Finite Group Databasebeta

← C3×#89545 #13055205

Subgroups — C3×#89545 #13055205

Order 1344.

396 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 90 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×D4×D7C211yesmaximal, characteristic, complemented
2C2×C3×#1141C211yesmaximal, characteristic, complemented
3C2×C3×D56C211yesmaximal, characteristic, complemented
4C3×#6458C211yesmaximal, characteristic, complemented
5C3×C7×#102C211yesmaximal, characteristic, complemented
6C3×(C2×D7)⋊C8C211yesmaximal, characteristic, complemented
7C3×#6383C211yesmaximal, characteristic, complemented
8#89545C311yesmaximal, characteristic, direct product factor, Hall
9C23×C3×D7—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C3×D4×D7—22no—
13C3×D4×D7—22no—
14C2×C3×D28—21no—
15C2×C3×D28C2211yescharacteristic
16C2×C3×D28C2211yescharacteristic, complemented
17C2×C3×C4×D7C2211yescharacteristic
18C2×C3×C7×C8C2211yescharacteristic, complemented
19C3×C7×C4⋊C4C2211yescharacteristic, complemented
20C3×#1141—22no—
21C3×(C2×D7)⋊C4—21no—
22C2×C3×C7⋊C8C2211yescharacteristic, complemented
23C3×D56—22no—
24C2×D4×D7C2×C311yescharacteristic, complemented
25C2×#1141C2×C311yescharacteristic, complemented
26C2×D56C2×C311yescharacteristic, complemented
27#6458C2×C311yescharacteristic, complemented
28C7×#102C2×C311yescharacteristic, complemented
29(C2×D7)⋊C8C2×C311yescharacteristic, complemented
30#6383C2×C311yescharacteristic, complemented
31C3×#447—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×C3×D7—24no—
34C22×C3×D7—21no—
35C22×C3×D7—21no—
36C22×C3×D7—42no—
37C22×C3×D7—42no—
38C22×C3×D7D411yescharacteristic
39C3×C7×D4—21no—
40C3×C7×D4D412yescomplemented
41C2×C3×C4×C7—21no—
42C2×C3×C4×C7C2311yescharacteristic
43C3×C22⋊D7—24no—
44C2×C3×Dic7D411yescharacteristic
45C3×D28—21no—
46C3×D28—21no—
47C3×D28—21no—
48C3×D28—42no—
49C3×D28D412yescomplemented
50C3×C4×D7—21no—
51C3×C4×D7—21no—
52C3×C7×C8—21no—
53C3×C7⋊C8—21no—
54C23×D7—21no—
55C2×C7×D4C22×C311yescharacteristic
56C2×C22⋊D7—21no—
57D4×D7—22no—
58D4×D7—22no—
59C2×D28—21no—
60C2×D28C22×C311yescharacteristic
61C2×D28C22×C311yescharacteristic, complemented
62C2×C4×D7C22×C311yescharacteristic
63C2×C7×C8C22×C311yescharacteristic, complemented
64C7×C4⋊C4C22×C311yescharacteristic, complemented
65#1141—22no—
66(C2×D7)⋊C4—21no—
67C2×C7⋊C8C22×C311yescharacteristic, complemented
68D56—22no—
69C22×C3×D4—71no—
70C2×C3×D8—71no—
71C2×C3×D8—71no—
72C3×#121—71no—
73C3×#102—71no—
74C3×#102D711yescharacteristic, complemented
75C3×C22⋊C8—71no—
76C22×C3×C7—22no—
77C22×C3×C7—41no—
78C22×C3×C7—41no—
79C22×C3×C7C2×D411yescharacteristic
80C2×C3×D7—21no—
81C2×C3×D7—21no—
82C2×C3×D7—22no—
83C2×C3×D7—48no—
84C2×C3×D7—41no—
85C2×C3×D7—41no—
86C2×C3×D7—41no—
87C2×C3×D7—41no—
88C2×C3×D7—41no—
89C2×C3×D7D812yescomplemented
90C3×C4×C7—41no—
91C3×C4×C7C2×D411yescharacteristic
92C3×C4×C7C2×D411yescharacteristic
93C3×Dic7—21no—
94#447—71noSylow
95C23×C7—21no—
96C22×D7—24no—
97C22×D7—21no—
98C22×D7—21no—
99C22×D7—42no—
100C22×D7—42no—
101C22×D7C3×D411yescharacteristic
102C7×D4—21no—
103C7×D4C3×D412yescomplemented
104C2×C4×C7—21no—
105C2×C4×C7C23×C311yescharacteristic
106C22⋊D7—24no—
107C2×Dic7C3×D411yescharacteristic
108D28—21no—
109D28—21no—
110D28—21no—
111D28—42no—
112D28C3×D412yescomplemented
113C4×D7—21no—
114C4×D7—21no—
115C7×C8—21no—
116C7⋊C8—21no—
117C24×C3—141no—
118C2×C3×D4—71no—
119C2×C3×D4—71no—
120C2×C3×D4—141no—
121C2×C3×D4—141no—
122C2×C3×D4—142no—
123C2×C3×D4—142no—
124C2×C3×D4C2×D711yescharacteristic
125C22×C3×C4—71no—
126C3×D8—142no—
127C3×D8—142no—
128C2×C3×C8—71no—
129C2×C3×C8C2×D711yescharacteristic, complemented
130C3×C4⋊C4C2×D711yescharacteristic, complemented
131C3×C22⋊C4—141no—
132C2×C3×C7—42no—
133C2×C3×C7#12011yescharacteristic
134C2×C3×C7C2×D811yescharacteristic
135C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
136C3×D7—22no—
137C3×D7—42no—
138C3×D7—81no—
139C22×D4—71no—
140C2×D8—71no—
141C2×D8—71no—
142#121—71no—
143#102—71no—
144#102C3×D711yescharacteristic, complemented
145C22⋊C8—71no—
146C22×C7—22no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7C2×C3×D411yescharacteristic
150C2×D7—21no—
151C2×D7—21no—
152C2×D7—22no—
153C2×D7—48no—
154C2×D7—41no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—41no—
158C2×D7—41no—
159C2×D7C3×D812yescomplemented
160C4×C7—41no—
161C4×C7C2×C3×D411yescharacteristic
162C4×C7C2×C3×D411yescharacteristic
163Dic7—21no—
164C23×C3—21no—
165C23×C3—71no—
166C23×C3—144no—
167C23×C3—141no—
168C23×C3—141no—
169C23×C3—282no—
170C23×C3—282no—
171C3×D4—21no—
172C3×D4—72no—
173C3×D4—144no—
174C3×D4—141no—
175C3×D4—141no—
176C3×D4—141no—
177C3×D4—282no—
178C3×D4D2812yescomplemented
179C2×C3×C4—21no—
180C2×C3×C4—71no—
181C2×C3×C4—141no—
182C2×C3×C4—141no—
183C2×C3×C4C22×D711yescharacteristic
184C3×C8—21no—
185C3×C8—141no—
186C3×C7#44711yescharacteristic, complemented, Hall
187C24—141no—
188C2×D4—71no—
189C2×D4—71no—
190C2×D4—141no—
191C2×D4—141no—
192C2×D4—142no—
193C2×D4—142no—
194C2×D4C2×C3×D711yescharacteristic
195C22×C4—71no—
196D8—142no—
197D8—142no—
198C2×C8—71no—
199C2×C8C2×C3×D711yescharacteristic, complemented
200C4⋊C4C2×C3×D711yescharacteristic, complemented
201C22⋊C4—141no—
202C2×C7—42no—
203C2×C7C3×#12011yescharacteristic
204C2×C7C2×C3×D811yescharacteristic
205C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
206D7—22no—
207D7—42no—
208D7—81no—
209C22×C3—22no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—72no—
213C22×C3—141no—
214C22×C3—141no—
215C22×C3—142no—
216C22×C3—288no—
217C22×C3—281no—
218C22×C3—281no—
219C22×C3—281no—
220C22×C3—281no—
221C22×C3—281no—
222C22×C3D4×D711yescharacteristic
223C3×C4—41no—
224C3×C4—141no—
225C3×C4C2×D2811yescharacteristic
226C3×C4D4×D711yescharacteristic
227C23—21no—
228C23—71no—
229C23—144no—
230C23—141no—
231C23—141no—
232C23—282no—
233C23—282no—
234D4—21no—
235D4—72no—
236D4—144no—
237D4—141no—
238D4—141no—
239D4—141no—
240D4—282no—
241D4C3×D2812yescomplemented
242C2×C4—21no—
243C2×C4—71no—
244C2×C4—141no—
245C2×C4—141no—
246C2×C4C22×C3×D711yescharacteristic
247C8—21no—
248C8—141no—
249C7C3×#44711yescharacteristic, complemented, Sylow
250C2×C3—42no—
251C2×C3—142no—
252C2×C3—282no—
253C2×C3—561no—
254C2×C3#644511yescharacteristic
255C2×C3D7×D811yescharacteristic
256C2×C3#647711yescharacteristic
257C22—22no—
258C22—41no—
259C22—41no—
260C22—72no—
261C22—141no—
262C22—141no—
263C22—142no—
264C22—288no—
265C22—281no—
266C22—281no—
267C22—281no—
268C22—281no—
269C22—281no—
270C22C3×D4×D711yescharacteristic
271C4—41no—
272C4—141no—
273C4C2×C3×D2811yescharacteristic
274C4C3×D4×D711yescharacteristic
275C3#8954511yescharacteristic, direct product factor, Sylow
276C2—42no—
277C2—142no—
278C2—282no—
279C2—561no—
280C2C3×#644511yescharacteristic
281C2C3×D7×D811yescharacteristic
282C2C3×#647711yescharacteristic
subgroupG/H
centerC22×C3D4×D7
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
hypercenterC3×#102D7
hypocenterC7C3×#447
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×D7C2semidirect
2. C2×C3×#1141C2semidirect
3. C2×C3×D56C2semidirect
4. C3×#6458C2semidirect
5. C3×C7×#102C2semidirect
6. C3×(C2×D7)⋊C8C2semidirect
7. C3×#6383C2semidirect
8. #89545C3direct (N × H)
16. C2×C3×D28C22semidirect
18. C2×C3×C7×C8C22semidirect
19. C3×C7×C4⋊C4C22semidirect
22. C2×C3×C7⋊C8C22semidirect
24. C2×D4×D7C2×C3semidirect
25. C2×#1141C2×C3semidirect
26. C2×D56C2×C3semidirect
27. #6458C2×C3semidirect
28. C7×#102C2×C3semidirect
29. (C2×D7)⋊C8C2×C3semidirect
30. #6383C2×C3semidirect
40. C3×C7×D4D4semidirect
49. C3×D28D4semidirect
61. C2×D28C22×C3semidirect
63. C2×C7×C8C22×C3semidirect
64. C7×C4⋊C4C22×C3semidirect
67. C2×C7⋊C8C22×C3semidirect
74. C3×#102D7semidirect
89. C2×C3×D7D8semidirect
103. C7×D4C3×D4semidirect
112. D28C3×D4semidirect
129. C2×C3×C8C2×D7semidirect
130. C3×C4⋊C4C2×D7semidirect
144. #102C3×D7semidirect
159. C2×D7C3×D8semidirect
178. C3×D4D28semidirect
186. C3×C7#447semidirect
199. C2×C8C2×C3×D7semidirect
200. C4⋊C4C2×C3×D7semidirect
241. D4C3×D28semidirect
249. C7C3×#447semidirect
275. C3#89545direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×D72
C2×C3×#11412
C2×C3×D562
C3×#64582
C3×C7×#1022
C3×(C2×D7)⋊C82
C3×#63832
#895453
C3×#4477