Finite Group Databasebeta

← #13058689

Subgroups — #13058689

Order 1344.

424 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
1#10621381C211yesmaximal, characteristic, complemented
2C7×#887C211yesmaximal, characteristic, complemented
3C3×#6461C211yesmaximal, characteristic, complemented
4#10621031C211yesmaximal, characteristic, complemented
5C4×#4248C211yesmaximal, characteristic, complemented
6#10620986C211yesmaximal, characteristic, complemented
7#10620983C211yesmaximal, characteristic, complemented
8#10620964C211yesmaximal, characteristic, complemented
9#10620957C211yesmaximal, characteristic
10#10620956C211yesmaximal, characteristic, complemented
11#10620924C211yesmaximal, characteristic, complemented
12#10620915C211yesmaximal, characteristic
13#10620915C211yesmaximal, characteristic
14#10620907C211yesmaximal, characteristic, complemented
15#10620886C211yesmaximal, characteristic, complemented
16#90275—31nomaximal, Hall
17C2×C7×Dic6C2211yescharacteristic
18C2×#4248C2211yescharacteristic
19C2×#4248C2211yescharacteristic
20C2×#4246C2211yescharacteristic
21C2×D21⋊C4C2211yescharacteristic
22C2×Dic3×D7C2211yescharacteristic
23C3×C7×C4⋊C4C2211yescharacteristic
24#66631C2211yescharacteristic
25#66631C2211yescharacteristic
26#66631C2211yescharacteristic
27C4⋊Dic21C2211yescharacteristic
28#66629C2211yescharacteristic
29C4×Dic21C2211yescharacteristic
30C7×Dic3⋊C4C2211yescharacteristic
31C7×Dic3⋊C4C2211yescharacteristic
32C4×C7×Dic3C2211yescharacteristic
33C4×C7×Dic3C2211yescharacteristic
34C4×C7×Dic3C2211yescharacteristic
35C3×(C2×D7)⋊C4C2211yescharacteristic
36C3×(C2×D7)⋊C4C2211yescharacteristic
37C3×(C2×D7)⋊C4C2211yescharacteristic
38C3×C4⋊Dic7C2211yescharacteristic
39C3×Dic7⋊C4C2211yescharacteristic
40C3×C4×Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42Dic3⋊Dic7C2211yescharacteristic
43Dic3⋊Dic7C2211yescharacteristic
44#66577C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46#66575C2211yescharacteristic
47#66575C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C4×C7×Q8—31no—
53#6509—31no—
54#6509—31no—
55#6461—31no—
56#6461—31no—
57#6461—31no—
58#6461D311yescharacteristic, complemented
59#6455—31no—
60#6455—31no—
61#6438—31no—
62#6438—31no—
63C4×D28—31no—
64#6432—31no—
65C4×Dic14—31no—
66C4×Dic14—31no—
67#5702—71nomaximal, Hall
68C22×D21C2311yescharacteristic
69C22×C3×D7C2311yescharacteristic
70C2×C3×C4×C7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×C3×C4×C7C2311yescharacteristic
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C2×C7×Dic3C2311yescharacteristic
77C2×C7×Dic3C2311yescharacteristic
78C2×C7×Dic3C2311yescharacteristic
79C2×C7×Dic3C2311yescharacteristic
80C7×Dic6—22no—
81C2×C3×Dic7C2311yescharacteristic
82C2×C3×Dic7C2311yescharacteristic
83C2×C3×Dic7C2311yescharacteristic
84#4248—22no—
85#4248—22no—
86#4246—22no—
87D21⋊C4—22no—
88Dic3×D7—22no—
89C2×C7×Q8—31no—
90C2×D28—31no—
91C2×C4×D7—31no—
92C2×C4×D7—31no—
93C2×Dic14—31no—
94C2×Dic14—31no—
95C7×C4⋊C4—31no—
96C7×C4⋊C4—31no—
97C7×C4⋊C4C2×D311yescharacteristic
98C42×C7—31no—
99C42×C7—31no—
100C42×C7—31no—
101(C2×D7)⋊C4—31no—
102(C2×D7)⋊C4—31no—
103(C2×D7)⋊C4—31no—
104(C2×D7)⋊C4—31no—
105(C2×D7)⋊C4—31no—
106(C2×D7)⋊C4—31no—
107(C2×D7)⋊C4—31no—
108(C2×D7)⋊C4C2×D311yescharacteristic
109(C2×D7)⋊C4C2×D311yescharacteristic
110(C2×D7)⋊C4C2×D311yescharacteristic
111C4⋊Dic7—31no—
112C4⋊Dic7—31no—
113C4⋊Dic7—31no—
114C4⋊Dic7—31no—
115C4⋊Dic7C2×D311yescharacteristic
116Dic7⋊C4—31no—
117Dic7⋊C4—31no—
118Dic7⋊C4—31no—
119Dic7⋊C4C2×D311yescharacteristic
120C4×Dic7—31no—
121C4×Dic7—31no—
122C4×Dic7—31no—
123C4×Dic7C2×D311yescharacteristic
124C3×#126—71no—
125#898—71no—
126#898—71no—
127#896—71no—
128#888—71no—
129#887—71no—
130#887D711yescharacteristic, complemented
131#885—71no—
132#885—71no—
133#67⋊C4—71no—
134#879—71no—
135#878—71no—
136#877—71no—
137#872—71no—
138C4×Dic6—71no—
139C22×C3×C7C2411yescharacteristic
140C2×D21—21no—
141C2×D21—21no—
142C2×D21—21no—
143C2×C3×D7—21no—
144C2×C3×D7—21no—
145C2×C3×D7—21no—
146C3×C4×C7—21no—
147C3×C4×C7—21no—
148C3×C4×C7—21no—
149Dic21—21no—
150Dic21—21no—
151Dic21—21no—
152C3×Dic7—21no—
153C3×Dic7—21no—
154C3×Dic7—21no—
155C7×Dic3—21no—
156C7×Dic3—21no—
157C7×Dic3#4112yes—
158C7×Dic3#4112yes—
159#556—211noSylow
160C22×D7—31no—
161C22×D7C22×D311yescharacteristic
162C7×Q8—62no—
163C2×C4×C7—31no—
164C2×C4×C7—31no—
165C2×C4×C7—31no—
166C2×C4×C7—31no—
167C2×C4×C7C22×D311yescharacteristic
168C2×C4×C7C22×D311yescharacteristic
169C2×C4×C7C22×D311yescharacteristic
170C2×Dic7—31no—
171C2×Dic7—31no—
172C2×Dic7—31no—
173C2×Dic7C22×D311yescharacteristic
174C2×Dic7C22×D311yescharacteristic
175C2×Dic7C22×D311yescharacteristic
176D28—62no—
177C4×D7—62no—
178C4×D7—62no—
179Dic14—62no—
180Dic14—62no—
181C2×#67—71no—
182C22×Dic3—71no—
183C2×C4×D3—71no—
184C2×Dic6—71no—
185C2×Dic6—71no—
186C2×Dic6C2×D711yescharacteristic
187C3×C4⋊C4—71no—
188C3×C4⋊C4—71no—
189C3×C4⋊C4C2×D711yescharacteristic
190C3×C22⋊C4—71no—
191C3×C22⋊C4—71no—
192C3×C22⋊C4—71no—
193C3×C42—71no—
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—
203Dic3⋊C4—71no—
204Dic3⋊C4—71no—
205Dic3⋊C4—71no—
206Dic3⋊C4—71no—
207Dic3⋊C4—71no—
208Dic3⋊C4C2×D711yescharacteristic
209Dic3⋊C4C2×D711yescharacteristic
210C4×Dic3—71no—
211C4×Dic3—71no—
212C4×Dic3—71no—
213C4×Dic3C2×D711yescharacteristic
214C4×Dic3C2×D711yescharacteristic
215C4×Dic3C2×D711yescharacteristic
216C2×C3×C7C2×#4111yescharacteristic
217C2×C3×C7C2×#4111yescharacteristic
218C2×C3×C72−1+411yescharacteristic
219D21—41no—
220C3×D7—41no—
221C4⋊Q8—211no—
222#126—71no—
223#126—211no—
224#126—211no—
225#126—211no—
226#124—211no—
227#124—211no—
228C22⋊Q8—211no—
229C22⋊Q8—211no—
230C4×Q8—31no—
231C4×Q8—211no—
232C4×Q8—211no—
233C4×D4—211no—
234#117—211no—
235#117—211no—
236C22×C7C23×D311yescharacteristic
237C2×D7—21no—
238C2×D7—21no—
239C2×D7—21no—
240C2×D7—61no—
241C2×D7—61no—
242C2×D7—61no—
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—32no—
247C4×C7—32no—
248C4×C7—61no—
249C4×C7—61no—
250Dic7—21no—
251Dic7—21no—
252Dic7—21no—
253Dic7—61no—
254Dic7—61no—
255Dic7—61no—
256C23×C3—71no—
257C22×D3—71no—
258C2×C3×C4—71no—
259C2×C3×C4—71no—
260C2×C3×C4—71no—
261C2×C3×C4C22×D711yescharacteristic
262C2×C3×C4C22×D711yescharacteristic
263C2×C3×C4C22×D711yescharacteristic
264#67—142no—
265C2×Dic3—71no—
266C2×Dic3—71no—
267C2×Dic3—71no—
268C2×Dic3—142no—
269C2×Dic3C22×D711yescharacteristic
270C2×Dic3C22×D711yescharacteristic
271C2×Dic3C22×D711yescharacteristic
272C2×Dic3C22×D711yescharacteristic
273C4×D3—142no—
274Dic6—22no—
275Dic6—142no—
276Dic6—142no—
277C3×C7#55611yescharacteristic, complemented, Hall
278C2×Q8—31no—
279C2×Q8—211no—
280C2×Q8—211no—
281C2×D4—211no—
282C22×C4—211no—
283C22×C4—211no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—71no—
287C4⋊C4—71no—
288C4⋊C4—211no—
289C4⋊C4—211no—
290C4⋊C4—211no—
291C4⋊C4—211no—
292C4⋊C4—211no—
293C4⋊C4—211no—
294C4⋊C4—211no—
295C4⋊C4D3×D711yescharacteristic
296C22⋊C4—71no—
297C22⋊C4—71no—
298C22⋊C4—71no—
299C22⋊C4—211no—
300C22⋊C4—211no—
301C22⋊C4—211no—
302C22⋊C4—211no—
303C22⋊C4—211no—
304C22⋊C4—211no—
305C22⋊C4—211no—
306C42—31no—
307C42—31no—
308C42—31no—
309C42—71no—
310C42—211no—
311C42—211no—
312C42—211no—
313C2×C7D3×#4111yescharacteristic
314C2×C7D3×#4111yescharacteristic
315C2×C7#101011yescharacteristic
316D7—41no—
317D7—121no—
318C22×C3—141no—
319C22×C3—141no—
320C22×C3—141no—
321C22×C3C23×D711yescharacteristic
322C2×D3—141no—
323C2×D3—141no—
324C2×D3—141no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—141no—
329C3×C4—141no—
330C3×C4—141no—
331Dic3—21no—
332Dic3—21no—
333Dic3—141no—
334Dic3—141no—
335Dic3—141no—
336Dic3#115712yes—
337Dic3#116112yes—
338C23—71no—
339C23—211no—
340Q8—62no—
341Q8—422no—
342Q8—422no—
343D4—422no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—71no—
349C2×C4—71no—
350C2×C4—71no—
351C2×C4—211no—
352C2×C4—211no—
353C2×C4—211no—
354C2×C4—422no—
355C2×C4—422no—
356C2×C4C2×D3×D711yescharacteristic
357C2×C4C2×D3×D711yescharacteristic
358C2×C4C2×D3×D711yescharacteristic
359C7#570211yescharacteristic, complemented, Sylow
360C2×C3—281no—
361C2×C3C2×#115711yescharacteristic
362C2×C3C2×#116111yescharacteristic
363C2×C3#655411yescharacteristic
364D3—281no—
365C22—141no—
366C22—141no—
367C22—141no—
368C22—421no—
369C22—421no—
370C22—421no—
371C22C22×D3×D711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—32no—
376C4—32no—
377C4—61no—
378C4—61no—
379C4—141no—
380C4—141no—
381C4—141no—
382C4—421no—
383C4—421no—
384C4—421no—
385C3#9027511yescharacteristic, complemented, Sylow
386C2—281no—
387C2—841no—
388C2D3×#115711yescharacteristic
389C2#1062156711yescharacteristic
390C2D3×#116111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC2×#4246C22
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#556
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621381C2semidirect
2. C7×#887C2semidirect
3. C3×#6461C2semidirect
4. #10621031C2semidirect
5. C4×#4248C2semidirect
6. #10620986C2semidirect
7. #10620983C2semidirect
8. #10620964C2semidirect
10. #10620956C2semidirect
11. #10620924C2semidirect
14. #10620907C2semidirect
15. #10620886C2semidirect
58. #6461D3semidirect
130. #887D7semidirect
277. C3×C7#556semidirect
359. C7#5702semidirect
385. C3#90275semidirect

Maximal subgroups

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

subgroupindex
#106213812
C7×#8872
C3×#64612
#106210312
C4×#42482
#106209862
#106209832
#106209642
#106209572
#106209562
#106209242
#106209152
#106209072
#106208862
#902753
#57027