Finite Group Databasebeta

← #13061742

Subgroups — #13061742

Order 1344.

428 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×#123C211yesmaximal, characteristic
2#10621418C211yesmaximal, characteristic
3D4×Dic21C211yesmaximal, characteristic
4#10621409C211yesmaximal, characteristic, complemented
5C2×#66629C211yesmaximal, characteristic, complemented
6#10621373C211yesmaximal, characteristic, complemented
7#10621372C211yesmaximal, characteristic, complemented
8#10621372C211yesmaximal, characteristic, complemented
9#10621370C211yesmaximal, characteristic, complemented
10#10621362C211yesmaximal, characteristic, complemented
11#10621362C211yesmaximal, characteristic, complemented
12#10621361C211yesmaximal, characteristic, complemented
13#10621361C211yesmaximal, characteristic, complemented
14#10621360C211yesmaximal, characteristic, complemented
15#10621360C211yesmaximal, characteristic, complemented
16#90341—31nomaximal, Hall
17C2×C3×C7×D4C2211yescharacteristic
18C22×C3×C4×C7C2211yescharacteristic
19C22×Dic21C2211yescharacteristic
20C22×Dic21C2211yescharacteristic
21C22×Dic21C2211yescharacteristic
22C2×Dic42C2211yescharacteristic
23C3×C7×C4⋊C4C2211yescharacteristic
24C3×C7×C4⋊C4C2211yescharacteristic
25C3×C7×C22⋊C4C2211yescharacteristic
26C3×C7×C22⋊C4C2211yescharacteristic
27C3×C7×C22⋊C4C2211yescharacteristic
28#66636C2211yescharacteristic
29#66636C2211yescharacteristic
30#66636C2211yescharacteristic
31#66636C2211yescharacteristic
32#66636C2211yescharacteristic
33C4⋊Dic21C2211yescharacteristic
34C4⋊Dic21C2211yescharacteristic
35C4⋊Dic21C2211yescharacteristic
36C4⋊Dic21C2211yescharacteristic
37#66629C2211yescharacteristic
38#66629C2211yescharacteristic
39#66629C2211yescharacteristic
40#66629C2211yescharacteristic
41#66629C2211yescharacteristic
42#66629C2211yescharacteristic
43#66629C2211yescharacteristic
44#66629C2211yescharacteristic
45#66629C2211yescharacteristic
46#66629C2211yescharacteristic
47C4×Dic21C2211yescharacteristic
48C4×Dic21C2211yescharacteristic
49C4×Dic21C2211yescharacteristic
50C4×Dic21C2211yescharacteristic
51C4×Dic21C2211yescharacteristic
52C7×#123D311yescharacteristic
53#6498—31no—
54D4×Dic7—31no—
55#6489—31no—
56C2×Dic7⋊C4—31no—
57#6453—31no—
58#6452—31no—
59#6452—31no—
60#6450—31no—
61#6442—31no—
62#6442—31no—
63#6441—31no—
64#6441—31no—
65#6440—31no—
66#6440—31no—
67#5650—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C23×C3×C7C2311yescharacteristic
70C3×C7×D4—22no—
71C2×C3×C4×C7—22no—
72C2×C3×C4×C7C2311yescharacteristic
73C2×C3×C4×C7C2311yescharacteristic
74C2×C3×C4×C7C2311yescharacteristic
75C2×C3×C4×C7C2311yescharacteristic
76C2×C3×C4×C7C2311yescharacteristic
77C2×Dic21—22no—
78C2×Dic21—22no—
79C2×Dic21—22no—
80C2×Dic21C2311yescharacteristic
81C2×Dic21C2311yescharacteristic
82C2×Dic21C2311yescharacteristic
83C2×Dic21C2311yescharacteristic
84C2×Dic21C2311yescharacteristic
85C2×Dic21C2311yescharacteristic
86C2×Dic21C2311yescharacteristic
87C2×Dic21C2311yescharacteristic
88Dic42—22no—
89C2×C7×D4C2×D311yescharacteristic
90C22×C4×C7C2×D311yescharacteristic
91C22×Dic7—31no—
92C22×Dic7—31no—
93C22×Dic7—31no—
94C2×Dic14—31no—
95C7×C4⋊C4C2×D311yescharacteristic
96C7×C4⋊C4C2×D311yescharacteristic
97C7×C22⋊C4C2×D311yescharacteristic
98C7×C22⋊C4C2×D311yescharacteristic
99C7×C22⋊C4C2×D311yescharacteristic
100C22⋊Dic7—31no—
101C22⋊Dic7—31no—
102C22⋊Dic7—31no—
103C22⋊Dic7—31no—
104C22⋊Dic7—31no—
105C4⋊Dic7—31no—
106C4⋊Dic7—31no—
107C4⋊Dic7—31no—
108C4⋊Dic7—31no—
109Dic7⋊C4—31no—
110Dic7⋊C4—31no—
111Dic7⋊C4—31no—
112Dic7⋊C4—31no—
113Dic7⋊C4—31no—
114Dic7⋊C4—31no—
115Dic7⋊C4—31no—
116Dic7⋊C4—31no—
117Dic7⋊C4—31no—
118Dic7⋊C4—31no—
119C4×Dic7—31no—
120C4×Dic7—31no—
121C4×Dic7—31no—
122C4×Dic7—31no—
123C4×Dic7—31no—
124C3×#123D711yescharacteristic
125#935—71no—
126D4×Dic3—71no—
127#926—71no—
128C2×Dic3⋊C4—71no—
129#890—71no—
130#889—71no—
131#889—71no—
132#887—71no—
133#879—71no—
134#879—71no—
135#878—71no—
136#878—71no—
137#877—71no—
138#877—71no—
139C22×C3×C7—21no—
140C22×C3×C7—21no—
141C22×C3×C7—21no—
142C22×C3×C7—21no—
143C22×C3×C7—21no—
144C22×C3×C7#4112yes—
145C22×C3×C7C2411yescharacteristic
146C3×C4×C7—21no—
147C3×C4×C7—21no—
148C3×C4×C7—21no—
149C3×C4×C7—21no—
150C3×C4×C7—21no—
151Dic21—21no—
152Dic21—21no—
153Dic21—21no—
154Dic21—21no—
155Dic21—21no—
156Dic21—21no—
157Dic21—21no—
158Dic21#4112yes—
159#552—211noSylow
160C23×C7C22×D311yescharacteristic
161C23×C7C22×D311yescharacteristic
162C7×D4—22no—
163C2×C4×C7—22no—
164C2×C4×C7C22×D311yescharacteristic
165C2×C4×C7C22×D311yescharacteristic
166C2×C4×C7C22×D311yescharacteristic
167C2×C4×C7C22×D311yescharacteristic
168C2×C4×C7C22×D311yescharacteristic
169C2×Dic7—31no—
170C2×Dic7—31no—
171C2×Dic7—31no—
172C2×Dic7—31no—
173C2×Dic7—31no—
174C2×Dic7—31no—
175C2×Dic7—31no—
176C2×Dic7—31no—
177C2×Dic7—62no—
178C2×Dic7—62no—
179C2×Dic7—62no—
180Dic14—62no—
181C2×C3×D4C2×D711yescharacteristic
182C22×C3×C4C2×D711yescharacteristic
183C22×Dic3—71no—
184C22×Dic3—71no—
185C22×Dic3—71no—
186C2×Dic6—71no—
187C3×C4⋊C4C2×D711yescharacteristic
188C3×C4⋊C4C2×D711yescharacteristic
189C3×C22⋊C4C2×D711yescharacteristic
190C3×C22⋊C4C2×D711yescharacteristic
191C3×C22⋊C4C2×D711yescharacteristic
192#217—71no—
193#217—71no—
194#217—71no—
195#217—71no—
196#217—71no—
197C4⋊Dic3—71no—
198C4⋊Dic3—71no—
199C4⋊Dic3—71no—
200C4⋊Dic3—71no—
201Dic3⋊C4—71no—
202Dic3⋊C4—71no—
203Dic3⋊C4—71no—
204Dic3⋊C4—71no—
205Dic3⋊C4—71no—
206Dic3⋊C4—71no—
207Dic3⋊C4—71no—
208Dic3⋊C4—71no—
209Dic3⋊C4—71no—
210Dic3⋊C4—71no—
211C4×Dic3—71no—
212C4×Dic3—71no—
213C4×Dic3—71no—
214C4×Dic3—71no—
215C4×Dic3—71no—
216C2×C3×C7—22no—
217C2×C3×C7—41no—
218C2×C3×C7C2×#4111yescharacteristic
219C2×C3×C7C2×#4111yescharacteristic
220C2×C3×C72−1+411yescharacteristic
221#126—211no—
222#126—211no—
223#125—211no—
224#125—211no—
225#125—211no—
226#123—211no—
227#123D2111yescharacteristic
228C22⋊Q8—211no—
229C22⋊Q8—211no—
230C4×Q8—211no—
231C4×D4—211no—
232#117—211no—
233#117—211no—
234#117—211no—
235C2×C4⋊C4—211no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—21no—
241C22×C7#23712yes—
242C22×C7C23×D311yescharacteristic
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248Dic7—32no—
249Dic7—61no—
250Dic7—61no—
251Dic7—61no—
252Dic7—61no—
253Dic7—61no—
254Dic7—61no—
255Dic7—61no—
256C23×C3C22×D711yescharacteristic
257C23×C3C22×D711yescharacteristic
258C3×D4—22no—
259C2×C3×C4—22no—
260C2×C3×C4C22×D711yescharacteristic
261C2×C3×C4C22×D711yescharacteristic
262C2×C3×C4C22×D711yescharacteristic
263C2×C3×C4C22×D711yescharacteristic
264C2×C3×C4C22×D711yescharacteristic
265C2×Dic3—71no—
266C2×Dic3—71no—
267C2×Dic3—71no—
268C2×Dic3—71no—
269C2×Dic3—71no—
270C2×Dic3—71no—
271C2×Dic3—71no—
272C2×Dic3—71no—
273C2×Dic3—142no—
274C2×Dic3—142no—
275C2×Dic3—142no—
276Dic6—142no—
277C3×C7#55211yescharacteristic, complemented, Hall
278C2×Q8—211no—
279C2×D4C2×D2111yescharacteristic
280C22×C4—211no—
281C22×C4—211no—
282C22×C4—211no—
283C22×C4C2×D2111yescharacteristic
284C4⋊C4—211no—
285C4⋊C4—211no—
286C4⋊C4—211no—
287C4⋊C4—211no—
288C4⋊C4—211no—
289C4⋊C4—211no—
290C4⋊C4—211no—
291C4⋊C4—211no—
292C4⋊C4—211no—
293C4⋊C4—211no—
294C4⋊C4—211no—
295C4⋊C4—211no—
296C4⋊C4—211no—
297C4⋊C4—211no—
298C4⋊C4C2×D2111yescharacteristic
299C4⋊C4C2×D2111yescharacteristic
300C22⋊C4—211no—
301C22⋊C4—211no—
302C22⋊C4—211no—
303C22⋊C4—211no—
304C22⋊C4—211no—
305C22⋊C4C2×D2111yescharacteristic
306C22⋊C4C2×D2111yescharacteristic
307C22⋊C4C2×D2111yescharacteristic
308C42—211no—
309C42—211no—
310C42—211no—
311C42—211no—
312C42—211no—
313C2×C7—22no—
314C2×C7—41no—
315C2×C7C2×#23711yescharacteristic
316C2×C7D3×#4111yescharacteristic
317C2×C7#101011yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3#115912yes—
324C22×C3C23×D711yescharacteristic
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330Dic3—72no—
331Dic3—141no—
332Dic3—141no—
333Dic3—141no—
334Dic3—141no—
335Dic3—141no—
336Dic3—141no—
337Dic3—141no—
338C23C22×D2111yescharacteristic
339C23C22×D2111yescharacteristic
340Q8—422no—
341D4—22no—
342C2×C4—22no—
343C2×C4—211no—
344C2×C4—211no—
345C2×C4—211no—
346C2×C4—211no—
347C2×C4—211no—
348C2×C4—211no—
349C2×C4—211no—
350C2×C4—211no—
351C2×C4—422no—
352C2×C4—422no—
353C2×C4—422no—
354C2×C4C22×D2111yescharacteristic
355C2×C4C22×D2111yescharacteristic
356C2×C4C22×D2111yescharacteristic
357C2×C4C22×D2111yescharacteristic
358C2×C4C22×D2111yescharacteristic
359C7#565011yescharacteristic, complemented, Sylow
360C2×C3—22no—
361C2×C3—41no—
362C2×C3C2×#115911yescharacteristic
363C2×C3D7×#4111yescharacteristic
364C2×C3#655411yescharacteristic
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22#6673012yes—
371C22C23×D2111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—212no—
378C4—421no—
379C4—421no—
380C4—421no—
381C4—421no—
382C4—421no—
383C4—421no—
384C4—421no—
385C3#9034111yescharacteristic, complemented, Sylow
386C2—22no—
387C2—41no—
388C2C2×#6673011yescharacteristic
389C2#41×D2111yescharacteristic
390C2#1062166311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#123C2
prime radicalC2×C3×C7×D4C22
hypercenter#123D21
hypocenterC3×C7#552
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
4. #10621409C2semidirect
5. C2×#66629C2semidirect
6. #10621373C2semidirect
7. #10621372C2semidirect
8. #10621372C2semidirect
9. #10621370C2semidirect
10. #10621362C2semidirect
11. #10621362C2semidirect
12. #10621361C2semidirect
13. #10621361C2semidirect
14. #10621360C2semidirect
15. #10621360C2semidirect
277. C3×C7#552semidirect
359. C7#5650semidirect
385. C3#90341semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1232
#106214182
D4×Dic212
#106214092
C2×#666292
#106213732
#106213722
#106213702
#106213622
#106213612
#106213602
#903413
#56507