Finite Group Databasebeta

← #13059517

Subgroups — #13059517

Order 1344.

540 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C22×C7×Dic6C211yesmaximal, characteristic, complemented
2C2×#66683C211yesmaximal, characteristic
3#10621412C211yesmaximal, characteristic, complemented
4C3×#6492C211yesmaximal, characteristic, complemented
5#10621104C211yesmaximal, characteristic, complemented
6#10621098C211yesmaximal, characteristic, complemented
7#10621091C211yesmaximal, characteristic, complemented
8#10621017C212yesmaximal, complemented
9#10620986C212yesmaximal, complemented
10#10620983C212yesmaximal, complemented
11#10620891C212yesmaximal, complemented
12#90507—31nomaximal, Hall
13C22×C3×C4×C7C2211yescharacteristic
14C2×#4268C2211yescharacteristic
15C22×C7×Dic3C2211yescharacteristic
16C22×C7×Dic3C2211yescharacteristic
17C2×C7×Dic6—24no—
18C2×C7×Dic6C2212yes—
19C2×C7×Dic6C2212yes—
20C2×C3×C22⋊D7C2211yescharacteristic
21C2×#4248C2211yescharacteristic
22C2×#4246C2211yescharacteristic
23C2×D21⋊C4C2211yescharacteristic
24#66683—24no—
25C2×Dic3×D7C2211yescharacteristic
26#66636C2211yescharacteristic
27#66631C2212yes—
28#66629C2212yes—
29C3×C22⋊Dic7C2211yescharacteristic
30C3×(C2×D7)⋊C4C2212yes—
31C3×Dic7⋊C4C2212yes—
32Dic3⋊Dic7C2212yes—
33#66577C2212yes—
34Dic7⋊Dic3C2212yes—
35#66575C2212yes—
36(C2×D7)⋊Dic3C2212yes—
37Dic3×Dic7C2212yes—
38C22×C7×Q8—31no—
39C2×#1157—31no—
40#6510—32no—
41#6509—32no—
42#6509—32no—
43#6507—32no—
44#6492—31no—
45#6492—31no—
46#6492—31no—
47#6492D311yescharacteristic, complemented
48#6489—31no—
49#5648—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21C2311yescharacteristic
52C22×C3×D7C2311yescharacteristic
53C2×C3×C4×C7—22no—
54C2×C3×C4×C7C2312yes—
55#4268—22no—
56C2×Dic21C2312yes—
57C2×Dic21C2311yescharacteristic
58C2×C7×Dic3—22no—
59C2×C7×Dic3D414yes—
60C2×C7×Dic3C2312yes—
61C2×C7×Dic3C2311yescharacteristic
62C2×C7×Dic3C2311yescharacteristic
63C7×Dic6—24no—
64C7×Dic6—24no—
65C3×C22⋊D7—22no—
66C2×C3×Dic7C2312yes—
67C2×C3×Dic7C2311yescharacteristic
68#4248—22no—
69#4246—22no—
70D21⋊C4—22no—
71Dic3×D7—22no—
72C2×C7×Q8—32no—
73C2×C7×Q8—32no—
74C2×C7×Q8—64no—
75C22×C4×C7—31no—
76C22×C4×C7—31no—
77C22×C4×C7C2×D311yescharacteristic
78C2×C22⋊D7—31no—
79C2×C22⋊D7C2×D311yescharacteristic
80#1157—64no—
81C2×D28—31no—
82C2×C4×D7—31no—
83C2×C4×D7—31no—
84C2×Dic14—31no—
85C22⋊Dic7—31no—
86C22⋊Dic7C2×D311yescharacteristic
87(C2×D7)⋊C4—32no—
88(C2×D7)⋊C4—32no—
89(C2×D7)⋊C4—32no—
90(C2×D7)⋊C4C2×D312yes—
91C4⋊Dic7—32no—
92Dic7⋊C4—32no—
93Dic7⋊C4—32no—
94Dic7⋊C4—32no—
95Dic7⋊C4C2×D312yes—
96C4×Dic7—32no—
97C2×#237—71no—
98C22×Dic6D711yescharacteristic, complemented
99C3×#123—71no—
100#935—71no—
101#929—71no—
102#896—72no—
103#888—72no—
104#886—71no—
105#885—72no—
106#878—72no—
107#877—71no—
108C22×C3×C7—22no—
109C22×C3×C7C2×D412yes—
110C22×C3×C7C2411yescharacteristic
111C2×D21—21no—
112C2×D21—22no—
113C2×C3×D7—21no—
114C2×C3×D7—22no—
115C3×C4×C7—22no—
116Dic21—21no—
117Dic21—22no—
118C3×Dic7—21no—
119C3×Dic7—22no—
120C7×Dic3—22no—
121C7×Dic3C2×D412yes—
122C7×Dic3C2×D412yes—
123#536—211noSylow
124C23×C7C22×D311yescharacteristic
125C22×D7—31no—
126C22×D7C22×D311yescharacteristic
127C7×Q8—64no—
128C7×Q8—64no—
129C2×C4×C7—22no—
130C2×C4×C7—34no—
131C2×C4×C7—31no—
132C2×C4×C7—31no—
133C2×C4×C7—32no—
134C2×C4×C7—62no—
135C2×C4×C7C22×D312yes—
136C22⋊D7—22no—
137C22⋊D7—62no—
138C2×Dic7—32no—
139C2×Dic7—31no—
140C2×Dic7C22×D312yes—
141C2×Dic7C22×D311yescharacteristic
142D28—62no—
143C4×D7—62no—
144C4×D7—62no—
145Dic14—62no—
146C2×C3×D4—71no—
147C22×C3×C4C2×D711yescharacteristic
148C2×#67—71no—
149C2×#67—71no—
150C22×Dic3—71no—
151C22×Dic3C2×D711yescharacteristic
152C22×Dic3C2×D711yescharacteristic
153#237—144no—
154C2×C4×D3—71no—
155C2×Dic6—24no—
156C2×Dic6—71no—
157C2×Dic6C2×D712yes—
158C2×Dic6C2×D712yes—
159C3×C4⋊C4—72no—
160C3×C22⋊C4—71no—
161C3×C22⋊C4—72no—
162#217—71no—
163#217—72no—
164(C2×D3)⋊C4—72no—
165(C2×D3)⋊C4—72no—
166C4⋊Dic3—72no—
167Dic3⋊C4—72no—
168Dic3⋊C4—72no—
169Dic3⋊C4—72no—
170C4×Dic3—72no—
171C2×C3×C7—22no—
172C2×C3×C7C22×D411yescharacteristic
173C2×C3×C72−1+412yes—
174D21—41no—
175C3×D7—41no—
176C2×#41—211no—
177C22×Q8—31no—
178C4⋊Q8—212no—
179#124—212no—
180#123—71no—
181#123—211no—
182#123—211no—
183#123—211no—
184C22⋊Q8—212no—
185C22⋊Q8—212no—
186#117—211no—
187C22×C7—22no—
188C22×C7D3×D412yes—
189C22×C7C23×D311yescharacteristic
190C2×D7—21no—
191C2×D7—22no—
192C2×D7—61no—
193C2×D7—62no—
194C4×C7—22no—
195C4×C7—32no—
196C4×C7—32no—
197C4×C7—62no—
198Dic7—21no—
199Dic7—22no—
200Dic7—61no—
201Dic7—62no—
202C23×C3—71no—
203C23×C3C22×D711yescharacteristic
204C22×D3—71no—
205C3×D4—142no—
206C2×C3×C4—22no—
207C2×C3×C4—72no—
208C2×C3×C4—71no—
209C2×C3×C4C22×D712yes—
210#67—142no—
211#67—142no—
212C2×Dic3—22no—
213C2×Dic3—72no—
214C2×Dic3—71no—
215C2×Dic3—142no—
216C2×Dic3C22⋊D714yes—
217C2×Dic3C22×D712yes—
218C2×Dic3C22×D711yescharacteristic
219C2×Dic3C22×D711yescharacteristic
220C4×D3—142no—
221Dic6—24no—
222Dic6—24no—
223Dic6—142no—
224C3×C7#53611yescharacteristic, complemented, Hall
225#41—424no—
226C2×Q8—32no—
227C2×Q8—32no—
228C2×Q8—64no—
229C2×Q8—211no—
230C2×D4—71no—
231C2×D4—211no—
232C2×D4—211no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—211no—
236C22×C4—211no—
237C22×C4D3×D711yescharacteristic
238C4⋊C4—72no—
239C4⋊C4—212no—
240C4⋊C4—212no—
241C4⋊C4—212no—
242C4⋊C4—212no—
243C22⋊C4—71no—
244C22⋊C4—72no—
245C22⋊C4—212no—
246C22⋊C4—212no—
247C22⋊C4—211no—
248C22⋊C4—212no—
249C42—212no—
250C2×C7—22no—
251C2×C7C2×D3×D411yescharacteristic
252C2×C7#101012yes—
253D7—41no—
254D7—121no—
255C22×C3—22no—
256C22×C3—142no—
257C22×C3—141no—
258C22×C3C2×C22⋊D712yes—
259C22×C3C23×D711yescharacteristic
260C2×D3—142no—
261C2×D3—141no—
262C3×C4—22no—
263C3×C4—142no—
264C3×C4—141no—
265Dic3—22no—
266Dic3—142no—
267Dic3—141no—
268Dic3C2×C22⋊D712yes—
269Dic3C2×C22⋊D712yes—
270C23—71no—
271C23—211no—
272C23C2×D3×D711yescharacteristic
273Q8—64no—
274Q8—64no—
275Q8—422no—
276D4—142no—
277D4—422no—
278D4—422no—
279C2×C4—22no—
280C2×C4—34no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—62no—
285C2×C4—72no—
286C2×C4—71no—
287C2×C4—212no—
288C2×C4—211no—
289C2×C4—422no—
290C2×C4—422no—
291C2×C4C2×D3×D712yes—
292C7#564811yescharacteristic, complemented, Sylow
293C2×C3—22no—
294C2×C3—281no—
295C2×C3#655112yes—
296C2×C3C22×C22⋊D711yescharacteristic
297D3—281no—
298C22—22no—
299C22—142no—
300C22—141no—
301C22—422no—
302C22—421no—
303C22D3×C22⋊D712yes—
304C22C22×D3×D711yescharacteristic
305C4—22no—
306C4—32no—
307C4—32no—
308C4—62no—
309C4—142no—
310C4—141no—
311C4—422no—
312C4—421no—
313C3#9050711yescharacteristic, complemented, Sylow
314C2—22no—
315C2—281no—
316C2—841no—
317C2#1062154712yes—
318C2C2×D3×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC2×#66683C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#536
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C7×Dic6C2semidirect
3. #10621412C2semidirect
4. C3×#6492C2semidirect
5. #10621104C2semidirect
6. #10621098C2semidirect
7. #10621091C2semidirect
8. #10621017C2semidirect
9. #10620986C2semidirect
10. #10620983C2semidirect
11. #10620891C2semidirect
47. #6492D3semidirect
98. C22×Dic6D7semidirect
224. C3×C7#536semidirect
292. C7#5648semidirect
313. C3#90507semidirect

Maximal subgroups

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

subgroupindex
C22×C7×Dic62
C2×#666832
#106214122
C3×#64922
#106211042
#106210982
#106210912
#106210172
#106209862
#106209832
#106208912
#905073
#56487