Finite Group Databasebeta

← #13058686

Subgroups — #13058686

Order 1344.

384 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621372C211yesmaximal, characteristic, complemented
2C7×#887C211yesmaximal, characteristic, complemented
3C3×#6461C211yesmaximal, characteristic
4C4×#4248C211yesmaximal, characteristic, complemented
5#10620984C211yesmaximal, characteristic
6#10620983C211yesmaximal, characteristic
7#10620955C211yesmaximal, characteristic
8#10620943C211yesmaximal, characteristic, complemented
9#10620941C211yesmaximal, characteristic, complemented
10#10620940C211yesmaximal, characteristic, complemented
11#10620928C211yesmaximal, characteristic
12#10620925C211yesmaximal, characteristic
13#10620925C211yesmaximal, characteristic
14#10620908C211yesmaximal, characteristic, complemented
15#10620891C211yesmaximal, characteristic, complemented
16#90274—31nomaximal, Hall
17C2×C7×Dic6C2211yescharacteristic
18C2×#4248C2211yescharacteristic
19C2×Dic3×D7C2211yescharacteristic
20C3×C7×C4⋊C4C2211yescharacteristic
21C4⋊Dic21C2211yescharacteristic
22#66629C2211yescharacteristic
23#66629C2211yescharacteristic
24#66629C2211yescharacteristic
25#66629C2211yescharacteristic
26C4×Dic21C2211yescharacteristic
27C7×Dic3⋊C4C2211yescharacteristic
28C7×Dic3⋊C4C2211yescharacteristic
29C4×C7×Dic3C2211yescharacteristic
30C4×C7×Dic3C2211yescharacteristic
31C4×C7×Dic3C2211yescharacteristic
32C3×(C2×D7)⋊C4C2211yescharacteristic
33C3×(C2×D7)⋊C4C2211yescharacteristic
34C3×(C2×D7)⋊C4C2211yescharacteristic
35C3×C4⋊Dic7C2211yescharacteristic
36C3×Dic7⋊C4C2211yescharacteristic
37C3×C4×Dic7C2211yescharacteristic
38Dic3⋊Dic7C2211yescharacteristic
39Dic3⋊Dic7C2211yescharacteristic
40Dic3⋊Dic7C2211yescharacteristic
41#66577C2211yescharacteristic
42#66577C2211yescharacteristic
43#66577C2211yescharacteristic
44Dic7⋊Dic3C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50(C2×D7)⋊Dic3C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C4×C7×Q8—31no—
53#6509—31no—
54#6507—31no—
55#6461—31no—
56#6461D311yescharacteristic
57#6460—31no—
58#6453—31no—
59#6452—31no—
60#6452—31no—
61#6439—31no—
62#6439—31no—
63#6435—31no—
64#6433—31no—
65#6433—31no—
66C4×Dic14—31no—
67#5701—71nomaximal, Hall
68C22×C3×D7C2311yescharacteristic
69C2×C3×C4×C7C2311yescharacteristic
70C2×C3×C4×C7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×Dic21C2311yescharacteristic
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C2×C7×Dic3C2311yescharacteristic
77C2×C7×Dic3C2311yescharacteristic
78C2×C7×Dic3C2311yescharacteristic
79C2×C7×Dic3C2311yescharacteristic
80C7×Dic6—22no—
81C2×C3×Dic7C2311yescharacteristic
82C2×C3×Dic7C2311yescharacteristic
83C2×C3×Dic7C2311yescharacteristic
84#4248—22no—
85Dic3×D7—22no—
86C2×C7×Q8—31no—
87C2×C4×D7—31no—
88C2×Dic14—31no—
89C7×C4⋊C4—31no—
90C7×C4⋊C4—31no—
91C7×C4⋊C4C2×D311yescharacteristic
92C42×C7—31no—
93C42×C7—31no—
94C42×C7—31no—
95(C2×D7)⋊C4—31no—
96(C2×D7)⋊C4—31no—
97(C2×D7)⋊C4—31no—
98(C2×D7)⋊C4C2×D311yescharacteristic
99(C2×D7)⋊C4C2×D311yescharacteristic
100(C2×D7)⋊C4C2×D311yescharacteristic
101C4⋊Dic7—31no—
102C4⋊Dic7—31no—
103C4⋊Dic7—31no—
104C4⋊Dic7—31no—
105C4⋊Dic7C2×D311yescharacteristic
106Dic7⋊C4—31no—
107Dic7⋊C4—31no—
108Dic7⋊C4—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
118C4×Dic7—31no—
119C4×Dic7—31no—
120C4×Dic7C2×D311yescharacteristic
121C3×#126—71no—
122#890—71no—
123#889—71no—
124#889—71no—
125#889—71no—
126#888—71no—
127#887D711yescharacteristic, complemented
128#879—71no—
129#879—71no—
130#879—71no—
131#878—71no—
132#878—71no—
133#877—71no—
134#870—71no—
135C4×Dic6—71no—
136C22×C3×C7C2411yescharacteristic
137C2×C3×D7—21no—
138C2×C3×D7—21no—
139C2×C3×D7—21no—
140C3×C4×C7—21no—
141C3×C4×C7—21no—
142C3×C4×C7—21no—
143Dic21—21no—
144Dic21—21no—
145Dic21—21no—
146Dic21—21no—
147C3×Dic7—21no—
148C3×Dic7—21no—
149C3×Dic7—21no—
150C7×Dic3—21no—
151C7×Dic3—21no—
152C7×Dic3—21no—
153C7×Dic3#4112yes—
154#541—211noSylow
155C22×D7C22×D311yescharacteristic
156C7×Q8—62no—
157C2×C4×C7—31no—
158C2×C4×C7—31no—
159C2×C4×C7—31no—
160C2×C4×C7—31no—
161C2×C4×C7C22×D311yescharacteristic
162C2×C4×C7C22×D311yescharacteristic
163C2×C4×C7C22×D311yescharacteristic
164C2×Dic7—31no—
165C2×Dic7—31no—
166C2×Dic7—31no—
167C2×Dic7—31no—
168C2×Dic7C22×D311yescharacteristic
169C2×Dic7C22×D311yescharacteristic
170C2×Dic7C22×D311yescharacteristic
171C4×D7—62no—
172Dic14—62no—
173C22×Dic3—71no—
174C2×Dic6—71no—
175C2×Dic6C2×D711yescharacteristic
176C3×C4⋊C4—71no—
177C3×C4⋊C4—71no—
178C3×C4⋊C4C2×D711yescharacteristic
179C3×C22⋊C4—71no—
180C3×C22⋊C4—71no—
181C3×C22⋊C4—71no—
182C3×C42—71no—
183#217—71no—
184#217—71no—
185#217—71no—
186C4⋊Dic3—71no—
187C4⋊Dic3—71no—
188C4⋊Dic3—71no—
189C4⋊Dic3—71no—
190C4⋊Dic3—71no—
191Dic3⋊C4—71no—
192Dic3⋊C4—71no—
193Dic3⋊C4—71no—
194Dic3⋊C4—71no—
195Dic3⋊C4—71no—
196Dic3⋊C4—71no—
197Dic3⋊C4—71no—
198Dic3⋊C4—71no—
199Dic3⋊C4—71no—
200Dic3⋊C4—71no—
201Dic3⋊C4C2×D711yescharacteristic
202Dic3⋊C4C2×D711yescharacteristic
203C4×Dic3—71no—
204C4×Dic3—71no—
205C4×Dic3C2×D711yescharacteristic
206C4×Dic3C2×D711yescharacteristic
207C4×Dic3C2×D711yescharacteristic
208C2×C3×C7C2×#4111yescharacteristic
209C2×C3×C72−1+411yescharacteristic
210C2×C3×C72−1+411yescharacteristic
211C3×D7—41no—
212C4⋊Q8—211no—
213#126—71no—
214#126—211no—
215#126—211no—
216#126—211no—
217#125—211no—
218#125—211no—
219#125—211no—
220#125—211no—
221#125—211no—
222C22⋊Q8—211no—
223C22⋊Q8—211no—
224C4×Q8—31no—
225C4×Q8—211no—
226#117—211no—
227C22×C7C23×D311yescharacteristic
228C2×D7—21no—
229C2×D7—21no—
230C2×D7—21no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—32no—
235C4×C7—61no—
236C4×C7—61no—
237C4×C7—61no—
238Dic7—21no—
239Dic7—21no—
240Dic7—21no—
241Dic7—61no—
242Dic7—61no—
243Dic7—61no—
244Dic7—61no—
245C23×C3—71no—
246C2×C3×C4—71no—
247C2×C3×C4—71no—
248C2×C3×C4—71no—
249C2×C3×C4C22×D711yescharacteristic
250C2×C3×C4C22×D711yescharacteristic
251C2×C3×C4C22×D711yescharacteristic
252C2×Dic3—71no—
253C2×Dic3—71no—
254C2×Dic3—71no—
255C2×Dic3—71no—
256C2×Dic3—142no—
257C2×Dic3C22×D711yescharacteristic
258C2×Dic3C22×D711yescharacteristic
259C2×Dic3C22×D711yescharacteristic
260C2×Dic3C22×D711yescharacteristic
261Dic6—22no—
262Dic6—142no—
263C3×C7#54111yescharacteristic, complemented, Hall
264C2×Q8—31no—
265C2×Q8—211no—
266C22×C4—211no—
267C4⋊C4—31no—
268C4⋊C4—31no—
269C4⋊C4—71no—
270C4⋊C4—71no—
271C4⋊C4—211no—
272C4⋊C4—211no—
273C4⋊C4—211no—
274C4⋊C4—211no—
275C4⋊C4—211no—
276C4⋊C4—211no—
277C4⋊C4—211no—
278C4⋊C4—211no—
279C4⋊C4—211no—
280C4⋊C4—211no—
281C4⋊C4—211no—
282C4⋊C4—211no—
283C4⋊C4—211no—
284C4⋊C4—211no—
285C4⋊C4—211no—
286C4⋊C4D3×D711yescharacteristic
287C22⋊C4—71no—
288C22⋊C4—71no—
289C22⋊C4—71no—
290C22⋊C4—211no—
291C22⋊C4—211no—
292C22⋊C4—211no—
293C42—31no—
294C42—31no—
295C42—31no—
296C42—71no—
297C42—211no—
298C42—211no—
299C2×C7D3×#4111yescharacteristic
300C2×C7#101011yescharacteristic
301C2×C7#101011yescharacteristic
302D7—41no—
303C22×C3—141no—
304C22×C3—141no—
305C22×C3—141no—
306C22×C3C23×D711yescharacteristic
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—141no—
311C3×C4—141no—
312C3×C4—141no—
313Dic3—21no—
314Dic3—21no—
315Dic3—21no—
316Dic3—141no—
317Dic3—141no—
318Dic3—141no—
319Dic3—141no—
320Dic3#115712yes—
321C23—71no—
322Q8—62no—
323Q8—422no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—71no—
329C2×C4—71no—
330C2×C4—71no—
331C2×C4—211no—
332C2×C4—211no—
333C2×C4—211no—
334C2×C4—211no—
335C2×C4—422no—
336C2×C4C2×D3×D711yescharacteristic
337C2×C4C2×D3×D711yescharacteristic
338C2×C4C2×D3×D711yescharacteristic
339C7#570111yescharacteristic, complemented, Sylow
340C2×C3—281no—
341C2×C3C2×#115711yescharacteristic
342C2×C3#655111yescharacteristic
343C2×C3#655411yescharacteristic
344C22—141no—
345C22—141no—
346C22—141no—
347C22C22×D3×D711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—32no—
352C4—61no—
353C4—61no—
354C4—61no—
355C4—141no—
356C4—141no—
357C4—141no—
358C4—421no—
359C4—421no—
360C4—421no—
361C4—421no—
362C3#9027411yescharacteristic, complemented, Sylow
363C2—281no—
364C2D3×#115711yescharacteristic
365C2#1062156711yescharacteristic
366C2#1062157511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC22×C3×D7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#541
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621372C2semidirect
2. C7×#887C2semidirect
4. C4×#4248C2semidirect
8. #10620943C2semidirect
9. #10620941C2semidirect
10. #10620940C2semidirect
14. #10620908C2semidirect
15. #10620891C2semidirect
127. #887D7semidirect
263. C3×C7#541semidirect
339. C7#5701semidirect
362. C3#90274semidirect

Maximal subgroups

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

subgroupindex
#106213722
C7×#8872
C3×#64612
C4×#42482
#106209842
#106209832
#106209552
#106209432
#106209412
#106209402
#106209282
#106209252
#106209082
#106208912
#902743
#57017