Finite Group Databasebeta

← #13058753

Subgroups — #13058753

Order 1344.

392 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#10621372C211yesmaximal, characteristic, complemented
2C7×#894C211yesmaximal, characteristic
3C3×#6451C211yesmaximal, characteristic, complemented
4#10620998C211yesmaximal, characteristic, complemented
5#10620985C211yesmaximal, characteristic
6#10620972C211yesmaximal, characteristic, complemented
7#10620944C211yesmaximal, characteristic, complemented
8#10620934C211yesmaximal, characteristic, complemented
9#10620934C211yesmaximal, characteristic, complemented
10#10620932C211yesmaximal, characteristic
11#10620931C211yesmaximal, characteristic, complemented
12#10620914C211yesmaximal, characteristic, complemented
13#10620904C211yesmaximal, characteristic, complemented
14#10620903C211yesmaximal, characteristic, complemented
15#10620889C211yesmaximal, characteristic, complemented
16#90342—31nomaximal, Hall
17C2×C7×D12C2211yescharacteristic
18C2×C4×D3×C7C2211yescharacteristic
19C2×C3×Dic14C2211yescharacteristic
20C2×C3×Dic14C2211yescharacteristic
21C2×#4248C2211yescharacteristic
22C2×D3×Dic7C2211yescharacteristic
23C3×C7×C4⋊C4C2211yescharacteristic
24C4⋊Dic21C2211yescharacteristic
25#66629C2211yescharacteristic
26#66629C2211yescharacteristic
27#66629C2211yescharacteristic
28#66629C2211yescharacteristic
29C4×Dic21C2211yescharacteristic
30C7×(C2×D3)⋊C4C2211yescharacteristic
31C7×(C2×D3)⋊C4C2211yescharacteristic
32C7×(C2×D3)⋊C4C2211yescharacteristic
33C7×Dic3⋊C4C2211yescharacteristic
34C3×C4⋊Dic7C2211yescharacteristic
35C3×Dic7⋊C4C2211yescharacteristic
36C3×Dic7⋊C4C2211yescharacteristic
37C3×C4×Dic7C2211yescharacteristic
38Dic3⋊Dic7C2211yescharacteristic
39Dic3⋊Dic7C2211yescharacteristic
40#66577C2211yescharacteristic
41#66577C2211yescharacteristic
42#66577C2211yescharacteristic
43Dic7⋊Dic3C2211yescharacteristic
44(C2×D3)⋊Dic7C2211yescharacteristic
45(C2×D3)⋊Dic7C2211yescharacteristic
46(C2×D3)⋊Dic7C2211yescharacteristic
47(C2×D3)⋊Dic7C2211yescharacteristic
48(C2×D3)⋊Dic7C2211yescharacteristic
49(C2×D3)⋊Dic7C2211yescharacteristic
50(C2×D3)⋊Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#123—31no—
53#6499—31no—
54#6498—31no—
55#6487—31no—
56#6487—31no—
57#6452—31no—
58#6452—31no—
59#6451—31no—
60#6451D311yescharacteristic, complemented
61#6442—31no—
62#6442—31no—
63#6442—31no—
64#6442—31no—
65#6441—31no—
66#6441—31no—
67#5735—71nomaximal, Hall
68C22×D3×C7C2311yescharacteristic
69C22×D3×C7C2311yescharacteristic
70C2×C3×C4×C7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×C3×C4×C7C2311yescharacteristic
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C2×Dic21C2311yescharacteristic
77C2×C7×Dic3C2311yescharacteristic
78C2×C7×Dic3C2311yescharacteristic
79C7×D12—41no—
80C4×D3×C7—41no—
81C2×C3×Dic7C2311yescharacteristic
82C2×C3×Dic7C2311yescharacteristic
83C2×C3×Dic7C2311yescharacteristic
84C2×C3×Dic7C2311yescharacteristic
85C3×Dic14—41no—
86C3×Dic14—41no—
87#4248—41no—
88D3×Dic7—41no—
89C2×C7×D4—31no—
90C22×C4×C7—31no—
91C22×Dic7—31no—
92C2×Dic14—31no—
93C2×Dic14C2×D311yescharacteristic
94C2×Dic14C2×D311yescharacteristic
95C7×C4⋊C4—31no—
96C7×C4⋊C4C2×D311yescharacteristic
97C7×C22⋊C4—31no—
98C7×C22⋊C4—31no—
99C7×C22⋊C4—31no—
100C22⋊Dic7—31no—
101C22⋊Dic7—31no—
102C22⋊Dic7—31no—
103C22⋊Dic7—31no—
104C22⋊Dic7—31no—
105C22⋊Dic7—31no—
106C22⋊Dic7—31no—
107C4⋊Dic7—31no—
108C4⋊Dic7—31no—
109C4⋊Dic7—31no—
110C4⋊Dic7C2×D311yescharacteristic
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×Dic7—31no—
123C4×Dic7C2×D311yescharacteristic
124C3×C4⋊Q8—71no—
125#947—71no—
126#946—71no—
127#946—71no—
128#944—71no—
129#898—71no—
130#898—71no—
131#896—71no—
132#896—71no—
133#894—71no—
134#894D711yescharacteristic
135#889—71no—
136#889—71no—
137#876—71no—
138#876—71no—
139C22×C3×C7C2411yescharacteristic
140C2×D3×C7—21no—
141C2×D3×C7—21no—
142C2×D3×C7—21no—
143C2×D3×C7—21no—
144C2×D3×C7—21no—
145C2×D3×C7—21no—
146C3×C4×C7—21no—
147C3×C4×C7—21no—
148C3×C4×C7—21no—
149Dic21—21no—
150Dic21—21no—
151Dic21—21no—
152Dic21—21no—
153C3×Dic7—21no—
154C3×Dic7—21no—
155C3×Dic7—21no—
156C3×Dic7—21no—
157C7×Dic3—21no—
158C7×Dic3—21no—
159#563—211noSylow
160C23×C7—31no—
161C23×C7—31no—
162C7×D4—121no—
163C2×C4×C7—31no—
164C2×C4×C7—31no—
165C2×C4×C7—121no—
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×Dic7—121no—
174C2×Dic7C22×D311yescharacteristic
175C2×Dic7C22×D311yescharacteristic
176C2×Dic7C22×D311yescharacteristic
177C2×Dic7C22×D311yescharacteristic
178Dic14—41no—
179Dic14—41no—
180Dic14—121no—
181C2×C3×Q8—71no—
182C2×C3×Q8—71no—
183C2×D12C2×D711yescharacteristic
184C2×C4×D3—71no—
185C2×C4×D3C2×D711yescharacteristic
186C2×Dic6—71no—
187C3×C4⋊C4—71no—
188C3×C4⋊C4—71no—
189C3×C4⋊C4—71no—
190C3×C4⋊C4C2×D711yescharacteristic
191C3×C42—71no—
192(C2×D3)⋊C4—71no—
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
201(C2×D3)⋊C4C2×D711yescharacteristic
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⋊C4—71no—
213Dic3⋊C4C2×D711yescharacteristic
214C4×Dic3—71no—
215C4×Dic3—71no—
216C2×C3×C7H(5,𝔽2)11yescharacteristic
217C2×C3×C72−1+411yescharacteristic
218C2×C3×C72−1+411yescharacteristic
219D3×C7—41no—
220D3×C7—41no—
221C4⋊Q8—71no—
222C4⋊Q8—211no—
223#126—211no—
224#126—211no—
225#126—211no—
226#126—211no—
227#125—211no—
228#125—211no—
229#124—211no—
230#123—31no—
231#123—211no—
232C22⋊Q8—211no—
233C22⋊Q8—211no—
234C22⋊Q8—211no—
235C22⋊Q8—211no—
236C22×C7—61no—
237C22×C7—61no—
238C22×C7—61no—
239C22×C7—61no—
240C22×C7—61no—
241C22×C7—61no—
242C22×C7C23×D311yescharacteristic
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—61no—
247C4×C7—61no—
248Dic7—21no—
249Dic7—21no—
250Dic7—21no—
251Dic7—21no—
252Dic7—61no—
253Dic7—61no—
254Dic7—61no—
255Dic7—61no—
256C22×D3C22×D711yescharacteristic
257C22×D3C22×D711yescharacteristic
258C3×Q8—281no—
259C3×Q8—281no—
260C2×C3×C4—71no—
261C2×C3×C4—71no—
262C2×C3×C4—71no—
263C2×C3×C4—71no—
264C2×C3×C4C22×D711yescharacteristic
265C2×C3×C4C22×D711yescharacteristic
266C2×C3×C4C22×D711yescharacteristic
267C2×Dic3—71no—
268C2×Dic3—71no—
269C2×Dic3—71no—
270C2×Dic3—71no—
271C2×Dic3C22×D711yescharacteristic
272C2×Dic3C22×D711yescharacteristic
273D12—41no—
274C4×D3—41no—
275C4×D3—281no—
276Dic6—281no—
277C3×C7#56311yescharacteristic, complemented, Hall
278C2×Q8—71no—
279C2×Q8—71no—
280C2×Q8—211no—
281C2×D4—31no—
282C22×C4—31no—
283C22×C4—211no—
284C4⋊C4—31no—
285C4⋊C4—71no—
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⋊C4D3×D711yescharacteristic
300C22⋊C4—31no—
301C22⋊C4—31no—
302C22⋊C4—31no—
303C22⋊C4—211no—
304C22⋊C4—211no—
305C22⋊C4—211no—
306C22⋊C4—211no—
307C22⋊C4—211no—
308C22⋊C4—211no—
309C22⋊C4—211no—
310C42—71no—
311C42—211no—
312C42—211no—
313C2×C7—121no—
314C2×C7—121no—
315C2×C7#100411yescharacteristic
316C2×C7#100711yescharacteristic
317C2×C7#100711yescharacteristic
318C22×C3C23×D711yescharacteristic
319C2×D3—21no—
320C2×D3—21no—
321C2×D3—21no—
322C2×D3—21no—
323C2×D3—21no—
324C2×D3—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—141no—
329C3×C4—141no—
330C3×C4—141no—
331C3×C4—141no—
332Dic3—21no—
333Dic3—21no—
334Dic3—141no—
335Dic3—141no—
336Dic3—141no—
337Dic3—141no—
338C23—31no—
339C23—31no—
340Q8—281no—
341Q8—281no—
342Q8—841no—
343D4—121no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—71no—
347C2×C4—71no—
348C2×C4—71no—
349C2×C4—71no—
350C2×C4—121no—
351C2×C4—211no—
352C2×C4—211no—
353C2×C4—211no—
354C2×C4—211no—
355C2×C4—841no—
356C2×C4C2×D3×D711yescharacteristic
357C2×C4C2×D3×D711yescharacteristic
358C2×C4C2×D3×D711yescharacteristic
359C7#573511yescharacteristic, complemented, Sylow
360C2×C3#654811yescharacteristic
361C2×C3#655411yescharacteristic
362C2×C3#655411yescharacteristic
363D3—41no—
364D3—41no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—61no—
370C22—61no—
371C22C22×D3×D711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—61no—
376C4—61no—
377C4—141no—
378C4—141no—
379C4—141no—
380C4—141no—
381C4—421no—
382C4—421no—
383C4—421no—
384C4—421no—
385C3#9034211yescharacteristic, complemented, Sylow
386C2—121no—
387C2—121no—
388C2#1062155111yescharacteristic
389C2#1062157211yescharacteristic
390C2#1062157711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC2×C7×D12C22
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#563
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
3. C3×#6451C2semidirect
4. #10620998C2semidirect
6. #10620972C2semidirect
7. #10620944C2semidirect
8. #10620934C2semidirect
9. #10620934C2semidirect
11. #10620931C2semidirect
12. #10620914C2semidirect
13. #10620904C2semidirect
14. #10620903C2semidirect
15. #10620889C2semidirect
60. #6451D3semidirect
277. C3×C7#563semidirect
359. C7#5735semidirect
385. C3#90342semidirect

Maximal subgroups

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

subgroupindex
#106213722
C7×#8942
C3×#64512
#106209982
#106209852
#106209722
#106209442
#106209342
#106209322
#106209312
#106209142
#106209042
#106209032
#106208892
#903423
#57357