Finite Group Databasebeta

← #13052644

Subgroups — #13052644

Order 1344.

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

normal subgroups only

This group has 240 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
1C7×#895C211yesmaximal, characteristic
2#10620931C211yesmaximal, characteristic, complemented
3C2×#66564C211yesmaximal, characteristic, complemented
4C2×#66558C211yesmaximal, characteristic, complemented
5#10620646C211yesmaximal, characteristic, complemented
6C3×#6384C211yesmaximal, characteristic, complemented
7#10620502C211yesmaximal, characteristic, complemented
8#89814—31nomaximal, Hall
9C2×C7×D12—21no—
10C2×C7×D12C2211yescharacteristic
11C2×C4×D3×C7C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C4⋊Dic21C2211yescharacteristic
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C7×(C2×D3)⋊C4—21no—
17C2×C3×C7⋊C8C2211yescharacteristic
18#66577—21no—
19(C2×D3)⋊Dic7—21no—
20#66564—22no—
21#66558—22no—
22C7×#121—31no—
23C2×#1142—31no—
24#6487—31no—
25C2×#1136—31no—
26#6406—31no—
27#6384D311yescharacteristic, complemented
28#6381—31no—
29#4812—71nomaximal, Hall
30C22×D3×C7—21no—
31C22×D3×C7D411yescharacteristic
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×C7×Dic3D411yescharacteristic
36C7×D12—21no—
37C7×D12—21no—
38C7×D12—42no—
39C4×D3×C7—21no—
40C4×D3×C7—21no—
41C2×C3×Dic7—21no—
42C3×Dic14—21no—
43C3×Dic14—21no—
44C3⋊(C7⋊C8)D412yescomplemented
45C3×C7⋊C8—21no—
46C2×C7×D4—31no—
47C2×C7×D4—61no—
48C22×C4×C7—31no—
49C2×Dic14C2×D311yescharacteristic
50C7×C4⋊C4C2×D311yescharacteristic
51C7×C22⋊C4—61no—
52C22⋊Dic7—61no—
53#1142—62no—
54C4⋊Dic7—31no—
55Dic7⋊C4—61no—
56#1136—62no—
57C2×C7⋊C8—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59#946—71no—
60C2×#215—71no—
61C2×ΓL(1,𝔽25)—71no—
62#895D711yescharacteristic
63C3×#103—71no—
64#821—71no—
65#807—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D3×C7—21no—
68C2×D3×C7—21no—
69C2×D3×C7—21no—
70C2×D3×C7—41no—
71C2×D3×C7—41no—
72C2×D3×C7—41no—
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—41no—
77C3×Dic7—41no—
78C7×Dic3—21no—
79#468—211noSylow
80C23×C7—31no—
81C23×C7—61no—
82C7×D4—61no—
83C7×D4—61no—
84C7×D4—122no—
85C2×C4×C7—21no—
86C2×C4×C7—31no—
87C2×C4×C7—61no—
88C2×C4×C7—61no—
89C2×C4×C7C22×D311yescharacteristic
90C2×Dic7—21no—
91C2×Dic7—61no—
92Dic14—21no—
93Dic14—21no—
94C7⋊C8—21no—
95C7⋊C8—32no—
96C2×C3×Q8—71no—
97C2×D12—21no—
98C2×D12C2×D711yescharacteristic
99C2×C4×D3C2×D711yescharacteristic
100C2×C3×C8—71no—
101C3×C4⋊C4C2×D711yescharacteristic
102#215—142no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3—71no—
106Dic3⋊C4—141no—
107C2×C3⋊C8—71no—
108ΓL(1,𝔽25)—142no—
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
111C2×C3×C7#13711yescharacteristic
112D3×C7—41no—
113D3×C7—81no—
114C2×SD16—211no—
115C2×M16—211no—
116C22⋊Q8—211no—
117#121—31no—
118#107—211no—
119#103—71no—
120#102—211no—
121C22×C7—61no—
122C22×C7—61no—
123C22×C7—61no—
124C22×C7—121no—
125C22×C7—121no—
126C22×C7—121no—
127C22×C7D3×D411yescharacteristic
128C4×C7—41no—
129C4×C7—61no—
130C4×C7#23511yescharacteristic
131C4×C7D3×D411yescharacteristic
132Dic7—41no—
133Dic7—121no—
134C22×D3—21no—
135C22×D3C22⋊D711yescharacteristic
136C3×Q8—141no—
137C3×Q8—141no—
138C2×C3×C4—21no—
139C2×C3×C4—141no—
140C2×C3×C4C22×D711yescharacteristic
141C2×Dic3—141no—
142C2×Dic3C22⋊D711yescharacteristic
143D12—21no—
144D12—21no—
145D12—42no—
146C4×D3—21no—
147C4×D3—21no—
148C3×C8—141no—
149C3⋊C8—72no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8—71no—
152C2×D4—31no—
153C2×D4—61no—
154C22×C4—31no—
155SD16—422no—
156M16—422no—
157C2×C8—71no—
158C2×C8—211no—
159C4⋊C4—211no—
160C4⋊C4—421no—
161C4⋊C4D3×D711yescharacteristic
162C22⋊C4—61no—
163C22⋊C4—421no—
164C2×C7—121no—
165C2×C7—241no—
166C2×C7#88411yescharacteristic
167C2×C7#91411yescharacteristic
168C2×C7#91811yescharacteristic
169C22×C3C2×C22⋊D711yescharacteristic
170C2×D3—21no—
171C2×D3—21no—
172C2×D3—21no—
173C2×D3—41no—
174C2×D3—41no—
175C2×D3—41no—
176C3×C4—41no—
177C3×C4—281no—
178C3×C4D4×D711yescharacteristic
179C3×C4#115911yescharacteristic
180Dic3—21no—
181Dic3—281no—
182C23—31no—
183C23—61no—
184Q8—141no—
185Q8—141no—
186D4—61no—
187D4—61no—
188D4—122no—
189C2×C4—21no—
190C2×C4—31no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—141no—
197C8—212no—
198C7#481211yescharacteristic, complemented, Sylow
199C2×C3#649511yescharacteristic
200C2×C3#650211yescharacteristic
201C2×C3#651411yescharacteristic
202D3—41no—
203D3—81no—
204C22—61no—
205C22—61no—
206C22—61no—
207C22—121no—
208C22—121no—
209C22—121no—
210C22D3×C22⋊D711yescharacteristic
211C4—41no—
212C4—61no—
213C4—281no—
214C4—841no—
215C4#6667611yescharacteristic
216C4#6668111yescharacteristic
217C3#8981411yescharacteristic, complemented, Sylow
218C2—121no—
219C2—241no—
220C2#1062101511yescharacteristic
221C2#1062105811yescharacteristic
222C2#1062107211yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC7×#895C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#468
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
2. #10620931C2semidirect
3. C2×#66564C2semidirect
4. C2×#66558C2semidirect
5. #10620646C2semidirect
6. C3×#6384C2semidirect
7. #10620502C2semidirect
27. #6384D3semidirect
44. C3⋊(C7⋊C8)D4semidirect
150. C3×C7#468semidirect
198. C7#4812semidirect
217. C3#89814semidirect

Maximal subgroups

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

subgroupindex
C7×#8952
#106209312
C2×#665642
C2×#665582
#106206462
C3×#63842
#106205022
#898143
#48127