Subgroups — D₇×#576 #11744270

Order 896.

1512 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₇×#576	C₂	1	1	yes	maximal, characteristic, complemented
3	#90526	C₂	1	2	yes	maximal, complemented
4	#90465	C₂	1	6	yes	maximal, complemented
5	D₇×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
6	#90460	C₂	1	3	yes	maximal, complemented
7	D₇×#135	C₂	1	3	yes	maximal, complemented
8	#90448	C₂	1	3	yes	maximal, complemented
9	C₂×D₇×D₈	C₂	1	3	yes	maximal, complemented
10	#90443	C₂	1	1	yes	maximal, characteristic, complemented
11	D₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×H(5,𝔽₂)	C₂²	1	2	yes	complemented
13	#6553	—	2	2	no	—
14	#6553	C₂²	1	6	yes	complemented
15	#6553	C₂²	1	2	yes	complemented
16	D₇×#41	—	2	2	no	—
17	D₇×#41	C₂²	1	1	yes	characteristic
18	D₇×#41	C₂²	1	6	yes	complemented
19	#6548	—	2	6	no	—
20	#6548	C₂²	1	6	yes	complemented
21	C₂×D₄×D₇	—	2	6	no	—
22	C₂×D₄×D₇	C₂²	1	6	yes	complemented
23	C₇×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
24	C₇×#135	C₂²	1	3	yes	complemented
25	C₂×C₇×D₈	C₂²	1	3	yes	complemented
26	C₇×#131	C₂²	1	1	yes	characteristic, complemented
27	#6513	C₂²	1	6	yes	complemented
28	#6512	C₂²	1	6	yes	complemented
29	#6511	C₂²	1	1	yes	characteristic, complemented
30	#6495	C₂²	1	6	yes	complemented
31	C₂×#1141	C₂²	1	6	yes	complemented
32	#6482	C₂²	1	3	yes	complemented
33	D₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
34	#6479	C₂²	1	6	yes	complemented
35	#6477	C₂²	1	12	yes	complemented
36	D₇×SD₁₆	C₂²	1	6	yes	complemented
37	#6475	C₂²	1	3	yes	complemented
38	#6474	C₂²	1	12	yes	complemented
39	D₇×D₈	C₂²	1	3	yes	complemented
40	D₇×D₈	C₂²	1	6	yes	complemented
41	D₇×D₈	C₂²	1	6	yes	complemented
42	D₇×D₈	C₂²	1	6	yes	complemented
43	#6471	C₂²	1	6	yes	complemented
44	#6470	C₂²	1	3	yes	complemented
45	D₇×M₁₆	C₂²	1	3	yes	complemented
46	#6467	C₂²	1	3	yes	complemented
47	C₂×D₅₆	C₂²	1	3	yes	complemented
48	#6464	C₂²	1	3	yes	complemented
49	C₂×C₈×D₇	C₂²	1	3	yes	complemented
50	C₂×#576	—	7	1	no	maximal, Sylow
51	C₂³×D₇	—	2	6	no	—
52	C₇×#41	—	2	2	no	—
53	C₇×#41	C₂³	1	1	yes	characteristic
54	C₇×#41	C₂³	1	6	yes	complemented
55	C₂×C₇×D₄	—	2	6	no	—
56	C₂×C₇×D₄	C₂³	1	6	yes	complemented
57	C₂×C₂²⋊D₇	—	2	12	no	—
58	C₂×C₂²⋊D₇	—	2	6	no	—
59	#1161	—	2	2	no	—
60	#1161	D₄	1	1	yes	characteristic
61	#1161	C₂³	1	6	yes	complemented
62	Q₈×D₇	D₄	1	1	yes	characteristic
63	Q₈×D₇	C₂³	1	2	yes	—
64	#1159	—	2	6	no	—
65	#1159	—	2	6	no	—
66	#1159	D₄	1	3	yes	—
67	#1159	C₂³	1	6	yes	complemented
68	D₄×D₇	—	2	6	no	—
69	D₄×D₇	—	2	6	no	—
70	D₄×D₇	—	2	12	no	—
71	D₄×D₇	—	2	6	no	—
72	D₄×D₇	—	2	6	no	—
73	D₄×D₇	—	2	6	no	—
74	D₄×D₇	D₄	1	3	yes	—
75	D₄×D₇	C₂³	1	6	yes	—
76	D₄×D₇	C₂³	1	12	yes	complemented
77	#1157	—	2	6	no	—
78	#1157	—	2	6	no	—
79	#1157	C₂³	1	3	yes	—
80	#1157	C₂³	1	6	yes	complemented
81	C₂×D₂₈	—	2	6	no	—
82	C₂×D₂₈	C₂³	1	6	yes	complemented
83	C₂×C₄×D₇	—	2	6	no	—
84	C₂×C₄×D₇	C₂³	1	3	yes	—
85	C₇×Q₁₆	C₂³	1	1	yes	characteristic, complemented
86	C₇×SD₁₆	C₂³	1	6	yes	complemented
87	C₇×D₈	C₂³	1	3	yes	complemented
88	C₇×D₈	C₂³	1	6	yes	complemented
89	C₇×M₁₆	C₂³	1	3	yes	complemented
90	C₂×C₇×C₈	C₂³	1	3	yes	complemented
91	#1144	C₂³	1	2	yes	complemented
92	#1143	C₂³	1	6	yes	complemented
93	#1142	C₂³	1	6	yes	complemented
94	#1141	C₂³	1	6	yes	complemented
95	#1141	C₂³	1	12	yes	complemented
96	#1136	C₂³	1	3	yes	complemented
97	C₂×C₇⋊C₈	C₂³	1	3	yes	complemented
98	Dic₂₈	C₂³	1	1	yes	characteristic, complemented
99	D₅₆	C₂³	1	3	yes	complemented
100	D₅₆	C₂³	1	6	yes	complemented
101	#1132	C₂³	1	6	yes	complemented
102	#1131	C₂³	1	6	yes	complemented
103	C₈×D₇	C₂³	1	1	yes	characteristic
104	C₈×D₇	C₂³	1	3	yes	—
105	C₈×D₇	C₂³	1	3	yes	complemented
106	C₈×D₇	C₂³	1	3	yes	complemented
107	C₂×H(5,𝔽₂)	—	7	2	no	—
108	#576	—	7	2	no	—
109	#576	—	7	6	no	—
110	#576	—	7	3	no	—
111	#576	—	7	3	no	—
112	#576	—	7	1	no	—
113	#576	D₇	1	1	yes	characteristic, direct product factor
114	C₂×AGL(1,ℤ/8)	—	7	6	no	—
115	C₂×#135	—	7	3	no	—
116	C₂²×D₈	—	7	3	no	—
117	C₂×#131	—	7	1	no	—
118	C₂³×C₇	—	2	6	no	—
119	C₂²×D₇	—	2	6	no	—
120	C₂²×D₇	—	2	6	no	—
121	C₂²×D₇	—	2	12	no	—
122	C₂²×D₇	—	8	12	no	—
123	C₂²×D₇	C₂×D₄	1	3	yes	—
124	C₇×Q₈	C₂×D₄	1	1	yes	characteristic
125	C₇×Q₈	C₂⁴	1	2	yes	—
126	C₇×D₄	—	2	6	no	—
127	C₇×D₄	—	2	6	no	—
128	C₇×D₄	C₂×D₄	1	3	yes	—
129	C₇×D₄	C₂⁴	1	6	yes	—
130	C₂×C₄×C₇	—	2	6	no	—
131	C₂×C₄×C₇	C₂⁴	1	3	yes	—
132	C₂²⋊D₇	—	2	6	no	—
133	C₂²⋊D₇	—	2	6	no	—
134	C₂²⋊D₇	—	2	6	no	—
135	C₂²⋊D₇	—	2	12	no	—
136	C₂²⋊D₇	C₂×D₄	1	6	yes	—
137	C₂×Dic₇	—	2	6	no	—
138	C₂×Dic₇	C₂×D₄	1	3	yes	—
139	D₂₈	—	2	6	no	—
140	D₂₈	—	2	6	no	—
141	D₂₈	C₂×D₄	1	3	yes	—
142	D₂₈	C₂⁴	1	6	yes	—
143	C₄×D₇	—	2	2	no	—
144	C₄×D₇	—	2	6	no	—
145	C₄×D₇	—	2	6	no	—
146	C₄×D₇	C₂×D₄	1	3	yes	—
147	C₄×D₇	C₂×D₄	1	3	yes	—
148	C₄×D₇	C₂⁴	1	1	yes	characteristic
149	C₄×D₇	C₂⁴	1	3	yes	—
150	Dic₁₄	—	2	2	no	—
151	Dic₁₄	C₂×D₄	1	3	yes	—
152	Dic₁₄	C₂⁴	1	2	yes	—
153	C₇×C₈	C₂⁴	1	1	yes	characteristic
154	C₇×C₈	C₂⁴	1	3	yes	—
155	C₇⋊C₈	C₂⁴	1	1	yes	characteristic
156	C₇⋊C₈	C₂⁴	1	3	yes	—
157	H(5,𝔽₂)	—	7	2	no	—
158	H(5,𝔽₂)	—	7	6	no	—
159	H(5,𝔽₂)	—	7	6	no	—
160	H(5,𝔽₂)	—	14	6	no	—
161	H(5,𝔽₂)	—	14	2	no	—
162	H(5,𝔽₂)	C₂×D₇	1	2	yes	complemented
163	C₂×#41	—	7	1	no	—
164	C₂×#41	—	7	6	no	—
165	C₂×#41	—	14	2	no	—
166	C₂²×D₄	—	7	6	no	—
167	C₂²×D₄	—	14	6	no	—
168	AGL(1,ℤ/8)	—	7	6	no	—
169	AGL(1,ℤ/8)	—	7	6	no	—
170	AGL(1,ℤ/8)	—	7	12	no	—
171	AGL(1,ℤ/8)	—	7	12	no	—
172	AGL(1,ℤ/8)	—	7	6	no	—
173	AGL(1,ℤ/8)	C₂×D₇	1	6	yes	complemented
174	#135	—	7	6	no	—
175	#135	—	7	3	no	—
176	#135	—	7	6	no	—
177	#135	—	7	3	no	—
178	#135	—	7	3	no	—
179	#135	C₂×D₇	1	3	yes	complemented
180	C₂×Q₁₆	—	7	1	no	—
181	C₂×SD₁₆	—	7	6	no	—
182	C₂×D₈	—	7	6	no	—
183	C₂×D₈	—	7	3	no	—
184	C₂×D₈	—	7	6	no	—
185	C₂×D₈	—	7	6	no	—
186	C₂×D₈	—	7	6	no	—
187	C₂×D₈	—	7	3	no	—
188	C₂×D₈	C₂×D₇	1	3	yes	complemented
189	#131	—	7	1	no	—
190	#131	—	7	3	no	—
191	#131	—	7	3	no	—
192	#131	C₂×D₇	1	1	yes	characteristic, complemented
193	C₂×M₁₆	—	7	3	no	—
194	C₂²×C₈	—	7	3	no	—
195	C₂²×C₇	—	2	6	no	—
196	C₂²×C₇	—	8	6	no	—
197	C₂²×C₇	C₂²×D₄	1	3	yes	—
198	C₂×D₇	—	2	6	no	—
199	C₂×D₇	—	2	6	no	—
200	C₂×D₇	—	4	12	no	—
201	C₂×D₇	—	8	12	no	—
202	C₂×D₇	—	8	6	no	—
203	C₂×D₇	C₂²×D₄	1	3	yes	—
204	C₂×D₇	C₂²×D₄	1	1	yes	characteristic
205	C₄×C₇	—	2	2	no	—
206	C₄×C₇	C₂²×D₄	1	3	yes	—
207	C₄×C₇	C₂⁵	1	1	yes	characteristic
208	Dic₇	—	2	2	no	—
209	Dic₇	C₂²×D₄	1	1	yes	characteristic
210	Dic₇	C₂²×D₄	1	3	yes	—
211	C₂⁴	—	14	6	no	—
212	#41	—	2	2	no	—
213	#41	—	7	1	no	—
214	#41	—	7	3	no	—
215	#41	—	7	3	no	—
216	#41	—	7	6	no	—
217	#41	—	7	6	no	—
218	#41	—	7	6	no	—
219	#41	—	14	6	no	—
220	#41	—	14	6	no	—
221	#41	—	14	6	no	—
222	#41	—	14	6	no	—
223	#41	—	14	2	no	—
224	#41	C₂²×D₇	1	1	yes	characteristic
225	#41	C₂²×D₇	1	6	yes	complemented
226	C₂×Q₈	—	7	1	no	—
227	C₂×Q₈	—	7	2	no	—
228	C₂×D₄	—	2	6	no	—
229	C₂×D₄	—	7	3	no	—
230	C₂×D₄	—	7	6	no	—
231	C₂×D₄	—	7	12	no	—
232	C₂×D₄	—	7	6	no	—
233	C₂×D₄	—	14	6	no	—
234	C₂×D₄	—	14	6	no	—
235	C₂×D₄	—	14	6	no	—
236	C₂×D₄	—	14	6	no	—
237	C₂×D₄	—	14	12	no	—
238	C₂×D₄	—	14	6	no	—
239	C₂×D₄	—	14	6	no	—
240	C₂×D₄	—	14	12	no	—
241	C₂×D₄	—	14	6	no	—
242	C₂×D₄	C₂²×D₇	1	6	yes	complemented
243	C₂²×C₄	—	7	3	no	—
244	C₂²×C₄	—	14	6	no	—
245	Q₁₆	—	7	2	no	—
246	Q₁₆	—	7	1	no	—
247	Q₁₆	C₂²×D₇	1	1	yes	characteristic, complemented
248	SD₁₆	—	7	6	no	—
249	SD₁₆	—	7	6	no	—
250	SD₁₆	—	7	6	no	—
251	SD₁₆	C₂²×D₇	1	6	yes	complemented
252	D₈	—	7	6	no	—
253	D₈	—	7	12	no	—
254	D₈	—	7	3	no	—
255	D₈	—	7	6	no	—
256	D₈	C₂²×D₇	1	3	yes	complemented
257	D₈	C₂²×D₇	1	6	yes	complemented
258	M₁₆	—	7	3	no	—
259	M₁₆	—	7	6	no	—
260	M₁₆	C₂²×D₇	1	3	yes	complemented
261	C₂×C₈	—	7	3	no	—
262	C₂×C₈	—	7	1	no	—
263	C₂×C₈	—	7	3	no	—
264	C₂×C₈	—	7	3	no	—
265	C₂×C₈	—	7	3	no	—
266	C₂×C₈	C₂²×D₇	1	3	yes	complemented
267	C₂×C₇	—	2	3	no	—
268	C₂×C₇	—	4	6	no	—
269	C₂×C₇	C₂³×D₄	1	1	yes	characteristic
270	D₇	—	2	3	no	—
271	D₇	—	4	6	no	—
272	D₇	#576	1	2	yes	direct product factor
273	C₂³	—	2	6	no	—
274	C₂³	—	7	3	no	—
275	C₂³	—	14	6	no	—
276	C₂³	—	14	12	no	—
277	C₂³	—	14	6	no	—
278	C₂³	—	56	12	no	—
279	Q₈	—	7	3	no	—
280	Q₈	—	7	2	no	—
281	Q₈	—	14	2	no	—
282	Q₈	D₄×D₇	1	1	yes	characteristic
283	Q₈	C₂³×D₇	1	2	yes	—
284	D₄	—	2	6	no	—
285	D₄	—	2	6	no	—
286	D₄	—	7	6	no	—
287	D₄	—	7	3	no	—
288	D₄	—	7	6	no	—
289	D₄	—	14	6	no	—
290	D₄	—	14	6	no	—
291	D₄	—	14	12	no	—
292	D₄	—	14	6	no	—
293	D₄	—	14	6	no	—
294	D₄	—	14	6	no	—
295	D₄	D₄×D₇	1	3	yes	—
296	D₄	C₂³×D₇	1	6	yes	—
297	C₂×C₄	—	2	6	no	—
298	C₂×C₄	—	7	3	no	—
299	C₂×C₄	—	7	3	no	—
300	C₂×C₄	—	7	3	no	—
301	C₂×C₄	—	7	1	no	—
302	C₂×C₄	—	7	3	no	—
303	C₂×C₄	—	14	6	no	—
304	C₂×C₄	—	14	2	no	—
305	C₂×C₄	—	14	6	no	—
306	C₂×C₄	—	14	6	no	—
307	C₂×C₄	C₂³×D₇	1	3	yes	—
308	C₈	—	7	1	no	—
309	C₈	—	7	3	no	—
310	C₈	C₂³×D₇	1	1	yes	characteristic
311	C₈	C₂³×D₇	1	3	yes	—
312	C₇	C₂×#576	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	6	no	—
314	C₂²	—	7	1	no	—
315	C₂²	—	7	3	no	—
316	C₂²	—	8	6	no	—
317	C₂²	—	14	6	no	—
318	C₂²	—	14	6	no	—
319	C₂²	—	28	12	no	—
320	C₂²	—	56	6	no	—
321	C₂²	—	56	12	no	—
322	C₂²	C₂×D₄×D₇	1	3	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	7	1	no	—
325	C₄	—	7	3	no	—
326	C₄	—	14	2	no	—
327	C₄	C₂×D₄×D₇	1	3	yes	—
328	C₄	C₂⁴×D₇	1	1	yes	characteristic
329	C₂	—	2	3	no	—
330	C₂	—	4	6	no	—
331	C₂	—	7	2	no	—
332	C₂	—	14	3	no	—
333	C₂	—	28	6	no	—
334	C₂	C₂²×D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₇
commutator, cosocle	C₄×C₇	C₂⁵
Frattini	C₄	C₂⁴×D₇
Fitting	C₇×#576	C₂
socle	C₂×C₇	C₂³×D₄
hypercenter	#576	D₇
hypocenter	C₇	C₂×#576
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. D₇×H(5,𝔽₂)	C₂	semidirect
2. C₇×#576	C₂	semidirect
3. #90526	C₂	semidirect
4. #90465	C₂	semidirect
5. D₇×AGL(1,ℤ/8)	C₂	semidirect
6. #90460	C₂	semidirect
7. D₇×#135	C₂	semidirect
8. #90448	C₂	semidirect
9. C₂×D₇×D₈	C₂	semidirect
10. #90443	C₂	semidirect
11. D₇×#131	C₂	semidirect
12. C₇×H(5,𝔽₂)	C₂²	semidirect
14. #6553	C₂²	semidirect
15. #6553	C₂²	semidirect
18. D₇×#41	C₂²	semidirect
20. #6548	C₂²	semidirect
22. C₂×D₄×D₇	C₂²	semidirect
23. C₇×AGL(1,ℤ/8)	C₂²	semidirect
24. C₇×#135	C₂²	semidirect
25. C₂×C₇×D₈	C₂²	semidirect
26. C₇×#131	C₂²	semidirect
27. #6513	C₂²	semidirect
28. #6512	C₂²	semidirect
29. #6511	C₂²	semidirect
30. #6495	C₂²	semidirect
31. C₂×#1141	C₂²	semidirect
32. #6482	C₂²	semidirect
33. D₇×Q₁₆	C₂²	semidirect
34. #6479	C₂²	semidirect
35. #6477	C₂²	semidirect
36. D₇×SD₁₆	C₂²	semidirect
37. #6475	C₂²	semidirect
38. #6474	C₂²	semidirect
39. D₇×D₈	C₂²	semidirect
40. D₇×D₈	C₂²	semidirect
41. D₇×D₈	C₂²	semidirect
42. D₇×D₈	C₂²	semidirect
43. #6471	C₂²	semidirect
44. #6470	C₂²	semidirect
45. D₇×M₁₆	C₂²	semidirect
46. #6467	C₂²	semidirect
47. C₂×D₅₆	C₂²	semidirect
48. #6464	C₂²	semidirect
49. C₂×C₈×D₇	C₂²	semidirect
54. C₇×#41	C₂³	semidirect
56. C₂×C₇×D₄	C₂³	semidirect
61. #1161	C₂³	semidirect
67. #1159	C₂³	semidirect
76. D₄×D₇	C₂³	semidirect
80. #1157	C₂³	semidirect
82. C₂×D₂₈	C₂³	semidirect
85. C₇×Q₁₆	C₂³	semidirect
86. C₇×SD₁₆	C₂³	semidirect
87. C₇×D₈	C₂³	semidirect
88. C₇×D₈	C₂³	semidirect
89. C₇×M₁₆	C₂³	semidirect
90. C₂×C₇×C₈	C₂³	semidirect
91. #1144	C₂³	semidirect
92. #1143	C₂³	semidirect
93. #1142	C₂³	semidirect
94. #1141	C₂³	semidirect
95. #1141	C₂³	semidirect
96. #1136	C₂³	semidirect
97. C₂×C₇⋊C₈	C₂³	semidirect
98. Dic₂₈	C₂³	semidirect
99. D₅₆	C₂³	semidirect
100. D₅₆	C₂³	semidirect
101. #1132	C₂³	semidirect
102. #1131	C₂³	semidirect
105. C₈×D₇	C₂³	semidirect
106. C₈×D₇	C₂³	semidirect
113. #576	D₇	direct (N × H)
162. H(5,𝔽₂)	C₂×D₇	semidirect
173. AGL(1,ℤ/8)	C₂×D₇	semidirect
179. #135	C₂×D₇	semidirect
188. C₂×D₈	C₂×D₇	semidirect
192. #131	C₂×D₇	semidirect
225. #41	C₂²×D₇	semidirect
242. C₂×D₄	C₂²×D₇	semidirect
247. Q₁₆	C₂²×D₇	semidirect
251. SD₁₆	C₂²×D₇	semidirect
256. D₈	C₂²×D₇	semidirect
257. D₈	C₂²×D₇	semidirect
260. M₁₆	C₂²×D₇	semidirect
266. C₂×C₈	C₂²×D₇	semidirect
272. D₇	#576	direct (N × H)
312. C₇	C₂×#576	semidirect

Maximal subgroups

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

subgroup	index
D₇×H(5,𝔽₂)	2
C₇×#576	2
#90526	2
#90465	2
D₇×AGL(1,ℤ/8)	2
#90460	2
D₇×#135	2
#90448	2
C₂×D₇×D₈	2
#90443	2
D₇×#131	2
C₂×#576	7

Subgroups — D7×#576 #11744270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#576 #11744270