Finite Group Databasebeta

← #13053096

Subgroups — #13053096

Order 1344.

216 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 66 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#10621030C211yesmaximal, characteristic, complemented
2#10620980C211yesmaximal, characteristic
3#10620660C211yesmaximal, characteristic
4C7×#817C211yesmaximal, characteristic, complemented
5C3×Dic7⋊C8C211yesmaximal, characteristic, complemented
6#10620520C211yesmaximal, characteristic, complemented
7#10620504C211yesmaximal, characteristic
8#89642—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×#4248—21no—
11C2×D21⋊C4—21no—
12C2×C3×C7×C8C2211yescharacteristic
13C4⋊Dic21C2211yescharacteristic
14C7×C4⋊Dic3C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C3×C4×Dic7C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18Dic7⋊Dic3—21no—
19#66575—21no—
20#6456—31no—
21#6451—31no—
22C7×#106—31no—
23#6395—31no—
24Dic7⋊C8D311yescharacteristic, complemented
25#6383—31no—
26#6382—31no—
27#4842—71nomaximal, Hall
28C22×D21—21no—
29C2×C3×C4×C7C2311yescharacteristic
30C2×Dic21—21no—
31D84—21no—
32D84Q812yescomplemented
33C2×C7×Dic3—21no—
34C2×C7×Dic3—21no—
35C2×C3×Dic7D411yescharacteristic
36C2×C3×Dic7D411yescharacteristic
37#4248—42no—
38D21⋊C4—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⋊C4—31no—
46C7×C4⋊C4—31no—
47(C2×D7)⋊C4—61no—
48C4⋊Dic7—31no—
49Dic7⋊C4—61no—
50C4×Dic7C2×D311yescharacteristic
51C2×C7⋊C8C2×D311yescharacteristic
52C4×D12—71no—
53#869—71no—
54C3×C4⋊C8—71no—
55#821—71no—
56#821—71no—
57#817—71no—
58#817D711yescharacteristic, complemented
59C22×C3×C7C2×D411yescharacteristic
60C2×D21—22no—
61C2×D21—41no—
62C2×D21—41no—
63C3×C4×C7C2×Q811yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—41no—
66C3×Dic7—21no—
67C3×Dic7SD1612yes—
68C7×Dic3—41no—
69C7×Dic3—41no—
70#476—211noSylow
71C22×D7—61no—
72C2×C4×C7—61no—
73C2×C4×C7—61no—
74C2×C4×C7C22×D311yescharacteristic
75C2×Dic7—61no—
76C2×Dic7D1211yescharacteristic
77C2×Dic7D1211yescharacteristic
78D28—32no—
79D28—61no—
80C4×D7—122no—
81Dic14—122no—
82C7×C8—21no—
83C7⋊C8—21no—
84C2×D12—71no—
85C2×C4×D3—141no—
86C2×Dic6—141no—
87C2×C3×C8—71no—
88C2×C3×C8C2×D711yescharacteristic
89C3×C42—71no—
90(C2×D3)⋊C4—141no—
91C4⋊Dic3—71no—
92C4⋊Dic3—141no—
93C4⋊Dic3C2×D711yescharacteristic
94C4⋊Dic3C2×D711yescharacteristic
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—31no—
102#106—211no—
103C4⋊C8—71no—
104#102—211no—
105#102—211no—
106C22×C7C2×D1211yescharacteristic
107C2×D7—62no—
108C2×D7—121no—
109C2×D7—121no—
110C4×C7—121no—
111C4×C7—121no—
112C4×C7#23711yescharacteristic
113C4×C7D3×Q811yescharacteristic
114Dic7—21no—
115Dic7—121no—
116Dic7#20412yes—
117C22×D3—141no—
118C2×C3×C4—71no—
119C2×C3×C4—71no—
120C2×C3×C4C22×D711yescharacteristic
121C2×Dic3—21no—
122C2×Dic3—21no—
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—71no—
135C2×C8D3×D711yescharacteristic
136C4⋊C4—31no—
137C4⋊C4—31no—
138C4⋊C4—211no—
139C4⋊C4—421no—
140C22⋊C4—421no—
141C42—71no—
142C2×C7#89711yescharacteristic
143C2×C7C2×#20411yescharacteristic
144C2×C7#90811yescharacteristic
145D7—122no—
146C22×C3D4×D711yescharacteristic
147C2×D3—142no—
148C2×D3—281no—
149C2×D3—281no—
150C3×C4—72no—
151C3×C4—141no—
152C3×C4#115711yescharacteristic
153C3×C4Q8×D711yescharacteristic
154Dic3—41no—
155Dic3—41no—
156Dic3—281no—
157C23—421no—
158Q8—842no—
159D4—212no—
160D4—421no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—71no—
164C2×C4—71no—
165C2×C4—421no—
166C2×C4—842no—
167C2×C4C2×D3×D711yescharacteristic
168C8—21no—
169C8—141no—
170C7#484211yescharacteristic, complemented, Sylow
171C2×C3#645911yescharacteristic
172C2×C3D7×SD1611yescharacteristic
173C2×C3#647711yescharacteristic
174D3—282no—
175C22—422no—
176C22—841no—
177C22—841no—
178C22D7×D1211yescharacteristic
179C4—72no—
180C4—121no—
181C4—121no—
182C4—141no—
183C4—841no—
184C4D21⋊Q811yescharacteristic
185C4#6668311yescharacteristic
186C3#8964211yescharacteristic, complemented, Sylow
187C2—842no—
188C2D7×#20411yescharacteristic
189C2#1062081111yescharacteristic
190C2#1062098111yescharacteristic
subgroupG/H
centerC22D7×D12
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×C8C22
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenterC2×C8D3×D7
hypocenterC3×C7#476
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #10621030C2semidirect
4. C7×#817C2semidirect
5. C3×Dic7⋊C8C2semidirect
6. #10620520C2semidirect
24. Dic7⋊C8D3semidirect
32. D84Q8semidirect
58. #817D7semidirect
130. C3×C7#476semidirect
170. C7#4842semidirect
186. C3#89642semidirect

Maximal subgroups

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

subgroupindex
#106210302
#106209802
#106206602
C7×#8172
C3×Dic7⋊C82
#106205202
#106205042
#896423
#48427