Finite Group Databasebeta

← #13061631

Subgroups — #13061631

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
1C3×C4×C7×D4C211yesmaximal, characteristic, complemented
2#10621422C211yesmaximal, characteristic, complemented
3#10621422C211yesmaximal, characteristic, complemented
4#10621413C211yesmaximal, characteristic, complemented
5#10621412C211yesmaximal, characteristic, complemented
6C4×#4268C211yesmaximal, characteristic, complemented
7C2×#66629C211yesmaximal, characteristic, complemented
8#10621377C211yesmaximal, characteristic, complemented
9#10621372C211yesmaximal, characteristic, complemented
10#10621367C211yesmaximal, characteristic, complemented
11#10621364C211yesmaximal, characteristic, complemented
12#10621364C211yesmaximal, characteristic, complemented
13#10621362C211yesmaximal, characteristic, complemented
14#10621359C211yesmaximal, characteristic, complemented
15#10621355C211yesmaximal, characteristic, complemented
16#90231—31nomaximal, Hall
17C2×C3×C7×D4C2211yescharacteristic
18C22×C3×C4×C7C2211yescharacteristic
19C22×C3×C4×C7C2211yescharacteristic
20C2×#4268C2211yescharacteristic
21C2×#4268C2211yescharacteristic
22C2×#4268C2211yescharacteristic
23C2×#4268C2211yescharacteristic
24C22×Dic21C2211yescharacteristic
25C22×Dic21C2211yescharacteristic
26C2×D84C2211yescharacteristic
27C2×C4×D21C2211yescharacteristic
28C2×C4×D21C2211yescharacteristic
29C3×C7×C4⋊C4C2211yescharacteristic
30C3×C7×C22⋊C4C2211yescharacteristic
31C3×C7×C22⋊C4C2211yescharacteristic
32C3×C42×C7C2211yescharacteristic
33#66636C2211yescharacteristic
34#66636C2211yescharacteristic
35#66631C2211yescharacteristic
36#66631C2211yescharacteristic
37#66631C2211yescharacteristic
38#66631C2211yescharacteristic
39#66631C2211yescharacteristic
40#66631C2211yescharacteristic
41C4⋊Dic21C2211yescharacteristic
42#66629C2211yescharacteristic
43#66629C2211yescharacteristic
44#66629C2211yescharacteristic
45#66629C2211yescharacteristic
46#66629C2211yescharacteristic
47#66629C2211yescharacteristic
48#66629C2211yescharacteristic
49#66629C2211yescharacteristic
50C4×Dic21C2211yescharacteristic
51C4×Dic21C2211yescharacteristic
52C4×C7×D4D311yescharacteristic, complemented
53#6502—31no—
54#6502—31no—
55#6493—31no—
56#6492—31no—
57C4×C22⋊D7—31no—
58C2×Dic7⋊C4—31no—
59#6457—31no—
60#6452—31no—
61#6447—31no—
62(C22⋊D7)⋊C4—31no—
63(C22⋊D7)⋊C4—31no—
64#6442—31no—
65#6439—31no—
66#6435—31no—
67#5540—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C23×C3×C7C2311yescharacteristic
70C22×D21C2311yescharacteristic
71C22×D21C2311yescharacteristic
72C3×C7×D4—22no—
73C2×C3×C4×C7—41no—
74C2×C3×C4×C7—41no—
75C2×C3×C4×C7C2311yescharacteristic
76C2×C3×C4×C7C2311yescharacteristic
77C2×C3×C4×C7C2311yescharacteristic
78C2×C3×C4×C7C2311yescharacteristic
79C2×C3×C4×C7C2311yescharacteristic
80#4268—22no—
81#4268—22no—
82#4268—22no—
83#4268—41no—
84C2×Dic21—22no—
85C2×Dic21—41no—
86C2×Dic21C2311yescharacteristic
87C2×Dic21C2311yescharacteristic
88C2×Dic21C2311yescharacteristic
89C2×Dic21C2311yescharacteristic
90C2×Dic21C2311yescharacteristic
91C2×Dic21C2311yescharacteristic
92D84—41no—
93C4×D21—22no—
94C4×D21—41no—
95C2×C7×D4C2×D311yescharacteristic
96C22×C4×C7C2×D311yescharacteristic
97C22×C4×C7C2×D311yescharacteristic
98C2×C22⋊D7—31no—
99C2×C22⋊D7—31no—
100C2×C22⋊D7—31no—
101C2×C22⋊D7—31no—
102C22×Dic7—31no—
103C22×Dic7—31no—
104C2×D28—31no—
105C2×C4×D7—31no—
106C2×C4×D7—31no—
107C7×C4⋊C4C2×D311yescharacteristic
108C7×C22⋊C4C2×D311yescharacteristic
109C7×C22⋊C4C2×D311yescharacteristic
110C42×C7C2×D311yescharacteristic
111C22⋊Dic7—31no—
112C22⋊Dic7—31no—
113(C2×D7)⋊C4—31no—
114(C2×D7)⋊C4—31no—
115(C2×D7)⋊C4—31no—
116(C2×D7)⋊C4—31no—
117(C2×D7)⋊C4—31no—
118(C2×D7)⋊C4—31no—
119C4⋊Dic7—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122Dic7⋊C4—31no—
123Dic7⋊C4—31no—
124Dic7⋊C4—31no—
125Dic7⋊C4—31no—
126Dic7⋊C4—31no—
127Dic7⋊C4—31no—
128C4×Dic7—31no—
129C4×Dic7—31no—
130C3×C4×D4D711yescharacteristic, complemented
131#939—71no—
132#939—71no—
133#930—71no—
134#929—71no—
135C4×#67—71no—
136C2×Dic3⋊C4—71no—
137#894—71no—
138#889—71no—
139#884—71no—
140#67⋊C4—71no—
141#67⋊C4—71no—
142#879—71no—
143#876—71no—
144#872—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×D21—21no—
153C2×D21—21no—
154C2×D21—21no—
155C2×D21—21no—
156C2×D21—21no—
157C2×D21—21no—
158C3×C4×C7—21no—
159C3×C4×C7—21no—
160C3×C4×C7—21no—
161C3×C4×C7—21no—
162C3×C4×C7—21no—
163Dic21—21no—
164Dic21—21no—
165Dic21—21no—
166Dic21—21no—
167Dic21—21no—
168Dic21#4112yes—
169#553—211noSylow
170C23×C7C22×D311yescharacteristic
171C23×C7C22×D311yescharacteristic
172C22×D7—31no—
173C22×D7—31no—
174C7×D4—22no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7C22×D311yescharacteristic
178C2×C4×C7C22×D311yescharacteristic
179C2×C4×C7C22×D311yescharacteristic
180C2×C4×C7C22×D311yescharacteristic
181C2×C4×C7C22×D311yescharacteristic
182C22⋊D7—62no—
183C22⋊D7—62no—
184C22⋊D7—62no—
185C22⋊D7—121no—
186C2×Dic7—31no—
187C2×Dic7—31no—
188C2×Dic7—31no—
189C2×Dic7—31no—
190C2×Dic7—31no—
191C2×Dic7—31no—
192C2×Dic7—62no—
193C2×Dic7—121no—
194D28—121no—
195C4×D7—62no—
196C4×D7—121no—
197C2×C3×D4C2×D711yescharacteristic
198C22×C3×C4C2×D711yescharacteristic
199C22×C3×C4C2×D711yescharacteristic
200C2×#67—71no—
201C2×#67—71no—
202C2×#67—71no—
203C2×#67—71no—
204C22×Dic3—71no—
205C22×Dic3—71no—
206C2×D12—71no—
207C2×C4×D3—71no—
208C2×C4×D3—71no—
209C3×C4⋊C4C2×D711yescharacteristic
210C3×C22⋊C4C2×D711yescharacteristic
211C3×C22⋊C4C2×D711yescharacteristic
212C3×C42C2×D711yescharacteristic
213#217—71no—
214#217—71no—
215(C2×D3)⋊C4—71no—
216(C2×D3)⋊C4—71no—
217(C2×D3)⋊C4—71no—
218(C2×D3)⋊C4—71no—
219(C2×D3)⋊C4—71no—
220(C2×D3)⋊C4—71no—
221C4⋊Dic3—71no—
222Dic3⋊C4—71no—
223Dic3⋊C4—71no—
224Dic3⋊C4—71no—
225Dic3⋊C4—71no—
226Dic3⋊C4—71no—
227Dic3⋊C4—71no—
228Dic3⋊C4—71no—
229Dic3⋊C4—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
237D21—41no—
238D21—41no—
239#126—211no—
240#126—211no—
241#125—211no—
242#123—211no—
243#123—211no—
244#121—211no—
245#121—211no—
246#121—211no—
247#121—211no—
248C4×D4—211no—
249C4×D4—211no—
250C4×D4—211no—
251C4×D4D2111yescharacteristic, complemented
252#117—211no—
253C2×C4⋊C4—211no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7#23512yes—
260C22×C7C23×D311yescharacteristic
261C2×D7—61no—
262C2×D7—61no—
263C2×D7—61no—
264C2×D7—61no—
265C2×D7—61no—
266C2×D7—61no—
267C4×C7—21no—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272Dic7—32no—
273Dic7—61no—
274Dic7—61no—
275Dic7—61no—
276Dic7—61no—
277Dic7—61no—
278C23×C3C22×D711yescharacteristic
279C23×C3C22×D711yescharacteristic
280C22×D3—71no—
281C22×D3—71no—
282C3×D4—22no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4C22×D711yescharacteristic
286C2×C3×C4C22×D711yescharacteristic
287C2×C3×C4C22×D711yescharacteristic
288C2×C3×C4C22×D711yescharacteristic
289C2×C3×C4C22×D711yescharacteristic
290#67—142no—
291#67—142no—
292#67—142no—
293#67—281no—
294C2×Dic3—71no—
295C2×Dic3—71no—
296C2×Dic3—71no—
297C2×Dic3—71no—
298C2×Dic3—71no—
299C2×Dic3—71no—
300C2×Dic3—142no—
301C2×Dic3—281no—
302D12—281no—
303C4×D3—142no—
304C4×D3—281no—
305C3×C7#55311yescharacteristic, complemented, Hall
306C2×D4—211no—
307C2×D4—211no—
308C2×D4—211no—
309C2×D4—211no—
310C2×D4—211no—
311C2×D4C2×D2111yescharacteristic
312C22×C4—211no—
313C22×C4—211no—
314C22×C4—211no—
315C22×C4—211no—
316C22×C4C2×D2111yescharacteristic
317C22×C4C2×D2111yescharacteristic
318C4⋊C4—211no—
319C4⋊C4—211no—
320C4⋊C4—211no—
321C4⋊C4—211no—
322C4⋊C4—211no—
323C4⋊C4—211no—
324C4⋊C4—211no—
325C4⋊C4—211no—
326C4⋊C4—211no—
327C4⋊C4C2×D2111yescharacteristic
328C22⋊C4—211no—
329C22⋊C4—211no—
330C22⋊C4—211no—
331C22⋊C4—211no—
332C22⋊C4—211no—
333C22⋊C4—211no—
334C22⋊C4—211no—
335C22⋊C4—211no—
336C22⋊C4C2×D2111yescharacteristic
337C22⋊C4C2×D2111yescharacteristic
338C42—211no—
339C42—211no—
340C42C2×D2111yescharacteristic
341C2×C7—22no—
342C2×C7—41no—
343C2×C7C2×#23511yescharacteristic
344C2×C7#100411yescharacteristic
345C2×C7D3×#4111yescharacteristic
346D7—121no—
347D7—121no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3#115712yes—
354C22×C3C23×D711yescharacteristic
355C2×D3—141no—
356C2×D3—141no—
357C2×D3—141no—
358C2×D3—141no—
359C2×D3—141no—
360C2×D3—141no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366Dic3—72no—
367Dic3—141no—
368Dic3—141no—
369Dic3—141no—
370Dic3—141no—
371Dic3—141no—
372C23—211no—
373C23—211no—
374C23C22×D2111yescharacteristic
375C23C22×D2111yescharacteristic
376D4—22no—
377D4—422no—
378D4—422no—
379D4—422no—
380D4—841no—
381D4—841no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—211no—
385C2×C4—211no—
386C2×C4—211no—
387C2×C4—211no—
388C2×C4—211no—
389C2×C4—211no—
390C2×C4—422no—
391C2×C4—422no—
392C2×C4—841no—
393C2×C4—841no—
394C2×C4C22×D2111yescharacteristic
395C2×C4C22×D2111yescharacteristic
396C2×C4C22×D2111yescharacteristic
397C2×C4C22×D2111yescharacteristic
398C2×C4C22×D2111yescharacteristic
399C7#554011yescharacteristic, complemented, Sylow
400C2×C3—22no—
401C2×C3—41no—
402C2×C3C2×#115711yescharacteristic
403C2×C3#654811yescharacteristic
404C2×C3D7×#4111yescharacteristic
405D3—281no—
406D3—281no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—421no—
413C22—421no—
414C22—421no—
415C22—421no—
416C22—421no—
417C22—421no—
418C22#6672812yes—
419C22C23×D2111yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—212no—
426C4—421no—
427C4—421no—
428C4—421no—
429C4—421no—
430C4—421no—
431C3#9023111yescharacteristic, complemented, Sylow
432C2—22no—
433C2—41no—
434C2—841no—
435C2—841no—
436C2C2×#6672811yescharacteristic
437C2#1062165711yescharacteristic
438C2#41×D2111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C4×C7×D4C2
hypercenterC4×D4D21
hypocenterC3×C7#553
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C7×D4C2semidirect
2. #10621422C2semidirect
3. #10621422C2semidirect
4. #10621413C2semidirect
5. #10621412C2semidirect
6. C4×#4268C2semidirect
7. C2×#66629C2semidirect
8. #10621377C2semidirect
9. #10621372C2semidirect
10. #10621367C2semidirect
11. #10621364C2semidirect
12. #10621364C2semidirect
13. #10621362C2semidirect
14. #10621359C2semidirect
15. #10621355C2semidirect
52. C4×C7×D4D3semidirect
130. C3×C4×D4D7semidirect
251. C4×D4D21semidirect
305. C3×C7#553semidirect
399. C7#5540semidirect
431. C3#90231semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C7×D42
#106214222
#106214132
#106214122
C4×#42682
C2×#666292
#106213772
#106213722
#106213672
#106213642
#106213622
#106213592
#106213552
#902313
#55407