Finite Group Databasebeta

← #13056397

Subgroups — #13056397

Order 1344.

256 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 65 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#10621423C211yesmaximal, characteristic
2C2×#66633C211yesmaximal, characteristic, complemented
3C2×#66623C211yesmaximal, characteristic, complemented
4#10621370C211yesmaximal, characteristic, complemented
5C3×C7×#102C211yesmaximal, characteristic, complemented
6#10620653C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic, complemented
8#89534—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×#4268—21no—
12C2×D84C2211yescharacteristic
13C2×Dic42C2211yescharacteristic
14C2×C3×C7×C8C2211yescharacteristic
15C3×C7×C4⋊C4C2211yescharacteristic
16#66633—22no—
17#66629—21no—
18C4×Dic21—21no—
19C4×Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21#66623—22no—
22#6503—31no—
23C2×#1142—31no—
24C2×#1132—31no—
25#6450—31no—
26C7×#102D311yescharacteristic, complemented
27Dic7⋊C8—31no—
28#6383—31no—
29#4762—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D21—21no—
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—42no—
37#4268—42no—
38C2×Dic21—21no—
39C2×Dic21D411yescharacteristic
40C2×Dic21D411yescharacteristic
41D84—21no—
42D84—21no—
43Dic42—21no—
44Dic42D412yescomplemented
45C3×C7×C8—21no—
46C3⋊(C7⋊C8)—21no—
47C2×C7×D4C2×D311yescharacteristic
48C2×C22⋊D7—61no—
49C2×C22⋊D7—61no—
50C2×D28—31no—
51C2×Dic14—31no—
52C2×C7×C8C2×D311yescharacteristic
53C7×C4⋊C4C2×D311yescharacteristic
54#1142—62no—
55Dic7⋊C4—61no—
56C4×Dic7—31no—
57C4×Dic7—61no—
58C2×C7⋊C8—31no—
59#1132—62no—
60#940—71no—
61C2×#214—71no—
62C2×#204—71no—
63#887—71no—
64C3×#102D711yescharacteristic, complemented
65Dic3⋊C8—71no—
66#809—71no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7—41no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D21—41no—
72C2×D21—41no—
73C2×D21—41no—
74C3×C4×C7—41no—
75C3×C4×C7C2×D411yescharacteristic
76C3×C4×C7#4111yescharacteristic
77Dic21—22no—
78Dic21—21no—
79Dic21SD1612yescomplemented
80#460—211noSylow
81C23×C7—21no—
82C22×D7—61no—
83C7×D4—21no—
84C7×D4—21no—
85C2×C4×C7—21no—
86C2×C4×C7C22×D311yescharacteristic
87C22⋊D7—122no—
88C22⋊D7—122no—
89C2×Dic7—31no—
90C2×Dic7—31no—
91C2×Dic7—61no—
92D28—61no—
93D28—61no—
94Dic14—32no—
95Dic14—61no—
96C7×C8—21no—
97C7⋊C8—61no—
98C2×C3×D4C2×D711yescharacteristic
99C2×#67—141no—
100C2×#67—141no—
101C2×D12—71no—
102C2×Dic6—71no—
103C2×C3×C8C2×D711yescharacteristic
104C3×C4⋊C4C2×D711yescharacteristic
105#214—142no—
106Dic3⋊C4—141no—
107C4×Dic3—71no—
108C4×Dic3—141no—
109C2×C3⋊C8—71no—
110#204—142no—
111C2×C3×C7—81no—
112C2×C3×C7#12111yescharacteristic
113C2×C3×C7C2×SD1611yescharacteristic
114C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
115D21—81no—
116C2×SD16—211no—
117C2×SD16—211no—
118#127—211no—
119C4×Q8—211no—
120C4⋊C8—211no—
121#102—211no—
122#102D2111yescharacteristic, complemented
123C22×C7—41no—
124C22×C7—41no—
125C22×C7—41no—
126C22×C7D3×D411yescharacteristic
127C2×D7—121no—
128C2×D7—121no—
129C2×D7—121no—
130C4×C7—41no—
131C4×C7#23511yescharacteristic
132C4×C7D3×D411yescharacteristic
133Dic7—32no—
134Dic7—62no—
135Dic7—61no—
136C23×C3—21no—
137C22×D3—141no—
138C3×D4—21no—
139C3×D4—21no—
140C2×C3×C4—21no—
141C2×C3×C4C22×D711yescharacteristic
142#67—282no—
143#67—282no—
144C2×Dic3—71no—
145C2×Dic3—71no—
146C2×Dic3—141no—
147D12—141no—
148D12—141no—
149Dic6—72no—
150Dic6—141no—
151C3×C8—21no—
152C3⋊C8—141no—
153C3×C7#46011yescharacteristic, complemented, Hall
154C2×Q8—211no—
155C2×D4—211no—
156C2×D4—421no—
157C2×D4—421no—
158C2×D4C2×D2111yescharacteristic
159SD16—422no—
160SD16—422no—
161C2×C8—211no—
162C2×C8C2×D2111yescharacteristic
163C4⋊C4—421no—
164C4⋊C4C2×D2111yescharacteristic
165C42—211no—
166C42—421no—
167C2×C7—81no—
168C2×C7#88411yescharacteristic
169C2×C7#91111yescharacteristic
170C2×C7D3×SD1611yescharacteristic
171D7—241no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3D4×D711yescharacteristic
176C2×D3—281no—
177C2×D3—281no—
178C2×D3—281no—
179C3×C4—41no—
180C3×C4#115711yescharacteristic
181C3×C4D4×D711yescharacteristic
182Dic3—72no—
183Dic3—142no—
184Dic3—141no—
185C23—21no—
186C23—421no—
187Q8—212no—
188Q8—421no—
189D4—21no—
190D4—21no—
191D4—421no—
192D4—421no—
193D4—842no—
194D4—842no—
195C2×C4—21no—
196C2×C4—211no—
197C2×C4—211no—
198C2×C4—421no—
199C2×C4C22×D2111yescharacteristic
200C8—21no—
201C8—421no—
202C7#476211yescharacteristic, complemented, Sylow
203C2×C3—81no—
204C2×C3#644711yescharacteristic
205C2×C3#647411yescharacteristic
206C2×C3D7×SD1611yescharacteristic
207D3—561no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—841no—
212C22—841no—
213C22—841no—
214C22D4×D2111yescharacteristic
215C4—41no—
216C4—212no—
217C4—422no—
218C4—421no—
219C4#6672811yescharacteristic
220C4D4×D2111yescharacteristic
221C3#8953411yescharacteristic, complemented, Sylow
222C2—81no—
223C2—1681no—
224C2#1062136711yescharacteristic
225C2#1062139411yescharacteristic
226C2SD16×D2111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radical#10621423C2
hypercenter#102D21
hypocenterC3×C7#460
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#66633C2semidirect
3. C2×#66623C2semidirect
4. #10621370C2semidirect
5. C3×C7×#102C2semidirect
6. #10620653C2semidirect
7. #10620648C2semidirect
26. C7×#102D3semidirect
44. Dic42D4semidirect
64. C3×#102D7semidirect
79. Dic21SD16semidirect
122. #102D21semidirect
153. C3×C7#460semidirect
202. C7#4762semidirect
221. C3#89534semidirect

Maximal subgroups

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

subgroupindex
#106214232
C2×#666332
C2×#666232
#106213702
C3×C7×#1022
#106206532
#106206482
#895343
#47627