Finite Group Databasebeta

← #13058130

Subgroups — #13058130

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#10621364C211yesmaximal, characteristic
2C7×#878C211yesmaximal, characteristic, complemented
3C3×#6441C211yesmaximal, characteristic, complemented
4#10621138C211yesmaximal, characteristic, complemented
5#10621118C211yesmaximal, characteristic
6#10621114C211yesmaximal, characteristic
7C2×Dic7⋊Dic3C211yesmaximal, characteristic, complemented
8#10620981C211yesmaximal, characteristic, complemented
9#10620973C211yesmaximal, characteristic, complemented
10#10620966C211yesmaximal, characteristic, complemented
11#10620945C211yesmaximal, characteristic, complemented
12#10620944C211yesmaximal, characteristic, complemented
13#10620901C211yesmaximal, characteristic, complemented
14#10620893C211yesmaximal, characteristic, complemented
15#10620891C211yesmaximal, characteristic, complemented
16#90327—31nomaximal, Hall
17C2×#4268C2211yescharacteristic
18C22×Dic21C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C22×C7×Dic3C2211yescharacteristic
21C2×C7×Dic6C2211yescharacteristic
22C22×C3×Dic7C2211yescharacteristic
23C2×C3×Dic14C2211yescharacteristic
24C2×#4248C2211yescharacteristic
25C2×D21⋊C4C2211yescharacteristic
26C2×D21⋊C4C2211yescharacteristic
27C3×C7×C22⋊C4C2211yescharacteristic
28#66631C2211yescharacteristic
29#66629C2211yescharacteristic
30C4×Dic21C2211yescharacteristic
31C7×#217C2211yescharacteristic
32C7×C4⋊Dic3C2211yescharacteristic
33C7×Dic3⋊C4C2211yescharacteristic
34C7×Dic3⋊C4C2211yescharacteristic
35C3×C22⋊Dic7C2211yescharacteristic
36C3×C4⋊Dic7C2211yescharacteristic
37C3×Dic7⋊C4C2211yescharacteristic
38C3×Dic7⋊C4C2211yescharacteristic
39Dic3⋊Dic7C2211yescharacteristic
40#66577C2211yescharacteristic
41#66577C2211yescharacteristic
42#66577C2211yescharacteristic
43Dic7⋊Dic3C2211yescharacteristic
44Dic7⋊Dic3C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48#66575C2211yescharacteristic
49#66575C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×C22⋊Q8—31no—
53#6509—31no—
54#6507—31no—
55C4×C22⋊D7—31no—
56C2×Dic7⋊C4—31no—
57#6459—31no—
58#6459—31no—
59D7×C4⋊C4—31no—
60#6452—31no—
61#6452—31no—
62#6450—31no—
63(C22⋊D7)⋊C4—31no—
64(C22⋊D7)⋊C4—31no—
65#6441—31no—
66#6441D311yescharacteristic, complemented
67#5635—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C2×C3×C4×C7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72#4268Q814yescomplemented
73C2×Dic21—41no—
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C2×Dic21C2311yescharacteristic
77C4×D21—41no—
78C2×C7×Dic3—41no—
79C2×C7×Dic3C2311yescharacteristic
80C2×C7×Dic3C2311yescharacteristic
81C2×C7×Dic3C2311yescharacteristic
82C2×C7×Dic3C2311yescharacteristic
83C7×Dic6—41no—
84C2×C3×Dic7—22no—
85C2×C3×Dic7C2311yescharacteristic
86C2×C3×Dic7C2311yescharacteristic
87C2×C3×Dic7C2311yescharacteristic
88C2×C3×Dic7C2311yescharacteristic
89C3×Dic14—22no—
90#4248—41no—
91D21⋊C4—22no—
92D21⋊C4—41no—
93C2×C7×Q8—31no—
94C22×C4×C7—31no—
95C2×C22⋊D7—31no—
96C22×Dic7—31no—
97C22×Dic7C2×D311yescharacteristic
98C2×C4×D7—31no—
99C2×C4×D7—31no—
100C2×C4×D7—31no—
101C2×Dic14—31no—
102C2×Dic14C2×D311yescharacteristic
103C7×C4⋊C4—31no—
104C7×C4⋊C4—31no—
105C7×C4⋊C4—31no—
106C7×C22⋊C4—31no—
107C7×C22⋊C4C2×D311yescharacteristic
108C22⋊Dic7C2×D311yescharacteristic
109(C2×D7)⋊C4—31no—
110(C2×D7)⋊C4—31no—
111(C2×D7)⋊C4—31no—
112C4⋊Dic7—31no—
113C4⋊Dic7C2×D311yescharacteristic
114Dic7⋊C4—31no—
115Dic7⋊C4—31no—
116Dic7⋊C4—31no—
117Dic7⋊C4—31no—
118Dic7⋊C4—31no—
119Dic7⋊C4—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122Dic7⋊C4—31no—
123Dic7⋊C4C2×D311yescharacteristic
124Dic7⋊C4C2×D311yescharacteristic
125C4×Dic7—31no—
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⋊C4—71no—
141#878—71no—
142#878D711yescharacteristic, complemented
143C22×C3×C7—21no—
144C22×C3×C7—21no—
145C22×C3×C7C2×Q812yes—
146C22×C3×C7C2411yescharacteristic
147C2×D21—21no—
148C2×D21—21no—
149C2×D21C2×Q812yes—
150C3×C4×C7—21no—
151C3×C4×C7—21no—
152Dic21—21no—
153Dic21—21no—
154Dic21C2×Q812yes—
155C3×Dic7—21no—
156C3×Dic7—21no—
157C3×Dic7—21no—
158C3×Dic7#4112yes—
159C7×Dic3—21no—
160C7×Dic3—21no—
161C7×Dic3—21no—
162C7×Dic3—21no—
163#554—211noSylow
164C23×C7C22×D311yescharacteristic
165C22×D7—31no—
166C7×Q8—121no—
167C2×C4×C7—31no—
168C2×C4×C7—31no—
169C2×C4×C7—31no—
170C2×C4×C7—31no—
171C2×C4×C7—121no—
172C2×C4×C7C22×D311yescharacteristic
173C2×C4×C7C22×D311yescharacteristic
174C22⋊D7—34no—
175C2×Dic7—22no—
176C2×Dic7—31no—
177C2×Dic7—31no—
178C2×Dic7—31no—
179C2×Dic7—121no—
180C2×Dic7C22×D311yescharacteristic
181C2×Dic7C22×D311yescharacteristic
182C2×Dic7C22×D311yescharacteristic
183C2×Dic7C22×D311yescharacteristic
184C4×D7—62no—
185C4×D7—121no—
186C4×D7—121no—
187Dic14—22no—
188Dic14—121no—
189C2×C3×Q8—71no—
190C22×C3×C4—71no—
191C2×#67—71no—
192C22×Dic3—71no—
193C22×Dic3C2×D711yescharacteristic
194C2×C4×D3—71no—
195C2×C4×D3—71no—
196C2×C4×D3—71no—
197C2×Dic6—71no—
198C2×Dic6C2×D711yescharacteristic
199C3×C4⋊C4—71no—
200C3×C4⋊C4—71no—
201C3×C4⋊C4—71no—
202C3×C22⋊C4—71no—
203C3×C22⋊C4C2×D711yescharacteristic
204#217C2×D711yescharacteristic
205(C2×D3)⋊C4—71no—
206(C2×D3)⋊C4—71no—
207(C2×D3)⋊C4—71no—
208C4⋊Dic3—71no—
209C4⋊Dic3—71no—
210C4⋊Dic3—71no—
211C4⋊Dic3—71no—
212C4⋊Dic3—71no—
213C4⋊Dic3C2×D711yescharacteristic
214Dic3⋊C4—71no—
215Dic3⋊C4—71no—
216Dic3⋊C4—71no—
217Dic3⋊C4—71no—
218Dic3⋊C4—71no—
219Dic3⋊C4C2×D711yescharacteristic
220Dic3⋊C4C2×D711yescharacteristic
221C4×Dic3—71no—
222C4×Dic3—71no—
223C4×Dic3—71no—
224C2×C3×C7—22no—
225C2×C3×C7C22×Q811yescharacteristic
226C2×C3×C7C2×#4111yescharacteristic
227C2×C3×C7H(5,𝔽2)11yescharacteristic
228D21—22no—
229C4⋊Q8—211no—
230#125—211no—
231#125—211no—
232C22⋊Q8—31no—
233C22⋊Q8—71no—
234C22⋊Q8—211no—
235C22⋊Q8—211no—
236C22⋊Q8—211no—
237C22⋊Q8—211no—
238C4×Q8—211no—
239C4×D4—211no—
240C4×D4—211no—
241C4×D4—211no—
242C2×C4⋊C4—211no—
243C2×C4⋊C4—211no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7D3×Q812yes—
247C22×C7C23×D311yescharacteristic
248C2×D7—32no—
249C2×D7—61no—
250C2×D7—61no—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—61no—
254C4×C7—61no—
255C4×C7—61no—
256C4×C7—61no—
257Dic7—21no—
258Dic7—21no—
259Dic7—21no—
260Dic7—32no—
261Dic7—61no—
262Dic7—61no—
263Dic7#23712yes—
264C23×C3C22×D711yescharacteristic
265C22×D3—71no—
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#67—74no—
275C2×Dic3—41no—
276C2×Dic3—71no—
277C2×Dic3—71no—
278C2×Dic3—71no—
279C2×Dic3—281no—
280C2×Dic3C22×D711yescharacteristic
281C2×Dic3C22×D711yescharacteristic
282C2×Dic3C22×D711yescharacteristic
283C2×Dic3C22×D711yescharacteristic
284C4×D3—142no—
285C4×D3—281no—
286C4×D3—281no—
287Dic6—41no—
288Dic6—281no—
289C3×C7#55411yescharacteristic, complemented, Hall
290C2×Q8—31no—
291C2×Q8—71no—
292C2×Q8—211no—
293C2×D4—211no—
294C22×C4—31no—
295C22×C4—71no—
296C22×C4—211no—
297C22×C4—211no—
298C22×C4—211no—
299C22×C4—211no—
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—71no—
304C4⋊C4—71no—
305C4⋊C4—71no—
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—211no—
323C42—211no—
324C42—211no—
325C2×C7—22no—
326C2×C7C2×#23711yescharacteristic
327C2×C7C2×D3×Q811yescharacteristic
328C2×C7#100911yescharacteristic
329D7—62no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3Q8×D712yes—
333C22×C3C23×D711yescharacteristic
334C2×D3—72no—
335C2×D3—141no—
336C2×D3—141no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—72no—
340C3×C4—141no—
341C3×C4—141no—
342C3×C4—141no—
343Dic3—21no—
344Dic3—21no—
345Dic3—21no—
346Dic3—21no—
347Dic3—72no—
348Dic3—141no—
349Dic3—141no—
350C23—211no—
351C23C2×D3×D711yescharacteristic
352Q8—121no—
353Q8—142no—
354Q8—841no—
355D4—214no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—71no—
361C2×C4—71no—
362C2×C4—71no—
363C2×C4—71no—
364C2×C4—121no—
365C2×C4—142no—
366C2×C4—211no—
367C2×C4—211no—
368C2×C4—211no—
369C2×C4—422no—
370C2×C4—841no—
371C2×C4—841no—
372C2×C4—841no—
373C2×C4C2×D3×D711yescharacteristic
374C2×C4C2×D3×D711yescharacteristic
375C7#563511yescharacteristic, complemented, Sylow
376C2×C3—22no—
377C2×C3#654811yescharacteristic
378C2×C3C2×Q8×D711yescharacteristic
379C2×C3D7×#4111yescharacteristic
380D3—142no—
381C22—21no—
382C22—21no—
383C22—212no—
384C22—421no—
385C22—421no—
386C22D21⋊Q812yes—
387C22C22×D3×D711yescharacteristic
388C4—21no—
389C4—21no—
390C4—61no—
391C4—61no—
392C4—61no—
393C4—61no—
394C4—72no—
395C4—141no—
396C4—141no—
397C4—141no—
398C4—212no—
399C4—421no—
400C4—421no—
401C3#9032711yescharacteristic, complemented, Sylow
402C2—22no—
403C2—422no—
404C2C2×D21⋊Q811yescharacteristic
405C2D7×#23711yescharacteristic
406C2#1062157411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radicalC2×#4268C22
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
2. C7×#878C2semidirect
3. C3×#6441C2semidirect
4. #10621138C2semidirect
7. C2×Dic7⋊Dic3C2semidirect
8. #10620981C2semidirect
9. #10620973C2semidirect
10. #10620966C2semidirect
11. #10620945C2semidirect
12. #10620944C2semidirect
13. #10620901C2semidirect
14. #10620893C2semidirect
15. #10620891C2semidirect
66. #6441D3semidirect
72. #4268Q8semidirect
142. #878D7semidirect
289. C3×C7#554semidirect
375. C7#5635semidirect
401. C3#90327semidirect

Maximal subgroups

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

subgroupindex
#106213642
C7×#8782
C3×#64412
#106211382
#106211182
#106211142
C2×Dic7⋊Dic32
#106209812
#106209732
#106209662
#106209452
#106209442
#106209012
#106208932
#106208912
#903273
#56357