Finite Group Databasebeta

← #13056417

Subgroups — #13056417

Order 1344.

324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#66730C211yesmaximal, characteristic, complemented
2C2×#66632C211yesmaximal, characteristic, complemented
3C2×#66623C211yesmaximal, characteristic, complemented
4#10621378C211yesmaximal, characteristic, complemented
5C3×C7×#102C211yesmaximal, characteristic, complemented
6#10620659C211yesmaximal, characteristic, complemented
7#10620649C211yesmaximal, characteristic, complemented
8#89553—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C22×Dic21—21no—
12#66730—22no—
13#66730—22no—
14C2×D84—21no—
15C2×D84C2211yescharacteristic
16C2×C4×D21C2211yescharacteristic
17C2×Dic42C2211yescharacteristic
18C2×C3×C7×C8C2211yescharacteristic
19C3×C7×C4⋊C4C2211yescharacteristic
20#66632—22no—
21#66631—21no—
22C2×C3⋊(C7⋊C8)C2211yescharacteristic
23#66623—22no—
24C2×#1159—31no—
25C2×#1141—31no—
26C2×#1132—31no—
27#6458—31no—
28C7×#102D311yescharacteristic, complemented
29(C2×D7)⋊C8—31no—
30#6384—31no—
31#4781—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21—21no—
34C22×D21D411yescharacteristic
35C3×C7×D4—21no—
36C3×C7×D4D412yescomplemented
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39#4268—24no—
40C2×Dic21—24no—
41C2×Dic21—21no—
42C2×Dic21D411yescharacteristic
43D84—21no—
44D84—21no—
45D84—42no—
46C4×D21—21no—
47C4×D21—21no—
48Dic42—21no—
49Dic42D412yescomplemented
50C3×C7×C8—21no—
51C3⋊(C7⋊C8)—21no—
52C2×C7×D4C2×D311yescharacteristic
53C2×C22⋊D7—61no—
54C22×Dic7—61no—
55#1159—62no—
56#1159—62no—
57C2×D28—31no—
58C2×D28—61no—
59C2×C4×D7—31no—
60C2×Dic14—31no—
61C2×C7×C8C2×D311yescharacteristic
62C7×C4⋊C4C2×D311yescharacteristic
63#1141—62no—
64(C2×D7)⋊C4—61no—
65C2×C7⋊C8—31no—
66#1132—62no—
67C2×#237—71no—
68C2×#213—71no—
69C2×#204—71no—
70#895—71no—
71C3×#102D711yescharacteristic, complemented
72(C2×D3)⋊C8—71no—
73#810—71no—
74C22×C3×C7—22no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D21—21no—
79C2×D21—21no—
80C2×D21—21no—
81C2×D21—41no—
82C2×D21—41no—
83C2×D21—41no—
84C3×C4×C7—41no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—22no—
88Dic21—21no—
89#449—211noSylow
90C23×C7—21no—
91C22×D7—31no—
92C22×D7—61no—
93C7×D4—21no—
94C7×D4D1212yescomplemented
95C2×C4×C7—21no—
96C2×C4×C7C22×D311yescharacteristic
97C22⋊D7—64no—
98C2×Dic7—31no—
99C2×Dic7—64no—
100C2×Dic7—61no—
101D28—61no—
102D28—61no—
103D28—122no—
104C4×D7—61no—
105C4×D7—61no—
106Dic14—32no—
107Dic14—61no—
108C7×C8—21no—
109C7⋊C8—61no—
110C2×C3×D4C2×D711yescharacteristic
111C2×#67—141no—
112C22×Dic3—141no—
113#237—142no—
114#237—142no—
115C2×D12—71no—
116C2×D12—141no—
117C2×C4×D3—71no—
118C2×Dic6—71no—
119C2×C3×C8C2×D711yescharacteristic
120C3×C4⋊C4C2×D711yescharacteristic
121#213—142no—
122(C2×D3)⋊C4—141no—
123C2×C3⋊C8—71no—
124#204—142no—
125C2×C3×C7—42no—
126C2×C3×C7#12011yescharacteristic
127C2×C3×C7#13511yescharacteristic
128C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
129D21—41no—
130D21—81no—
131C2×#41—211no—
132C2×SD16—211no—
133C2×D8—211no—
134#121—211no—
135#103—211no—
136#102D2111yescharacteristic, complemented
137C22⋊C8—211no—
138C22×C7—22no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7D3×D411yescharacteristic
142C2×D7—61no—
143C2×D7—61no—
144C2×D7—61no—
145C2×D7—121no—
146C2×D7—121no—
147C2×D7—121no—
148C4×C7—41no—
149C4×C7C2×D1211yescharacteristic
150C4×C7D3×D411yescharacteristic
151Dic7—62no—
152Dic7—61no—
153C23×C3—21no—
154C22×D3—71no—
155C22×D3—141no—
156C3×D4—21no—
157C3×D4D2812yescomplemented
158C2×C3×C4—21no—
159C2×C3×C4C22×D711yescharacteristic
160#67—144no—
161C2×Dic3—71no—
162C2×Dic3—144no—
163C2×Dic3—141no—
164D12—141no—
165D12—141no—
166D12—282no—
167C4×D3—141no—
168C4×D3—141no—
169Dic6—72no—
170Dic6—141no—
171C3×C8—21no—
172C3⋊C8—141no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—422no—
175#41—422no—
176C2×Q8—211no—
177C2×D4—211no—
178C2×D4—421no—
179C2×D4—421no—
180C2×D4C2×D2111yescharacteristic
181C22×C4—211no—
182C22×C4—421no—
183SD16—422no—
184D8—422no—
185C2×C8—211no—
186C2×C8C2×D2111yescharacteristic
187C4⋊C4C2×D2111yescharacteristic
188C22⋊C4—421no—
189C2×C7—42no—
190C2×C7#88211yescharacteristic
191C2×C7#91111yescharacteristic
192C2×C7#91611yescharacteristic
193D7—121no—
194D7—241no—
195C22×C3—22no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3D4×D711yescharacteristic
199C2×D3—141no—
200C2×D3—141no—
201C2×D3—141no—
202C2×D3—281no—
203C2×D3—281no—
204C2×D3—281no—
205C3×C4—41no—
206C3×C4C2×D2811yescharacteristic
207C3×C4D4×D711yescharacteristic
208Dic3—142no—
209Dic3—141no—
210C23—21no—
211C23—211no—
212C23—421no—
213Q8—212no—
214Q8—421no—
215D4—21no—
216D4—424no—
217D4—421no—
218D4—421no—
219D4—842no—
220D4D8412yescomplemented
221C2×C4—21no—
222C2×C4—211no—
223C2×C4—424no—
224C2×C4—421no—
225C2×C4—421no—
226C2×C4—421no—
227C2×C4C22×D2111yescharacteristic
228C8—21no—
229C8—421no—
230C7#478111yescharacteristic, complemented, Sylow
231C2×C3—42no—
232C2×C3#644511yescharacteristic
233C2×C3#647411yescharacteristic
234C2×C3#647911yescharacteristic
235D3—281no—
236D3—561no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—421no—
241C22—421no—
242C22—421no—
243C22—841no—
244C22—841no—
245C22—841no—
246C22D4×D2111yescharacteristic
247C4—41no—
248C4—421no—
249C4—422no—
250C4C2×D8411yescharacteristic
251C4D4×D2111yescharacteristic
252C3#8955311yescharacteristic, complemented, Sylow
253C2—42no—
254C2—841no—
255C2—1681no—
256C2#1062136511yescharacteristic
257C2#1062139411yescharacteristic
258C2#1062139911yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
hypercenter#102D21
hypocenterC3×C7#449
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
1. C2×#66730C2semidirect
2. C2×#66632C2semidirect
3. C2×#66623C2semidirect
4. #10621378C2semidirect
5. C3×C7×#102C2semidirect
6. #10620659C2semidirect
7. #10620649C2semidirect
28. C7×#102D3semidirect
36. C3×C7×D4D4semidirect
49. Dic42D4semidirect
71. C3×#102D7semidirect
94. C7×D4D12semidirect
136. #102D21semidirect
157. C3×D4D28semidirect
173. C3×C7#449semidirect
220. D4D84semidirect
230. C7#4781semidirect
252. C3#89553semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C2×#666322
C2×#666232
#106213782
C3×C7×#1022
#106206592
#106206492
#895533
#47817