Finite Group Databasebeta

← #13058085

Subgroups — #13058085

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#10621362C211yesmaximal, characteristic, complemented
2C7×#883C211yesmaximal, characteristic
3C3×#6440C211yesmaximal, characteristic, complemented
4#10621136C211yesmaximal, characteristic, complemented
5#10621117C211yesmaximal, characteristic
6#67×Dic7C211yesmaximal, characteristic
7#10621093C211yesmaximal, characteristic, complemented
8#10620979C211yesmaximal, characteristic, complemented
9D3×Dic7⋊C4C211yesmaximal, characteristic, complemented
10#10620960C211yesmaximal, characteristic, complemented
11#10620950C211yesmaximal, characteristic, complemented
12#10620940C211yesmaximal, characteristic, complemented
13#10620933C211yesmaximal, characteristic, complemented
14#10620897C211yesmaximal, characteristic, complemented
15#10620887C211yesmaximal, characteristic, complemented
16#90341—31nomaximal, Hall
17C2×C7×#67C2211yescharacteristic
18C2×C4×D3×C7C2211yescharacteristic
19C22×C3×Dic7C2211yescharacteristic
20C2×#4248C2211yescharacteristic
21C2×D3×Dic7C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
33C7×Dic3⋊C4C2211yescharacteristic
34C3×C22⋊Dic7C2211yescharacteristic
35C3×Dic7⋊C4C2211yescharacteristic
36C3×Dic7⋊C4C2211yescharacteristic
37C3×C4×Dic7C2211yescharacteristic
38C3×C4×Dic7C2211yescharacteristic
39Dic3⋊Dic7C2211yescharacteristic
40Dic3⋊Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42#66577C2211yescharacteristic
43#66577C2211yescharacteristic
44#66577C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic7C2211yescharacteristic
49(C2×D3)⋊Dic7C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#123—31no—
53#6498—31no—
54D4×Dic7—31no—
55#6489—31no—
56C2×Dic7⋊C4—31no—
57#6453—31no—
58#6452—31no—
59#6452—31no—
60#6450—31no—
61#6442—31no—
62#6442—31no—
63#6441—31no—
64#6441—31no—
65#6440—31no—
66#6440D311yescharacteristic, complemented
67#5529—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
80C4×D3×C7—22no—
81C2×C3×Dic7—22no—
82C2×C3×Dic7C2311yescharacteristic
83C2×C3×Dic7C2311yescharacteristic
84C2×C3×Dic7C2311yescharacteristic
85C2×C3×Dic7C2311yescharacteristic
86#4248—22no—
87D3×Dic7—22no—
88D3×Dic7—22no—
89C2×C7×D4—31no—
90C22×C4×C7—31no—
91C22×Dic7—31no—
92C22×Dic7—31no—
93C22×Dic7C2×D311yescharacteristic
94C2×Dic14—31no—
95C7×C4⋊C4—31no—
96C7×C4⋊C4—31no—
97C7×C22⋊C4—31no—
98C7×C22⋊C4—31no—
99C7×C22⋊C4C2×D311yescharacteristic
100C22⋊Dic7—31no—
101C22⋊Dic7—31no—
102C22⋊Dic7—31no—
103C22⋊Dic7—31no—
104C22⋊Dic7C2×D311yescharacteristic
105C4⋊Dic7—31no—
106C4⋊Dic7—31no—
107C4⋊Dic7—31no—
108C4⋊Dic7—31no—
109Dic7⋊C4—31no—
110Dic7⋊C4—31no—
111Dic7⋊C4—31no—
112Dic7⋊C4—31no—
113Dic7⋊C4—31no—
114Dic7⋊C4—31no—
115Dic7⋊C4—31no—
116Dic7⋊C4—31no—
117Dic7⋊C4C2×D311yescharacteristic
118Dic7⋊C4C2×D311yescharacteristic
119C4×Dic7—31no—
120C4×Dic7—31no—
121C4×Dic7—31no—
122C4×Dic7C2×D311yescharacteristic
123C4×Dic7C2×D311yescharacteristic
124C3×#117—71no—
125C4×#67—71no—
126#924—71no—
127#897—71no—
128D3×C4⋊C4—71no—
129#890—71no—
130#887—71no—
131#883—71no—
132#883D711yescharacteristic
133#879—71no—
134#879—71no—
135#872—71no—
136#872—71no—
137#870—71no—
138#870—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#4112yes—
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—21no—
159#552—211noSylow
160C23×C7—31no—
161C23×C7C22×D311yescharacteristic
162C7×D4—62no—
163C2×C4×C7—31no—
164C2×C4×C7—31no—
165C2×C4×C7—31no—
166C2×C4×C7—62no—
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×Dic7—62no—
176C2×Dic7C22×D311yescharacteristic
177C2×Dic7C22×D311yescharacteristic
178C2×Dic7C22×D311yescharacteristic
179C2×Dic7C22×D311yescharacteristic
180Dic14—62no—
181C22×C3×C4—71no—
182C2×#67C2×D711yescharacteristic
183C2×C4×D3—71no—
184C2×C4×D3—71no—
185C2×C4×D3C2×D711yescharacteristic
186C2×Dic6—71no—
187C3×C4⋊C4—71no—
188C3×C4⋊C4—71no—
189C3×C22⋊C4—71no—
190C3×C22⋊C4C2×D711yescharacteristic
191C3×C42—71no—
192C3×C42—71no—
193#217—71no—
194#217—71no—
195#217C2×D711yescharacteristic
196(C2×D3)⋊C4—71no—
197(C2×D3)⋊C4—71no—
198(C2×D3)⋊C4C2×D711yescharacteristic
199C4⋊Dic3—71no—
200C4⋊Dic3—71no—
201C4⋊Dic3—71no—
202C4⋊Dic3—71no—
203C4⋊Dic3C2×D711yescharacteristic
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
213C4×Dic3—71no—
214C4×Dic3—71no—
215C4×Dic3—71no—
216C2×C3×C7—41no—
217C2×C3×C7C2×#4111yescharacteristic
218C2×C3×C7C2×#4111yescharacteristic
219C2×C3×C72−1+411yescharacteristic
220D3×C7—22no—
221#126—211no—
222#126—211no—
223#125—211no—
224#125—211no—
225#125—211no—
226#123—31no—
227#123—211no—
228C22⋊Q8—211no—
229C22⋊Q8—211no—
230C4×Q8—211no—
231C4×D4—211no—
232#117—71no—
233#117—211no—
234#117—211no—
235C2×C4⋊C4—211no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—32no—
240C22×C7—61no—
241C22×C7—61no—
242C22×C7C23×D311yescharacteristic
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—61no—
246C4×C7—61no—
247C4×C7—61no—
248Dic7—21no—
249Dic7—21no—
250Dic7—21no—
251Dic7—61no—
252Dic7—61no—
253Dic7—61no—
254Dic7—61no—
255Dic7#23512yes—
256C23×C3C22×D711yescharacteristic
257C22×D3C22×D711yescharacteristic
258C2×C3×C4—71no—
259C2×C3×C4—71no—
260C2×C3×C4—71no—
261C2×C3×C4—71no—
262C2×C3×C4—142no—
263C2×C3×C4C22×D711yescharacteristic
264C2×C3×C4C22×D711yescharacteristic
265#67—22no—
266C2×Dic3—71no—
267C2×Dic3—71no—
268C2×Dic3—71no—
269C2×Dic3—71no—
270C2×Dic3C22×D711yescharacteristic
271C2×Dic3C22×D711yescharacteristic
272C2×Dic3C22×D711yescharacteristic
273C4×D3—22no—
274C4×D3—142no—
275C4×D3—142no—
276Dic6—142no—
277C3×C7#55211yescharacteristic, complemented, Hall
278C2×Q8—211no—
279C2×D4—31no—
280C22×C4—31no—
281C22×C4—71no—
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⋊C4—211no—
296C4⋊C4—211no—
297C4⋊C4—211no—
298C4⋊C4—211no—
299C4⋊C4—211no—
300C22⋊C4—31no—
301C22⋊C4—31no—
302C22⋊C4—71no—
303C22⋊C4—211no—
304C22⋊C4—211no—
305C22⋊C4—211no—
306C22⋊C4—211no—
307C22⋊C4D3×D711yescharacteristic
308C42—71no—
309C42—71no—
310C42—211no—
311C42—211no—
312C42—211no—
313C2×C7—41no—
314C2×C7—62no—
315C2×C7C2×#23511yescharacteristic
316C2×C7D3×#4111yescharacteristic
317C2×C7#101011yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3C23×D711yescharacteristic
322C2×D3—21no—
323C2×D3—21no—
324C2×D3#115912yes—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—72no—
328C3×C4—141no—
329C3×C4—141no—
330C3×C4—141no—
331Dic3—21no—
332Dic3—21no—
333Dic3—21no—
334Dic3—141no—
335Dic3—141no—
336Dic3—141no—
337Dic3—141no—
338C23—31no—
339C23C2×D3×D711yescharacteristic
340Q8—422no—
341D4—62no—
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—142no—
351C2×C4—211no—
352C2×C4—211no—
353C2×C4—211no—
354C2×C4—211no—
355C2×C4—422no—
356C2×C4—422no—
357C2×C4C2×D3×D711yescharacteristic
358C2×C4C2×D3×D711yescharacteristic
359C7#552911yescharacteristic, complemented, Sylow
360C2×C3—41no—
361C2×C3C2×#115911yescharacteristic
362C2×C3D7×#4111yescharacteristic
363C2×C3#655411yescharacteristic
364D3—22no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—32no—
369C22—61no—
370C22—61no—
371C22C22×D3×D711yescharacteristic
372C4—21no—
373C4—21no—
374C4—61no—
375C4—61no—
376C4—61no—
377C4—72no—
378C4—141no—
379C4—141no—
380C4—141no—
381C4—421no—
382C4—421no—
383C4—421no—
384C4—421no—
385C3#9034111yescharacteristic, complemented, Sylow
386C2—41no—
387C2—62no—
388C2D7×#23511yescharacteristic
389C2#1062156711yescharacteristic
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#552
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×#6440C2semidirect
4. #10621136C2semidirect
7. #10621093C2semidirect
8. #10620979C2semidirect
9. D3×Dic7⋊C4C2semidirect
10. #10620960C2semidirect
11. #10620950C2semidirect
12. #10620940C2semidirect
13. #10620933C2semidirect
14. #10620897C2semidirect
15. #10620887C2semidirect
66. #6440D3semidirect
277. C3×C7#552semidirect
359. C7#5529semidirect
385. C3#90341semidirect

Maximal subgroups

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

subgroupindex
#106213622
C7×#8832
C3×#64402
#106211362
#106211172
#67×Dic72
#106210932
#106209792
D3×Dic7⋊C42
#106209602
#106209502
#106209402
#106209332
#106208972
#106208872
#903413
#55297