Finite Group Databasebeta

← #13060147

Subgroups — #13060147

Order 1344.

472 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 132 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#10621421C211yesmaximal, characteristic, complemented
2C7×#938C211yesmaximal, characteristic, complemented
3C3×D4×Dic7C211yesmaximal, characteristic, complemented
4#10621128C212yesmaximal, complemented
5#10621090C212yesmaximal, complemented
6#10621088C212yesmaximal, complemented
7#10621087C212yesmaximal, complemented
8#10621030C211yesmaximal, characteristic, complemented
9C4×#4247C211yesmaximal, characteristic, complemented
10#10620909C211yesmaximal, characteristic, complemented
11#10620900C211yesmaximal, characteristic, complemented
12#90307—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15C2×C4×D21C2211yescharacteristic
16C2×C7×#67C2212yes—
17C2×C4×D3×C7C2211yescharacteristic
18C22×C3×Dic7C2212yes—
19C2×#4248C2212yes—
20C2×#4247C2211yescharacteristic
21#66636C2212yes—
22C4⋊Dic21C2211yescharacteristic
23C7×#217C2212yes—
24C7×C4⋊Dic3C2211yescharacteristic
25C3×C22⋊Dic7C2212yes—
26C3×C4⋊Dic7C2211yescharacteristic
27C3×C4×Dic7C2211yescharacteristic
28Dic7⋊Dic3C2211yescharacteristic
29Dic7⋊Dic3C2212yes—
30#66575C2211yescharacteristic
31#66575C2212yes—
32(C2×D3)⋊Dic7C2211yescharacteristic
33(C2×D3)⋊Dic7C2212yes—
34Dic3×Dic7C2212yes—
35Dic3×Dic7C2212yes—
36C7×#121—31no—
37#6502—32no—
38#6501—31no—
39#6499—32no—
40D4×Dic7D311yescharacteristic, complemented
41C4×C22⋊D7—31no—
42#6489—31no—
43#6455—31no—
44#6451—31no—
45#6448—32no—
46#6440—32no—
47#5563—71nomaximal, Hall
48C23×C3×C7C2312yes—
49C22×D21C2311yescharacteristic
50C22×D3×C7C2311yescharacteristic
51C3×C7×D4—22no—
52C2×C3×C4×C7C2311yescharacteristic
53#4268—42no—
54C2×Dic21C2311yescharacteristic
55C2×Dic21C2312yes—
56C4×D21—22no—
57C7×#67—42no—
58C2×C7×Dic3C2311yescharacteristic
59C2×C7×Dic3C2312yes—
60C4×D3×C7—22no—
61C2×C3×Dic7—24no—
62C2×C3×Dic7C2312yes—
63C2×C3×Dic7C2311yescharacteristic
64C2×C3×Dic7C2311yescharacteristic
65#4248—42no—
66#4247—22no—
67C2×C7×D4—32no—
68C2×C7×D4C2×D311yescharacteristic
69C22×C4×C7—31no—
70C2×C22⋊D7—32no—
71C2×C22⋊D7—31no—
72C22×Dic7C2×D312yes—
73C2×C4×D7—31no—
74C2×Dic14—32no—
75C7×C4⋊C4—31no—
76C7×C22⋊C4—32no—
77C22⋊Dic7—32no—
78C22⋊Dic7—32no—
79C22⋊Dic7—31no—
80C22⋊Dic7C2×D312yes—
81(C2×D7)⋊C4—31no—
82(C2×D7)⋊C4—32no—
83C4⋊Dic7—31no—
84C4⋊Dic7C2×D311yescharacteristic
85Dic7⋊C4—31no—
86Dic7⋊C4—32no—
87C4×Dic7—32no—
88C4×Dic7—32no—
89C4×Dic7C2×D311yescharacteristic
90C3×C4×D4—71no—
91#938—71no—
92#938D711yescharacteristic, complemented
93#930—72no—
94#926—72no—
95#892—71no—
96#892—71no—
97#885—72no—
98#885—72no—
99C4×D12—71no—
100#869—71no—
101C22×C3×C7—22no—
102C22×C3×C7—22no—
103C22×C3×C7—22no—
104C22×C3×C7C2411yescharacteristic
105C2×D21—21no—
106C2×D21—21no—
107C2×D21—21no—
108C2×D3×C7—21no—
109C2×D3×C7—21no—
110C2×D3×C7—21no—
111C3×C4×C7#4112yes—
112Dic21—21no—
113Dic21—22no—
114C3×Dic7—22no—
115C3×Dic7—21no—
116C3×Dic7#4112yes—
117C7×Dic3—21no—
118C7×Dic3—22no—
119#555—211noSylow
120C23×C7—31no—
121C23×C7C22×D312yes—
122C22×D7—31no—
123C7×D4—22no—
124C7×D4—122no—
125C2×C4×C7—31no—
126C2×C4×C7—32no—
127C2×C4×C7—62no—
128C2×C4×C7C22×D311yescharacteristic
129C22⋊D7—62no—
130C22⋊D7—122no—
131C2×Dic7—24no—
132C2×Dic7—31no—
133C2×Dic7—32no—
134C2×Dic7C22×D312yes—
135C2×Dic7C22×D311yescharacteristic
136C2×Dic7C22×D311yescharacteristic
137C4×D7—62no—
138Dic14—122no—
139C2×C3×D4C2×D711yescharacteristic
140C22×C3×C4—72no—
141C2×#67—72no—
142C2×#67C2×D712yes—
143C2×D12—71no—
144C2×C4×D3—71no—
145C2×C4×D3C2×D711yescharacteristic
146C2×Dic6—72no—
147C3×C4⋊C4—71no—
148C3×C22⋊C4—72no—
149C3×C42—71no—
150#217—72no—
151#217C2×D712yes—
152(C2×D3)⋊C4—71no—
153(C2×D3)⋊C4—72no—
154(C2×D3)⋊C4—71no—
155(C2×D3)⋊C4—72no—
156C4⋊Dic3—71no—
157C4⋊Dic3—72no—
158C4⋊Dic3—71no—
159C4⋊Dic3C2×D711yescharacteristic
160C4×Dic3—72no—
161C4×Dic3—72no—
162C2×C3×C7—42no—
163C2×C3×C7C2×#4111yescharacteristic
164C2×C3×C7C2×#4111yescharacteristic
165C2×C3×C7H(5,𝔽2)11yescharacteristic
166D21—41no—
167D3×C7—41no—
168C4⋊Q8—211no—
169#124—212no—
170#124—212no—
171#121—31no—
172#121—212no—
173#121—211no—
174C4×D4—71no—
175C4×D4—211no—
176#117—212no—
177#117—211no—
178#117—211no—
179C22×C7—22no—
180C22×C7—22no—
181C22×C7—22no—
182C22×C7—61no—
183C22×C7—61no—
184C22×C7—61no—
185C22×C7C23×D311yescharacteristic
186C2×D7—61no—
187C2×D7—61no—
188C2×D7—61no—
189C4×C7—61no—
190C4×C7—62no—
191C4×C7#23512yes—
192Dic7—22no—
193Dic7—21no—
194Dic7—61no—
195Dic7—62no—
196Dic7#23712yes—
197C23×C3C22×D712yes—
198C22×D3—71no—
199C22×D3C22×D711yescharacteristic
200C3×D4—22no—
201C2×C3×C4—71no—
202C2×C3×C4—72no—
203C2×C3×C4—71no—
204C2×C3×C4—144no—
205C2×C3×C4C22×D711yescharacteristic
206#67—42no—
207#67—282no—
208C2×Dic3—71no—
209C2×Dic3—72no—
210C2×Dic3C22×D711yescharacteristic
211C2×Dic3C22×D712yes—
212D12—142no—
213C4×D3—22no—
214C4×D3—142no—
215Dic6—282no—
216C3×C7#55511yescharacteristic, complemented, Hall
217C2×Q8—212no—
218C2×D4—32no—
219C2×D4—211no—
220C2×D4—212no—
221C2×D4D3×D711yescharacteristic
222C22×C4—31no—
223C22×C4—72no—
224C22×C4—211no—
225C4⋊C4—31no—
226C4⋊C4—71no—
227C4⋊C4—211no—
228C4⋊C4—212no—
229C4⋊C4—211no—
230C22⋊C4—32no—
231C22⋊C4—72no—
232C22⋊C4—212no—
233C22⋊C4—211no—
234C22⋊C4—212no—
235C22⋊C4—211no—
236C22⋊C4—212no—
237C42—71no—
238C42—212no—
239C42—212no—
240C2×C7—42no—
241C2×C7—121no—
242C2×C7C2×#23511yescharacteristic
243C2×C7C2×#23711yescharacteristic
244C2×C7#100911yescharacteristic
245D7—121no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3C23×D711yescharacteristic
250C2×D3—21no—
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—141no—
254C2×D3—141no—
255C2×D3—141no—
256C3×C4—72no—
257C3×C4—141no—
258C3×C4—142no—
259C3×C4#115912yes—
260Dic3—21no—
261Dic3—22no—
262Dic3—141no—
263Dic3—142no—
264C23—31no—
265C23—211no—
266C23C2×D3×D712yes—
267Q8—842no—
268D4—22no—
269D4—122no—
270D4—422no—
271D4—842no—
272C2×C4—31no—
273C2×C4—32no—
274C2×C4—62no—
275C2×C4—71no—
276C2×C4—71no—
277C2×C4—72no—
278C2×C4—144no—
279C2×C4—211no—
280C2×C4—212no—
281C2×C4—422no—
282C2×C4C2×D3×D711yescharacteristic
283C7#556311yescharacteristic, complemented, Sylow
284C2×C3—42no—
285C2×C3C2×#115911yescharacteristic
286C2×C3#654811yescharacteristic
287C2×C3D7×#4111yescharacteristic
288D3—41no—
289D3—281no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—61no—
294C22—61no—
295C22—61no—
296C22—421no—
297C22—421no—
298C22—421no—
299C22C22×D3×D711yescharacteristic
300C4—62no—
301C4—61no—
302C4—72no—
303C4—142no—
304C4—141no—
305C4—422no—
306C4—421no—
307C4#6668612yes—
308C3#9030711yescharacteristic, complemented, Sylow
309C2—42no—
310C2—121no—
311C2—841no—
312C2D7×#23711yescharacteristic
313C2#1062157411yescharacteristic
314C2C2×#6668611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#555
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. #10621421C2semidirect
2. C7×#938C2semidirect
3. C3×D4×Dic7C2semidirect
4. #10621128C2semidirect
5. #10621090C2semidirect
6. #10621088C2semidirect
7. #10621087C2semidirect
8. #10621030C2semidirect
9. C4×#4247C2semidirect
10. #10620909C2semidirect
11. #10620900C2semidirect
40. D4×Dic7D3semidirect
92. #938D7semidirect
216. C3×C7#555semidirect
283. C7#5563semidirect
308. C3#90307semidirect

Maximal subgroups

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

subgroupindex
#106214212
C7×#9382
C3×D4×Dic72
#106211282
#106210902
#106210882
#106210872
#106210302
C4×#42472
#106209092
#106209002
#903073
#55637