Finite Group Databasebeta

← #13053807

Subgroups — #13053807

Order 1344.

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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#10621407C211yesmaximal, characteristic
2C3×#6489C211yesmaximal, characteristic
3C7×#830C211yesmaximal, characteristic
4#10620525C211yesmaximal, characteristic, complemented
5#10620522C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7#10620509C211yesmaximal, characteristic, complemented
8#89495—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11#66636—21no—
12C4⋊Dic21C2211yescharacteristic
13#66629—21no—
14C2×C7×C3⋊C8C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C22⋊Dic7—21no—
17C3×C4⋊Dic7C2211yescharacteristic
18C3×C4⋊Dic7C2211yescharacteristic
19C3×C4×Dic7—21no—
20#6489D311yescharacteristic
21#6487—31no—
22C7×C22⋊C8—31no—
23#6392—31no—
24#6391—31no—
25#6390—31no—
26#6390—31no—
27#4996—71nomaximal, Hall
28C23×C3×C7D411yescharacteristic
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7D411yescharacteristic
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21—21no—
35Dic42—21no—
36Dic42—21no—
37C2×C3×Dic7—21no—
38C2×C3×Dic7—21no—
39C7×C3⋊C8—21no—
40C7×C3⋊C8—21no—
41C22×C4×C7C2×D311yescharacteristic
42C2×Dic14—31no—
43C2×C7×C8—31no—
44C2×C7×C8—31no—
45C22⋊Dic7—21no—
46C22⋊Dic7—61no—
47C4⋊Dic7—31no—
48C4⋊Dic7C2×D311yescharacteristic
49C4⋊Dic7C2×D311yescharacteristic
50Dic7⋊C4—61no—
51C4×Dic7—21no—
52C3×#117—71no—
53#924—71no—
54#830D711yescharacteristic
55#810—71no—
56#810—71no—
57#808—71no—
58#807—71no—
59C22×C3×C7—21no—
60C22×C3×C7—21no—
61C22×C3×C7—21no—
62C22×C3×C7C2×D411yescharacteristic
63C3×C4×C7—21no—
64C3×C4×C7#4111yescharacteristic
65C3×C4×C7#4111yescharacteristic
66Dic21—41no—
67Dic21—41no—
68C3×Dic7—41no—
69C3×Dic7—41no—
70#485—211noSylow
71C23×C7#6711yescharacteristic
72C2×C4×C7—21no—
73C2×C4×C7—21no—
74C2×C4×C7#6711yescharacteristic
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—21no—
77C2×Dic7—21no—
78C2×Dic7—61no—
79C2×Dic7—61no—
80Dic14—61no—
81Dic14—61no—
82C7×C8—61no—
83C7×C8—61no—
84C22×C3×C4C2×D711yescharacteristic
85C2×Dic6—71no—
86C3×C4⋊C4—71no—
87C3×C4⋊C4—71no—
88C3×C22⋊C4—141no—
89C3×C42—141no—
90#217—141no—
91C4⋊Dic3—71no—
92Dic3⋊C4—141no—
93C2×C3⋊C8C2×D711yescharacteristic
94C2×C3⋊C8C2×D711yescharacteristic
95C2×C3×C7—41no—
96C2×C3×C7#12311yescharacteristic
97C2×C3×C7#13511yescharacteristic
98C2×C3×C7#13711yescharacteristic
99C22⋊Q8—211no—
100#117—71no—
101#107—211no—
102#106—211no—
103#103—211no—
104#103—211no—
105C22⋊C8—31no—
106C22×C7—21no—
107C22×C7—21no—
108C22×C7—21no—
109C22×C7C2×#6711yescharacteristic
110C4×C7—21no—
111C4×C7#23511yescharacteristic
112C4×C7#23511yescharacteristic
113Dic7—41no—
114Dic7—41no—
115Dic7—121no—
116Dic7—121no—
117C23×C3D2811yescharacteristic
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—141no—
121C2×C3×C4—141no—
122C2×C3×C4D2811yescharacteristic
123C2×C3×C4C22×D711yescharacteristic
124C2×Dic3—141no—
125C2×Dic3—141no—
126Dic6—141no—
127Dic6—141no—
128C3⋊C8—21no—
129C3⋊C8—21no—
130C3×C7#48511yescharacteristic, complemented, Hall
131C2×Q8—211no—
132C22×C4D3×D711yescharacteristic
133C2×C8—31no—
134C2×C8—31no—
135C4⋊C4—71no—
136C4⋊C4—71no—
137C4⋊C4—211no—
138C4⋊C4—421no—
139C22⋊C4—141no—
140C22⋊C4—421no—
141C42—141no—
142C2×C7—41no—
143C2×C7#92911yescharacteristic
144C2×C7#95011yescharacteristic
145C2×C7#95111yescharacteristic
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3C2×D2811yescharacteristic
150C3×C4—21no—
151C3×C4—281no—
152C3×C4—281no—
153C3×C4#115911yescharacteristic
154C3×C4#115911yescharacteristic
155Dic3—281no—
156Dic3—281no—
157C23#424611yescharacteristic
158Q8—421no—
159Q8—421no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—141no—
163C2×C4—141no—
164C2×C4—421no—
165C2×C4—421no—
166C2×C4#424611yescharacteristic
167C2×C4C2×D3×D711yescharacteristic
168C8—61no—
169C8—61no—
170C7#499611yescharacteristic, complemented, Sylow
171C2×C3—41no—
172C2×C3#644911yescharacteristic
173C2×C3#646711yescharacteristic
174C2×C3#647211yescharacteristic
175C22—21no—
176C22—21no—
177C22—21no—
178C22C2×#424611yescharacteristic
179C4—21no—
180C4—281no—
181C4—281no—
182C4—841no—
183C4—841no—
184C4#6668611yescharacteristic
185C4#6668611yescharacteristic
186C3#8949511yescharacteristic, complemented, Sylow
187C2—41no—
188C2#1062087311yescharacteristic
189C2#1062087511yescharacteristic
190C2#1062110811yescharacteristic
subgroupG/H
centerC22C2×#4246
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC22×C4D3×D7
hypocenterC3×C7#485
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #10620525C2semidirect
5. #10620522C2semidirect
6. #10620509C2semidirect
7. #10620509C2semidirect
130. C3×C7#485semidirect
170. C7#4996semidirect
186. C3#89495semidirect

Maximal subgroups

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

subgroupindex
#106214072
C3×#64892
C7×#8302
#106205252
#106205222
#106205092
#894953
#49967