Finite Group Databasebeta

← #13060169

Subgroups — #13060169

Order 1344.

476 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic21C211yesmaximal, characteristic, complemented
2C7×#939C211yesmaximal, characteristic, complemented
3C3×#6502C211yesmaximal, characteristic, complemented
4#10621133C211yesmaximal, characteristic, complemented
5#10621133C211yesmaximal, characteristic, complemented
6#10621117C211yesmaximal, characteristic, complemented
7Dic3×C22⋊D7C211yesmaximal, characteristic, complemented
8#67×Dic7C211yesmaximal, characteristic, complemented
9#10621110C211yesmaximal, characteristic, complemented
10C2×#66577C211yesmaximal, characteristic, complemented
11#10621094C211yesmaximal, characteristic, complemented
12#10621004C211yesmaximal, characteristic, complemented
13#10620944C211yesmaximal, characteristic, complemented
14#10620924C211yesmaximal, characteristic, complemented
15#10620904C211yesmaximal, characteristic, complemented
16#90292—31nomaximal, Hall
17C2×C3×C7×D4C2211yescharacteristic
18C22×Dic21C2211yescharacteristic
19C22×Dic21C2211yescharacteristic
20C2×C7×#67C2211yescharacteristic
21C2×C7×#67C2211yescharacteristic
22C22×C7×Dic3C2211yescharacteristic
23C2×C3×C22⋊D7C2211yescharacteristic
24C2×C3×C22⋊D7C2211yescharacteristic
25C22×C3×Dic7C2211yescharacteristic
26C2×#4245C2211yescharacteristic
27C2×D3×Dic7C2211yescharacteristic
28C2×Dic3×D7C2211yescharacteristic
29#66636C2211yescharacteristic
30#66636C2211yescharacteristic
31C4⋊Dic21C2211yescharacteristic
32C4×Dic21C2211yescharacteristic
33C7×#217C2211yescharacteristic
34C7×(C2×D3)⋊C4C2211yescharacteristic
35C7×Dic3⋊C4C2211yescharacteristic
36C3×C22⋊Dic7C2211yescharacteristic
37C3×(C2×D7)⋊C4C2211yescharacteristic
38C3×Dic7⋊C4C2211yescharacteristic
39Dic3⋊Dic7C2211yescharacteristic
40#66577C2211yescharacteristic
41#66577C2211yescharacteristic
42#66577C2211yescharacteristic
43#66577C2211yescharacteristic
44#66577C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46(C2×D3)⋊Dic7C2211yescharacteristic
47(C2×D3)⋊Dic7C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#121—31no—
53#6502—31no—
54#6502—31no—
55#6502D311yescharacteristic, complemented
56#6498—31no—
57D4×Dic7—31no—
58D4×Dic7—31no—
59C4×C22⋊D7—31no—
60C2×Dic7⋊C4—31no—
61#6461—31no—
62#6452—31no—
63#6446—31no—
64(C22⋊D7)⋊C4—31no—
65#6442—31no—
66#6440—31no—
67#5601—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C23×C3×C7C2311yescharacteristic
70C22×D3×C7C2311yescharacteristic
71C22×C3×D7C2311yescharacteristic
72C3×C7×D4—22no—
73C2×C3×C4×C7C2311yescharacteristic
74C2×Dic21—22no—
75C2×Dic21—22no—
76C2×Dic21C2311yescharacteristic
77C2×Dic21C2311yescharacteristic
78C2×Dic21C2311yescharacteristic
79C2×Dic21C2311yescharacteristic
80C7×#67—22no—
81C7×#67—41no—
82C2×C7×Dic3—41no—
83C2×C7×Dic3C2311yescharacteristic
84C2×C7×Dic3C2311yescharacteristic
85C2×C7×Dic3C2311yescharacteristic
86C3×C22⋊D7—22no—
87C3×C22⋊D7—41no—
88C2×C3×Dic7—41no—
89C2×C3×Dic7C2311yescharacteristic
90C2×C3×Dic7C2311yescharacteristic
91C2×C3×Dic7C2311yescharacteristic
92#4245—22no—
93D3×Dic7—41no—
94Dic3×D7—41no—
95C2×C7×D4—31no—
96C2×C7×D4—31no—
97C2×C7×D4C2×D311yescharacteristic
98C22×C4×C7—31no—
99C2×C22⋊D7—31no—
100C2×C22⋊D7C2×D311yescharacteristic
101C2×C22⋊D7C2×D311yescharacteristic
102C22×Dic7—31no—
103C22×Dic7—31no—
104C22×Dic7—31no—
105C22×Dic7C2×D311yescharacteristic
106C2×C4×D7—31no—
107C7×C4⋊C4—31no—
108C7×C22⋊C4—31no—
109C7×C22⋊C4—31no—
110C22⋊Dic7—31no—
111C22⋊Dic7—31no—
112C22⋊Dic7—31no—
113C22⋊Dic7—31no—
114C22⋊Dic7C2×D311yescharacteristic
115(C2×D7)⋊C4—31no—
116(C2×D7)⋊C4—31no—
117(C2×D7)⋊C4C2×D311yescharacteristic
118C4⋊Dic7—31no—
119C4⋊Dic7—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122Dic7⋊C4—31no—
123Dic7⋊C4—31no—
124Dic7⋊C4—31no—
125Dic7⋊C4—31no—
126Dic7⋊C4C2×D311yescharacteristic
127C4×Dic7—31no—
128C4×Dic7—31no—
129C4×Dic7—31no—
130C3×#121—71no—
131#939—71no—
132#939—71no—
133#939D711yescharacteristic, complemented
134#935—71no—
135D4×Dic3—71no—
136D4×Dic3—71no—
137C4×#67—71no—
138C2×Dic3⋊C4—71no—
139#898—71no—
140#889—71no—
141#883—71no—
142#67⋊C4—71no—
143#879—71no—
144#877—71no—
145C22×C3×C7—21no—
146C22×C3×C7—21no—
147C22×C3×C7—21no—
148C22×C3×C7—21no—
149C22×C3×C7—21no—
150C22×C3×C7#4112yes—
151C22×C3×C7C2411yescharacteristic
152C2×D3×C7—21no—
153C2×D3×C7—21no—
154C2×D3×C7—21no—
155C2×C3×D7—21no—
156C2×C3×D7—21no—
157C2×C3×D7—21no—
158C3×C4×C7—21no—
159Dic21—21no—
160Dic21—21no—
161Dic21—21no—
162Dic21#4112yes—
163C3×Dic7—21no—
164C3×Dic7—21no—
165C3×Dic7—21no—
166C7×Dic3—21no—
167C7×Dic3—21no—
168C7×Dic3—21no—
169#553—211noSylow
170C23×C7—31no—
171C23×C7C22×D311yescharacteristic
172C23×C7C22×D311yescharacteristic
173C22×D7C22×D311yescharacteristic
174C7×D4—22no—
175C7×D4—62no—
176C7×D4—121no—
177C2×C4×C7—31no—
178C2×C4×C7—31no—
179C2×C4×C7—31no—
180C2×C4×C7—121no—
181C2×C4×C7C22×D311yescharacteristic
182C22⋊D7—22no—
183C22⋊D7—41no—
184C22⋊D7—62no—
185C2×Dic7—31no—
186C2×Dic7—31no—
187C2×Dic7—31no—
188C2×Dic7—31no—
189C2×Dic7—41no—
190C2×Dic7—62no—
191C2×Dic7—62no—
192C2×Dic7—121no—
193C2×Dic7C22×D311yescharacteristic
194C2×Dic7C22×D311yescharacteristic
195C2×Dic7C22×D311yescharacteristic
196C4×D7—121no—
197C2×C3×D4—71no—
198C2×C3×D4—71no—
199C2×C3×D4C2×D711yescharacteristic
200C22×C3×C4—71no—
201C2×#67—71no—
202C2×#67C2×D711yescharacteristic
203C2×#67C2×D711yescharacteristic
204C22×Dic3—71no—
205C22×Dic3—71no—
206C22×Dic3—71no—
207C22×Dic3C2×D711yescharacteristic
208C2×C4×D3—71no—
209C3×C4⋊C4—71no—
210C3×C22⋊C4—71no—
211C3×C22⋊C4—71no—
212#217—71no—
213#217—71no—
214#217—71no—
215#217—71no—
216#217C2×D711yescharacteristic
217(C2×D3)⋊C4—71no—
218(C2×D3)⋊C4—71no—
219(C2×D3)⋊C4C2×D711yescharacteristic
220C4⋊Dic3—71no—
221C4⋊Dic3—71no—
222Dic3⋊C4—71no—
223Dic3⋊C4—71no—
224Dic3⋊C4—71no—
225Dic3⋊C4—71no—
226Dic3⋊C4—71no—
227Dic3⋊C4—71no—
228Dic3⋊C4C2×D711yescharacteristic
229C4×Dic3—71no—
230C4×Dic3—71no—
231C4×Dic3—71no—
232C2×C3×C7—22no—
233C2×C3×C7—41no—
234C2×C3×C7C2×#4111yescharacteristic
235C2×C3×C7C2×#4111yescharacteristic
236C2×C3×C7H(5,𝔽2)11yescharacteristic
237C3×D7—41no—
238D3×C7—41no—
239#126—211no—
240#126—211no—
241#125—211no—
242#123—211no—
243#123—211no—
244#121—31no—
245#121—71no—
246#121—211no—
247#121—211no—
248C4×D4—211no—
249C4×D4—211no—
250C4×D4—211no—
251C4×D4—211no—
252#117—211no—
253C2×C4⋊C4—211no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—61no—
260C22×C7—61no—
261C22×C7—61no—
262C22×C7#23712yes—
263C22×C7C23×D311yescharacteristic
264C2×D7—21no—
265C2×D7—21no—
266C2×D7—21no—
267C4×C7—21no—
268C4×C7—61no—
269C4×C7—61no—
270C4×C7—61no—
271Dic7—21no—
272Dic7—21no—
273Dic7—21no—
274Dic7—32no—
275Dic7—61no—
276Dic7—61no—
277Dic7—61no—
278C23×C3—71no—
279C23×C3C22×D711yescharacteristic
280C23×C3C22×D711yescharacteristic
281C22×D3C22×D711yescharacteristic
282C3×D4—22no—
283C3×D4—142no—
284C3×D4—281no—
285C2×C3×C4—71no—
286C2×C3×C4—71no—
287C2×C3×C4—71no—
288C2×C3×C4—281no—
289C2×C3×C4C22×D711yescharacteristic
290#67—22no—
291#67—41no—
292#67—142no—
293C2×Dic3—41no—
294C2×Dic3—71no—
295C2×Dic3—71no—
296C2×Dic3—71no—
297C2×Dic3—71no—
298C2×Dic3—142no—
299C2×Dic3—142no—
300C2×Dic3—281no—
301C2×Dic3C22×D711yescharacteristic
302C2×Dic3C22×D711yescharacteristic
303C2×Dic3C22×D711yescharacteristic
304C4×D3—281no—
305C3×C7#55311yescharacteristic, complemented, Hall
306C2×D4—31no—
307C2×D4—31no—
308C2×D4—71no—
309C2×D4—71no—
310C2×D4—211no—
311C2×D4D3×D711yescharacteristic
312C22×C4—31no—
313C22×C4—71no—
314C22×C4—211no—
315C22×C4—211no—
316C22×C4—211no—
317C22×C4—211no—
318C4⋊C4—31no—
319C4⋊C4—71no—
320C4⋊C4—211no—
321C4⋊C4—211no—
322C4⋊C4—211no—
323C4⋊C4—211no—
324C4⋊C4—211no—
325C4⋊C4—211no—
326C4⋊C4—211no—
327C4⋊C4—211no—
328C22⋊C4—31no—
329C22⋊C4—31no—
330C22⋊C4—71no—
331C22⋊C4—71no—
332C22⋊C4—211no—
333C22⋊C4—211no—
334C22⋊C4—211no—
335C22⋊C4—211no—
336C22⋊C4—211no—
337C22⋊C4—211no—
338C42—211no—
339C42—211no—
340C42—211no—
341C2×C7—22no—
342C2×C7—41no—
343C2×C7—121no—
344C2×C7C2×#23711yescharacteristic
345C2×C7#100411yescharacteristic
346C2×C7D3×#4111yescharacteristic
347D7—41no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—141no—
354C22×C3—141no—
355C22×C3—141no—
356C22×C3#115912yes—
357C22×C3C23×D711yescharacteristic
358C2×D3—21no—
359C2×D3—21no—
360C2×D3—21no—
361C3×C4—21no—
362C3×C4—141no—
363C3×C4—141no—
364C3×C4—141no—
365Dic3—21no—
366Dic3—21no—
367Dic3—21no—
368Dic3—72no—
369Dic3—141no—
370Dic3—141no—
371Dic3—141no—
372C23—31no—
373C23—71no—
374C23C2×D3×D711yescharacteristic
375C23C2×D3×D711yescharacteristic
376D4—22no—
377D4—62no—
378D4—121no—
379D4—142no—
380D4—281no—
381D4—422no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—71no—
386C2×C4—71no—
387C2×C4—71no—
388C2×C4—121no—
389C2×C4—211no—
390C2×C4—211no—
391C2×C4—211no—
392C2×C4—211no—
393C2×C4—281no—
394C2×C4—422no—
395C2×C4—422no—
396C2×C4—841no—
397C2×C4—841no—
398C2×C4C2×D3×D711yescharacteristic
399C7#560111yescharacteristic, complemented, Sylow
400C2×C3—22no—
401C2×C3—41no—
402C2×C3—281no—
403C2×C3C2×#115911yescharacteristic
404C2×C3#654811yescharacteristic
405C2×C3D7×#4111yescharacteristic
406D3—41no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—61no—
413C22—61no—
414C22—61no—
415C22—141no—
416C22—141no—
417C22—141no—
418C22#6668412yes—
419C22C22×D3×D711yescharacteristic
420C4—21no—
421C4—61no—
422C4—61no—
423C4—61no—
424C4—141no—
425C4—141no—
426C4—141no—
427C4—212no—
428C4—421no—
429C4—421no—
430C4—421no—
431C3#9029211yescharacteristic, complemented, Sylow
432C2—22no—
433C2—41no—
434C2—121no—
435C2—281no—
436C2#1062157111yescharacteristic
437C2#1062157211yescharacteristic
438C2C2×#6668411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#553
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. D4×Dic21C2semidirect
2. C7×#939C2semidirect
3. C3×#6502C2semidirect
4. #10621133C2semidirect
5. #10621133C2semidirect
6. #10621117C2semidirect
7. Dic3×C22⋊D7C2semidirect
8. #67×Dic7C2semidirect
9. #10621110C2semidirect
10. C2×#66577C2semidirect
11. #10621094C2semidirect
12. #10621004C2semidirect
13. #10620944C2semidirect
14. #10620924C2semidirect
15. #10620904C2semidirect
55. #6502D3semidirect
133. #939D7semidirect
305. C3×C7#553semidirect
399. C7#5601semidirect
431. C3#90292semidirect

Maximal subgroups

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

subgroupindex
D4×Dic212
C7×#9392
C3×#65022
#106211332
#106211172
Dic3×C22⋊D72
#67×Dic72
#106211102
C2×#665772
#106210942
#106210042
#106209442
#106209242
#106209042
#902923
#56017