Finite Group Databasebeta

← #13057778

Subgroups — #13057778

Order 1344.

428 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#10621359C211yesmaximal, characteristic, complemented
2C7×#872C211yesmaximal, characteristic, complemented
3C3×C4×Dic14C211yesmaximal, characteristic, complemented
4#10620989C211yesmaximal, characteristic
5#10620973C211yesmaximal, characteristic, complemented
6#10620972C211yesmaximal, characteristic, complemented
7#10620967C211yesmaximal, characteristic
8#10620965C211yesmaximal, characteristic, complemented
9D3×Dic7⋊C4C211yesmaximal, characteristic, complemented
10#10620960C211yesmaximal, characteristic, complemented
11#10620949C211yesmaximal, characteristic, complemented
12#10620941C211yesmaximal, characteristic, complemented
13#10620940C211yesmaximal, characteristic, complemented
14#10620913C211yesmaximal, characteristic
15#10620901C211yesmaximal, characteristic, complemented
16#90222—31nomaximal, Hall
17C2×C4×D3×C7C2211yescharacteristic
18C2×C3×Dic14C2211yescharacteristic
19C2×#4247C2211yescharacteristic
20C2×D21⋊C4C2211yescharacteristic
21C2×D3×Dic7C2211yescharacteristic
22C2×D3×Dic7C2211yescharacteristic
23C3×C42×C7C2211yescharacteristic
24#66631C2211yescharacteristic
25#66631C2211yescharacteristic
26#66631C2211yescharacteristic
27#66629C2211yescharacteristic
28#66629C2211yescharacteristic
29#66629C2211yescharacteristic
30C7×(C2×D3)⋊C4C2211yescharacteristic
31C7×(C2×D3)⋊C4C2211yescharacteristic
32C7×Dic3⋊C4C2211yescharacteristic
33C7×Dic3⋊C4C2211yescharacteristic
34C4×C7×Dic3C2211yescharacteristic
35C3×C4⋊Dic7C2211yescharacteristic
36C3×Dic7⋊C4C2211yescharacteristic
37C3×Dic7⋊C4C2211yescharacteristic
38C3×C4×Dic7C2211yescharacteristic
39C3×C4×Dic7C2211yescharacteristic
40Dic3⋊Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42#66577C2211yescharacteristic
43#66577C2211yescharacteristic
44#66577C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48#66575C2211yescharacteristic
49#66575C2211yescharacteristic
50(C2×D3)⋊Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#117—31no—
53#6492—31no—
54C2×Dic7⋊C4—31no—
55#6461—31no—
56#6459—31no—
57#6452—31no—
58#6452—31no—
59#6449—31no—
60(C22⋊D7)⋊C4—31no—
61#6441—31no—
62#6440—31no—
63#6439—31no—
64#6435—31no—
65#6433—31no—
66C4×Dic14D311yescharacteristic, complemented
67#5581—71nomaximal, Hall
68C22×D21C2311yescharacteristic
69C22×D3×C7C2311yescharacteristic
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
79C4×D3×C7—22no—
80C2×C3×Dic7C2311yescharacteristic
81C2×C3×Dic7C2311yescharacteristic
82C2×C3×Dic7C2311yescharacteristic
83C2×C3×Dic7C2311yescharacteristic
84C3×Dic14—22no—
85#4247—22no—
86D21⋊C4—22no—
87D3×Dic7—22no—
88D3×Dic7—22no—
89C22×C4×C7—31no—
90C2×C22⋊D7—31no—
91C22×Dic7—31no—
92C22×Dic7—31no—
93C2×C4×D7—31no—
94C2×Dic14C2×D311yescharacteristic
95C7×C4⋊C4—31no—
96C7×C4⋊C4—31no—
97C7×C22⋊C4—31no—
98C7×C22⋊C4—31no—
99C42×C7—31no—
100C42×C7C2×D311yescharacteristic
101C22⋊Dic7—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—
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⋊C4—31no—
117Dic7⋊C4—31no—
118Dic7⋊C4—31no—
119Dic7⋊C4C2×D311yescharacteristic
120Dic7⋊C4C2×D311yescharacteristic
121C4×Dic7—31no—
122C4×Dic7C2×D311yescharacteristic
123C4×Dic7C2×D311yescharacteristic
124C3×C4×Q8—71no—
125#946—71no—
126#946—71no—
127#894—71no—
128#894—71no—
129#892—71no—
130D3×C4⋊C4—71no—
131#889—71no—
132#889—71no—
133#876—71no—
134#876—71no—
135C4×D12—71no—
136#872—71no—
137#872D711yescharacteristic, complemented
138#870—71no—
139C22×C3×C7C2411yescharacteristic
140C2×D21—21no—
141C2×D21—21no—
142C2×D21—21no—
143C2×D3×C7—21no—
144C2×D3×C7—21no—
145C2×D3×C7#4112yes—
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—
155C3×Dic7#4112yes—
156C7×Dic3—21no—
157C7×Dic3—21no—
158C7×Dic3—21no—
159#552—211noSylow
160C23×C7—31no—
161C22×D7—31no—
162C2×C4×C7—31no—
163C2×C4×C7—31no—
164C2×C4×C7—31no—
165C2×C4×C7—62no—
166C2×C4×C7C22×D311yescharacteristic
167C2×C4×C7C22×D311yescharacteristic
168C2×C4×C7C22×D311yescharacteristic
169C22⋊D7—62no—
170C2×Dic7—31no—
171C2×Dic7—31no—
172C2×Dic7—31no—
173C2×Dic7—62no—
174C2×Dic7—62no—
175C2×Dic7C22×D311yescharacteristic
176C2×Dic7C22×D311yescharacteristic
177C2×Dic7C22×D311yescharacteristic
178C2×Dic7C22×D311yescharacteristic
179C4×D7—62no—
180Dic14—22no—
181C2×C3×Q8—71no—
182C2×D12—71no—
183C2×C4×D3—71no—
184C2×C4×D3—71no—
185C2×C4×D3—71no—
186C2×C4×D3C2×D711yescharacteristic
187C3×C4⋊C4—71no—
188C3×C4⋊C4—71no—
189C3×C4⋊C4—71no—
190C3×C42—71no—
191C3×C42—71no—
192C3×C42C2×D711yescharacteristic
193(C2×D3)⋊C4—71no—
194(C2×D3)⋊C4—71no—
195(C2×D3)⋊C4—71no—
196(C2×D3)⋊C4—71no—
197(C2×D3)⋊C4—71no—
198(C2×D3)⋊C4—71no—
199(C2×D3)⋊C4C2×D711yescharacteristic
200(C2×D3)⋊C4C2×D711yescharacteristic
201C4⋊Dic3—71no—
202C4⋊Dic3—71no—
203C4⋊Dic3—71no—
204Dic3⋊C4—71no—
205Dic3⋊C4—71no—
206Dic3⋊C4—71no—
207Dic3⋊C4—71no—
208Dic3⋊C4—71no—
209Dic3⋊C4—71no—
210Dic3⋊C4—71no—
211Dic3⋊C4—71no—
212Dic3⋊C4C2×D711yescharacteristic
213Dic3⋊C4C2×D711yescharacteristic
214C4×Dic3—71no—
215C4×Dic3C2×D711yescharacteristic
216C2×C3×C7C2×#4111yescharacteristic
217C2×C3×C7C2×#4111yescharacteristic
218C2×C3×C72−1+411yescharacteristic
219D21—41no—
220D3×C7—22no—
221#126—211no—
222#126—211no—
223#125—211no—
224#125—211no—
225#125—211no—
226#123—211no—
227#123—211no—
228C22⋊Q8—211no—
229C22⋊Q8—211no—
230C4×Q8—71no—
231C4×D4—211no—
232#117—31no—
233#117—211no—
234#117—211no—
235C2×C4⋊C4—211no—
236C22×C7—32no—
237C22×C7—61no—
238C22×C7—61no—
239C22×C7C23×D311yescharacteristic
240C2×D7—61no—
241C2×D7—61no—
242C2×D7—61no—
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—61no—
247C4×C7—61no—
248C4×C7—61no—
249Dic7—21no—
250Dic7—21no—
251Dic7—21no—
252Dic7—61no—
253Dic7—61no—
254Dic7—61no—
255Dic7#23512yes—
256C22×D3—71no—
257C22×D3C22×D711yescharacteristic
258C3×Q8—142no—
259C2×C3×C4—71no—
260C2×C3×C4—71no—
261C2×C3×C4—71no—
262C2×C3×C4—71no—
263C2×C3×C4C22×D711yescharacteristic
264C2×C3×C4C22×D711yescharacteristic
265C2×C3×C4C22×D711yescharacteristic
266C2×Dic3—71no—
267C2×Dic3—71no—
268C2×Dic3—71no—
269C2×Dic3C22×D711yescharacteristic
270C2×Dic3C22×D711yescharacteristic
271C2×Dic3C22×D711yescharacteristic
272D12—142no—
273C4×D3—22no—
274C4×D3—142no—
275C4×D3—142no—
276C4×D3—142no—
277C3×C7#55211yescharacteristic, complemented, Hall
278C2×Q8—71no—
279C2×D4—211no—
280C22×C4—31no—
281C22×C4—211no—
282C22×C4—211no—
283C22×C4—211no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—71no—
287C4⋊C4—71no—
288C4⋊C4—71no—
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⋊C4—211no—
298C4⋊C4—211no—
299C4⋊C4—211no—
300C22⋊C4—31no—
301C22⋊C4—31no—
302C22⋊C4—211no—
303C22⋊C4—211no—
304C22⋊C4—211no—
305C22⋊C4—211no—
306C22⋊C4—211no—
307C22⋊C4—211no—
308C42—31no—
309C42—71no—
310C42—71no—
311C42—211no—
312C42D3×D711yescharacteristic
313C2×C7—62no—
314C2×C7C2×#23511yescharacteristic
315C2×C7#100711yescharacteristic
316C2×C7D3×#4111yescharacteristic
317D7—121no—
318C22×C3C23×D711yescharacteristic
319C2×D3—21no—
320C2×D3—21no—
321C2×D3—141no—
322C2×D3—141no—
323C2×D3—141no—
324C2×D3#115712yes—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—72no—
329C3×C4—141no—
330C3×C4—141no—
331C3×C4—141no—
332Dic3—21no—
333Dic3—21no—
334Dic3—21no—
335Dic3—141no—
336Dic3—141no—
337Dic3—141no—
338C23—31no—
339C23—211no—
340Q8—142no—
341D4—422no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—62no—
346C2×C4—71no—
347C2×C4—71no—
348C2×C4—71no—
349C2×C4—71no—
350C2×C4—211no—
351C2×C4—211no—
352C2×C4—211no—
353C2×C4—422no—
354C2×C4—422no—
355C2×C4—422no—
356C2×C4C2×D3×D711yescharacteristic
357C2×C4C2×D3×D711yescharacteristic
358C2×C4C2×D3×D711yescharacteristic
359C7#558111yescharacteristic, complemented, Sylow
360C2×C3C2×#115711yescharacteristic
361C2×C3D7×#4111yescharacteristic
362C2×C3#655411yescharacteristic
363D3—22no—
364D3—281no—
365C22—32no—
366C22—61no—
367C22—61no—
368C22—421no—
369C22—421no—
370C22—421no—
371C22C22×D3×D711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—61no—
376C4—61no—
377C4—61no—
378C4—72no—
379C4—141no—
380C4—141no—
381C4—141no—
382C4—421no—
383C4—421no—
384C4—421no—
385C3#9022211yescharacteristic, complemented, Sylow
386C2—62no—
387C2—841no—
388C2#1062155111yescharacteristic
389C2D7×#23511yescharacteristic
390C2D3×#115711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C42×C7C22
prime radicalC2×#4247C22
hypercenterC42D3×D7
hypocenterC3×C7#552
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #10621359C2semidirect
2. C7×#872C2semidirect
3. C3×C4×Dic14C2semidirect
5. #10620973C2semidirect
6. #10620972C2semidirect
8. #10620965C2semidirect
9. D3×Dic7⋊C4C2semidirect
10. #10620960C2semidirect
11. #10620949C2semidirect
12. #10620941C2semidirect
13. #10620940C2semidirect
15. #10620901C2semidirect
66. C4×Dic14D3semidirect
137. #872D7semidirect
277. C3×C7#552semidirect
359. C7#5581semidirect
385. C3#90222semidirect

Maximal subgroups

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

subgroupindex
#106213592
C7×#8722
C3×C4×Dic142
#106209892
#106209732
#106209722
#106209672
#106209652
D3×Dic7⋊C42
#106209602
#106209492
#106209412
#106209402
#106209132
#106209012
#902223
#55817