Finite Group Databasebeta

← #13058126

Subgroups — #13058126

Order 1344.

432 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621369C211yesmaximal, characteristic, complemented
2C7×#879C211yesmaximal, characteristic, complemented
3C3×#6448C211yesmaximal, characteristic, complemented
4#10621138C211yesmaximal, characteristic, complemented
5(C22⋊D7)⋊Dic3C211yesmaximal, characteristic, complemented
6#10621095C211yesmaximal, characteristic
7#10621088C211yesmaximal, characteristic, complemented
8#10621030C211yesmaximal, characteristic, complemented
9#10620984C211yesmaximal, characteristic, complemented
10#10620975C211yesmaximal, characteristic, complemented
11#10620973C211yesmaximal, characteristic, complemented
12#10620964C211yesmaximal, characteristic, complemented
13#10620924C211yesmaximal, characteristic, complemented
14#10620915C211yesmaximal, characteristic, complemented
15Dic3×Dic14C211yesmaximal, characteristic, complemented
16#90401—31nomaximal, Hall
17C2×#4268C2211yescharacteristic
18C22×Dic21C2211yescharacteristic
19C2×C3×C22⋊D7C2211yescharacteristic
20C2×C3×Dic14C2211yescharacteristic
21C2×#4248C2211yescharacteristic
22C2×#4248C2211yescharacteristic
23C2×#4246C2211yescharacteristic
24C2×D21⋊C4C2211yescharacteristic
25C2×Dic3×D7C2211yescharacteristic
26C3×C7×C22⋊C4C2211yescharacteristic
27#66631C2211yescharacteristic
28#66631C2211yescharacteristic
29C4⋊Dic21C2211yescharacteristic
30C4⋊Dic21C2211yescharacteristic
31C7×#217C2211yescharacteristic
32C7×#217C2211yescharacteristic
33C7×C4⋊Dic3C2211yescharacteristic
34C7×Dic3⋊C4C2211yescharacteristic
35C7×Dic3⋊C4C2211yescharacteristic
36C4×C7×Dic3C2211yescharacteristic
37C3×C22⋊Dic7C2211yescharacteristic
38C3×(C2×D7)⋊C4C2211yescharacteristic
39C3×(C2×D7)⋊C4C2211yescharacteristic
40C3×C4×Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42#66577C2211yescharacteristic
43Dic7⋊Dic3C2211yescharacteristic
44Dic7⋊Dic3C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46#66575C2211yescharacteristic
47#66575C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#126—31no—
53#6461—31no—
54#6460—31no—
55#6459—31no—
56#6457—31no—
57#6455—31no—
58#6451—31no—
59#6449—31no—
60#6448—31no—
61#6448D311yescharacteristic, complemented
62#6447—31no—
63(C22⋊D7)⋊C4—31no—
64#6441—31no—
65#6438—31no—
66C4×Dic14—31no—
67#5682—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C22×C3×D7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×C3×C4×C7C2311yescharacteristic
73#4268—41no—
74C2×Dic21—41no—
75C2×Dic21C2311yescharacteristic
76C2×Dic21C2311yescharacteristic
77C2×Dic21C2311yescharacteristic
78C2×C7×Dic3C2311yescharacteristic
79C2×C7×Dic3C2311yescharacteristic
80C2×C7×Dic3C2311yescharacteristic
81C2×C7×Dic3C2311yescharacteristic
82C3×C22⋊D7—22no—
83C2×C3×Dic7C2311yescharacteristic
84C2×C3×Dic7C2311yescharacteristic
85C2×C3×Dic7C2311yescharacteristic
86C3×Dic14—22no—
87#4248—41no—
88#4248—41no—
89#4246—41no—
90D21⋊C4—22no—
91Dic3×D7—41no—
92C2×C22⋊D7—31no—
93C2×C22⋊D7C2×D311yescharacteristic
94C22×Dic7—31no—
95C2×D28—31no—
96C2×C4×D7—31no—
97C2×C4×D7—31no—
98C2×Dic14—31no—
99C2×Dic14—31no—
100C2×Dic14C2×D311yescharacteristic
101C7×C4⋊C4—31no—
102C7×C4⋊C4—31no—
103C7×C4⋊C4—31no—
104C7×C22⋊C4—31no—
105C7×C22⋊C4—31no—
106C7×C22⋊C4C2×D311yescharacteristic
107C42×C7—31no—
108C22⋊Dic7C2×D311yescharacteristic
109(C2×D7)⋊C4—31no—
110(C2×D7)⋊C4—31no—
111(C2×D7)⋊C4—31no—
112(C2×D7)⋊C4—31no—
113(C2×D7)⋊C4—31no—
114(C2×D7)⋊C4—31no—
115(C2×D7)⋊C4C2×D311yescharacteristic
116(C2×D7)⋊C4C2×D311yescharacteristic
117C4⋊Dic7—31no—
118C4⋊Dic7—31no—
119C4⋊Dic7—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122Dic7⋊C4—31no—
123Dic7⋊C4—31no—
124C4×Dic7—31no—
125C4×Dic7—31no—
126C4×Dic7C2×D311yescharacteristic
127C3×#124—71no—
128#946—71no—
129Q8×Dic3—71no—
130#938—71no—
131D4×Dic3—71no—
132#886—71no—
133#886—71no—
134#885—71no—
135#885—71no—
136#879—71no—
137#879D711yescharacteristic, complemented
138#878—71no—
139#878—71no—
140#872—71no—
141#869—71no—
142C22×C3×C7—21no—
143C22×C3×C7—21no—
144C22×C3×C7—21no—
145C22×C3×C7C2411yescharacteristic
146C2×D21—21no—
147C2×D21—21no—
148C2×D21—21no—
149C2×C3×D7—21no—
150C2×C3×D7—21no—
151C2×C3×D7—21no—
152C3×C4×C7—21no—
153C3×C4×C7—21no—
154Dic21—21no—
155Dic21—21no—
156Dic21—21no—
157C3×Dic7—21no—
158C3×Dic7—21no—
159C3×Dic7#4112yes—
160C7×Dic3—21no—
161C7×Dic3—21no—
162C7×Dic3—21no—
163C7×Dic3—21no—
164#542—211noSylow
165C23×C7C22×D311yescharacteristic
166C22×D7—31no—
167C22×D7C22×D311yescharacteristic
168C2×C4×C7—31no—
169C2×C4×C7—31no—
170C2×C4×C7—31no—
171C2×C4×C7—31no—
172C2×C4×C7C22×D311yescharacteristic
173C2×C4×C7C22×D311yescharacteristic
174C22⋊D7—22no—
175C22⋊D7—121no—
176C2×Dic7—31no—
177C2×Dic7—31no—
178C2×Dic7—31no—
179C2×Dic7—121no—
180C2×Dic7C22×D311yescharacteristic
181C2×Dic7C22×D311yescharacteristic
182C2×Dic7C22×D311yescharacteristic
183D28—121no—
184C4×D7—62no—
185C4×D7—121no—
186Dic14—22no—
187Dic14—121no—
188Dic14—121no—
189C2×C3×Q8—71no—
190C2×C3×D4—71no—
191C2×#67—71no—
192C2×#67—71no—
193C22×Dic3—71no—
194C22×Dic3—71no—
195C2×C4×D3—71no—
196C2×Dic6—71no—
197C2×Dic6—71no—
198C3×C22⋊C4—71no—
199C3×C22⋊C4—71no—
200C3×C22⋊C4—71no—
201C3×C22⋊C4C2×D711yescharacteristic
202C3×C42—71no—
203#217—71no—
204#217—71no—
205#217C2×D711yescharacteristic
206#217C2×D711yescharacteristic
207(C2×D3)⋊C4—71no—
208(C2×D3)⋊C4—71no—
209(C2×D3)⋊C4—71no—
210(C2×D3)⋊C4—71no—
211C4⋊Dic3—71no—
212C4⋊Dic3—71no—
213C4⋊Dic3—71no—
214C4⋊Dic3—71no—
215C4⋊Dic3—71no—
216C4⋊Dic3C2×D711yescharacteristic
217Dic3⋊C4—71no—
218Dic3⋊C4—71no—
219Dic3⋊C4C2×D711yescharacteristic
220Dic3⋊C4C2×D711yescharacteristic
221C4×Dic3—71no—
222C4×Dic3—71no—
223C4×Dic3C2×D711yescharacteristic
224C2×C3×C7—41no—
225C2×C3×C7C2×#4111yescharacteristic
226C2×C3×C7H(5,𝔽2)11yescharacteristic
227C2×C3×C72−1+411yescharacteristic
228D21—41no—
229C3×D7—41no—
230C4⋊Q8—211no—
231#126—31no—
232#126—211no—
233#124—71no—
234#124—211no—
235#124—211no—
236#123—211no—
237#123—211no—
238C22⋊Q8—211no—
239C22⋊Q8—211no—
240C22⋊Q8—211no—
241#121—211no—
242C4×Q8—211no—
243C4×D4—211no—
244#117—211no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7C23×D311yescharacteristic
249C2×D7—21no—
250C2×D7—21no—
251C2×D7—21no—
252C2×D7—61no—
253C2×D7—61no—
254C2×D7—61no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—61no—
258C4×C7—61no—
259C4×C7—61no—
260C4×C7—61no—
261Dic7—21no—
262Dic7—21no—
263Dic7—61no—
264Dic7—61no—
265Dic7—61no—
266Dic7#23712yes—
267C23×C3—71no—
268C23×C3C22×D711yescharacteristic
269C22×D3—71no—
270C3×Q8—142no—
271C3×D4—142no—
272C2×C3×C4—71no—
273C2×C3×C4—71no—
274C2×C3×C4—71no—
275C2×C3×C4C22×D711yescharacteristic
276C2×C3×C4C22×D711yescharacteristic
277#67—281no—
278#67—281no—
279C2×Dic3—71no—
280C2×Dic3—71no—
281C2×Dic3—71no—
282C2×Dic3—281no—
283C2×Dic3—281no—
284C2×Dic3C22×D711yescharacteristic
285C2×Dic3C22×D711yescharacteristic
286C2×Dic3C22×D711yescharacteristic
287C2×Dic3C22×D711yescharacteristic
288C4×D3—142no—
289Dic6—281no—
290Dic6—281no—
291C3×C7#54211yescharacteristic, complemented, Hall
292C2×Q8—71no—
293C2×Q8—211no—
294C2×Q8—211no—
295C2×D4—71no—
296C2×D4—211no—
297C2×D4—211no—
298C22×C4—211no—
299C22×C4—211no—
300C22×C4—211no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—31no—
304C4⋊C4—211no—
305C4⋊C4—211no—
306C4⋊C4—211no—
307C4⋊C4—211no—
308C4⋊C4—211no—
309C4⋊C4—211no—
310C4⋊C4—211no—
311C22⋊C4—31no—
312C22⋊C4—31no—
313C22⋊C4—71no—
314C22⋊C4—71no—
315C22⋊C4—71no—
316C22⋊C4—211no—
317C22⋊C4—211no—
318C22⋊C4—211no—
319C22⋊C4—211no—
320C22⋊C4—211no—
321C22⋊C4—211no—
322C22⋊C4D3×D711yescharacteristic
323C42—31no—
324C42—71no—
325C42—211no—
326C42—211no—
327C2×C7—41no—
328C2×C7C2×#23711yescharacteristic
329C2×C7#100911yescharacteristic
330C2×C7#101011yescharacteristic
331D7—41no—
332D7—121no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—141no—
337C22×C3—141no—
338C22×C3—141no—
339C22×C3C23×D711yescharacteristic
340C2×D3—141no—
341C2×D3—141no—
342C2×D3—141no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—72no—
346C3×C4—141no—
347C3×C4—141no—
348Dic3—21no—
349Dic3—21no—
350Dic3—21no—
351Dic3—21no—
352Dic3—141no—
353Dic3—141no—
354Dic3—141no—
355C23—71no—
356C23—211no—
357C23C2×D3×D711yescharacteristic
358Q8—142no—
359Q8—841no—
360Q8—841no—
361D4—142no—
362D4—841no—
363D4—841no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—71no—
369C2×C4—71no—
370C2×C4—71no—
371C2×C4—211no—
372C2×C4—211no—
373C2×C4—211no—
374C2×C4—422no—
375C2×C4—841no—
376C2×C4—841no—
377C2×C4C2×D3×D711yescharacteristic
378C2×C4C2×D3×D711yescharacteristic
379C7#568211yescharacteristic, complemented, Sylow
380C2×C3—41no—
381C2×C3—281no—
382C2×C3D7×#4111yescharacteristic
383C2×C3#655311yescharacteristic
384C2×C3#655411yescharacteristic
385D3—281no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—141no—
390C22—141no—
391C22—141no—
392C22—421no—
393C22—421no—
394C22—421no—
395C22C22×D3×D711yescharacteristic
396C4—21no—
397C4—21no—
398C4—61no—
399C4—61no—
400C4—61no—
401C4—61no—
402C4—72no—
403C4—141no—
404C4—141no—
405C4—421no—
406C4—421no—
407C4—421no—
408C3#9040111yescharacteristic, complemented, Sylow
409C2—41no—
410C2—281no—
411C2—841no—
412C2#1062156611yescharacteristic
413C2#1062156711yescharacteristic
414C2D7×#23711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radical#10621095C2
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#542
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. #10621369C2semidirect
2. C7×#879C2semidirect
3. C3×#6448C2semidirect
4. #10621138C2semidirect
5. (C22⋊D7)⋊Dic3C2semidirect
7. #10621088C2semidirect
8. #10621030C2semidirect
9. #10620984C2semidirect
10. #10620975C2semidirect
11. #10620973C2semidirect
12. #10620964C2semidirect
13. #10620924C2semidirect
14. #10620915C2semidirect
15. Dic3×Dic14C2semidirect
61. #6448D3semidirect
137. #879D7semidirect
291. C3×C7#542semidirect
379. C7#5682semidirect
408. C3#90401semidirect

Maximal subgroups

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

subgroupindex
#106213692
C7×#8792
C3×#64482
#106211382
(C22⋊D7)⋊Dic32
#106210952
#106210882
#106210302
#106209842
#106209752
#106209732
#106209642
#106209242
#106209152
Dic3×Dic142
#904013
#56827