Finite Group Databasebeta

← #13054587

Subgroups — #13054587

Order 1344.

192 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 58 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#10621429C211yesmaximal, characteristic
2#10620909C211yesmaximal, characteristic
3C7×#835C211yesmaximal, characteristic, complemented
4C3×#6409C211yesmaximal, characteristic, complemented
5#10620523C211yesmaximal, characteristic, complemented
6#10620522C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic
8#89587—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C4×D21C2211yescharacteristic
11#66631—21no—
12C4⋊Dic21C2211yescharacteristic
13#66629—21no—
14C7×C4⋊Dic3C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C4⋊Dic7C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18#66575—21no—
19Dic3×Dic7—21no—
20#6509—31no—
21#6455—31no—
22C7×#103—31no—
23#6409D311yescharacteristic, complemented
24(C2×D7)⋊C8—31no—
25#6391—31no—
26#6381—31no—
27#4814—71nomaximal, Hall
28C22×D21D411yescharacteristic
29C3×C7×Q8—21no—
30C3×C7×Q8—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21D411yescharacteristic
35C4×D21—21no—
36C4×D21—21no—
37C2×C7×Dic3—21no—
38C2×C3×Dic7—21no—
39C3×C7⋊C8—21no—
40C7×C3⋊C8—21no—
41C2×C7×Q8C2×D311yescharacteristic
42C2×C4×D7—31no—
43C2×C7×C8—31no—
44C7×C4⋊C4—31no—
45(C2×D7)⋊C4—61no—
46(C2×D7)⋊C4—61no—
47C4⋊Dic7—31no—
48C4⋊Dic7C2×D311yescharacteristic
49Dic7⋊C4—61no—
50C4×Dic7—61no—
51C2×C7⋊C8C2×D311yescharacteristic
52#946—71no—
53#892—71no—
54C3×#103—71no—
55#835D711yescharacteristic, complemented
56(C2×D3)⋊C8—71no—
57#818—71no—
58#808—71no—
59C22×C3×C7C2×D411yescharacteristic
60C2×D21—21no—
61C2×D21—21no—
62C2×D21—21no—
63C3×C4×C7—41no—
64C3×C4×C7#4111yescharacteristic
65C3×C4×C7#4111yescharacteristic
66Dic21—21no—
67Dic21—41no—
68C3×Dic7—41no—
69C7×Dic3—41no—
70#485—211noSylow
71C22×D7—31no—
72C7×Q8—21no—
73C7×Q8—21no—
74C2×C4×C7—21no—
75C2×C4×C7—61no—
76C2×C4×C7C22×D311yescharacteristic
77C2×Dic7—21no—
78C2×Dic7—31no—
79C2×Dic7—61no—
80C4×D7—61no—
81C4×D7—61no—
82C7×C8—61no—
83C7⋊C8—21no—
84C2×C3×Q8C2×D711yescharacteristic
85C2×C4×D3—71no—
86C2×C3×C8—71no—
87C3×C4⋊C4—71no—
88(C2×D3)⋊C4—141no—
89(C2×D3)⋊C4—141no—
90C4⋊Dic3—71no—
91C4⋊Dic3C2×D711yescharacteristic
92Dic3⋊C4—141no—
93C4×Dic3—141no—
94C2×C3⋊C8C2×D711yescharacteristic
95C2×C3×C7#12311yescharacteristic
96C2×C3×C7#13511yescharacteristic
97C2×C3×C7#13711yescharacteristic
98D21—41no—
99C22⋊Q8—211no—
100#117—211no—
101#107—211no—
102#106—211no—
103#103—31no—
104#103—71no—
105C22⋊C8—211no—
106C22×C7D3×D411yescharacteristic
107C2×D7—61no—
108C2×D7—61no—
109C2×D7—61no—
110C4×C7—41no—
111C4×C7—121no—
112C4×C7#23511yescharacteristic
113C4×C7#23711yescharacteristic
114Dic7—41no—
115Dic7—61no—
116Dic7—121no—
117C22×D3—71no—
118C3×Q8—21no—
119C3×Q8—21no—
120C2×C3×C4—21no—
121C2×C3×C4—141no—
122C2×C3×C4C22×D711yescharacteristic
123C2×Dic3—21no—
124C2×Dic3—71no—
125C2×Dic3—141no—
126C4×D3—141no—
127C4×D3—141no—
128C3×C8—141no—
129C3⋊C8—21no—
130C3×C7#48511yescharacteristic, complemented, Hall
131C2×Q8D3×D711yescharacteristic
132C22×C4—211no—
133C2×C8—31no—
134C2×C8—71no—
135C4⋊C4—31no—
136C4⋊C4—71no—
137C4⋊C4—211no—
138C4⋊C4—421no—
139C22⋊C4—421no—
140C22⋊C4—421no—
141C42—421no—
142C2×C7#88311yescharacteristic
143C2×C7#91511yescharacteristic
144C2×C7#91911yescharacteristic
145D7—121no—
146C22×C3D4×D711yescharacteristic
147C2×D3—141no—
148C2×D3—141no—
149C2×D3—141no—
150C3×C4—41no—
151C3×C4—281no—
152C3×C4#115711yescharacteristic
153C3×C4#115911yescharacteristic
154Dic3—41no—
155Dic3—141no—
156Dic3—281no—
157C23—211no—
158Q8—21no—
159Q8—21no—
160C2×C4—21no—
161C2×C4—61no—
162C2×C4—141no—
163C2×C4—211no—
164C2×C4—421no—
165C2×C4—421no—
166C2×C4—421no—
167C2×C4C2×D3×D711yescharacteristic
168C8—61no—
169C8—141no—
170C7#481411yescharacteristic, complemented, Sylow
171C2×C3#644611yescharacteristic
172C2×C3#647911yescharacteristic
173C2×C3#648111yescharacteristic
174D3—281no—
175C22—421no—
176C22—421no—
177C22—421no—
178C22#6669411yescharacteristic
179C4—41no—
180C4—121no—
181C4—281no—
182C4—421no—
183C4—841no—
184C4#6668311yescharacteristic
185C4#6668611yescharacteristic
186C3#8958711yescharacteristic, complemented, Sylow
187C2—841no—
188C2#1062108411yescharacteristic
189C2#1062108511yescharacteristic
190C2#1062112411yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#485
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C7×#835C2semidirect
4. C3×#6409C2semidirect
5. #10620523C2semidirect
6. #10620522C2semidirect
23. #6409D3semidirect
55. #835D7semidirect
130. C3×C7#485semidirect
170. C7#4814semidirect
186. C3#89587semidirect

Maximal subgroups

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

subgroupindex
#106214292
#106209092
C7×#8352
C3×#64092
#106205232
#106205222
#106204922
#895873
#48147