—
141C42—211no—
142C2×C7#89611yescharacteristic
143C2×C7D3×SD1611yescharacteristic
144C2×C7#91411yescharacteristic
145D7—122no—
146C22×C3D4×D711yescharacteristic
147C2×D3—142no—
148C2×D3—281no—
149C2×D3—281no—
150C3×C4—41no—
151C3×C4—41no—
152C3×C4#115711yescharacteristic
153C3×C4Q8×D711yescharacteristic
154Dic3—72no—
155Dic3—141no—
156Dic3—281no—
157C23—421no—
158Q8—842no—
159D4—212no—
160D4—421no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—211no—
164C2×C4—211no—
165C2×C4—421no—
166C2×C4—842no—
167C2×C4C22×D2111yescharacteristic
168C8—21no—
169C8—421no—
170C7#487011yescharacteristic, complemented, Sylow
171C2×C3#645911yescharacteristic
172C2×C3D7×SD1611yescharacteristic
173C2×C3#647711yescharacteristic
174D3—282no—
175C22—422no—
176C22—841no—
177C22—841no—
178C22D4×D2111yescharacteristic
179C4—41no—
180C4—41no—
181C4—212no—
182C4—421no—
183C4—841no—
184C4#6672811yescharacteristic
185C4Q8×D2111yescharacteristic
186C3#8964211yescharacteristic, complemented, Sylow
187C2—842no—
188C2#1062137911yescharacteristic
189C2SD16×D2111yescharacteristic
190C2#1062139711yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#106C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenter#106D21
hypocenterC3×C7#476
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
2. #10621371C2semidirect
3. C3×C7×#106C2semidirect
5. #10620653C2semidirect
7. #10620647C2semidirect
22. C7×#106D3semidirect
36. D84Q8semidirect
54. C3×#106D7semidirect
102. #106D21semidirect
130. C3×C7#476semidirect
170. C7#4870semidirect
186. C3#89642semidirect

Maximal subgroups

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

subgroupindex
#106213762
#106213712
C3×C7×#1062
#106206602
#106206532
#106206482
#106206472
#896423
#48707