Finite Group Databasebeta

← #13058116

Subgroups — #13058116

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621361C211yesmaximal, characteristic, complemented
2C7×#67⋊C4C211yesmaximal, characteristic
3C3×#6441C211yesmaximal, characteristic, complemented
4#10621137C211yesmaximal, characteristic, complemented
5#67⋊Dic7C211yesmaximal, characteristic
6#67×Dic7C211yesmaximal, characteristic
7C2×Dic7⋊Dic3C211yesmaximal, characteristic, complemented
8#10620979C211yesmaximal, characteristic, complemented
9#10620972C211yesmaximal, characteristic, complemented
10D3×Dic7⋊C4C211yesmaximal, characteristic, complemented
11#10620941C211yesmaximal, characteristic, complemented
12#10620927C211yesmaximal, characteristic, complemented
13#10620908C211yesmaximal, characteristic, complemented
14Dic3×Dic14C211yesmaximal, characteristic, complemented
15#10620890C211yesmaximal, characteristic, complemented
16#90230—31nomaximal, Hall
17C22×Dic21C2211yescharacteristic
18C2×Dic42C2211yescharacteristic
19C2×C7×#67C2211yescharacteristic
20C22×C7×Dic3C2211yescharacteristic
21C2×C4×D3×C7C2211yescharacteristic
22C22×C3×Dic7C2211yescharacteristic
23C2×C3×Dic14C2211yescharacteristic
24C2×#4248C2211yescharacteristic
25C2×D3×Dic7C2211yescharacteristic
26C2×D3×Dic7C2211yescharacteristic
27C3×C7×C22⋊C4C2211yescharacteristic
28#66636C2211yescharacteristic
29C4⋊Dic21C2211yescharacteristic
30#66629C2211yescharacteristic
31#66629C2211yescharacteristic
32C7×(C2×D3)⋊C4C2211yescharacteristic
33C7×Dic3⋊C4C2211yescharacteristic
34C4×C7×Dic3C2211yescharacteristic
35C3×C22⋊Dic7C2211yescharacteristic
36C3×C4⋊Dic7C2211yescharacteristic
37C3×Dic7⋊C4C2211yescharacteristic
38C3×Dic7⋊C4C2211yescharacteristic
39Dic3⋊Dic7C2211yescharacteristic
40Dic3⋊Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42#66577C2211yescharacteristic
43Dic7⋊Dic3C2211yescharacteristic
44Dic7⋊Dic3C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic7C2211yescharacteristic
49(C2×D3)⋊Dic7C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C4×C7×D4—31no—
53D4×Dic7—31no—
54D4×Dic7—31no—
55#6487—31no—
56#6487—31no—
57C2×Dic7⋊C4—31no—
58C2×Dic7⋊C4—31no—
59#6453—31no—
60#6451—31no—
61#6441—31no—
62#6441—31no—
63#6441—31no—
64#6441D311yescharacteristic, complemented
65#6433—31no—
66C4×Dic14—31no—
67#5635—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D3×C7C2311yescharacteristic
70C2×C3×C4×C7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×Dic21—41no—
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C2×Dic21C2311yescharacteristic
77Dic42—41no—
78C7×#67Q814yescomplemented
79C2×C7×Dic3—41no—
80C2×C7×Dic3C2311yescharacteristic
81C2×C7×Dic3C2311yescharacteristic
82C2×C7×Dic3C2311yescharacteristic
83C4×D3×C7—41no—
84C2×C3×Dic7—22no—
85C2×C3×Dic7C2311yescharacteristic
86C2×C3×Dic7C2311yescharacteristic
87C2×C3×Dic7C2311yescharacteristic
88C2×C3×Dic7C2311yescharacteristic
89C3×Dic14—22no—
90#4248—41no—
91D3×Dic7—22no—
92D3×Dic7—41no—
93C2×C7×D4—31no—
94C22×C4×C7—31no—
95C22×C4×C7—31no—
96C22×Dic7—31no—
97C22×Dic7—31no—
98C22×Dic7—31no—
99C22×Dic7C2×D311yescharacteristic
100C2×Dic14—31no—
101C2×Dic14—31no—
102C2×Dic14C2×D311yescharacteristic
103C7×C4⋊C4—31no—
104C7×C22⋊C4—31no—
105C7×C22⋊C4C2×D311yescharacteristic
106C42×C7—31no—
107C22⋊Dic7—31no—
108C22⋊Dic7—31no—
109C22⋊Dic7—31no—
110C22⋊Dic7C2×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⋊C4—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122Dic7⋊C4—31no—
123Dic7⋊C4—31no—
124Dic7⋊C4C2×D311yescharacteristic
125Dic7⋊C4C2×D311yescharacteristic
126C4×Dic7—31no—
127C4×Dic7—31no—
128C3×C22⋊Q8—71no—
129#946—71no—
130Q8×Dic3—71no—
131C4×#67—71no—
132C2×C4⋊Dic3—71no—
133#897—71no—
134#897—71no—
135D3×C4⋊C4—71no—
136#889—71no—
137#889—71no—
138#888—71no—
139#67⋊C4—71no—
140#67⋊C4D711yescharacteristic
141#878—71no—
142#878—71no—
143C22×C3×C7—21no—
144C22×C3×C7—21no—
145C22×C3×C7C2×Q812yes—
146C22×C3×C7C2411yescharacteristic
147C2×D3×C7—21no—
148C2×D3×C7—21no—
149C2×D3×C7C2×Q812yes—
150C3×C4×C7—21no—
151C3×C4×C7—21no—
152Dic21—21no—
153Dic21—21no—
154Dic21—21no—
155Dic21—21no—
156C3×Dic7—21no—
157C3×Dic7—21no—
158C3×Dic7—21no—
159C3×Dic7#4112yes—
160C7×Dic3—21no—
161C7×Dic3—21no—
162C7×Dic3C2×Q812yes—
163#554—211noSylow
164C23×C7—31no—
165C23×C7C22×D311yescharacteristic
166C7×D4—34no—
167C2×C4×C7—31no—
168C2×C4×C7—31no—
169C2×C4×C7—31no—
170C2×C4×C7—121no—
171C2×C4×C7—121no—
172C2×C4×C7C22×D311yescharacteristic
173C2×C4×C7C22×D311yescharacteristic
174C2×Dic7—22no—
175C2×Dic7—31no—
176C2×Dic7—31no—
177C2×Dic7—31no—
178C2×Dic7—31no—
179C2×Dic7—62no—
180C2×Dic7—121no—
181C2×Dic7—121no—
182C2×Dic7C22×D311yescharacteristic
183C2×Dic7C22×D311yescharacteristic
184C2×Dic7C22×D311yescharacteristic
185C2×Dic7C22×D311yescharacteristic
186Dic14—22no—
187Dic14—121no—
188Dic14—121no—
189C2×C3×Q8—71no—
190C22×C3×C4—71no—
191C2×#67C2×D711yescharacteristic
192C22×Dic3—71no—
193C22×Dic3C2×D711yescharacteristic
194C2×C4×D3—71no—
195C2×C4×D3—71no—
196C2×C4×D3C2×D711yescharacteristic
197C2×Dic6—71no—
198C2×Dic6—71no—
199C3×C4⋊C4—71no—
200C3×C4⋊C4—71no—
201C3×C4⋊C4—71no—
202C3×C22⋊C4—71no—
203C3×C22⋊C4C2×D711yescharacteristic
204#217—71no—
205(C2×D3)⋊C4—71no—
206(C2×D3)⋊C4—71no—
207(C2×D3)⋊C4C2×D711yescharacteristic
208C4⋊Dic3—71no—
209C4⋊Dic3—71no—
210C4⋊Dic3—71no—
211C4⋊Dic3—71no—
212C4⋊Dic3—71no—
213C4⋊Dic3—71no—
214Dic3⋊C4—71no—
215Dic3⋊C4—71no—
216Dic3⋊C4—71no—
217Dic3⋊C4—71no—
218Dic3⋊C4—71no—
219Dic3⋊C4—71no—
220Dic3⋊C4C2×D711yescharacteristic
221C4×Dic3—71no—
222C4×Dic3—71no—
223C4×Dic3C2×D711yescharacteristic
224C2×C3×C7—22no—
225C2×C3×C7C22×Q811yescharacteristic
226C2×C3×C7C2×#4111yescharacteristic
227C2×C3×C7H(5,𝔽2)11yescharacteristic
228D3×C7—22no—
229C4⋊Q8—211no—
230#125—211no—
231#125—211no—
232C22⋊Q8—71no—
233C22⋊Q8—211no—
234C22⋊Q8—211no—
235C22⋊Q8—211no—
236C22⋊Q8—211no—
237C22⋊Q8—211no—
238C4×Q8—211no—
239C4×D4—31no—
240C4×D4—211no—
241C4×D4—211no—
242C2×C4⋊C4—211no—
243C2×C4⋊C4—211no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—32no—
247C22×C7—61no—
248C22×C7—61no—
249C22×C7D3×Q812yes—
250C22×C7C23×D311yescharacteristic
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—32no—
254C4×C7—61no—
255C4×C7—61no—
256Dic7—21no—
257Dic7—21no—
258Dic7—21no—
259Dic7—61no—
260Dic7—61no—
261Dic7—61no—
262Dic7—61no—
263Dic7#23712yes—
264C23×C3C22×D711yescharacteristic
265C22×D3C22×D711yescharacteristic
266C3×Q8—142no—
267C2×C3×C4—71no—
268C2×C3×C4—71no—
269C2×C3×C4—71no—
270C2×C3×C4—71no—
271C2×C3×C4—142no—
272C2×C3×C4C22×D711yescharacteristic
273C2×C3×C4C22×D711yescharacteristic
274#67Dic1414yescomplemented
275C2×Dic3—41no—
276C2×Dic3—71no—
277C2×Dic3—71no—
278C2×Dic3—71no—
279C2×Dic3—71no—
280C2×Dic3—281no—
281C2×Dic3C22×D711yescharacteristic
282C2×Dic3C22×D711yescharacteristic
283C2×Dic3C22×D711yescharacteristic
284C4×D3—41no—
285C4×D3—142no—
286C4×D3—281no—
287Dic6—281no—
288Dic6—281no—
289C3×C7#55411yescharacteristic, complemented, Hall
290C2×Q8—71no—
291C2×Q8—211no—
292C2×Q8—211no—
293C2×D4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—71no—
297C22×C4—211no—
298C22×C4—211no—
299C22×C4—211no—
300C4⋊C4—31no—
301C4⋊C4—71no—
302C4⋊C4—71no—
303C4⋊C4—71no—
304C4⋊C4—211no—
305C4⋊C4—211no—
306C4⋊C4—211no—
307C4⋊C4—211no—
308C4⋊C4—211no—
309C4⋊C4—211no—
310C4⋊C4—211no—
311C4⋊C4—211no—
312C4⋊C4—211no—
313C4⋊C4—211no—
314C4⋊C4—211no—
315C4⋊C4—211no—
316C22⋊C4—31no—
317C22⋊C4—71no—
318C22⋊C4—211no—
319C22⋊C4—211no—
320C22⋊C4—211no—
321C22⋊C4D3×D711yescharacteristic
322C42—31no—
323C42—211no—
324C42—211no—
325C2×C7—22no—
326C2×C7—62no—
327C2×C7C2×#23711yescharacteristic
328C2×C7C2×D3×Q811yescharacteristic
329C2×C7#100911yescharacteristic
330C22×C3—21no—
331C22×C3—21no—
332C22×C3C2×Dic1412yes—
333C22×C3C23×D711yescharacteristic
334C2×D3—21no—
335C2×D3—21no—
336C2×D3C2×Dic1412yes—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—72no—
340C3×C4—141no—
341C3×C4—141no—
342C3×C4—141no—
343Dic3—21no—
344Dic3—21no—
345Dic3—141no—
346Dic3—141no—
347Dic3—141no—
348Dic3—141no—
349Dic3C2×Dic1412yes—
350C23—31no—
351C23C2×D3×D711yescharacteristic
352Q8—142no—
353Q8—841no—
354Q8—841no—
355D4—34no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—71no—
360C2×C4—71no—
361C2×C4—71no—
362C2×C4—71no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—142no—
366C2×C4—211no—
367C2×C4—211no—
368C2×C4—211no—
369C2×C4—211no—
370C2×C4—422no—
371C2×C4—841no—
372C2×C4—841no—
373C2×C4C2×D3×D711yescharacteristic
374C2×C4C2×D3×D711yescharacteristic
375C7#563511yescharacteristic, complemented, Sylow
376C2×C3—22no—
377C2×C3C22×Dic1411yescharacteristic
378C2×C3#654811yescharacteristic
379C2×C3D7×#4111yescharacteristic
380D3—22no—
381C22—21no—
382C22—21no—
383C22—32no—
384C22—61no—
385C22—61no—
386C22D3×Dic1412yes—
387C22C22×D3×D711yescharacteristic
388C4—21no—
389C4—21no—
390C4—32no—
391C4—61no—
392C4—61no—
393C4—72no—
394C4—141no—
395C4—141no—
396C4—141no—
397C4—421no—
398C4—421no—
399C4—421no—
400C4—421no—
401C3#9023011yescharacteristic, complemented, Sylow
402C2—22no—
403C2—62no—
404C2C2×D3×Dic1411yescharacteristic
405C2D7×#23711yescharacteristic
406C2#1062157411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radicalC2×C7×#67C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#554
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. #10621361C2semidirect
3. C3×#6441C2semidirect
4. #10621137C2semidirect
7. C2×Dic7⋊Dic3C2semidirect
8. #10620979C2semidirect
9. #10620972C2semidirect
10. D3×Dic7⋊C4C2semidirect
11. #10620941C2semidirect
12. #10620927C2semidirect
13. #10620908C2semidirect
14. Dic3×Dic14C2semidirect
15. #10620890C2semidirect
64. #6441D3semidirect
78. C7×#67Q8semidirect
274. #67Dic14semidirect
289. C3×C7#554semidirect
375. C7#5635semidirect
401. C3#90230semidirect

Maximal subgroups

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

subgroupindex
#106213612
C7×#67⋊C42
C3×#64412
#106211372
#67⋊Dic72
#67×Dic72
C2×Dic7⋊Dic32
#106209792
#106209722
D3×Dic7⋊C42
#106209412
#106209272
#106209082
Dic3×Dic142
#106208902
#902303
#56357