Finite Group Databasebeta

← #13054626

Subgroups — #13054626

Order 1344.

316 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×Q8×D7C211yesmaximal, characteristic, complemented
2C2×#66634C211yesmaximal, characteristic, complemented
3#10621012C211yesmaximal, characteristic
4C2×#66566C211yesmaximal, characteristic, complemented
5C7×#835C211yesmaximal, characteristic, complemented
6#10620505C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89578—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C3×Q8×D7—22no—
12C3×Q8×D7—22no—
13C2×C3×C4×D7—21no—
14C2×C3×C4×D7C2211yescharacteristic
15C2×C3×Dic14—21no—
16C2×C3×Dic14C2211yescharacteristic
17C2×#4246—21no—
18#66634—22no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×C4⋊Dic3C2211yescharacteristic
21C2×C7×C3⋊C8C2211yescharacteristic
22#66575—21no—
23#66566—22no—
24C2×Q8×D7D311yescharacteristic, complemented
25C2×#1143—31no—
26C2×#1132—31no—
27#6458—31no—
28C7×#103—31no—
29(C2×D7)⋊C8—31no—
30#6384—31no—
31#5227—71nomaximal, Hall
32C22×D21—21no—
33C22×C3×D7D411yescharacteristic
34C3×C7×Q8—21no—
35C3×C7×Q8D412yescomplemented
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38D84—21no—
39D84—21no—
40C2×C7×Dic3—21no—
41C2×C3×Dic7—21no—
42C2×C3×Dic7D411yescharacteristic
43C3×C4×D7—24no—
44C3×C4×D7—21no—
45C3×C4×D7—21no—
46C3×Dic14—24no—
47C3×Dic14—21no—
48C3×Dic14D412yescomplemented
49#4246—42no—
50C3⋊(C7⋊C8)—21no—
51C7×C3⋊C8—21no—
52C2×C7×Q8C2×D311yescharacteristic
53Q8×D7—22no—
54Q8×D7—22no—
55C2×D28—31no—
56C2×D28—61no—
57C2×C4×D7—21no—
58C2×C4×D7C2×D311yescharacteristic
59C2×Dic14—21no—
60C2×Dic14C2×D311yescharacteristic
61C2×C7×C8—31no—
62C7×C4⋊C4—31no—
63#1143—62no—
64(C2×D7)⋊C4—61no—
65C2×C7⋊C8—31no—
66#1132—62no—
67C22×C3×Q8—71no—
68C2×#215—71no—
69C2×#215—71no—
70#930—71no—
71#835—71no—
72#835D711yescharacteristic, complemented
73#830—71no—
74C22×C3×C7C2×D411yescharacteristic
75C2×D21—41no—
76C2×D21—41no—
77C2×D21—41no—
78C2×C3×D7—21no—
79C2×C3×D7—21no—
80C2×C3×D7SD1612yescomplemented
81C3×C4×C7—22no—
82C3×C4×C7C2×D411yescharacteristic
83C3×C4×C7C2×D411yescharacteristic
84C3×Dic7—21no—
85C3×Dic7—22no—
86C7×Dic3—41no—
87#448—211noSylow
88C22×D7—61no—
89C22×D7#6711yescharacteristic
90C7×Q8—21no—
91C7×Q8#6712yescomplemented
92C2×C4×C7—21no—
93C2×C4×C7—61no—
94C2×C4×C7C22×D311yescharacteristic
95C2×Dic7—21no—
96C2×Dic7#6711yescharacteristic
97D28—61no—
98D28—61no—
99D28—122no—
100C4×D7—24no—
101C4×D7—21no—
102C4×D7—21no—
103Dic14—24no—
104Dic14—21no—
105Dic14#6712yescomplemented
106C7×C8—61no—
107C7⋊C8—61no—
108C2×C3×Q8—71no—
109C2×C3×Q8—141no—
110C2×C3×Q8—142no—
111C2×C3×Q8—142no—
112C2×C3×Q8C2×D711yescharacteristic
113C22×C3×C4—71no—
114C22×C3×C4—141no—
115C2×#67—141no—
116C2×D12—71no—
117#215—142no—
118#215—142no—
119(C2×D3)⋊C4—141no—
120C4⋊Dic3C2×D711yescharacteristic
121C2×C3⋊C8—71no—
122C2×C3⋊C8C2×D711yescharacteristic
123C2×C3×C7#12011yescharacteristic
124C2×C3×C7C2×SD1611yescharacteristic
125C2×C3×C7#13711yescharacteristic
126D21—81no—
127C3×D7—22no—
128C22×Q8—71no—
129C2×SD16—211no—
130C2×SD16—211no—
131#121—211no—
132#103—31no—
133#103—211no—
134C22⋊C8—211no—
135C22×C7C2×#6711yescharacteristic
136C2×D7—21no—
137C2×D7—21no—
138C2×D7—121no—
139C2×D7—121no—
140C2×D7—121no—
141C2×D7#21512yescomplemented
142C4×C7—22no—
143C4×C7—121no—
144C4×C7C2×#6711yescharacteristic
145C4×C7C2×#6711yescharacteristic
146Dic7—21no—
147Dic7—22no—
148C23×C3—71no—
149C22×D3—141no—
150C3×Q8—21no—
151C3×Q8—72no—
152C3×Q8—144no—
153C3×Q8—141no—
154C3×Q8D2812yescomplemented
155C2×C3×C4—21no—
156C2×C3×C4—71no—
157C2×C3×C4—144no—
158C2×C3×C4—141no—
159C2×C3×C4—141no—
160C2×C3×C4—141no—
161C2×C3×C4C22×D711yescharacteristic
162#67—282no—
163C2×Dic3—21no—
164D12—141no—
165D12—141no—
166C3⋊C8—21no—
167C3⋊C8—141no—
168C3×C7#44811yescharacteristic, complemented, Hall
169C2×Q8—71no—
170C2×Q8—141no—
171C2×Q8—142no—
172C2×Q8—142no—
173C2×Q8D3×D711yescharacteristic
174C2×D4—211no—
175C2×D4—421no—
176C22×C4—71no—
177C22×C4—141no—
178SD16—422no—
179SD16—422no—
180C2×C8—31no—
181C2×C8—211no—
182C4⋊C4—31no—
183C22⋊C4—421no—
184C2×C7C2×#21511yescharacteristic
185C2×C7#94211yescharacteristic
186C2×C7#95311yescharacteristic
187D7—22no—
188D7—241no—
189C22×C3—72no—
190C22×C3—141no—
191C22×C3—141no—
192C22×C3D4×D711yescharacteristic
193C2×D3—281no—
194C2×D3—281no—
195C2×D3—281no—
196C3×C4—22no—
197C3×C4—141no—
198C3×C4—142no—
199C3×C4C2×D2811yescharacteristic
200C3×C4D4×D711yescharacteristic
201Dic3—41no—
202C23—71no—
203C23—421no—
204Q8—21no—
205Q8—72no—
206Q8—144no—
207Q8—141no—
208Q8#424612yescomplemented
209D4—421no—
210D4—421no—
211D4—842no—
212C2×C4—21no—
213C2×C4—61no—
214C2×C4—71no—
215C2×C4—144no—
216C2×C4—141no—
217C2×C4—141no—
218C2×C4—141no—
219C2×C4C2×D3×D711yescharacteristic
220C8—61no—
221C8—421no—
222C7#522711yescharacteristic, complemented, Sylow
223C2×C3—142no—
224C2×C3#644511yescharacteristic
225C2×C3D7×SD1611yescharacteristic
226C2×C3#648111yescharacteristic
227D3—561no—
228C22—72no—
229C22—141no—
230C22—141no—
231C22—841no—
232C22—841no—
233C22—841no—
234C22D7×#6711yescharacteristic
235C4—22no—
236C4—121no—
237C4—141no—
238C4—142no—
239C4C2×#424611yescharacteristic
240C4D7×#6711yescharacteristic
241C3#8957811yescharacteristic, complemented, Sylow
242C2—142no—
243C2—1681no—
244C2D7×#21511yescharacteristic
245C2#1062107711yescharacteristic
246C2#1062112611yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radical#10621012C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#448
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×D7C2semidirect
2. C2×#66634C2semidirect
4. C2×#66566C2semidirect
5. C7×#835C2semidirect
6. #10620505C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
24. C2×Q8×D7D3semidirect
35. C3×C7×Q8D4semidirect
48. C3×Dic14D4semidirect
72. #835D7semidirect
80. C2×C3×D7SD16semidirect
91. C7×Q8#67semidirect
105. Dic14#67semidirect
141. C2×D7#215semidirect
154. C3×Q8D28semidirect
168. C3×C7#448semidirect
208. Q8#4246semidirect
222. C7#5227semidirect
241. C3#89578semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D72
C2×#666342
#106210122
C2×#665662
C7×#8352
#106205052
(C2×D7)⋊(C3⋊C8)2
#895783
#52277