Finite Group Databasebeta

← #13058121

Subgroups — #13058121

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#10621362C211yesmaximal, characteristic, complemented
2C7×#885C211yesmaximal, characteristic
3C3×#6441C211yesmaximal, characteristic, complemented
4#67×Dic7C211yesmaximal, characteristic
5#10621093C211yesmaximal, characteristic, complemented
6#10621090C211yesmaximal, characteristic, complemented
7#10621087C211yesmaximal, characteristic
8#10620972C211yesmaximal, characteristic, complemented
9#10620962C211yesmaximal, characteristic, complemented
10#10620906C211yesmaximal, characteristic, complemented
11#10620904C211yesmaximal, characteristic, complemented
12#10620903C211yesmaximal, characteristic, complemented
13Dic3×Dic14C211yesmaximal, characteristic, complemented
14#10620891C211yesmaximal, characteristic, complemented
15#10620886C211yesmaximal, characteristic, complemented
16#90362—31nomaximal, Hall
17C2×C7×#67C2211yescharacteristic
18C2×C7×Dic6C2211yescharacteristic
19C22×C3×Dic7C2211yescharacteristic
20C2×C3×Dic14C2211yescharacteristic
21C2×#4248C2211yescharacteristic
22C2×D3×Dic7C2211yescharacteristic
23C3×C7×C22⋊C4C2211yescharacteristic
24#66636C2211yescharacteristic
25#66636C2211yescharacteristic
26C4⋊Dic21C2211yescharacteristic
27#66629C2211yescharacteristic
28#66629C2211yescharacteristic
29C4×Dic21C2211yescharacteristic
30C7×#217C2211yescharacteristic
31C7×(C2×D3)⋊C4C2211yescharacteristic
32C7×(C2×D3)⋊C4C2211yescharacteristic
33C4×C7×Dic3C2211yescharacteristic
34C3×C22⋊Dic7C2211yescharacteristic
35C3×C4⋊Dic7C2211yescharacteristic
36C3×Dic7⋊C4C2211yescharacteristic
37C3×Dic7⋊C4C2211yescharacteristic
38Dic3⋊Dic7C2211yescharacteristic
39Dic3⋊Dic7C2211yescharacteristic
40#66577C2211yescharacteristic
41Dic7⋊Dic3C2211yescharacteristic
42Dic7⋊Dic3C2211yescharacteristic
43Dic7⋊Dic3C2211yescharacteristic
44(C2×D3)⋊Dic7C2211yescharacteristic
45(C2×D3)⋊Dic7C2211yescharacteristic
46(C2×D3)⋊Dic7C2211yescharacteristic
47Dic3×Dic7C2211yescharacteristic
48Dic3×Dic7C2211yescharacteristic
49Dic3×Dic7C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#124—31no—
53Q8×Dic7—31no—
54#6507—31no—
55#6499—31no—
56D4×Dic7—31no—
57#6442—31no—
58#6442—31no—
59#6442—31no—
60#6442—31no—
61#6441—31no—
62#6441D311yescharacteristic, complemented
63#6440—31no—
64#6440—31no—
65C4×Dic14—31no—
66C4×Dic14—31no—
67#5641—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D3×C7C2311yescharacteristic
70C2×C3×C4×C7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×Dic21C2311yescharacteristic
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C7×#67—22no—
77C2×C7×Dic3C2311yescharacteristic
78C2×C7×Dic3C2311yescharacteristic
79C2×C7×Dic3C2311yescharacteristic
80C7×Dic6—22no—
81C2×C3×Dic7—22no—
82C2×C3×Dic7C2311yescharacteristic
83C2×C3×Dic7C2311yescharacteristic
84C2×C3×Dic7C2311yescharacteristic
85C2×C3×Dic7C2311yescharacteristic
86C3×Dic14—22no—
87#4248—22no—
88D3×Dic7—22no—
89C2×C7×Q8—31no—
90C2×C7×D4—31no—
91C22×Dic7—31no—
92C22×Dic7C2×D311yescharacteristic
93C2×Dic14—31no—
94C2×Dic14C2×D311yescharacteristic
95C7×C22⋊C4—31no—
96C7×C22⋊C4—31no—
97C7×C22⋊C4—31no—
98C7×C22⋊C4C2×D311yescharacteristic
99C42×C7—31no—
100C22⋊Dic7—31no—
101C22⋊Dic7—31no—
102C22⋊Dic7—31no—
103C22⋊Dic7—31no—
104C22⋊Dic7—31no—
105C22⋊Dic7C2×D311yescharacteristic
106C4⋊Dic7—31no—
107C4⋊Dic7—31no—
108C4⋊Dic7—31no—
109C4⋊Dic7C2×D311yescharacteristic
110Dic7⋊C4—31no—
111Dic7⋊C4—31no—
112Dic7⋊C4—31no—
113Dic7⋊C4—31no—
114Dic7⋊C4—31no—
115Dic7⋊C4—31no—
116Dic7⋊C4C2×D311yescharacteristic
117Dic7⋊C4C2×D311yescharacteristic
118C4×Dic7—31no—
119C4×Dic7—31no—
120C4×Dic7—31no—
121C4×Dic7—31no—
122C4×Dic7—31no—
123C4×Dic7—31no—
124C3×C22⋊Q8—71no—
125#946—71no—
126Q8×Dic3—71no—
127C4×#67—71no—
128#926—71no—
129#898—71no—
130#898—71no—
131#892—71no—
132#888—71no—
133#887—71no—
134#887—71no—
135#885—71no—
136#885D711yescharacteristic
137#879—71no—
138#879—71no—
139C22×C3×C7—21no—
140C22×C3×C7—21no—
141C22×C3×C7—21no—
142C22×C3×C7C2411yescharacteristic
143C2×D3×C7—21no—
144C2×D3×C7—21no—
145C2×D3×C7—21no—
146C3×C4×C7—21no—
147C3×C4×C7—21no—
148Dic21—21no—
149Dic21—21no—
150Dic21—21no—
151Dic21—21no—
152C3×Dic7—21no—
153C3×Dic7—21no—
154C3×Dic7—21no—
155C3×Dic7#4112yes—
156C7×Dic3—21no—
157C7×Dic3—21no—
158C7×Dic3#4112yes—
159#556—211noSylow
160C23×C7—31no—
161C23×C7C22×D311yescharacteristic
162C7×Q8—62no—
163C7×D4—62no—
164C2×C4×C7—31no—
165C2×C4×C7—31no—
166C2×C4×C7—31no—
167C2×C4×C7C22×D311yescharacteristic
168C2×C4×C7C22×D311yescharacteristic
169C2×Dic7—22no—
170C2×Dic7—31no—
171C2×Dic7—31no—
172C2×Dic7—31no—
173C2×Dic7—31no—
174C2×Dic7—62no—
175C2×Dic7C22×D311yescharacteristic
176C2×Dic7C22×D311yescharacteristic
177C2×Dic7C22×D311yescharacteristic
178C2×Dic7C22×D311yescharacteristic
179Dic14—22no—
180Dic14—62no—
181C2×C3×Q8—71no—
182C22×C3×C4—71no—
183C2×#67C2×D711yescharacteristic
184C2×C4×D3—71no—
185C2×Dic6—71no—
186C2×Dic6C2×D711yescharacteristic
187C3×C4⋊C4—71no—
188C3×C4⋊C4—71no—
189C3×C4⋊C4—71no—
190C3×C22⋊C4—71no—
191C3×C22⋊C4C2×D711yescharacteristic
192#217—71no—
193#217—71no—
194#217C2×D711yescharacteristic
195(C2×D3)⋊C4—71no—
196(C2×D3)⋊C4—71no—
197(C2×D3)⋊C4—71no—
198(C2×D3)⋊C4C2×D711yescharacteristic
199(C2×D3)⋊C4C2×D711yescharacteristic
200C4⋊Dic3—71no—
201C4⋊Dic3—71no—
202C4⋊Dic3—71no—
203C4⋊Dic3—71no—
204Dic3⋊C4—71no—
205Dic3⋊C4—71no—
206Dic3⋊C4—71no—
207Dic3⋊C4—71no—
208Dic3⋊C4—71no—
209C4×Dic3—71no—
210C4×Dic3—71no—
211C4×Dic3—71no—
212C4×Dic3—71no—
213C4×Dic3—71no—
214C4×Dic3—71no—
215C4×Dic3C2×D711yescharacteristic
216C2×C3×C7—41no—
217C2×C3×C7C2×#4111yescharacteristic
218C2×C3×C7C2×#4111yescharacteristic
219C2×C3×C72−1+411yescharacteristic
220D3×C7—41no—
221C4⋊Q8—211no—
222#126—211no—
223#126—211no—
224#126—211no—
225#126—211no—
226#124—31no—
227#124—211no—
228C22⋊Q8—71no—
229C22⋊Q8—211no—
230C4×Q8—211no—
231C4×Q8—211no—
232C4×Q8—211no—
233C4×D4—211no—
234#117—211no—
235#117—211no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—61no—
240C22×C7—61no—
241C22×C7—61no—
242C22×C7C23×D311yescharacteristic
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—32no—
246C4×C7—61no—
247C4×C7—61no—
248Dic7—21no—
249Dic7—21no—
250Dic7—21no—
251Dic7—61no—
252Dic7—61no—
253Dic7—61no—
254Dic7—61no—
255Dic7#23712yes—
256C23×C3C22×D711yescharacteristic
257C22×D3C22×D711yescharacteristic
258C3×Q8—142no—
259C2×C3×C4—71no—
260C2×C3×C4—71no—
261C2×C3×C4—71no—
262C2×C3×C4—71no—
263C2×C3×C4—142no—
264C2×C3×C4C22×D711yescharacteristic
265C2×C3×C4C22×D711yescharacteristic
266#67—22no—
267C2×Dic3—71no—
268C2×Dic3—71no—
269C2×Dic3—71no—
270C2×Dic3—71no—
271C2×Dic3C22×D711yescharacteristic
272C2×Dic3C22×D711yescharacteristic
273C2×Dic3C22×D711yescharacteristic
274C4×D3—142no—
275Dic6—22no—
276Dic6—142no—
277C3×C7#55611yescharacteristic, complemented, Hall
278C2×Q8—31no—
279C2×Q8—71no—
280C2×Q8—211no—
281C2×D4—31no—
282C22×C4—71no—
283C22×C4—211no—
284C4⋊C4—71no—
285C4⋊C4—71no—
286C4⋊C4—71no—
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—
296C22⋊C4—31no—
297C22⋊C4—31no—
298C22⋊C4—31no—
299C22⋊C4—71no—
300C22⋊C4—211no—
301C22⋊C4—211no—
302C22⋊C4—211no—
303C22⋊C4—211no—
304C22⋊C4—211no—
305C22⋊C4D3×D711yescharacteristic
306C42—31no—
307C42—211no—
308C42—211no—
309C42—211no—
310C42—211no—
311C42—211no—
312C42—211no—
313C2×C7—41no—
314C2×C7—121no—
315C2×C7C2×#23711yescharacteristic
316C2×C7#100711yescharacteristic
317C2×C7D3×#4111yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3C23×D711yescharacteristic
322C2×D3—21no—
323C2×D3—21no—
324C2×D3—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—72no—
328C3×C4—141no—
329C3×C4—141no—
330C3×C4—141no—
331Dic3—21no—
332Dic3—21no—
333Dic3—141no—
334Dic3—141no—
335Dic3—141no—
336Dic3—141no—
337Dic3#115912yes—
338C23—31no—
339C23C2×D3×D711yescharacteristic
340Q8—62no—
341Q8—142no—
342Q8—422no—
343D4—62no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—71no—
348C2×C4—71no—
349C2×C4—71no—
350C2×C4—71no—
351C2×C4—142no—
352C2×C4—211no—
353C2×C4—211no—
354C2×C4—211no—
355C2×C4—211no—
356C2×C4—422no—
357C2×C4C2×D3×D711yescharacteristic
358C2×C4C2×D3×D711yescharacteristic
359C7#564111yescharacteristic, complemented, Sylow
360C2×C3—41no—
361C2×C3C2×#115911yescharacteristic
362C2×C3D7×#4111yescharacteristic
363C2×C3#655411yescharacteristic
364D3—41no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—61no—
369C22—61no—
370C22—61no—
371C22C22×D3×D711yescharacteristic
372C4—21no—
373C4—21no—
374C4—32no—
375C4—61no—
376C4—61no—
377C4—72no—
378C4—141no—
379C4—141no—
380C4—141no—
381C4—421no—
382C4—421no—
383C4—421no—
384C4—421no—
385C3#9036211yescharacteristic, complemented, Sylow
386C2—41no—
387C2—121no—
388C2#1062155111yescharacteristic
389C2D7×#23711yescharacteristic
390C2D3×#115911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radicalC2×C7×#67C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#556
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. #10621362C2semidirect
3. C3×#6441C2semidirect
5. #10621093C2semidirect
6. #10621090C2semidirect
8. #10620972C2semidirect
9. #10620962C2semidirect
10. #10620906C2semidirect
11. #10620904C2semidirect
12. #10620903C2semidirect
13. Dic3×Dic14C2semidirect
14. #10620891C2semidirect
15. #10620886C2semidirect
62. #6441D3semidirect
277. C3×C7#556semidirect
359. C7#5641semidirect
385. C3#90362semidirect

Maximal subgroups

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

subgroupindex
#106213622
C7×#8852
C3×#64412
#67×Dic72
#106210932
#106210902
#106210872
#106209722
#106209622
#106209062
#106209042
#106209032
Dic3×Dic142
#106208912
#106208862
#903623
#56417