Finite Group Databasebeta

← #11732829

Subgroups — #11732829

Order 896.

420 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
1C7×#387C211yesmaximal, characteristic, complemented
2#89766C211yesmaximal, characteristic, complemented
3#89764C211yesmaximal, characteristic, complemented
4#89763C211yesmaximal, characteristic
5#89756C211yesmaximal, characteristic, complemented
6#89711C211yesmaximal, characteristic
7C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
8#89447C211yesmaximal, characteristic, complemented
9#89446C211yesmaximal, characteristic, complemented
10#89444C211yesmaximal, characteristic
11#89442C211yesmaximal, characteristic, complemented
12#89442C211yesmaximal, characteristic, complemented
13#89441C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89433C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C42×C7C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41C7×#95C2211yescharacteristic
42C7×#95C2211yescharacteristic
43C7×#95C2211yescharacteristic
44C7×#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2651—71nomaximal, Sylow
52C23×D7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C2×D28—22no—
68C2×D28—41no—
69C2×D28—41no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—41no—
72C42×C7—41no—
73(C2×D7)⋊C4—22no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—22no—
76(C2×D7)⋊C4—22no—
77(C2×D7)⋊C4—41no—
78(C2×D7)⋊C4—41no—
79(C2×D7)⋊C4—41no—
80(C2×D7)⋊C4—41no—
81(C2×D7)⋊C4—41no—
82(C2×D7)⋊C4—41no—
83(C2×D7)⋊C4—41no—
84(C2×D7)⋊C4—41no—
85C4⋊Dic7—41no—
86C4⋊Dic7—41no—
87C4⋊Dic7—41no—
88Dic7⋊C4—22no—
89Dic7⋊C4—22no—
90Dic7⋊C4—41no—
91Dic7⋊C4—41no—
92C4×Dic7—22no—
93C4×Dic7—41no—
94#399—71no—
95#398—71no—
96#398—71no—
97#397—71no—
98#397—71no—
99#396—71no—
100#394—71no—
101#394—71no—
102#390—71no—
103#388—71no—
104#388—71no—
105#388—71no—
106#388—71no—
107#387D711yescharacteristic, complemented
108C4×C4⋊C4—71no—
109C23×C7C2411yescharacteristic
110C22×D7—21no—
111C22×D7—21no—
112C22×D7—21no—
113C22×D7—21no—
114C22×D7—21no—
115C22×D7—21no—
116C22×D7—21no—
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—21no—
120C22×D7—21no—
121C22×D7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7#4112yes—
144C2×C4×C7#4112yes—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7#4112yes—
163D28—44no—
164C22×D4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4C2×D711yescharacteristic
173C2×C4⋊C4C2×D711yescharacteristic
174C2×C22⋊C4—71no—
175C2×C22⋊C4—71no—
176C2×C22⋊C4—71no—
177C2×C22⋊C4—71no—
178C2×C22⋊C4—71no—
179C2×C22⋊C4—71no—
180C2×C22⋊C4—71no—
181C2×C22⋊C4—71no—
182C2×C22⋊C4—71no—
183C2×C22⋊C4—71no—
184C2×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C42—71no—
187C2×C42—71no—
188C2×C42C2×D711yescharacteristic
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95—71no—
194#95—71no—
195#95C2×D711yescharacteristic
196#95C2×D711yescharacteristic
197#95C2×D711yescharacteristic
198#95C2×D711yescharacteristic
199C22×C7C2×#4111yescharacteristic
200C22×C7C2×#4111yescharacteristic
201C22×C7C2×#4111yescharacteristic
202C22×C7H(5,𝔽2)11yescharacteristic
203C22×C7H(5,𝔽2)11yescharacteristic
204C22×C7H(5,𝔽2)11yescharacteristic
205C22×C72−1+411yescharacteristic
206C2×D7—41no—
207C2×D7—41no—
208C2×D7—41no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—22no—
221C4×C7—22no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227Dic7—22no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233C24—71no—
234C24—71no—
235C2×D4—142no—
236C2×D4—281no—
237C2×D4—281no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4C22×D711yescharacteristic
245C22×C4C22×D711yescharacteristic
246C22×C4C22×D711yescharacteristic
247C22×C4C22×D711yescharacteristic
248C22×C4C22×D711yescharacteristic
249C22×C4C22×D711yescharacteristic
250C22×C4C22×D711yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—142no—
254C4⋊C4—142no—
255C4⋊C4—281no—
256C4⋊C4—281no—
257C4⋊C4—281no—
258C4⋊C4—281no—
259C4⋊C4—281no—
260C22⋊C4—142no—
261C22⋊C4—142no—
262C22⋊C4—142no—
263C22⋊C4—142no—
264C22⋊C4—281no—
265C22⋊C4—281no—
266C22⋊C4—281no—
267C22⋊C4—281no—
268C22⋊C4—281no—
269C22⋊C4—281no—
270C22⋊C4—281no—
271C22⋊C4—281no—
272C42—41no—
273C42—142no—
274C42—281no—
275C2×C7#52911yescharacteristic
276C2×C7#53811yescharacteristic
277C2×C7#53911yescharacteristic
278C2×C7#54011yescharacteristic
279C2×C7#54011yescharacteristic
280C2×C7#54211yescharacteristic
281C2×C7#54211yescharacteristic
282D7—81no—
283D7—81no—
284C23—141no—
285C23—141no—
286C23—141no—
287C23—141no—
288C23—141no—
289C23—141no—
290C23—141no—
291C23—141no—
292C23—141no—
293C23—141no—
294C23—141no—
295C23—141no—
296C23—141no—
297C23—141no—
298C23C23×D711yescharacteristic
299D4—284no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—72no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4#115712yes—
338C2×C4#116112yes—
339C7#265111yescharacteristic, complemented, Sylow
340C22—281no—
341C22—281no—
342C22—281no—
343C22—281no—
344C22—281no—
345C22—281no—
346C22—281no—
347C22—281no—
348C22—281no—
349C22—281no—
350C22—281no—
351C22—281no—
352C22—281no—
353C22—281no—
354C22C2×#115711yescharacteristic
355C22#654811yescharacteristic
356C22C2×#116111yescharacteristic
357C22D7×#4111yescharacteristic
358C22#655311yescharacteristic
359C22#655311yescharacteristic
360C22#655411yescharacteristic
361C4—22no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—142no—
369C4—281no—
370C4—281no—
371C4—281no—
372C4—281no—
373C4—281no—
374C2—561no—
375C2—561no—
376C2#9022511yescharacteristic
377C2#9025111yescharacteristic
378C2#9027111yescharacteristic
379C2#9033611yescharacteristic
380C2#9035411yescharacteristic
381C2#9038911yescharacteristic
382C2#9040111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#387C2
prime radicalC22×D28C22
hypercenter#387D7
hypocenterC7#2651
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C7×#387C2semidirect
2. #89766C2semidirect
3. #89764C2semidirect
5. #89756C2semidirect
7. C4×Dic7⋊C4C2semidirect
8. #89447C2semidirect
9. #89446C2semidirect
11. #89442C2semidirect
12. #89442C2semidirect
13. #89441C2semidirect
14. #89441C2semidirect
15. #89433C2semidirect
107. #387D7semidirect
339. C7#2651semidirect

Maximal subgroups

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

subgroupindex
C7×#3872
#897662
#897642
#897632
#897562
#897112
C4×Dic7⋊C42
#894472
#894462
#894442
#894422
#894412
#894332
#26517