Finite Group Databasebeta

← #11735461

Subgroups — #11735461

Order 896.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#398C211yesmaximal, characteristic, complemented
2#89766C211yesmaximal, characteristic
3#89766C211yesmaximal, characteristic
4#89764C211yesmaximal, characteristic
5#89759C211yesmaximal, characteristic
6#89756C211yesmaximal, characteristic, complemented
7#89752C211yesmaximal, characteristic, complemented
8#89751C211yesmaximal, characteristic, complemented
9#89751C211yesmaximal, characteristic, complemented
10#89445C211yesmaximal, characteristic
11#89439C211yesmaximal, characteristic
12#89436C211yesmaximal, characteristic
13#89433C211yesmaximal, characteristic, complemented
14#89432C211yesmaximal, characteristic, complemented
15#89431C211yesmaximal, characteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊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×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C7×#95C2211yescharacteristic
38C7×#95C2211yescharacteristic
39C7×#95C2211yescharacteristic
40C7⋊#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2841—71nomaximal, Sylow
52C23×D7C2311yescharacteristic
53C22×C4×C7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×Dic7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C2×C4×D7—41no—
68C2×C4×D7—41no—
69C2×C4×D7Q814yescomplemented
70C2×Dic14—41no—
71C2×Dic14—41no—
72C2×Dic14—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
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—
83C4⋊Dic7—41no—
84C4⋊Dic7—41no—
85C4⋊Dic7—41no—
86Dic7⋊C4—41no—
87Dic7⋊C4—41no—
88Dic7⋊C4—41no—
89Dic7⋊C4—41no—
90Dic7⋊C4—41no—
91Dic7⋊C4—41no—
92#400—71no—
93#400—71no—
94#399—71no—
95#398—71no—
96#398—71no—
97#398—71no—
98#398D711yescharacteristic, complemented
99#396—71no—
100#396—71no—
101#395—71no—
102#395—71no—
103#393—71no—
104#381—71no—
105#380—71no—
106#380—71no—
107C23×C7C2411yescharacteristic
108C22×D7—21no—
109C22×D7—21no—
110C22×D7—21no—
111C22×D7—21no—
112C22×D7—21no—
113C22×D7—21no—
114C22×D7C2×Q812yes—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—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×C7C2×Q812yes—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
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×Dic7C2×Q812yes—
157C4×D7—44no—
158Dic14—44no—
159C22×Q8—71no—
160C23×C4—71no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4C2×D711yescharacteristic
171C2×C4⋊C4C2×D711yescharacteristic
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—
180#95—71no—
181#95—71no—
182#95—71no—
183#95—71no—
184#95—71no—
185#95—71no—
186#95—71no—
187#95—71no—
188#95—71no—
189#95—71no—
190#95—71no—
191#95C2×D711yescharacteristic
192#95C2×D711yescharacteristic
193#95C2×D711yescharacteristic
194C22×C7C22×Q811yescharacteristic
195C22×C7H(5,𝔽2)11yescharacteristic
196C22×C7H(5,𝔽2)11yescharacteristic
197C22×C7H(5,𝔽2)11yescharacteristic
198C22×C7H(5,𝔽2)11yescharacteristic
199C22×C72−1+411yescharacteristic
200C22×C72−1+411yescharacteristic
201C2×D7—22no—
202C2×D7—22no—
203C2×D7—22no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C2×D7—41no—
208C4×C7—22no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215Dic7—22no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222C24—71no—
223C2×Q8—281no—
224C2×Q8—281no—
225C2×Q8—281no—
226C22×C4—71no—
227C22×C4—74no—
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—281no—
235C22×C4—281no—
236C22×C4C22×D711yescharacteristic
237C22×C4C22×D711yescharacteristic
238C22×C4C22×D711yescharacteristic
239C22×C4C22×D711yescharacteristic
240C22×C4C22×D711yescharacteristic
241C22×C4C22×D711yescharacteristic
242C22×C4C22×D711yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—281no—
248C4⋊C4—281no—
249C4⋊C4—281no—
250C4⋊C4—281no—
251C4⋊C4—281no—
252C4⋊C4—281no—
253C4⋊C4—281no—
254C4⋊C4—281no—
255C4⋊C4—281no—
256C22⋊C4—281no—
257C22⋊C4—281no—
258C22⋊C4—281no—
259C22⋊C4—281no—
260C22⋊C4—281no—
261C22⋊C4—281no—
262C2×C7#54311yescharacteristic
263C2×C7#54411yescharacteristic
264C2×C7#54411yescharacteristic
265C2×C7#56011yescharacteristic
266C2×C7#56211yescharacteristic
267C2×C7#56211yescharacteristic
268C2×C7#56311yescharacteristic
269D7—42no—
270C23—72no—
271C23—141no—
272C23—141no—
273C23—141no—
274C23—141no—
275C23—141no—
276C23—141no—
277C23C23×D711yescharacteristic
278Q8—284no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—72no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—284no—
321C2×C4Q8×D712yes—
322C7#284111yescharacteristic, complemented, Sylow
323C22—142no—
324C22—142no—
325C22—142no—
326C22—281no—
327C22—281no—
328C22—281no—
329C22—281no—
330C22#654811yescharacteristic
331C22#654811yescharacteristic
332C22C2×Q8×D711yescharacteristic
333C22#655111yescharacteristic
334C22#655311yescharacteristic
335C22#655311yescharacteristic
336C22#655411yescharacteristic
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—142no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4—281no—
350C4—281no—
351C2—282no—
352C2#9031111yescharacteristic
353C2#9032511yescharacteristic
354C2#9033711yescharacteristic
355C2#9035811yescharacteristic
356C2#9038011yescharacteristic
357C2#9039111yescharacteristic
358C2#9042011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC23×D7C23
hypercenter#398D7
hypocenterC7#2841
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C7×#398C2semidirect
6. #89756C2semidirect
7. #89752C2semidirect
8. #89751C2semidirect
9. #89751C2semidirect
13. #89433C2semidirect
14. #89432C2semidirect
15. #89431C2semidirect
69. C2×C4×D7Q8semidirect
98. #398D7semidirect
322. C7#2841semidirect

Maximal subgroups

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

subgroupindex
C7×#3982
#897662
#897642
#897592
#897562
#897522
#897512
#894452
#894392
#894362
#894332
#894322
#894312
#28417