Finite Group Databasebeta

← #13053557

Subgroups — #13053557

Order 1344.

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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#10621392C211yesmaximal, characteristic
2#10620883C211yesmaximal, characteristic
3C2×#66571C211yesmaximal, characteristic, complemented
4#10620852C211yesmaximal, characteristic
5C7×#824C211yesmaximal, characteristic, complemented
6C3×#6396C211yesmaximal, characteristic, complemented
7#10620498C211yesmaximal, characteristic, complemented
8#89527—31nomaximal, Hall
9#66728C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13Dic84—21no—
14#66623—21no—
15C7×#208C2211yescharacteristic
16C3×#1136C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18#66571—21no—
19#66571—21no—
20#66571—21no—
21#66568—21no—
22#66559—21no—
23D21⋊C8—21no—
24C2×#1144—31no—
25#6505—31no—
26#6472—31no—
27#6470—31no—
28C7×#101—31no—
29#6398—31no—
30#6396D311yescharacteristic, complemented
31#4896—71nomaximal, Hall
32C2×C3×C4×C7C2311yescharacteristic
33#4268—21no—
34C2×Dic21—21no—
35D84D411yescharacteristic
36C4×D21—21no—
37Dic42—21no—
38Dic42—21no—
39Dic42D411yescharacteristic
40C2×C7×Dic3—21no—
41C7×Dic6—21no—
42C7×Dic6—21no—
43C3×C7×C8—21no—
44C3×C7⋊C8—21no—
45C3×C7⋊C8D411yescharacteristic
46C3×C7⋊C8D411yescharacteristic
47C7×C3⋊C8—21no—
48C2×C7×Q8—31no—
49#1157—31no—
50C2×Dic14—31no—
51C7×M16—31no—
52C7×M16C2×D311yescharacteristic
53#1144—61no—
54#1144—61no—
55#1144—61no—
56#1143—61no—
57#1136C2×D311yescharacteristic
58C2×C7⋊C8C2×D311yescharacteristic
59Dic28—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63#909—71no—
64#909—71no—
65C2×Dic12—71no—
66#901—71no—
67C3×#108—71no—
68#824—71no—
69#824D711yescharacteristic, complemented
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C3×C4×C7C2×D411yescharacteristic
73C3×C4×C7C2×D411yescharacteristic
74Dic21—21no—
75Dic21—41no—
76C7×Dic3—41no—
77#473—211noSylow
78C7×Q8—61no—
79C7×Q8—61no—
80C2×C4×C7—61no—
81C2×C4×C7C22×D311yescharacteristic
82C22⋊D7—61no—
83C2×Dic7—61no—
84D28—31no—
85C4×D7—61no—
86Dic14—31no—
87Dic14—61no—
88Dic14—61no—
89C7×C8—21no—
90C7×C8—61no—
91C7⋊C8—21no—
92C7⋊C8D1211yescharacteristic
93C7⋊C8D1211yescharacteristic
94#235—71no—
95C2×Dic6—71no—
96C2×Dic6C2×D711yescharacteristic
97C3×M16—71no—
98C3×M16C2×D711yescharacteristic
99C2×C3×C8—71no—
100#208C2×D711yescharacteristic
101Dic12—141no—
102Dic12—141no—
103Dic12—141no—
104Dic12—141no—
105#204—141no—
106#204—141no—
107ΓL(1,𝔽25)—141no—
108D3×C8—141no—
109C2×C3×C7—21no—
110C2×C3×C7#12111yescharacteristic
111D21—41no—
112#137—211no—
113#137—211no—
114C2×Q16—211no—
115#131—211no—
116#108—71no—
117#101—31no—
118#101—211no—
119C22×C7#23911yescharacteristic
120C2×D7—61no—
121C4×C7—121no—
122C4×C7C2×D1211yescharacteristic
123C4×C7D3×D411yescharacteristic
124Dic7—61no—
125Dic7—121no—
126C2×C3×C4C22×D711yescharacteristic
127#67—141no—
128C2×Dic3—21no—
129C2×Dic3—141no—
130D12—71no—
131C4×D3—141no—
132Dic6—21no—
133Dic6—21no—
134Dic6—71no—
135Dic6—141no—
136Dic6—141no—
137C3×C8—21no—
138C3×C8—71no—
139C3×C8—71no—
140C3×C8—141no—
141C3⋊C8—21no—
142C3×C7#47311yescharacteristic, complemented, Hall
143#41—211no—
144C2×Q8—31no—
145C2×Q8—211no—
146Q16—421no—
147Q16—421no—
148Q16—421no—
149Q16—421no—
150SD16—421no—
151SD16—421no—
152M16—31no—
153M16—71no—
154M16—421no—
155M16D3×D711yescharacteristic
156C2×C8—71no—
157C2×C8—421no—
158C2×C7—21no—
159C2×C7#89511yescharacteristic
160D7—121no—
161C22×C3#115711yescharacteristic
162C2×D3—141no—
163C3×C4D4×D711yescharacteristic
164C3×C4D4×D711yescharacteristic
165Dic3—41no—
166Dic3—141no—
167Dic3—281no—
168Q8—61no—
169Q8—61no—
170Q8—211no—
171Q8—421no—
172Q8—421no—
173D4—211no—
174D4—421no—
175C2×C4—61no—
176C2×C4—421no—
177C2×C4—421no—
178C2×C4C2×D3×D711yescharacteristic
179C8—21no—
180C8—61no—
181C8—71no—
182C8—71no—
183C8—141no—
184C7#489611yescharacteristic, complemented, Sylow
185C2×C3—21no—
186C2×C3#644711yescharacteristic
187D3—281no—
188C22—421no—
189C22#6667311yescharacteristic
190C4—121no—
191C4—421no—
192C4—841no—
193C4D7×D1211yescharacteristic
194C4#6669411yescharacteristic
195C3#8952711yescharacteristic, complemented, Sylow
196C2—21no—
197C2—841no—
198C2#1062098211yescharacteristic
subgroupG/H
centerC2#10620982
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#121
prime radical#66728C22
hypercenterM16D3×D7
hypocenterC3×C7#473
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
3. C2×#66571C2semidirect
5. C7×#824C2semidirect
6. C3×#6396C2semidirect
7. #10620498C2semidirect
30. #6396D3semidirect
69. #824D7semidirect
142. C3×C7#473semidirect
184. C7#4896semidirect
195. C3#89527semidirect

Maximal subgroups

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

subgroupindex
#106213922
#106208832
C2×#665712
#106208522
C7×#8242
C3×#63962
#106204982
#895273
#48967