Finite Group Databasebeta

← #13061802

Subgroups — #13061802

Order 1344.

392 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×#126C211yesmaximal, characteristic, complemented
2#10621381C211yesmaximal, characteristic, complemented
3#10621380C211yesmaximal, characteristic, complemented
4#10621379C211yesmaximal, characteristic, complemented
5#10621373C211yesmaximal, characteristic, complemented
6#10621372C211yesmaximal, characteristic, complemented
7#10621371C211yesmaximal, characteristic, complemented
8#10621369C211yesmaximal, characteristic
9#10621368C211yesmaximal, characteristic
10#10621366C211yesmaximal, characteristic
11#10621362C211yesmaximal, characteristic, complemented
12#10621361C211yesmaximal, characteristic, complemented
13#10621361C211yesmaximal, characteristic, complemented
14#10621359C211yesmaximal, characteristic, complemented
15#10621352C211yesmaximal, characteristic, complemented
16#90403—31nomaximal, Hall
17C2×#4268C2211yescharacteristic
18C22×Dic21C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C2×Dic42C2211yescharacteristic
21C2×Dic42C2211yescharacteristic
22C2×Dic42C2211yescharacteristic
23C3×C7×C4⋊C4C2211yescharacteristic
24C3×C7×C4⋊C4C2211yescharacteristic
25C3×C7×C4⋊C4C2211yescharacteristic
26C3×C7×C22⋊C4C2211yescharacteristic
27C3×C7×C22⋊C4C2211yescharacteristic
28C3×C7×C22⋊C4C2211yescharacteristic
29C3×C42×C7C2211yescharacteristic
30#66636C2211yescharacteristic
31#66636C2211yescharacteristic
32#66631C2211yescharacteristic
33#66631C2211yescharacteristic
34#66631C2211yescharacteristic
35#66631C2211yescharacteristic
36#66631C2211yescharacteristic
37C4⋊Dic21C2211yescharacteristic
38C4⋊Dic21C2211yescharacteristic
39C4⋊Dic21C2211yescharacteristic
40C4⋊Dic21C2211yescharacteristic
41C4⋊Dic21C2211yescharacteristic
42C4⋊Dic21C2211yescharacteristic
43#66629C2211yescharacteristic
44#66629C2211yescharacteristic
45#66629C2211yescharacteristic
46#66629C2211yescharacteristic
47#66629C2211yescharacteristic
48#66629C2211yescharacteristic
49#66629C2211yescharacteristic
50C4×Dic21C2211yescharacteristic
51C4×Dic21C2211yescharacteristic
52C7×#126D311yescharacteristic, complemented
53#6461—31no—
54#6460—31no—
55#6459—31no—
56#6453—31no—
57#6452—31no—
58#6451—31no—
59#6449—31no—
60#6448—31no—
61#6446—31no—
62#6442—31no—
63#6441—31no—
64#6441—31no—
65#6439—31no—
66#6432—31no—
67#5712—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C2×C3×C4×C7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×C3×C4×C7C2311yescharacteristic
73C2×C3×C4×C7C2311yescharacteristic
74C2×C3×C4×C7C2311yescharacteristic
75C2×C3×C4×C7C2311yescharacteristic
76#4268—41no—
77C2×Dic21—41no—
78C2×Dic21C2311yescharacteristic
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81C2×Dic21C2311yescharacteristic
82C2×Dic21C2311yescharacteristic
83C2×Dic21C2311yescharacteristic
84C2×Dic21C2311yescharacteristic
85C4×D21—41no—
86Dic42—41no—
87Dic42—41no—
88Dic42—41no—
89C2×C22⋊D7—31no—
90C22×Dic7—31no—
91C2×C4×D7—31no—
92C2×Dic14—31no—
93C2×Dic14—31no—
94C2×Dic14—31no—
95C7×C4⋊C4C2×D311yescharacteristic
96C7×C4⋊C4C2×D311yescharacteristic
97C7×C4⋊C4C2×D311yescharacteristic
98C7×C22⋊C4C2×D311yescharacteristic
99C7×C22⋊C4C2×D311yescharacteristic
100C7×C22⋊C4C2×D311yescharacteristic
101C42×C7C2×D311yescharacteristic
102C22⋊Dic7—31no—
103C22⋊Dic7—31no—
104(C2×D7)⋊C4—31no—
105(C2×D7)⋊C4—31no—
106(C2×D7)⋊C4—31no—
107(C2×D7)⋊C4—31no—
108(C2×D7)⋊C4—31no—
109C4⋊Dic7—31no—
110C4⋊Dic7—31no—
111C4⋊Dic7—31no—
112C4⋊Dic7—31no—
113C4⋊Dic7—31no—
114C4⋊Dic7—31no—
115Dic7⋊C4—31no—
116Dic7⋊C4—31no—
117Dic7⋊C4—31no—
118Dic7⋊C4—31no—
119Dic7⋊C4—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122C4×Dic7—31no—
123C4×Dic7—31no—
124C3×#126D711yescharacteristic, complemented
125#898—71no—
126#897—71no—
127#896—71no—
128#890—71no—
129#889—71no—
130#888—71no—
131#886—71no—
132#885—71no—
133#883—71no—
134#879—71no—
135#878—71no—
136#878—71no—
137#876—71no—
138#869—71no—
139C22×C3×C7—21no—
140C22×C3×C7—21no—
141C22×C3×C7—21no—
142C22×C3×C7C2411yescharacteristic
143C2×D21—21no—
144C2×D21—21no—
145C2×D21—21no—
146C3×C4×C7—21no—
147C3×C4×C7—21no—
148C3×C4×C7—21no—
149C3×C4×C7—21no—
150C3×C4×C7—21no—
151C3×C4×C7—21no—
152Dic21—21no—
153Dic21—21no—
154Dic21—21no—
155Dic21—21no—
156Dic21—21no—
157Dic21—21no—
158Dic21—21no—
159#563—211noSylow
160C23×C7C22×D311yescharacteristic
161C22×D7—31no—
162C2×C4×C7C22×D311yescharacteristic
163C2×C4×C7C22×D311yescharacteristic
164C2×C4×C7C22×D311yescharacteristic
165C2×C4×C7C22×D311yescharacteristic
166C2×C4×C7C22×D311yescharacteristic
167C2×C4×C7C22×D311yescharacteristic
168C22⋊D7—121no—
169C2×Dic7—31no—
170C2×Dic7—31no—
171C2×Dic7—31no—
172C2×Dic7—31no—
173C2×Dic7—31no—
174C2×Dic7—31no—
175C2×Dic7—31no—
176C2×Dic7—121no—
177C4×D7—121no—
178Dic14—121no—
179Dic14—121no—
180Dic14—121no—
181C2×#67—71no—
182C22×Dic3—71no—
183C2×C4×D3—71no—
184C2×Dic6—71no—
185C2×Dic6—71no—
186C2×Dic6—71no—
187C3×C4⋊C4C2×D711yescharacteristic
188C3×C4⋊C4C2×D711yescharacteristic
189C3×C4⋊C4C2×D711yescharacteristic
190C3×C22⋊C4C2×D711yescharacteristic
191C3×C22⋊C4C2×D711yescharacteristic
192C3×C22⋊C4C2×D711yescharacteristic
193C3×C42C2×D711yescharacteristic
194#217—71no—
195#217—71no—
196(C2×D3)⋊C4—71no—
197(C2×D3)⋊C4—71no—
198(C2×D3)⋊C4—71no—
199(C2×D3)⋊C4—71no—
200(C2×D3)⋊C4—71no—
201C4⋊Dic3—71no—
202C4⋊Dic3—71no—
203C4⋊Dic3—71no—
204C4⋊Dic3—71no—
205C4⋊Dic3—71no—
206C4⋊Dic3—71no—
207Dic3⋊C4—71no—
208Dic3⋊C4—71no—
209Dic3⋊C4—71no—
210Dic3⋊C4—71no—
211Dic3⋊C4—71no—
212Dic3⋊C4—71no—
213Dic3⋊C4—71no—
214C4×Dic3—71no—
215C4×Dic3—71no—
216C2×C3×C7—41no—
217C2×C3×C7H(5,𝔽2)11yescharacteristic
218C2×C3×C72−1+411yescharacteristic
219C2×C3×C72−1+411yescharacteristic
220D21—41no—
221C4⋊Q8—211no—
222C4⋊Q8—211no—
223#126—211no—
224#126—211no—
225#126—211no—
226#126D2111yescharacteristic, complemented
227#125—211no—
228#125—211no—
229#124—211no—
230#123—211no—
231#123—211no—
232C22⋊Q8—211no—
233C22⋊Q8—211no—
234C22⋊Q8—211no—
235C22⋊Q8—211no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7C23×D311yescharacteristic
240C2×D7—61no—
241C2×D7—61no—
242C2×D7—61no—
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249Dic7—61no—
250Dic7—61no—
251Dic7—61no—
252Dic7—61no—
253Dic7—61no—
254Dic7—61no—
255Dic7—61no—
256C23×C3C22×D711yescharacteristic
257C22×D3—71no—
258C2×C3×C4C22×D711yescharacteristic
259C2×C3×C4C22×D711yescharacteristic
260C2×C3×C4C22×D711yescharacteristic
261C2×C3×C4C22×D711yescharacteristic
262C2×C3×C4C22×D711yescharacteristic
263C2×C3×C4C22×D711yescharacteristic
264#67—281no—
265C2×Dic3—71no—
266C2×Dic3—71no—
267C2×Dic3—71no—
268C2×Dic3—71no—
269C2×Dic3—71no—
270C2×Dic3—71no—
271C2×Dic3—71no—
272C2×Dic3—281no—
273C4×D3—281no—
274Dic6—281no—
275Dic6—281no—
276Dic6—281no—
277C3×C7#56311yescharacteristic, complemented, Hall
278C2×Q8—211no—
279C2×Q8—211no—
280C2×Q8—211no—
281C2×D4—211no—
282C22×C4—211no—
283C22×C4—211no—
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⋊C4C2×D2111yescharacteristic
298C4⋊C4C2×D2111yescharacteristic
299C4⋊C4C2×D2111yescharacteristic
300C22⋊C4—211no—
301C22⋊C4—211no—
302C22⋊C4—211no—
303C22⋊C4—211no—
304C22⋊C4—211no—
305C22⋊C4—211no—
306C22⋊C4—211no—
307C22⋊C4C2×D2111yescharacteristic
308C22⋊C4C2×D2111yescharacteristic
309C22⋊C4C2×D2111yescharacteristic
310C42—211no—
311C42—211no—
312C42C2×D2111yescharacteristic
313C2×C7—41no—
314C2×C7#100911yescharacteristic
315C2×C7#101011yescharacteristic
316C2×C7#101011yescharacteristic
317D7—121no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3C23×D711yescharacteristic
322C2×D3—141no—
323C2×D3—141no—
324C2×D3—141no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331Dic3—141no—
332Dic3—141no—
333Dic3—141no—
334Dic3—141no—
335Dic3—141no—
336Dic3—141no—
337Dic3—141no—
338C23—211no—
339C23C22×D2111yescharacteristic
340Q8—841no—
341Q8—841no—
342Q8—841no—
343D4—841no—
344C2×C4—211no—
345C2×C4—211no—
346C2×C4—211no—
347C2×C4—211no—
348C2×C4—211no—
349C2×C4—211no—
350C2×C4—211no—
351C2×C4—841no—
352C2×C4—841no—
353C2×C4C22×D2111yescharacteristic
354C2×C4C22×D2111yescharacteristic
355C2×C4C22×D2111yescharacteristic
356C2×C4C22×D2111yescharacteristic
357C2×C4C22×D2111yescharacteristic
358C2×C4C22×D2111yescharacteristic
359C7#571211yescharacteristic, complemented, Sylow
360C2×C3—41no—
361C2×C3#655311yescharacteristic
362C2×C3#655411yescharacteristic
363C2×C3#655411yescharacteristic
364D3—281no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—421no—
369C22—421no—
370C22—421no—
371C22C23×D2111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—421no—
379C4—421no—
380C4—421no—
381C4—421no—
382C4—421no—
383C4—421no—
384C4—421no—
385C3#9040311yescharacteristic, complemented, Sylow
386C2—41no—
387C2—841no—
388C2#1062166211yescharacteristic
389C2#1062166311yescharacteristic
390C2#1062166311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#126C2
prime radicalC2×#4268C22
hypercenter#126D21
hypocenterC3×C7#563
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C3×C7×#126C2semidirect
2. #10621381C2semidirect
3. #10621380C2semidirect
4. #10621379C2semidirect
5. #10621373C2semidirect
6. #10621372C2semidirect
7. #10621371C2semidirect
11. #10621362C2semidirect
12. #10621361C2semidirect
13. #10621361C2semidirect
14. #10621359C2semidirect
15. #10621352C2semidirect
52. C7×#126D3semidirect
124. C3×#126D7semidirect
226. #126D21semidirect
277. C3×C7#563semidirect
359. C7#5712semidirect
385. C3#90403semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1262
#106213812
#106213802
#106213792
#106213732
#106213722
#106213712
#106213692
#106213682
#106213662
#106213622
#106213612
#106213592
#106213522
#904033
#57127