Finite Group Databasebeta

← #13054698

Subgroups — #13054698

Order 1344.

196 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 62 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#10621427C211yesmaximal, characteristic
2#10621032C211yesmaximal, characteristic
3C7×#835C211yesmaximal, characteristic
4C3×#6409C211yesmaximal, characteristic
5#10620510C211yesmaximal, characteristic
6#10620509C211yesmaximal, characteristic
7#10620486C211yesmaximal, characteristic
8#89564—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×#4248—21no—
12#66629—21no—
13#66629—21no—
14C4×Dic21C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19#66577—21no—
20#6507—31no—
21#6451—31no—
22C7×#103—31no—
23#6409D311yescharacteristic
24#6390—31no—
25C8×Dic7—31no—
26#6384—31no—
27#4792—71nomaximal, Hall
28C3×C7×Q8—21no—
29C3×C7×Q8—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C2×Dic21D411yescharacteristic
35Dic42—21no—
36Dic42—21no—
37C2×C7×Dic3—21no—
38C2×C3×Dic7—21no—
39#4248—42no—
40C3×C7⋊C8—21no—
41C7×C3⋊C8—21no—
42C2×C7×Q8C2×D311yescharacteristic
43C2×Dic14—31no—
44C2×Dic14—61no—
45C2×C7×C8—31no—
46C7×C4⋊C4—31no—
47C4⋊Dic7C2×D311yescharacteristic
48Dic7⋊C4—61no—
49Dic7⋊C4—61no—
50Dic7⋊C4—61no—
51C4×Dic7—31no—
52C2×C7⋊C8C2×D311yescharacteristic
53#944—71no—
54#888—71no—
55C3×#103—71no—
56#835D711yescharacteristic
57#816—71no—
58C8×Dic3—71no—
59#810—71no—
60C22×C3×C7C2×D411yescharacteristic
61C3×C4×C7—41no—
62C3×C4×C7#4111yescharacteristic
63C3×C4×C7#4111yescharacteristic
64Dic21—41no—
65Dic21SD1612yes—
66Dic21Q1612yes—
67C3×Dic7—41no—
68C7×Dic3—41no—
69#487—211noSylow
70C7×Q8—21no—
71C7×Q8—21no—
72C2×C4×C7—21no—
73C2×C4×C7—61no—
74C2×C4×C7C22×D311yescharacteristic
75C2×Dic7—21no—
76C2×Dic7—31no—
77C2×Dic7—31no—
78C2×Dic7—61no—
79Dic14—61no—
80Dic14—61no—
81Dic14—122no—
82C7×C8—61no—
83C7⋊C8—21no—
84C2×C3×Q8C2×D711yescharacteristic
85C2×Dic6—71no—
86C2×Dic6—141no—
87C2×C3×C8—71no—
88C3×C4⋊C4—71no—
89C4⋊Dic3C2×D711yescharacteristic
90Dic3⋊C4—141no—
91Dic3⋊C4—141no—
92Dic3⋊C4—141no—
93C4×Dic3—71no—
94C2×C3⋊C8C2×D711yescharacteristic
95C2×C3×C7#12411yescharacteristic
96C2×C3×C7C2×SD1611yescharacteristic
97C2×C3×C7C2×Q1611yescharacteristic
98C4⋊Q8—211no—
99C4⋊Q8—211no—
100#103—31no—
101#103—71no—
102#103—211no—
103#103—211no—
104C4×C8—211no—
105C22×C7D3×D411yescharacteristic
106C4×C7—41no—
107C4×C7—121no—
108C4×C7#23511yescharacteristic
109C4×C7#23711yescharacteristic
110Dic7—32no—
111Dic7—32no—
112Dic7—41no—
113Dic7—121no—
114C3×Q8—21no—
115C3×Q8—21no—
116C2×C3×C4—21no—
117C2×C3×C4—141no—
118C2×C3×C4C22×D711yescharacteristic
119C2×Dic3—21no—
120C2×Dic3—71no—
121C2×Dic3—71no—
122C2×Dic3—141no—
123Dic6—141no—
124Dic6—141no—
125Dic6—282no—
126C3×C8—141no—
127C3⋊C8—21no—
128C3×C7#48711yescharacteristic, complemented, Hall
129C2×Q8—211no—
130C2×Q8—421no—
131C2×Q8D3×D711yescharacteristic
132C2×C8—31no—
133C2×C8—71no—
134C4⋊C4—31no—
135C4⋊C4—71no—
136C4⋊C4—421no—
137C4⋊C4—421no—
138C4⋊C4—421no—
139C42—211no—
140C2×C7#88511yescharacteristic
141C2×C7D3×SD1611yescharacteristic
142C2×C7D3×Q1611yescharacteristic
143C22×C3D4×D711yescharacteristic
144C3×C4—41no—
145C3×C4—281no—
146C3×C4#115711yescharacteristic
147C3×C4#115911yescharacteristic
148Dic3—41no—
149Dic3—72no—
150Dic3—72no—
151Dic3—281no—
152Q8—21no—
153Q8—21no—
154Q8—421no—
155Q8—421no—
156Q8—842no—
157C2×C4—21no—
158C2×C4—61no—
159C2×C4—141no—
160C2×C4—211no—
161C2×C4—211no—
162C2×C4—421no—
163C2×C4C2×D3×D711yescharacteristic
164C8—61no—
165C8—141no—
166C7#479211yescharacteristic, complemented, Sylow
167C2×C3#644811yescharacteristic
168C2×C3D7×SD1611yescharacteristic
169C2×C3D7×Q1611yescharacteristic
170C22#6669411yescharacteristic
171C4—41no—
172C4—121no—
173C4—212no—
174C4—212no—
175C4—281no—
176C4—841no—
177C4#6668311yescharacteristic
178C4#6668611yescharacteristic
179C3#8956411yescharacteristic, complemented, Sylow
180C2#1062106711yescharacteristic
181C2#1062107311yescharacteristic
182C2#1062108811yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socle, prime radicalC22×C3×C7C2×D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#487
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
128. C3×C7#487semidirect
166. C7#4792semidirect
179. C3#89564semidirect

Maximal subgroups

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

subgroupindex
#106214272
#106210322
C7×#8352
C3×#64092
#106205102
#106205092
#106204862
#895643
#47927