Subgroups — #31891287

Order 18176.

556 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#22588320	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985750	C₂	1	2	yes	maximal, complemented
4	#22586175	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇₁⋊C₈	—	2	1	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
8	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
13	#17985754	—	2	1	no	—
14	#17985750	—	2	4	no	—
15	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₇₁	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₇₁	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
19	#336×C₇₁	C₄	1	2	yes	—
20	C₂²×C₇₁⋊C₈	—	2	4	no	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	C₂²×C₇₁⋊C₈	—	2	1	no	—
23	C₂²×C₇₁⋊C₈	—	2	1	no	—
24	C₂²×C₇₁⋊C₈	—	4	2	no	—
25	C₂²×C₇₁⋊C₈	—	4	2	no	—
26	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
27	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
28	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
29	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
30	C₄⋊(C₇₁⋊C₈)	D₄	1	4	yes	—
31	C₂²⋊(C₇₁⋊C₈)	—	4	2	no	—
32	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₇₁	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₁	—	2	2	no	—
38	C₂×C₄²×C₇₁	—	2	2	no	—
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
43	C₂²⋊C₈×C₇₁	—	2	2	no	—
44	C₂²⋊C₈×C₇₁	C₈	1	4	yes	complemented
45	C₄×C₈×C₇₁	—	2	2	no	—
46	C₄×C₈×C₇₁	—	2	2	no	—
47	C₂×C₇₁⋊C₈	—	2	4	no	—
48	C₂×C₇₁⋊C₈	—	4	8	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	1	no	—
52	C₂×C₇₁⋊C₈	—	4	1	no	—
53	C₂×C₇₁⋊C₈	—	4	1	no	—
54	C₂×C₇₁⋊C₈	—	4	1	no	—
55	C₂×C₇₁⋊C₈	—	4	1	no	—
56	C₂×C₇₁⋊C₈	—	4	1	no	—
57	C₂⁴×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	—	2	2	no	—
59	C₂²×C₄×C₇₁	—	2	1	no	—
60	C₂²×C₄×C₇₁	—	2	1	no	—
61	C₂²×C₄×C₇₁	—	2	2	no	—
62	C₂²×C₄×C₇₁	—	2	1	no	—
63	C₂²×C₄×C₇₁	—	2	1	no	—
64	C₂²×C₄×C₇₁	—	2	2	no	—
65	C₂²×C₄×C₇₁	—	2	2	no	—
66	C₂²×C₄×C₇₁	—	4	1	no	—
67	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
72	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₇₁	—	2	2	no	—
78	C₂×C₈×C₇₁	—	2	2	no	—
79	C₂×C₈×C₇₁	—	2	2	no	—
80	C₂×C₈×C₇₁	—	2	2	no	—
81	C₂×C₈×C₇₁	—	2	2	no	—
82	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	complemented
83	C₄²×C₇₁	—	2	2	no	—
84	C₄²×C₇₁	—	2	1	no	—
85	C₄²×C₇₁	—	2	2	no	—
86	C₄²×C₇₁	—	2	1	no	—
87	C₄²×C₇₁	C₂×D₄	1	2	yes	—
88	C₄²×C₇₁	#41	1	2	yes	—
89	C₂³×C₇₁	—	2	1	no	—
90	C₂³×C₇₁	—	2	1	no	—
91	C₂³×C₇₁	—	2	1	no	—
92	C₂³×C₇₁	—	2	1	no	—
93	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
94	C₂³×C₇₁	#106	1	2	yes	—
95	C₂³×C₇₁	#108	1	2	yes	—
96	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₁	—	2	1	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	2	1	no	—
102	C₂×C₄×C₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	4	no	—
104	C₂×C₄×C₇₁	—	2	4	no	—
105	C₂×C₄×C₇₁	—	2	2	no	—
106	C₂×C₄×C₇₁	—	2	2	no	—
107	C₂×C₄×C₇₁	—	2	2	no	—
108	C₂×C₄×C₇₁	—	2	2	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	4	4	no	—
113	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₇₁	#106	1	2	yes	—
115	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₇₁	#131	1	2	yes	—
122	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
126	C₈×C₇₁	—	2	4	no	—
127	C₈×C₇₁	—	4	2	no	—
128	C₇₁⋊C₈	—	4	4	no	—
129	C₇₁⋊C₈	—	8	1	no	—
130	C₇₁⋊C₈	—	8	1	no	—
131	C₂²×C₇₁	—	2	2	no	—
132	C₂²×C₇₁	—	2	2	no	—
133	C₂²×C₇₁	—	2	2	no	—
134	C₂²×C₇₁	—	2	2	no	—
135	C₂²×C₇₁	—	2	2	no	—
136	C₂²×C₇₁	—	2	2	no	—
137	C₂²×C₇₁	#334	1	4	yes	complemented
138	C₂²×C₇₁	#385	1	1	yes	characteristic
139	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₂²×C₇₁	#424	1	1	yes	characteristic
141	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
142	C₂²×C₇₁	#428	1	1	yes	characteristic
143	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
144	C₂²×C₇₁	#430	1	1	yes	characteristic
145	C₄×C₇₁	—	2	2	no	—
146	C₄×C₇₁	—	2	2	no	—
147	C₄×C₇₁	—	2	2	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	C₈×D₄	1	2	yes	—
150	C₄×C₇₁	#435	1	2	yes	—
151	C₄×C₇₁	#448	1	1	yes	characteristic
152	C₄×C₇₁	#450	1	1	yes	characteristic
153	C₄×C₇₁	#481	1	1	yes	characteristic
154	C₄×C₇₁	#483	1	1	yes	characteristic
155	#8830	—	71	1	no	maximal, Sylow
156	C₂×C₇₁	—	2	4	no	—
157	C₂×C₇₁	C₂×#334	1	1	yes	characteristic
158	C₂×C₇₁	#1565	1	1	yes	characteristic
159	C₂×C₇₁	#1570	1	1	yes	characteristic
160	C₂×C₇₁	#1573	1	1	yes	characteristic
161	C₂×C₇₁	#1850	1	1	yes	characteristic
162	C₂×C₇₁	#1853	1	1	yes	characteristic
163	C₂×C₇₁	#1855	1	1	yes	characteristic
164	C₂²×C₄⋊C₈	—	71	1	no	—
165	#1797	—	71	1	no	—
166	C₄×C₂²⋊C₈	D₇₁	1	1	yes	characteristic
167	C₂×#334	—	71	2	no	—
168	#1277	—	71	2	no	—
169	C₇₁	#8830	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₈	—	142	1	no	—
171	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
172	C₂×C₄⋊C₈	—	71	2	no	—
173	C₂×C₄⋊C₈	—	71	1	no	—
174	C₂×C₄⋊C₈	—	71	1	no	—
175	C₂×C₄⋊C₈	—	142	4	no	—
176	C₂×C₄⋊C₈	—	142	1	no	—
177	C₂×C₂²⋊C₈	—	142	1	no	—
178	C₂×C₂²⋊C₈	Dic₇₁	1	1	yes	characteristic
179	C₂×C₂²⋊C₈	Dic₇₁	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
181	#336	—	142	1	no	—
182	#336	Dic₇₁	1	2	yes	—
183	#334	—	142	4	no	—
184	C₂³×C₄	Dic₁₄₂	1	1	yes	characteristic
185	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
187	C₂²×C₈	—	284	2	no	—
188	C₂²×C₈	—	284	2	no	—
189	C₂²×C₈	—	142	1	no	—
190	C₂²×C₈	—	142	1	no	—
191	C₂²×C₈	—	142	4	no	—
192	C₂²×C₈	—	142	2	no	—
193	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
194	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	Dic₁₄₂	1	1	yes	characteristic
198	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
201	C₄⋊C₈	—	71	4	no	—
202	C₄⋊C₈	—	142	4	no	—
203	C₄⋊C₈	—	142	2	no	—
204	C₄⋊C₈	—	142	2	no	—
205	C₄⋊C₈	—	142	2	no	—
206	C₂²⋊C₈	—	2	2	no	—
207	C₂²⋊C₈	—	284	2	no	—
208	C₂²⋊C₈	C₇₁⋊C₈	1	4	yes	complemented
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	—	2	2	no	—
211	C₂⁴	C₄⋊Dic₇₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
223	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
225	C₂²×C₄	#11765849	1	2	yes	—
226	C₂²×C₄	#11765854	1	2	yes	—
227	C₂²×C₄	#11765854	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	284	8	no	—
237	C₂×C₈	—	284	2	no	—
238	C₂×C₈	—	284	2	no	—
239	C₂×C₈	—	284	1	no	—
240	C₂×C₈	—	284	1	no	—
241	C₂×C₈	—	284	1	no	—
242	C₂×C₈	—	284	1	no	—
243	C₂×C₈	—	284	1	no	—
244	C₂×C₈	—	284	1	no	—
245	C₂×C₈	—	142	4	no	—
246	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	complemented
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	D₄×D₇₁	1	2	yes	—
252	C₄²	#11765864	1	2	yes	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
258	C₂³	#14556060	1	2	yes	—
259	C₂³	#14556064	1	2	yes	—
260	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
278	C₂×C₄	#14556060	1	2	yes	—
279	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
280	C₂×C₄	#14556098	1	1	yes	characteristic
281	C₂×C₄	#14556105	1	2	yes	—
282	C₂×C₄	#14556112	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
284	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
285	C₂×C₄	#14556135	1	2	yes	—
286	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
287	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
288	C₂×C₄	#14556156	1	1	yes	characteristic
289	C₂×C₄	#14556161	1	1	yes	characteristic
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	284	4	no	—
293	C₈	—	568	1	no	—
294	C₈	—	568	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#17985750	1	4	yes	complemented
302	C₂²	#17985874	1	1	yes	characteristic
303	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
304	C₂²	#17986026	1	1	yes	characteristic
305	C₂²	C₂×#14556060	1	1	yes	characteristic
306	C₂²	#17986041	1	1	yes	characteristic
307	C₂²	C₂×#14556064	1	1	yes	characteristic
308	C₂²	#17986048	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	D₄×C₇₁⋊C₈	1	2	yes	—
314	C₄	#17986071	1	2	yes	—
315	C₄	#17986140	1	1	yes	characteristic
316	C₄	#17986152	1	1	yes	characteristic
317	C₄	#17986318	1	1	yes	characteristic
318	C₄	#17986330	1	1	yes	characteristic
319	C₂	—	2	4	no	—
320	C₂	C₂×#17985750	1	1	yes	characteristic
321	C₂	#22587092	1	1	yes	characteristic
322	C₂	#22587111	1	1	yes	characteristic
323	C₂	#22587125	1	1	yes	characteristic
324	C₂	#22588520	1	1	yes	characteristic
325	C₂	#22588535	1	1	yes	characteristic
326	C₂	#22588547	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×C₂²⋊C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₁	C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×C₂²⋊C₈	D₇₁
hypocenter	C₇₁	#8830
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#17985750	C₂	semidirect
4. #22586175	C₂	semidirect
44. C₂²⋊C₈×C₇₁	C₈	semidirect
82. C₂×C₈×C₇₁	C₂×C₈	semidirect
137. C₂²×C₇₁	#334	semidirect
169. C₇₁	#8830	semidirect
208. C₂²⋊C₈	C₇₁⋊C₈	semidirect
246. C₂×C₈	C₂×C₇₁⋊C₈	semidirect
301. C₂²	#17985750	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₇₁⋊C₈)	2
#22588320	2
C₂×#17985750	2
#22586175	2
C₄×C₂²⋊C₈×C₇₁	2
#8830	71