Finite Group Databasebeta

← #13058597

Subgroups — #13058597

Order 1344.

412 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621372C211yesmaximal, characteristic, complemented
2C7×#888C211yesmaximal, characteristic, complemented
3C3×#6459C211yesmaximal, characteristic
4#10620992C211yesmaximal, characteristic
5#10620983C211yesmaximal, characteristic
6#10620956C211yesmaximal, characteristic
7#10620955C211yesmaximal, characteristic
8D7×Dic3⋊C4C211yesmaximal, characteristic
9#10620947C211yesmaximal, characteristic, complemented
10#10620941C211yesmaximal, characteristic, complemented
11#10620921C211yesmaximal, characteristic
12#10620905C211yesmaximal, characteristic, complemented
13#10620898C211yesmaximal, characteristic, complemented
14#10620891C211yesmaximal, characteristic, complemented
15#10620889C211yesmaximal, characteristic, complemented
16#90420—31nomaximal, Hall
17C2×C7×Dic6C2211yescharacteristic
18C2×C7×Dic6C2211yescharacteristic
19C2×C3×C4×D7C2211yescharacteristic
20C2×C3×Dic14C2211yescharacteristic
21C2×#4248C2211yescharacteristic
22C2×Dic3×D7C2211yescharacteristic
23C2×Dic3×D7C2211yescharacteristic
24C3×C7×C4⋊C4C2211yescharacteristic
25C4⋊Dic21C2211yescharacteristic
26#66629C2211yescharacteristic
27#66629C2211yescharacteristic
28#66629C2211yescharacteristic
29#66629C2211yescharacteristic
30C4×Dic21C2211yescharacteristic
31C7×C4⋊Dic3C2211yescharacteristic
32C7×Dic3⋊C4C2211yescharacteristic
33C7×Dic3⋊C4C2211yescharacteristic
34C4×C7×Dic3C2211yescharacteristic
35C3×(C2×D7)⋊C4C2211yescharacteristic
36C3×(C2×D7)⋊C4C2211yescharacteristic
37C3×Dic7⋊C4C2211yescharacteristic
38C3×Dic7⋊C4C2211yescharacteristic
39Dic3⋊Dic7C2211yescharacteristic
40Dic3⋊Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42#66577C2211yescharacteristic
43#66577C2211yescharacteristic
44#66577C2211yescharacteristic
45#66577C2211yescharacteristic
46#66577C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×C4⋊Q8—31no—
53#6509—31no—
54#6509—31no—
55#6507—31no—
56#6507—31no—
57#6459—31no—
58#6459D311yescharacteristic
59#6455—31no—
60D7×C4⋊C4—31no—
61#6453—31no—
62#6452—31no—
63#6452—31no—
64#6451—31no—
65#6435—31no—
66#6433—31no—
67#5623—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—41no—
81C7×Dic6—41no—
82C2×C3×Dic7C2311yescharacteristic
83C2×C3×Dic7C2311yescharacteristic
84C2×C3×Dic7C2311yescharacteristic
85C3×C4×D7—41no—
86C3×Dic14—41no—
87#4248—41no—
88Dic3×D7—41no—
89Dic3×D7Q814yescomplemented
90C2×C7×Q8—31no—
91C2×C7×Q8—31no—
92C2×C4×D7—31no—
93C2×C4×D7—31no—
94C2×C4×D7C2×D311yescharacteristic
95C2×Dic14—31no—
96C2×Dic14C2×D311yescharacteristic
97C7×C4⋊C4—31no—
98C7×C4⋊C4—31no—
99C7×C4⋊C4—31no—
100C7×C4⋊C4C2×D311yescharacteristic
101C42×C7—31no—
102(C2×D7)⋊C4—31no—
103(C2×D7)⋊C4—31no—
104(C2×D7)⋊C4C2×D311yescharacteristic
105(C2×D7)⋊C4C2×D311yescharacteristic
106C4⋊Dic7—31no—
107C4⋊Dic7—31no—
108C4⋊Dic7—31no—
109C4⋊Dic7—31no—
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⋊C4—31no—
120Dic7⋊C4C2×D311yescharacteristic
121Dic7⋊C4C2×D311yescharacteristic
122C4×Dic7—31no—
123C4×Dic7—31no—
124C4×Dic7—31no—
125C3×C22⋊Q8—71no—
126#944—71no—
127#944—71no—
128#924—71no—
129C2×Dic3⋊C4—71no—
130#889—71no—
131#889—71no—
132#889—71no—
133#889—71no—
134#888—71no—
135#888D711yescharacteristic, complemented
136#878—71no—
137#878—71no—
138#877—71no—
139#877—71no—
140C22×C3×C7C2411yescharacteristic
141C2×C3×D7—21no—
142C2×C3×D7—21no—
143C2×C3×D7C2×Q812yes—
144C3×C4×C7—21no—
145C3×C4×C7—21no—
146C3×C4×C7—21no—
147Dic21—21no—
148Dic21—21no—
149Dic21—21no—
150Dic21C2×Q812yes—
151C3×Dic7—21no—
152C3×Dic7—21no—
153C3×Dic7—21no—
154C7×Dic3—21no—
155C7×Dic3—21no—
156C7×Dic3—21no—
157C7×Dic3C2×Q812yes—
158#544—211noSylow
159C22×D7C22×D311yescharacteristic
160C7×Q8—121no—
161C7×Q8—121no—
162C2×C4×C7—31no—
163C2×C4×C7—31no—
164C2×C4×C7—31no—
165C2×C4×C7—31no—
166C2×C4×C7C22×D311yescharacteristic
167C2×C4×C7C22×D311yescharacteristic
168C2×C4×C7C22×D311yescharacteristic
169C2×Dic7—31no—
170C2×Dic7—31no—
171C2×Dic7—31no—
172C2×Dic7—31no—
173C2×Dic7C22×D311yescharacteristic
174C2×Dic7C22×D311yescharacteristic
175C2×Dic7C22×D311yescharacteristic
176C4×D7—34no—
177C4×D7—41no—
178C4×D7—121no—
179Dic14—41no—
180Dic14—121no—
181C2×C3×Q8—71no—
182C22×C3×C4—71no—
183C22×Dic3—71no—
184C22×Dic3—71no—
185C2×Dic6—71no—
186C2×Dic6C2×D711yescharacteristic
187C2×Dic6C2×D711yescharacteristic
188C3×C4⋊C4—71no—
189C3×C4⋊C4—71no—
190C3×C4⋊C4C2×D711yescharacteristic
191C3×C22⋊C4—71no—
192C3×C22⋊C4—71no—
193#217—71no—
194#217—71no—
195C4⋊Dic3—71no—
196C4⋊Dic3—71no—
197C4⋊Dic3C2×D711yescharacteristic
198Dic3⋊C4—71no—
199Dic3⋊C4—71no—
200Dic3⋊C4—71no—
201Dic3⋊C4—71no—
202Dic3⋊C4—71no—
203Dic3⋊C4—71no—
204Dic3⋊C4—71no—
205Dic3⋊C4—71no—
206Dic3⋊C4—71no—
207Dic3⋊C4—71no—
208Dic3⋊C4—71no—
209Dic3⋊C4—71no—
210Dic3⋊C4C2×D711yescharacteristic
211Dic3⋊C4C2×D711yescharacteristic
212C4×Dic3—71no—
213C4×Dic3—71no—
214C4×Dic3—71no—
215C4×Dic3C2×D711yescharacteristic
216C2×C3×C7C22×Q811yescharacteristic
217C2×C3×C7H(5,𝔽2)11yescharacteristic
218C2×C3×C72−1+411yescharacteristic
219C3×D7—22no—
220C4⋊Q8—31no—
221C4⋊Q8—211no—
222C4⋊Q8—211no—
223C4⋊Q8—211no—
224#125—211no—
225#125—211no—
226#125—211no—
227#125—211no—
228C22⋊Q8—71no—
229C22⋊Q8—211no—
230C22⋊Q8—211no—
231C22⋊Q8—211no—
232#117—211no—
233#117—211no—
234C2×C4⋊C4—211no—
235C22×C7C23×D311yescharacteristic
236C2×D7—21no—
237C2×D7—21no—
238C2×D7D3×Q812yes—
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—32no—
243C4×C7—61no—
244C4×C7—61no—
245C4×C7—61no—
246Dic7—21no—
247Dic7—21no—
248Dic7—21no—
249Dic7—32no—
250Dic7—61no—
251Dic7—61no—
252Dic7—61no—
253C23×C3—71no—
254C3×Q8—281no—
255C2×C3×C4—71no—
256C2×C3×C4—71no—
257C2×C3×C4—71no—
258C2×C3×C4—281no—
259C2×C3×C4C22×D711yescharacteristic
260C2×C3×C4C22×D711yescharacteristic
261C2×C3×C4C22×D711yescharacteristic
262C2×Dic3—71no—
263C2×Dic3—74no—
264C2×Dic3—71no—
265C2×Dic3—71no—
266C2×Dic3—71no—
267C2×Dic3—281no—
268C2×Dic3C22×D711yescharacteristic
269C2×Dic3C22×D711yescharacteristic
270C2×Dic3C22×D711yescharacteristic
271C2×Dic3C22×D711yescharacteristic
272Dic6—41no—
273Dic6—41no—
274Dic6—281no—
275C3×C7#54411yescharacteristic, complemented, Hall
276C2×Q8—31no—
277C2×Q8—31no—
278C2×Q8—71no—
279C2×Q8—211no—
280C22×C4—71no—
281C22×C4—211no—
282C22×C4—211no—
283C4⋊C4—31no—
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—
300C4⋊C4—211no—
301C4⋊C4—211no—
302C4⋊C4D3×D711yescharacteristic
303C22⋊C4—71no—
304C22⋊C4—71no—
305C22⋊C4—211no—
306C22⋊C4—211no—
307C42—31no—
308C42—211no—
309C42—211no—
310C42—211no—
311C2×C7#100411yescharacteristic
312C2×C7C2×D3×Q811yescharacteristic
313C2×C7#101011yescharacteristic
314D7—22no—
315C22×C3—72no—
316C22×C3—141no—
317C22×C3—141no—
318C22×C3C23×D711yescharacteristic
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—141no—
323C3×C4—141no—
324C3×C4—141no—
325Dic3—21no—
326Dic3—21no—
327Dic3—21no—
328Dic3—72no—
329Dic3—141no—
330Dic3—141no—
331Dic3—141no—
332Dic3Q8×D712yes—
333C23—71no—
334Q8—121no—
335Q8—121no—
336Q8—281no—
337Q8—841no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—71no—
343C2×C4—71no—
344C2×C4—71no—
345C2×C4—214no—
346C2×C4—211no—
347C2×C4—211no—
348C2×C4—211no—
349C2×C4—211no—
350C2×C4—281no—
351C2×C4—841no—
352C2×C4C2×D3×D711yescharacteristic
353C2×C4C2×D3×D711yescharacteristic
354C2×C4C2×D3×D711yescharacteristic
355C7#562311yescharacteristic, complemented, Sylow
356C2×C3—142no—
357C2×C3#654811yescharacteristic
358C2×C3C2×Q8×D711yescharacteristic
359C2×C3#655111yescharacteristic
360C22—72no—
361C22—141no—
362C22—141no—
363C22C22×D3×D711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—32no—
368C4—61no—
369C4—61no—
370C4—61no—
371C4—141no—
372C4—141no—
373C4—141no—
374C4—212no—
375C4—421no—
376C4—421no—
377C4—421no—
378C3#9042011yescharacteristic, complemented, Sylow
379C2—142no—
380C2#1062154711yescharacteristic
381C2#1062157211yescharacteristic
382C2D3×Q8×D711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC22×C3×D7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#544
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×#888C2semidirect
9. #10620947C2semidirect
10. #10620941C2semidirect
12. #10620905C2semidirect
13. #10620898C2semidirect
14. #10620891C2semidirect
15. #10620889C2semidirect
89. Dic3×D7Q8semidirect
135. #888D7semidirect
275. C3×C7#544semidirect
355. C7#5623semidirect
378. C3#90420semidirect

Maximal subgroups

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

subgroupindex
#106213722
C7×#8882
C3×#64592
#106209922
#106209832
#106209562
#106209552
D7×Dic3⋊C42
#106209472
#106209412
#106209212
#106209052
#106208982
#106208912
#106208892
#904203
#56237