Finite Group Databasebeta

← #13060061

Subgroups — #13060061

Order 1344.

480 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 128 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, complemented
2C7×D4×Dic3C211yesmaximal, characteristic, complemented
3C3×#6501C211yesmaximal, characteristic, complemented
4#10621131C212yesmaximal, complemented
5#10621124C212yesmaximal, complemented
6#10621104C212yesmaximal, complemented
7#10621095C212yesmaximal, complemented
8#10621012C211yesmaximal, characteristic, complemented
9#10620990C211yesmaximal, characteristic, complemented
10#10620946C211yesmaximal, characteristic, complemented
11#10620919C211yesmaximal, characteristic, complemented
12#90253—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15C2×#4268C2212yes—
16C2×D84C2211yescharacteristic
17C22×C7×Dic3C2212yes—
18C2×C3×C22⋊D7C2212yes—
19C2×C3×C4×D7C2211yescharacteristic
20C2×#4246C2212yes—
21C2×D21⋊C4C2212yes—
22C4×Dic21C2211yescharacteristic
23C7×#217C2212yes—
24C7×C4⋊Dic3C2211yescharacteristic
25C4×C7×Dic3C2211yescharacteristic
26C3×C22⋊Dic7C2212yes—
27C3×C4⋊Dic7C2211yescharacteristic
28Dic3⋊Dic7C2212yes—
29Dic7⋊Dic3C2212yes—
30Dic7⋊Dic3C2211yescharacteristic
31Dic7⋊Dic3C2211yescharacteristic
32#66575C2212yes—
33#66575C2212yes—
34(C2×D7)⋊Dic3C2211yescharacteristic
35(C2×D7)⋊Dic3C2211yescharacteristic
36C4×C7×D4—31no—
37#6503—31no—
38#6501D311yescharacteristic, complemented
39#6493—32no—
40#6492—32no—
41#6458—31no—
42#6453—31no—
43#6447—32no—
44#6446—32no—
45C4×D28—31no—
46#6435—31no—
47#5619—71nomaximal, Hall
48C23×C3×C7C2312yes—
49C22×D21C2312yes—
50C22×C3×D7C2311yescharacteristic
51C3×C7×D4—22no—
52C2×C3×C4×C7C2311yescharacteristic
53#4268—42no—
54#4268—42no—
55C2×Dic21C2311yescharacteristic
56C2×Dic21C2311yescharacteristic
57D84—22no—
58C2×C7×Dic3—42no—
59C2×C7×Dic3C2311yescharacteristic
60C2×C7×Dic3C2311yescharacteristic
61C2×C7×Dic3C2312yes—
62C3×C22⋊D7—42no—
63C2×C3×Dic7C2311yescharacteristic
64C2×C3×Dic7C2312yes—
65C3×C4×D7—22no—
66#4246—42no—
67D21⋊C4—42no—
68C2×C7×D4C2×D311yescharacteristic
69C22×C4×C7—32no—
70C2×C22⋊D7—32no—
71C2×C22⋊D7—32no—
72C2×C22⋊D7C2×D312yes—
73C2×D28—31no—
74C2×D28—32no—
75C2×C4×D7—32no—
76C2×C4×D7C2×D311yescharacteristic
77C7×C4⋊C4—31no—
78C7×C22⋊C4—32no—
79C42×C7—31no—
80C22⋊Dic7C2×D312yes—
81(C2×D7)⋊C4—31no—
82(C2×D7)⋊C4—31no—
83(C2×D7)⋊C4—32no—
84(C2×D7)⋊C4—32no—
85C4⋊Dic7—32no—
86C4⋊Dic7C2×D311yescharacteristic
87Dic7⋊C4—31no—
88Dic7⋊C4—31no—
89Dic7⋊C4—32no—
90C4×Dic7—31no—
91C3×#121—71no—
92#940—71no—
93#939—72no—
94#938—72no—
95D4×Dic3D711yescharacteristic, complemented
96#930—71no—
97#926—71no—
98#893—71no—
99#890—71no—
100#886—72no—
101#883—72no—
102C22×C3×C7—22no—
103C22×C3×C7—22no—
104C22×C3×C7—22no—
105C22×C3×C7C2411yescharacteristic
106C2×D21—22no—
107C2×D21—22no—
108C2×D21—22no—
109C2×C3×D7—21no—
110C2×C3×D7—21no—
111C2×C3×D7—21no—
112C3×C4×C7#4112yes—
113Dic21—21no—
114Dic21—21no—
115C3×Dic7—21no—
116C3×Dic7—22no—
117C7×Dic3—21no—
118C7×Dic3—22no—
119C7×Dic3—21no—
120#540—211noSylow
121C23×C7C22×D312yes—
122C22×D7—32no—
123C22×D7C22×D311yescharacteristic
124C7×D4—22no—
125C2×C4×C7—32no—
126C2×C4×C7—31no—
127C2×C4×C7—31no—
128C2×C4×C7—122no—
129C2×C4×C7C22×D311yescharacteristic
130C22⋊D7—42no—
131C22⋊D7—122no—
132C22⋊D7—122no—
133C2×Dic7—31no—
134C2×Dic7—31no—
135C2×Dic7C22×D311yescharacteristic
136C2×Dic7C22×D312yes—
137D28—62no—
138D28—122no—
139C4×D7—22no—
140C4×D7—122no—
141C2×C3×D4—72no—
142C2×C3×D4C2×D711yescharacteristic
143C22×C3×C4—71no—
144C2×#67—72no—
145C2×#67—72no—
146C2×#67—72no—
147C22×Dic3C2×D712yes—
148C2×D12—71no—
149C2×C4×D3—72no—
150C3×C4⋊C4—71no—
151C3×C22⋊C4—72no—
152#217—71no—
153#217—71no—
154#217C2×D712yes—
155(C2×D3)⋊C4—72no—
156(C2×D3)⋊C4—72no—
157C4⋊Dic3—71no—
158C4⋊Dic3—71no—
159C4⋊Dic3—72no—
160C4⋊Dic3C2×D711yescharacteristic
161Dic3⋊C4—72no—
162C4×Dic3—71no—
163C4×Dic3C2×D711yescharacteristic
164C2×C3×C7—42no—
165C2×C3×C7C2×#4111yescharacteristic
166C2×C3×C7H(5,𝔽2)11yescharacteristic
167C2×C3×C7H(5,𝔽2)11yescharacteristic
168D21—42no—
169C3×D7—41no—
170#127—211no—
171#125—211no—
172#123—212no—
173#123—212no—
174#121—71no—
175#121—212no—
176#121—211no—
177#121—212no—
178C4×D4—31no—
179C4×D4—211no—
180#117—211no—
181C22×C7—22no—
182C22×C7—22no—
183C22×C7—22no—
184C22×C7C23×D311yescharacteristic
185C2×D7—21no—
186C2×D7—21no—
187C2×D7—21no—
188C2×D7—62no—
189C2×D7—62no—
190C2×D7—62no—
191C4×C7—61no—
192C4×C7—62no—
193C4×C7—61no—
194C4×C7#23712yes—
195Dic7—21no—
196Dic7—22no—
197Dic7—61no—
198Dic7—61no—
199C23×C3—71no—
200C23×C3C22×D712yes—
201C22×D3—72no—
202C3×D4—22no—
203C3×D4—282no—
204C2×C3×C4—72no—
205C2×C3×C4—71no—
206C2×C3×C4—142no—
207C2×C3×C4C22×D711yescharacteristic
208#67—282no—
209#67—282no—
210#67—282no—
211C2×Dic3—42no—
212C2×Dic3—71no—
213C2×Dic3—71no—
214C2×Dic3C22×D711yescharacteristic
215C2×Dic3C22×D711yescharacteristic
216C2×Dic3C22×D712yes—
217D12—142no—
218C4×D3—282no—
219C3×C7#54011yescharacteristic, complemented, Hall
220C2×D4—72no—
221C2×D4—212no—
222C2×D4—212no—
223C2×D4—212no—
224C2×D4—211no—
225C2×D4D3×D711yescharacteristic
226C22×C4—32no—
227C22×C4—71no—
228C22×C4—212no—
229C4⋊C4—31no—
230C4⋊C4—71no—
231C4⋊C4—211no—
232C4⋊C4—211no—
233C4⋊C4—212no—
234C4⋊C4—212no—
235C22⋊C4—32no—
236C22⋊C4—72no—
237C22⋊C4—212no—
238C22⋊C4—212no—
239C22⋊C4—211no—
240C22⋊C4—211no—
241C42—31no—
242C42—211no—
243C2×C7—42no—
244C2×C7C2×#23711yescharacteristic
245C2×C7#100411yescharacteristic
246C2×C7#100911yescharacteristic
247D7—41no—
248D7—122no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—141no—
253C22×C3—141no—
254C22×C3—141no—
255C22×C3C23×D711yescharacteristic
256C2×D3—142no—
257C2×D3—142no—
258C2×D3—142no—
259C3×C4—141no—
260C3×C4—142no—
261C3×C4#115712yes—
262Dic3—21no—
263Dic3—22no—
264Dic3—21no—
265Dic3—141no—
266Dic3—141no—
267C23—71no—
268C23—212no—
269C23C2×D3×D712yes—
270D4—22no—
271D4—282no—
272D4—422no—
273D4—842no—
274D4—842no—
275D4—842no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—32no—
279C2×C4—71no—
280C2×C4—72no—
281C2×C4—122no—
282C2×C4—142no—
283C2×C4—211no—
284C2×C4—211no—
285C2×C4—842no—
286C2×C4C2×D3×D711yescharacteristic
287C7#561911yescharacteristic, complemented, Sylow
288C2×C3—42no—
289C2×C3—281no—
290C2×C3C2×#115711yescharacteristic
291C2×C3#654811yescharacteristic
292C2×C3#655311yescharacteristic
293D3—282no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—141no—
298C22—141no—
299C22—141no—
300C22—422no—
301C22—422no—
302C22—422no—
303C22C22×D3×D711yescharacteristic
304C4—61no—
305C4—61no—
306C4—62no—
307C4—142no—
308C4—141no—
309C4—421no—
310C4—421no—
311C4#6668312yes—
312C3#9025311yescharacteristic, complemented, Sylow
313C2—42no—
314C2—281no—
315C2—842no—
316C2#1062157311yescharacteristic
317C2#1062157411yescharacteristic
318C2C2×#6668311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#540
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621423C2semidirect
2. C7×D4×Dic3C2semidirect
3. C3×#6501C2semidirect
4. #10621131C2semidirect
5. #10621124C2semidirect
6. #10621104C2semidirect
7. #10621095C2semidirect
8. #10621012C2semidirect
9. #10620990C2semidirect
10. #10620946C2semidirect
11. #10620919C2semidirect
38. #6501D3semidirect
95. D4×Dic3D7semidirect
219. C3×C7#540semidirect
287. C7#5619semidirect
312. C3#90253semidirect

Maximal subgroups

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

subgroupindex
#106214232
C7×D4×Dic32
C3×#65012
#106211312
#106211242
#106211042
#106210952
#106210122
#106209902
#106209462
#106209192
#902533
#56197