Finite Group Databasebeta

← #13052629

Subgroups — #13052629

Order 1344.

192 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C3×#6455C211yesmaximal, characteristic
2#10620922C211yesmaximal, characteristic
3#10620649C211yesmaximal, characteristic, complemented
4C7×#807C211yesmaximal, characteristic, complemented
5#10620521C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic
8#89660—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C3×C4×D7C2211yescharacteristic
11C3×C7×C4⋊C4C2211yescharacteristic
12C2×C3⋊(C7⋊C8)C2211yescharacteristic
13C7×C4⋊Dic3C2211yescharacteristic
14C2×C7×C3⋊C8C2211yescharacteristic
15C3×(C2×D7)⋊C4—21no—
16C3×C4⋊Dic7C2211yescharacteristic
17C3×C4×Dic7—21no—
18Dic3⋊Dic7—21no—
19(C2×D7)⋊Dic3—21no—
20#6460—31no—
21#6455D311yescharacteristic
22C7×#107—31no—
23(C2×D7)⋊C8—31no—
24#6390—31no—
25#6384—31no—
26#6382—31no—
27#4996—71nomaximal, Hall
28C22×C3×D7D411yescharacteristic
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7C2311yescharacteristic
31C2×Dic21—21no—
32Dic42—21no—
33Dic42—21no—
34C2×C7×Dic3—21no—
35C2×C3×Dic7—21no—
36C2×C3×Dic7D411yescharacteristic
37C3×C4×D7—21no—
38C3×C4×D7—21no—
39C3⋊(C7⋊C8)—21no—
40C7×C3⋊C8—21no—
41C2×C4×D7C2×D311yescharacteristic
42C2×Dic14—31no—
43C2×C7×C8—31no—
44C7×C4⋊C4—31no—
45C7×C4⋊C4C2×D311yescharacteristic
46(C2×D7)⋊C4—21no—
47(C2×D7)⋊C4—61no—
48C4⋊Dic7—61no—
49C4⋊Dic7C2×D311yescharacteristic
50C4×Dic7—21no—
51C2×C7⋊C8—31no—
52C3×#117—71no—
53#924—71no—
54#830—71no—
55#810—71no—
56#810—71no—
57#808—71no—
58#807D711yescharacteristic, complemented
59C22×C3×C7C2×D411yescharacteristic
60C2×C3×D7—21no—
61C2×C3×D7—21no—
62C2×C3×D7—21no—
63C3×C4×C7—41no—
64C3×C4×C7#4111yescharacteristic
65C3×C4×C7#4111yescharacteristic
66Dic21—41no—
67C3×Dic7—21no—
68C3×Dic7—41no—
69C7×Dic3—41no—
70#485—211noSylow
71C22×D7#6711yescharacteristic
72C2×C4×C7—21no—
73C2×C4×C7—61no—
74C2×C4×C7C22×D311yescharacteristic
75C2×Dic7—21no—
76C2×Dic7—61no—
77C2×Dic7#6711yescharacteristic
78C4×D7—21no—
79C4×D7—21no—
80Dic14—61no—
81Dic14—61no—
82C7×C8—61no—
83C7⋊C8—61no—
84C22×C3×C4—71no—
85C2×Dic6—71no—
86C3×C4⋊C4—71no—
87C3×C4⋊C4C2×D711yescharacteristic
88C3×C22⋊C4—141no—
89C3×C42—141no—
90#217—141no—
91C4⋊Dic3C2×D711yescharacteristic
92Dic3⋊C4—141no—
93C2×C3⋊C8—71no—
94C2×C3⋊C8C2×D711yescharacteristic
95C2×C3×C7#12311yescharacteristic
96C2×C3×C7#13511yescharacteristic
97C2×C3×C7#13711yescharacteristic
98C3×D7—41no—
99C22⋊Q8—211no—
100#117—71no—
101#107—31no—
102#106—211no—
103#103—211no—
104#103—211no—
105C22⋊C8—211no—
106C22×C7C2×#6711yescharacteristic
107C2×D7—21no—
108C2×D7—21no—
109C2×D7—21no—
110C4×C7—41no—
111C4×C7—121no—
112C4×C7#23511yescharacteristic
113C4×C7#23511yescharacteristic
114Dic7—21no—
115Dic7—41no—
116Dic7—121no—
117C23×C3—71no—
118C2×C3×C4—21no—
119C2×C3×C4—71no—
120C2×C3×C4—141no—
121C2×C3×C4—141no—
122C2×C3×C4—141no—
123C2×C3×C4C22×D711yescharacteristic
124C2×Dic3—21no—
125C2×Dic3—141no—
126Dic6—141no—
127Dic6—141no—
128C3⋊C8—21no—
129C3⋊C8—141no—
130C3×C7#48511yescharacteristic, complemented, Hall
131C2×Q8—211no—
132C22×C4—71no—
133C2×C8—31no—
134C2×C8—211no—
135C4⋊C4—31no—
136C4⋊C4—71no—
137C4⋊C4—421no—
138C4⋊C4D3×D711yescharacteristic
139C22⋊C4—141no—
140C22⋊C4—421no—
141C42—141no—
142C2×C7#92911yescharacteristic
143C2×C7#95011yescharacteristic
144C2×C7#95111yescharacteristic
145D7—41no—
146C22×C3—141no—
147C22×C3—141no—
148C22×C3—141no—
149C22×C3D4×D711yescharacteristic
150C3×C4—41no—
151C3×C4—141no—
152C3×C4—281no—
153C3×C4#115711yescharacteristic
154C3×C4#116111yescharacteristic
155Dic3—41no—
156Dic3—281no—
157C23—71no—
158Q8—421no—
159Q8—421no—
160C2×C4—21no—
161C2×C4—61no—
162C2×C4—71no—
163C2×C4—141no—
164C2×C4—141no—
165C2×C4—141no—
166C2×C4—421no—
167C2×C4C2×D3×D711yescharacteristic
168C8—61no—
169C8—421no—
170C7#499611yescharacteristic, complemented, Sylow
171C2×C3—281no—
172C2×C3#645711yescharacteristic
173C2×C3#647511yescharacteristic
174C2×C3#648111yescharacteristic
175C22—141no—
176C22—141no—
177C22—141no—
178C22D7×#6711yescharacteristic
179C4—41no—
180C4—121no—
181C4—141no—
182C4—281no—
183C4—841no—
184C4#6667511yescharacteristic
185C4#6667711yescharacteristic
186C3#8966011yescharacteristic, complemented, Sylow
187C2—281no—
188C2#1062091011yescharacteristic
189C2#1062105211yescharacteristic
190C2#1062107011yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC22×C3×D7D4
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#485
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #10620649C2semidirect
4. C7×#807C2semidirect
5. #10620521C2semidirect
6. #10620509C2semidirect
58. #807D7semidirect
130. C3×C7#485semidirect
170. C7#4996semidirect
186. C3#89660semidirect

Maximal subgroups

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

subgroupindex
C3×#64552
#106209222
#106206492
C7×#8072
#106205212
#106205092
(C2×D7)⋊(C3⋊C8)2
#896603
#49967