Finite Group Databasebeta

← #13061709

Subgroups — #13061709

Order 1344.

480 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×#121C211yesmaximal, characteristic, complemented
2#10621423C211yesmaximal, characteristic, complemented
3#10621422C212yesmaximal, complemented
4#10621421C212yesmaximal, complemented
5D4×Dic21C211yesmaximal, characteristic, complemented
6#10621413C211yesmaximal, characteristic, complemented
7#10621409C211yesmaximal, characteristic, complemented
8#10621376C211yesmaximal, characteristic, complemented
9#10621373C211yesmaximal, characteristic, complemented
10#10621369C212yesmaximal, complemented
11#10621366C212yesmaximal, complemented
12#90310—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C2×C3×C7×D4C2212yes—
15C22×C3×C4×C7C2211yescharacteristic
16C2×#4268C2212yes—
17C2×#4268C2212yes—
18C2×#4268C2212yes—
19C22×Dic21C2212yes—
20C2×D84C2211yescharacteristic
21C2×C4×D21C2212yes—
22C3×C7×C4⋊C4C2211yescharacteristic
23C3×C7×C22⋊C4C2212yes—
24#66636C2212yes—
25#66636C2211yescharacteristic
26#66636C2211yescharacteristic
27#66631C2212yes—
28#66631C2212yes—
29C4⋊Dic21C2211yescharacteristic
30C4⋊Dic21C2211yescharacteristic
31C4⋊Dic21C2212yes—
32C4⋊Dic21C2211yescharacteristic
33#66629C2212yes—
34C4×Dic21C2211yescharacteristic
35C4×Dic21C2211yescharacteristic
36C7×#121D311yescharacteristic, complemented
37#6503—31no—
38#6502—32no—
39#6501—32no—
40D4×Dic7—31no—
41#6493—31no—
42#6489—31no—
43#6456—31no—
44#6453—31no—
45#6449—32no—
46#6446—32no—
47#5619—71nomaximal, Hall
48C23×C3×C7C2311yescharacteristic
49C23×C3×C7C2312yes—
50C22×D21C2312yes—
51C3×C7×D4—22no—
52C3×C7×D4—42no—
53C2×C3×C4×C7—22no—
54C2×C3×C4×C7C2312yes—
55C2×C3×C4×C7C2311yescharacteristic
56C2×C3×C4×C7C2311yescharacteristic
57#4268—42no—
58#4268—42no—
59#4268—42no—
60C2×Dic21—42no—
61C2×Dic21C2311yescharacteristic
62C2×Dic21C2311yescharacteristic
63C2×Dic21C2311yescharacteristic
64C2×Dic21C2311yescharacteristic
65C2×Dic21C2312yes—
66D84—22no—
67C4×D21—42no—
68C2×C7×D4C2×D311yescharacteristic
69C2×C7×D4C2×D312yes—
70C22×C4×C7C2×D311yescharacteristic
71C2×C22⋊D7—32no—
72C2×C22⋊D7—32no—
73C2×C22⋊D7—32no—
74C22×Dic7—32no—
75C2×D28—31no—
76C2×C4×D7—32no—
77C7×C4⋊C4C2×D311yescharacteristic
78C7×C22⋊C4C2×D312yes—
79C22⋊Dic7—32no—
80C22⋊Dic7—31no—
81C22⋊Dic7—31no—
82(C2×D7)⋊C4—32no—
83(C2×D7)⋊C4—32no—
84C4⋊Dic7—31no—
85C4⋊Dic7—31no—
86C4⋊Dic7—32no—
87C4⋊Dic7—31no—
88Dic7⋊C4—32no—
89C4×Dic7—31no—
90C4×Dic7—31no—
91C3×#121D711yescharacteristic, complemented
92#940—71no—
93#939—72no—
94#938—72no—
95D4×Dic3—71no—
96#930—71no—
97#926—71no—
98#893—71no—
99#890—71no—
100#886—72no—
101#883—72no—
102C22×C3×C7—21no—
103C22×C3×C7—22no—
104C22×C3×C7—22no—
105C22×C3×C7—21no—
106C22×C3×C7—22no—
107C22×C3×C7—21no—
108C22×C3×C7C2411yescharacteristic
109C2×D21—22no—
110C2×D21—22no—
111C2×D21—22no—
112C3×C4×C7—21no—
113C3×C4×C7—22no—
114C3×C4×C7#4112yes—
115Dic21—21no—
116Dic21—21no—
117Dic21—22no—
118Dic21—21no—
119Dic21—21no—
120#540—211noSylow
121C23×C7C22×D311yescharacteristic
122C23×C7C22×D312yes—
123C22×D7—32no—
124C7×D4—22no—
125C7×D4—42no—
126C2×C4×C7—22no—
127C2×C4×C7C22×D312yes—
128C2×C4×C7C22×D311yescharacteristic
129C2×C4×C7C22×D311yescharacteristic
130C22⋊D7—122no—
131C22⋊D7—122no—
132C22⋊D7—122no—
133C2×Dic7—31no—
134C2×Dic7—31no—
135C2×Dic7—31no—
136C2×Dic7—31no—
137C2×Dic7—32no—
138C2×Dic7—122no—
139D28—62no—
140C4×D7—122no—
141C2×C3×D4C2×D711yescharacteristic
142C2×C3×D4C2×D712yes—
143C22×C3×C4C2×D711yescharacteristic
144C2×#67—72no—
145C2×#67—72no—
146C2×#67—72no—
147C22×Dic3—72no—
148C2×D12—71no—
149C2×C4×D3—72no—
150C3×C4⋊C4C2×D711yescharacteristic
151C3×C22⋊C4C2×D712yes—
152#217—72no—
153#217—71no—
154#217—71no—
155(C2×D3)⋊C4—72no—
156(C2×D3)⋊C4—72no—
157C4⋊Dic3—71no—
158C4⋊Dic3—71no—
159C4⋊Dic3—72no—
160C4⋊Dic3—71no—
161Dic3⋊C4—72no—
162C4×Dic3—71no—
163C4×Dic3—71no—
164C2×C3×C7—41no—
165C2×C3×C7—42no—
166C2×C3×C7C2×#4111yescharacteristic
167C2×C3×C7H(5,𝔽2)11yescharacteristic
168C2×C3×C7H(5,𝔽2)11yescharacteristic
169D21—42no—
170#127—211no—
171#125—211no—
172#123—212no—
173#123—212no—
174#121—212no—
175#121—212no—
176#121—211no—
177#121D2111yescharacteristic, complemented
178C4×D4—211no—
179C4×D4—211no—
180#117—211no—
181C22×C7—21no—
182C22×C7—22no—
183C22×C7—22no—
184C22×C7—21no—
185C22×C7—22no—
186C22×C7—21no—
187C22×C7C23×D311yescharacteristic
188C2×D7—62no—
189C2×D7—62no—
190C2×D7—62no—
191C4×C7—21no—
192C4×C7—22no—
193C4×C7#23712yes—
194Dic7—61no—
195Dic7—61no—
196Dic7—61no—
197Dic7—62no—
198Dic7—61no—
199C23×C3C22×D711yescharacteristic
200C23×C3C22×D712yes—
201C22×D3—72no—
202C3×D4—22no—
203C3×D4—42no—
204C2×C3×C4—22no—
205C2×C3×C4C22×D712yes—
206C2×C3×C4C22×D711yescharacteristic
207C2×C3×C4C22×D711yescharacteristic
208#67—282no—
209#67—282no—
210#67—282no—
211C2×Dic3—71no—
212C2×Dic3—71no—
213C2×Dic3—71no—
214C2×Dic3—71no—
215C2×Dic3—72no—
216C2×Dic3—282no—
217D12—142no—
218C4×D3—282no—
219C3×C7#54011yescharacteristic, complemented, Hall
220C2×D4—212no—
221C2×D4—212no—
222C2×D4—212no—
223C2×D4—211no—
224C2×D4C2×D2111yescharacteristic
225C2×D4C2×D2112yes—
226C22×C4—212no—
227C22×C4—212no—
228C22×C4C2×D2111yescharacteristic
229C4⋊C4—212no—
230C4⋊C4—211no—
231C4⋊C4—211no—
232C4⋊C4—211no—
233C4⋊C4—212no—
234C4⋊C4C2×D2111yescharacteristic
235C22⋊C4—211no—
236C22⋊C4—211no—
237C22⋊C4—212no—
238C22⋊C4—212no—
239C22⋊C4—212no—
240C22⋊C4C2×D2112yes—
241C42—211no—
242C42—211no—
243C2×C7—41no—
244C2×C7—42no—
245C2×C7C2×#23711yescharacteristic
246C2×C7#100411yescharacteristic
247C2×C7#100911yescharacteristic
248D7—122no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—21no—
253C22×C3—22no—
254C22×C3—21no—
255C22×C3C23×D711yescharacteristic
256C2×D3—142no—
257C2×D3—142no—
258C2×D3—142no—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4#115912yes—
262Dic3—141no—
263Dic3—141no—
264Dic3—141no—
265Dic3—142no—
266Dic3—141no—
267C23—212no—
268C23C22×D2111yescharacteristic
269C23C22×D2112yes—
270D4—22no—
271D4—42no—
272D4—422no—
273D4—842no—
274D4—842no—
275D4—842no—
276C2×C4—22no—
277C2×C4—211no—
278C2×C4—211no—
279C2×C4—211no—
280C2×C4—211no—
281C2×C4—212no—
282C2×C4—842no—
283C2×C4—842no—
284C2×C4C22×D2111yescharacteristic
285C2×C4C22×D2112yes—
286C2×C4C22×D2111yescharacteristic
287C7#561911yescharacteristic, complemented, Sylow
288C2×C3—41no—
289C2×C3—42no—
290C2×C3C2×#115911yescharacteristic
291C2×C3#654811yescharacteristic
292C2×C3#655311yescharacteristic
293D3—282no—
294C22—21no—
295C22—22no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—21no—
300C22—422no—
301C22—422no—
302C22—422no—
303C22C23×D2111yescharacteristic
304C4—21no—
305C4—22no—
306C4—421no—
307C4—421no—
308C4—421no—
309C4—422no—
310C4—421no—
311C4#6673012yes—
312C3#9031011yescharacteristic, complemented, Sylow
313C2—41no—
314C2—42no—
315C2—842no—
316C2C2×#6673011yescharacteristic
317C2#1062165711yescharacteristic
318C2#1062166211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#121C2
hypercenter#121D21
hypocenterC3×C7#540
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C3×C7×#121C2semidirect
2. #10621423C2semidirect
3. #10621422C2semidirect
4. #10621421C2semidirect
5. D4×Dic21C2semidirect
6. #10621413C2semidirect
7. #10621409C2semidirect
8. #10621376C2semidirect
9. #10621373C2semidirect
10. #10621369C2semidirect
11. #10621366C2semidirect
36. C7×#121D3semidirect
91. C3×#121D7semidirect
177. #121D21semidirect
219. C3×C7#540semidirect
287. C7#5619semidirect
312. C3#90310semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
#106214232
#106214222
#106214212
D4×Dic212
#106214132
#106214092
#106213762
#106213732
#106213692
#106213662
#903103
#56197