Finite Group Databasebeta

← #13054235

Subgroups — #13054235

Order 1344.

220 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 68 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
2C2×C7×#208C211yesmaximal, characteristic
3C3×#6487C211yesmaximal, characteristic
4C2×#66570C211yesmaximal, characteristic, complemented
5#10620522C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7#10620506C211yesmaximal, characteristic, complemented
8#89909—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66636—21no—
13C4⋊Dic21C2211yescharacteristic
14#66629—21no—
15C7×#208—22no—
16C2×C7×C3⋊C8C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C22⋊Dic7—21no—
19C3×C4⋊Dic7C2211yescharacteristic
20C3×Dic7⋊C4—21no—
21#66570—22no—
22C2×C7×M16—31no—
23#6487—31no—
24#6487D311yescharacteristic
25C2×Dic28—31no—
26#6391—31no—
27#6390—31no—
28#6390—31no—
29#5052—71nomaximal, Hall
30C23×C3×C7D411yescharacteristic
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7D411yescharacteristic
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21—21no—
37Dic42—21no—
38Dic42—21no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C7×C3⋊C8—21no—
44C7×C3⋊C8D412yes—
45C22×C4×C7C2×D311yescharacteristic
46C2×Dic14—31no—
47C2×Dic14C2×D311yescharacteristic
48C7×M16—62no—
49C2×C7×C8—31no—
50C2×C7×C8—31no—
51C22⋊Dic7—21no—
52C22⋊Dic7—61no—
53C4⋊Dic7—31no—
54C4⋊Dic7C2×D311yescharacteristic
55Dic7⋊C4—21no—
56Dic7⋊C4—61no—
57Dic28—62no—
58C3×C22⋊Q8—71no—
59C2×#216—71no—
60#924—71no—
61C2×#208D711yescharacteristic
62#835—71no—
63#810—71no—
64#808—71no—
65C22×C3×C7—21no—
66C22×C3×C7—21no—
67C22×C3×C7—21no—
68C22×C3×C7C2×D411yescharacteristic
69C3×C4×C7—21no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—41no—
73Dic21—41no—
74C3×Dic7—41no—
75C3×Dic7—41no—
76#470—211noSylow
77C23×C7#6711yescharacteristic
78C2×C4×C7—21no—
79C2×C4×C7—21no—
80C2×C4×C7#6711yescharacteristic
81C2×C4×C7C22×D311yescharacteristic
82C2×Dic7—21no—
83C2×Dic7—21no—
84C2×Dic7—61no—
85C2×Dic7—61no—
86Dic14—21no—
87Dic14—21no—
88Dic14—61no—
89Dic14—61no—
90C7×C8—32no—
91C7×C8—61no—
92C2×C3×Q8—71no—
93C22×C3×C4C2×D711yescharacteristic
94C2×Dic6—71no—
95C3×C4⋊C4—71no—
96C3×C4⋊C4—141no—
97C3×C22⋊C4—141no—
98#217—141no—
99#216—142no—
100C4⋊Dic3—71no—
101Dic3⋊C4—141no—
102#208—22no—
103C2×C3⋊C8C2×D711yescharacteristic
104C2×C3⋊C8C2×D711yescharacteristic
105C2×C3×C7—41no—
106C2×C3×C7#12111yescharacteristic
107C2×C3×C7#13711yescharacteristic
108C2×C3×C7#13711yescharacteristic
109C2×Q16—211no—
110C2×M16—31no—
111C22⋊Q8—71no—
112C22⋊Q8—211no—
113#106—211no—
114#103—211no—
115#103—211no—
116C22×C7—21no—
117C22×C7—21no—
118C22×C7—21no—
119C22×C7C2×#6711yescharacteristic
120C4×C7—21no—
121C4×C7D3×D411yescharacteristic
122C4×C7#23711yescharacteristic
123Dic7—41no—
124Dic7—41no—
125Dic7—121no—
126Dic7—121no—
127C23×C3D2811yescharacteristic
128C3×Q8—141no—
129C3×Q8—141no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—141no—
133C2×C3×C4—141no—
134C2×C3×C4D2811yescharacteristic
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—141no—
137C2×Dic3—141no—
138Dic6—141no—
139Dic6—141no—
140C3⋊C8—21no—
141C3⋊C8C22⋊D712yes—
142C3×C7#47011yescharacteristic, complemented, Hall
143C2×Q8—71no—
144C2×Q8—211no—
145C22×C4D3×D711yescharacteristic
146Q16—422no—
147M16—62no—
148C2×C8—31no—
149C2×C8—31no—
150C4⋊C4—71no—
151C4⋊C4—141no—
152C4⋊C4—211no—
153C4⋊C4—421no—
154C22⋊C4—141no—
155C22⋊C4—421no—
156C2×C7—41no—
157C2×C7#93911yescharacteristic
158C2×C7#94211yescharacteristic
159C2×C7#95111yescharacteristic
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3C2×D2811yescharacteristic
164C3×C4—21no—
165C3×C4—281no—
166C3×C4—281no—
167C3×C4#115711yescharacteristic
168C3×C4C2×C22⋊D711yescharacteristic
169Dic3—281no—
170Dic3—281no—
171C23#424611yescharacteristic
172Q8—141no—
173Q8—141no—
174Q8—421no—
175Q8—421no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—141no—
179C2×C4—141no—
180C2×C4—421no—
181C2×C4—421no—
182C2×C4#424611yescharacteristic
183C2×C4C2×D3×D711yescharacteristic
184C8—32no—
185C8—61no—
186C7#505211yescharacteristic, complemented, Sylow
187C2×C3—41no—
188C2×C3#647211yescharacteristic
189C2×C3#647211yescharacteristic
190C2×C3#649311yescharacteristic
191C22—21no—
192C22—21no—
193C22—21no—
194C22C2×#424611yescharacteristic
195C4—21no—
196C4—281no—
197C4—281no—
198C4—841no—
199C4—841no—
200C4#6668311yescharacteristic
201C4D3×C22⋊D711yescharacteristic
202C3#8990911yescharacteristic, complemented, Sylow
203C2—41no—
204C2#1062087511yescharacteristic
205C2#1062087711yescharacteristic
206C2#1062113111yescharacteristic
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#470
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. C2×#66570C2semidirect
5. #10620522C2semidirect
6. #10620509C2semidirect
7. #10620506C2semidirect
142. C3×C7#470semidirect
186. C7#5052semidirect
202. C3#89909semidirect

Maximal subgroups

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

subgroupindex
#106214072
C2×C7×#2082
C3×#64872
C2×#665702
#106205222
#106205092
#106205062
#899093
#50527