Finite Group Databasebeta

← #13058097

Subgroups — #13058097

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
1#10621367C211yesmaximal, characteristic, complemented
2C7×#879C211yesmaximal, characteristic, complemented
3C3×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
4#10621114C211yesmaximal, characteristic, complemented
5#10621108C211yesmaximal, characteristic, complemented
6#10621095C211yesmaximal, characteristic, complemented
7#10621095C211yesmaximal, characteristic, complemented
8#10621021C211yesmaximal, characteristic, complemented
9C4×#4246C211yesmaximal, characteristic, complemented
10#10620980C211yesmaximal, characteristic, complemented
11#10620966C211yesmaximal, characteristic, complemented
12#10620965C211yesmaximal, characteristic, complemented
13#10620940C211yesmaximal, characteristic, complemented
14#10620923C211yesmaximal, characteristic, complemented
15#10620910C211yesmaximal, characteristic, complemented
16#90400—31nomaximal, Hall
17C2×#4268C2211yescharacteristic
18C2×#4268C2211yescharacteristic
19C2×D84C2211yescharacteristic
20C2×C4×D21C2211yescharacteristic
21C2×C3×C22⋊D7C2211yescharacteristic
22C22×C3×Dic7C2211yescharacteristic
23C2×C3×C4×D7C2211yescharacteristic
24C2×#4246C2211yescharacteristic
25C2×#4246C2211yescharacteristic
26C2×D21⋊C4C2211yescharacteristic
27C2×D21⋊C4C2211yescharacteristic
28C2×D21⋊C4C2211yescharacteristic
29C3×C7×C22⋊C4C2211yescharacteristic
30#66631C2211yescharacteristic
31#66629C2211yescharacteristic
32C7×#217C2211yescharacteristic
33C7×#217C2211yescharacteristic
34C7×C4⋊Dic3C2211yescharacteristic
35C7×Dic3⋊C4C2211yescharacteristic
36C7×Dic3⋊C4C2211yescharacteristic
37C4×C7×Dic3C2211yescharacteristic
38C3×(C2×D7)⋊C4C2211yescharacteristic
39C3×Dic7⋊C4C2211yescharacteristic
40C3×C4×Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42Dic3⋊Dic7C2211yescharacteristic
43#66577C2211yescharacteristic
44Dic7⋊Dic3C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46#66575C2211yescharacteristic
47#66575C2211yescharacteristic
48#66575C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50(C2×D7)⋊Dic3C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#126—31no—
53#6461—31no—
54#6458—31no—
55#6457—31no—
56#6456—31no—
57D7×C4⋊C4—31no—
58#6452—31no—
59#6449—31no—
60#6447—31no—
61#6447—31no—
62#6447—31no—
63(C22⋊D7)⋊C4—31no—
64(C22⋊D7)⋊C4D311yescharacteristic, complemented
65C4×D28—31no—
66#6435—31no—
67#5558—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C22×D21C2311yescharacteristic
71C22×C3×D7C2311yescharacteristic
72C2×C3×C4×C7C2311yescharacteristic
73C2×C3×C4×C7C2311yescharacteristic
74#4268—22no—
75#4268—41no—
76C2×Dic21C2311yescharacteristic
77C2×Dic21C2311yescharacteristic
78D84—22no—
79C4×D21—41no—
80C2×C7×Dic3C2311yescharacteristic
81C2×C7×Dic3C2311yescharacteristic
82C2×C7×Dic3C2311yescharacteristic
83C2×C7×Dic3C2311yescharacteristic
84C3×C22⋊D7—22no—
85C2×C3×Dic7—41no—
86C2×C3×Dic7C2311yescharacteristic
87C2×C3×Dic7C2311yescharacteristic
88C2×C3×Dic7C2311yescharacteristic
89C3×C4×D7—41no—
90#4246—22no—
91#4246—41no—
92D21⋊C4—22no—
93D21⋊C4—22no—
94D21⋊C4—41no—
95C2×C22⋊D7—31no—
96C2×C22⋊D7—31no—
97C2×C22⋊D7C2×D311yescharacteristic
98C22×Dic7C2×D311yescharacteristic
99C2×D28—31no—
100C2×D28—31no—
101C2×D28—31no—
102C2×C4×D7—31no—
103C2×C4×D7—31no—
104C2×C4×D7—31no—
105C2×C4×D7—31no—
106C2×C4×D7C2×D311yescharacteristic
107C7×C4⋊C4—31no—
108C7×C4⋊C4—31no—
109C7×C4⋊C4—31no—
110C7×C22⋊C4—31no—
111C7×C22⋊C4—31no—
112C7×C22⋊C4C2×D311yescharacteristic
113C42×C7—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—
119(C2×D7)⋊C4—31no—
120(C2×D7)⋊C4C2×D311yescharacteristic
121C4⋊Dic7—31no—
122C4⋊Dic7—31no—
123Dic7⋊C4—31no—
124Dic7⋊C4—31no—
125Dic7⋊C4—31no—
126Dic7⋊C4—31no—
127Dic7⋊C4C2×D311yescharacteristic
128C4×Dic7—31no—
129C4×Dic7C2×D311yescharacteristic
130C3×C4×D4—71no—
131#938—71no—
132#938—71no—
133#929—71no—
134#929—71no—
135C4×#67—71no—
136C4×#67—71no—
137#892—71no—
138D3×C4⋊C4—71no—
139#884—71no—
140#884—71no—
141#879—71no—
142#879D711yescharacteristic, complemented
143C4×D12—71no—
144#870—71no—
145C22×C3×C7—21no—
146C22×C3×C7—21no—
147C22×C3×C7—21no—
148C22×C3×C7C2411yescharacteristic
149C2×D21—21no—
150C2×D21—21no—
151C2×D21—21no—
152C2×D21—21no—
153C2×D21—21no—
154C2×D21#4112yes—
155C2×C3×D7—21no—
156C2×C3×D7—21no—
157C2×C3×D7—21no—
158C3×C4×C7—21no—
159C3×C4×C7—21no—
160Dic21—21no—
161Dic21—21no—
162C3×Dic7—21no—
163C3×Dic7—21no—
164C3×Dic7#4112yes—
165C7×Dic3—21no—
166C7×Dic3—21no—
167C7×Dic3—21no—
168C7×Dic3—21no—
169#553—211noSylow
170C23×C7C22×D311yescharacteristic
171C22×D7—31no—
172C22×D7—31no—
173C22×D7C22×D311yescharacteristic
174C2×C4×C7—31no—
175C2×C4×C7—31no—
176C2×C4×C7—31no—
177C2×C4×C7—31no—
178C2×C4×C7C22×D311yescharacteristic
179C2×C4×C7C22×D311yescharacteristic
180C22⋊D7—22no—
181C22⋊D7—62no—
182C22⋊D7—121no—
183C2×Dic7—31no—
184C2×Dic7—31no—
185C2×Dic7—41no—
186C2×Dic7C22×D311yescharacteristic
187C2×Dic7C22×D311yescharacteristic
188C2×Dic7C22×D311yescharacteristic
189D28—62no—
190D28—62no—
191D28—121no—
192C4×D7—41no—
193C4×D7—62no—
194C4×D7—62no—
195C4×D7—121no—
196C4×D7—121no—
197C2×C3×D4—71no—
198C22×C3×C4—71no—
199C22×C3×C4—71no—
200C2×#67—71no—
201C2×#67—71no—
202C2×#67—71no—
203C2×#67—71no—
204C2×D12—71no—
205C2×C4×D3—71no—
206C2×C4×D3—71no—
207C2×C4×D3—71no—
208C2×C4×D3—71no—
209C3×C4⋊C4—71no—
210C3×C22⋊C4—71no—
211C3×C22⋊C4C2×D711yescharacteristic
212C3×C42—71no—
213#217—71no—
214#217—71no—
215#217C2×D711yescharacteristic
216#217C2×D711yescharacteristic
217(C2×D3)⋊C4—71no—
218(C2×D3)⋊C4—71no—
219(C2×D3)⋊C4—71no—
220(C2×D3)⋊C4—71no—
221C4⋊Dic3—71no—
222C4⋊Dic3—71no—
223C4⋊Dic3C2×D711yescharacteristic
224Dic3⋊C4—71no—
225Dic3⋊C4—71no—
226Dic3⋊C4—71no—
227Dic3⋊C4—71no—
228Dic3⋊C4C2×D711yescharacteristic
229Dic3⋊C4C2×D711yescharacteristic
230C4×Dic3—71no—
231C4×Dic3C2×D711yescharacteristic
232C2×C3×C7—41no—
233C2×C3×C7C2×#4111yescharacteristic
234C2×C3×C7C2×#4111yescharacteristic
235C2×C3×C7H(5,𝔽2)11yescharacteristic
236D21—22no—
237D21—41no—
238C3×D7—41no—
239#126—31no—
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—71no—
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×C7C23×D311yescharacteristic
258C2×D7—21no—
259C2×D7—21no—
260C2×D7—21no—
261C2×D7—32no—
262C2×D7—61no—
263C2×D7—61no—
264C2×D7—61no—
265C2×D7—61no—
266C2×D7—61no—
267C4×C7—21no—
268C4×C7—21no—
269C4×C7—61no—
270C4×C7—61no—
271C4×C7—61no—
272C4×C7—61no—
273Dic7—21no—
274Dic7—21no—
275Dic7—61no—
276Dic7—61no—
277Dic7#23512yes—
278C23×C3—71no—
279C23×C3C22×D711yescharacteristic
280C22×D3—71no—
281C22×D3—71no—
282C3×D4—142no—
283C2×C3×C4—71no—
284C2×C3×C4—71no—
285C2×C3×C4—71no—
286C2×C3×C4—281no—
287C2×C3×C4—281no—
288C2×C3×C4C22×D711yescharacteristic
289C2×C3×C4C22×D711yescharacteristic
290#67—142no—
291#67—142no—
292#67—281no—
293#67—281no—
294C2×Dic3—71no—
295C2×Dic3—71no—
296C2×Dic3C22×D711yescharacteristic
297C2×Dic3C22×D711yescharacteristic
298C2×Dic3C22×D711yescharacteristic
299C2×Dic3C22×D711yescharacteristic
300D12—142no—
301C4×D3—142no—
302C4×D3—142no—
303C4×D3—281no—
304C4×D3—281no—
305C3×C7#55311yescharacteristic, complemented, Hall
306C2×D4—71no—
307C2×D4—211no—
308C2×D4—211no—
309C2×D4—211no—
310C2×D4—211no—
311C2×D4—211no—
312C22×C4—71no—
313C22×C4—71no—
314C22×C4—211no—
315C22×C4—211no—
316C22×C4—211no—
317C22×C4—211no—
318C4⋊C4—31no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—71no—
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—211no—
332C22⋊C4—211no—
333C22⋊C4—211no—
334C22⋊C4—211no—
335C22⋊C4—211no—
336C22⋊C4—211no—
337C22⋊C4D3×D711yescharacteristic
338C42—31no—
339C42—71no—
340C42—211no—
341C2×C7—41no—
342C2×C7C2×#23511yescharacteristic
343C2×C7#100411yescharacteristic
344C2×C7D3×#4111yescharacteristic
345D7—41no—
346D7—62no—
347D7—121no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—141no—
352C22×C3—141no—
353C22×C3—141no—
354C22×C3C23×D711yescharacteristic
355C2×D3—72no—
356C2×D3—141no—
357C2×D3—141no—
358C2×D3—141no—
359C2×D3—141no—
360C2×D3—141no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—72no—
364C3×C4—141no—
365C3×C4—141no—
366Dic3—21no—
367Dic3—21no—
368Dic3—21no—
369Dic3—21no—
370Dic3—141no—
371Dic3—141no—
372C23—71no—
373C23—211no—
374C23—211no—
375C23C2×D3×D711yescharacteristic
376D4—142no—
377D4—422no—
378D4—422no—
379D4—422no—
380D4—841no—
381D4—841no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—71no—
387C2×C4—71no—
388C2×C4—71no—
389C2×C4—211no—
390C2×C4—211no—
391C2×C4—281no—
392C2×C4—281no—
393C2×C4—422no—
394C2×C4—422no—
395C2×C4—841no—
396C2×C4—841no—
397C2×C4C2×D3×D711yescharacteristic
398C2×C4C2×D3×D711yescharacteristic
399C7#555811yescharacteristic, complemented, Sylow
400C2×C3—41no—
401C2×C3—281no—
402C2×C3D7×#4111yescharacteristic
403C2×C3D7×#4111yescharacteristic
404C2×C3#655311yescharacteristic
405D3—142no—
406D3—281no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—141no—
411C22—141no—
412C22—141no—
413C22—212no—
414C22—421no—
415C22—421no—
416C22—421no—
417C22—421no—
418C22—421no—
419C22C22×D3×D711yescharacteristic
420C4—21no—
421C4—21no—
422C4—61no—
423C4—61no—
424C4—61no—
425C4—61no—
426C4—72no—
427C4—141no—
428C4—141no—
429C4—421no—
430C4—421no—
431C3#9040011yescharacteristic, complemented, Sylow
432C2—41no—
433C2—281no—
434C2—422no—
435C2—841no—
436C2D7×#23511yescharacteristic
437C2#1062157111yescharacteristic
438C2#1062157311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#553
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. #10621367C2semidirect
2. C7×#879C2semidirect
3. C3×(C22⋊D7)⋊C4C2semidirect
4. #10621114C2semidirect
5. #10621108C2semidirect
6. #10621095C2semidirect
7. #10621095C2semidirect
8. #10621021C2semidirect
9. C4×#4246C2semidirect
10. #10620980C2semidirect
11. #10620966C2semidirect
12. #10620965C2semidirect
13. #10620940C2semidirect
14. #10620923C2semidirect
15. #10620910C2semidirect
64. (C22⋊D7)⋊C4D3semidirect
142. #879D7semidirect
305. C3×C7#553semidirect
399. C7#5558semidirect
431. C3#90400semidirect

Maximal subgroups

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

subgroupindex
#106213672
C7×#8792
C3×(C22⋊D7)⋊C42
#106211142
#106211082
#106210952
#106210212
C4×#42462
#106209802
#106209662
#106209652
#106209402
#106209232
#106209102
#904003
#55587