Finite Group Databasebeta

← #13052512

Subgroups — #13052512

Order 1344.

320 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 58 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#10621596C211yesmaximal, characteristic
2C3×#6449C211yesmaximal, characteristic, complemented
3#10621108C211yesmaximal, characteristic, complemented
4#10621101C211yesmaximal, characteristic, complemented
5#10620645C211yesmaximal, characteristic, complemented
6C7×#806C211yesmaximal, characteristic, complemented
7#10620530C211yesmaximal, characteristic, complemented
8#89514—31nomaximal, Hall
9C2×#4268C2211yescharacteristic
10C2×C7×#67C2211yescharacteristic
11C2×C3×C22⋊D7C2211yescharacteristic
12C22×C3×Dic7C2211yescharacteristic, complemented
13#66694—21no—
14D3×C22⋊D7—21no—
15D7×#67—21no—
16#66686—21no—
17#66684—21no—
18#66683—21no—
19C3×C7×C22⋊C4C2211yescharacteristic, complemented
20#66636C2211yescharacteristic, complemented
21C7×#217C2211yescharacteristic, complemented
22C3×(C2×D7)⋊C4—21no—
23C3×C4⋊Dic7—21no—
24Dic3⋊Dic7—21no—
25#66577—21no—
26#66575—21no—
27(C2×D3)⋊Dic7—21no—
28#6548—31no—
29#6498—31no—
30#6449—31no—
31#6449D311yescharacteristic, complemented
32C7×#99—31no—
33#6380—31no—
34#6380—31no—
35#5053—71nomaximal, Hall
36C23×C3×C7C2311yescharacteristic
37C22×D21D411yescharacteristic
38C22×D3×C7D411yescharacteristic
39C22×C3×D7D411yescharacteristic
40C2×D3×D7—21no—
41C2×C3×C4×C7—21no—
42#4268—21no—
43#4268—21no—
44C2×Dic21—21no—
45C2×Dic21D411yescharacteristic
46C7×#67—21no—
47C7×#67—21no—
48C2×C7×Dic3—21no—
49C2×C7×Dic3D411yescharacteristic
50C3×C22⋊D7—21no—
51C3×C22⋊D7—21no—
52C2×C3×Dic7—21no—
53C2×C3×Dic7—21no—
54C2×C3×Dic7—21no—
55C2×C3×Dic7D411yescharacteristic
56#4248—21no—
57#4247—21no—
58#4246—21no—
59#4245—21no—
60D21⋊C4—21no—
61D3×Dic7—21no—
62Dic3×D7—21no—
63C2×C7×D4—31no—
64C2×C22⋊D7—31no—
65C2×C22⋊D7—61no—
66C2×C22⋊D7C2×D311yescharacteristic
67C22×Dic7C2×D311yescharacteristic, complemented
68#1159—61no—
69#1159—61no—
70D4×D7—61no—
71D4×D7—61no—
72#1157—61no—
73C7×C22⋊C4—31no—
74C7×C22⋊C4C2×D311yescharacteristic, complemented
75C22⋊Dic7—31no—
76C22⋊Dic7—61no—
77(C2×D7)⋊C4—21no—
78(C2×D7)⋊C4—61no—
79C4⋊Dic7—21no—
80C4⋊Dic7—61no—
81Dic7⋊C4—61no—
82#1004—71no—
83C3×#123—71no—
84#929—71no—
85#929—71no—
86#834—71no—
87#806—71no—
88#806D711yescharacteristic, complemented
89C22×C3×C7—41no—
90C22×C3×C7C2×D411yescharacteristic
91C22×C3×C7C2×D411yescharacteristic
92C22×C3×C7C2×D411yescharacteristic
93C2×D21—21no—
94C2×D21—41no—
95C2×D3×C7—21no—
96C2×D3×C7—41no—
97C2×C3×D7—21no—
98C2×C3×D7—41no—
99D3×D7—81no—
100C3×C4×C7—41no—
101Dic21—21no—
102Dic21—41no—
103C3×Dic7—21no—
104C3×Dic7—41no—
105C7×Dic3—21no—
106C7×Dic3—41no—
107#458—211noSylow
108C23×C7—31no—
109C23×C7C22×D311yescharacteristic
110C22×D7—31no—
111C22×D7—61no—
112C22×D7#6711yescharacteristic
113C7×D4—61no—
114C7×D4—61no—
115C2×C4×C7—21no—
116C2×C4×C7—31no—
117C2×C4×C7—61no—
118C22⋊D7—21no—
119C22⋊D7—21no—
120C22⋊D7—61no—
121C22⋊D7—61no—
122C22⋊D7—61no—
123C22⋊D7—61no—
124C2×Dic7—21no—
125C2×Dic7—21no—
126C2×Dic7—21no—
127C2×Dic7—31no—
128C2×Dic7—61no—
129C2×Dic7—61no—
130C2×Dic7#6711yescharacteristic
131D28—61no—
132C4×D7—61no—
133C4×D7—61no—
134Dic14—61no—
135C2×C3×D4—71no—
136C22×C3×C4—71no—
137C2×#67—71no—
138C2×#67—141no—
139C2×#67C2×D711yescharacteristic
140#237—141no—
141#237—141no—
142D3×D4—141no—
143D3×D4—141no—
144#235—141no—
145C3×C4⋊C4—141no—
146C3×C22⋊C4—141no—
147C3×C22⋊C4C2×D711yescharacteristic, complemented
148#217—71no—
149#217C2×D711yescharacteristic, complemented
150(C2×D3)⋊C4—141no—
151(C2×D3)⋊C4—141no—
152Dic3⋊C4—141no—
153Dic3⋊C4—141no—
154C2×C3×C7—21no—
155C2×C3×C7—21no—
156C2×C3×C7—21no—
157C2×C3×C7#12011yescharacteristic
158D21—41no—
159C3×D7—41no—
160D3×C7—41no—
161H(5,𝔽2)—211no—
162#123—71no—
163#123—211no—
164#123—211no—
165#99—31no—
166#99—211no—
167#99—211no—
168C22×C7—41no—
169C22×C7—61no—
170C22×C7—121no—
171C22×C7D3×D411yescharacteristic
172C22×C7D3×D411yescharacteristic
173C22×C7C2×#6711yescharacteristic
174C2×D7—21no—
175C2×D7—41no—
176C2×D7—61no—
177C2×D7—121no—
178C2×D7—241no—
179C4×C7—41no—
180C4×C7—61no—
181C4×C7—121no—
182Dic7—21no—
183Dic7—41no—
184Dic7—61no—
185Dic7—121no—
186C23×C3—71no—
187C23×C3C22×D711yescharacteristic
188C22×D3—71no—
189C22×D3—141no—
190C22×D3D2811yescharacteristic
191C3×D4—141no—
192C3×D4—141no—
193C2×C3×C4—21no—
194C2×C3×C4—71no—
195C2×C3×C4—141no—
196C2×C3×C4—141no—
197C2×C3×C4—141no—
198#67—21no—
199#67—21no—
200#67—141no—
201#67—141no—
202#67—141no—
203#67—141no—
204C2×Dic3—21no—
205C2×Dic3—71no—
206C2×Dic3—141no—
207C2×Dic3—141no—
208C2×Dic3D2811yescharacteristic
209D12—141no—
210C4×D3—141no—
211C4×D3—141no—
212Dic6—141no—
213C3×C7#45811yescharacteristic, complemented, Hall
214#41—421no—
215#41—421no—
216#41—421no—
217C2×D4—31no—
218C2×D4—71no—
219C2×D4—211no—
220C2×D4—421no—
221C2×D4—421no—
222C2×D4—421no—
223C22×C4—71no—
224C4⋊C4—141no—
225C4⋊C4—421no—
226C4⋊C4—421no—
227C22⋊C4—31no—
228C22⋊C4—141no—
229C22⋊C4—211no—
230C22⋊C4—421no—
231C22⋊C4—421no—
232C22⋊C4D3×D711yescharacteristic, complemented
233C2×C7—21no—
234C2×C7—21no—
235C2×C7—21no—
236C2×C7—121no—
237C2×C7#93711yescharacteristic
238D7—41no—
239D7—121no—
240C22×C3—41no—
241C22×C3—141no—
242C22×C3—281no—
243C22×C3C2×D2811yescharacteristic
244C22×C3D4×D711yescharacteristic
245C22×C3D4×D711yescharacteristic
246C2×D3—21no—
247C2×D3—41no—
248C2×D3—141no—
249C2×D3—281no—
250C2×D3—561no—
251C3×C4—41no—
252C3×C4—141no—
253C3×C4—281no—
254Dic3—21no—
255Dic3—41no—
256Dic3—141no—
257Dic3—281no—
258C23—31no—
259C23—71no—
260C23—211no—
261C23—421no—
262C23C2×D3×D711yescharacteristic
263Q8—421no—
264D4—61no—
265D4—61no—
266D4—141no—
267D4—141no—
268D4—421no—
269D4—421no—
270D4—421no—
271D4—421no—
272D4—421no—
273C2×C4—21no—
274C2×C4—31no—
275C2×C4—61no—
276C2×C4—71no—
277C2×C4—141no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—211no—
281C2×C4—421no—
282C2×C4—421no—
283C2×C4—421no—
284C2×C4—421no—
285C7#505311yescharacteristic, complemented, Sylow
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—281no—
290C2×C3#644511yescharacteristic
291D3—41no—
292D3—281no—
293C22—41no—
294C22—61no—
295C22—121no—
296C22—141no—
297C22—281no—
298C22—421no—
299C22—841no—
300C22—1681no—
301C22D3×D2811yescharacteristic
302C22#6668111yescharacteristic
303C22D7×#6711yescharacteristic
304C4—41no—
305C4—61no—
306C4—121no—
307C4—141no—
308C4—281no—
309C4—421no—
310C4—841no—
311C3#8951411yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2—21no—
315C2—121no—
316C2—281no—
317C2—841no—
318C2#1062103611yescharacteristic
subgroupG/H
centerC2#10621036
commutator, cosocleC23×C3×C7C23
FrattiniC23C2×D3×D7
FittingC3×C7×C22⋊C4C22
socleC2×C3×C7#120
prime radical#10621596C2
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#458
LCSG ▷ C23×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#6449C2semidirect
3. #10621108C2semidirect
4. #10621101C2semidirect
5. #10620645C2semidirect
6. C7×#806C2semidirect
7. #10620530C2semidirect
12. C22×C3×Dic7C22semidirect
19. C3×C7×C22⋊C4C22semidirect
20. #66636C22semidirect
21. C7×#217C22semidirect
31. #6449D3semidirect
67. C22×Dic7C2×D3semidirect
74. C7×C22⋊C4C2×D3semidirect
88. #806D7semidirect
147. C3×C22⋊C4C2×D7semidirect
149. #217C2×D7semidirect
213. C3×C7#458semidirect
232. C22⋊C4D3×D7semidirect
285. C7#5053semidirect
311. C3#89514semidirect

Maximal subgroups

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

subgroupindex
#106215962
C3×#64492
#106211082
#106211012
#106206452
C7×#8062
#106205302
#895143
#50537