Subgroups — C₂×C₃₁×#1669 #21475859

Order 7936.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#474	C₂	1	2	yes	maximal
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1669	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#332	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₃₁	—	2	1	no	—
8	C₂²×Q₁₆×C₃₁	—	2	1	no	—
9	C₂³×C₈×C₃₁	—	2	1	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#494	—	2	2	no	—
13	C₃₁×#494	C₂²	1	4	yes	complemented
14	C₃₁×#474	—	2	4	no	—
15	C₃₁×#474	C₂²	1	8	yes	—
16	C₈×D₄×C₃₁	—	2	2	no	—
17	C₈×D₄×C₃₁	C₂²	1	4	yes	—
18	C₂×C₃₁×#107	—	2	2	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	—	2	2	no	—
22	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#335	C₂²	1	4	yes	complemented
25	C₃₁×#332	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	—	2	1	no	—
29	C₂×Q₁₆×C₃₁	—	2	4	no	—
30	C₂×Q₁₆×C₃₁	—	2	4	no	—
31	C₂×Q₁₆×C₃₁	—	4	4	no	—
32	C₂×Q₁₆×C₃₁	—	4	4	no	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	—	4	4	no	—
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
43	C₄×D₄×C₃₁	—	2	1	no	—
44	C₄×D₄×C₃₁	—	2	1	no	—
45	C₄×D₄×C₃₁	D₄	1	2	yes	—
46	C₄×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	—	2	8	no	—
53	C₃₁×C₄⋊C₈	—	2	1	no	—
54	C₃₁×C₄⋊C₈	—	2	2	no	—
55	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
56	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
57	C₃₁×#102	—	2	8	no	—
58	C₃₁×C₂²⋊C₈	—	2	4	no	—
59	C₄×C₈×C₃₁	D₄	1	2	yes	—
60	C₄×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂⁴×C₃₁	—	2	1	no	—
62	C₂×Q₈×C₃₁	—	2	4	no	—
63	C₂×Q₈×C₃₁	—	4	2	no	—
64	C₂×Q₈×C₃₁	—	4	4	no	—
65	C₂×Q₈×C₃₁	—	4	4	no	—
66	C₂×D₄×C₃₁	—	2	1	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	Q₁₆	1	4	yes	complemented
69	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	4	2	no	—
74	C₂²×C₄×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	—	4	2	no	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
79	Q₁₆×C₃₁	—	4	8	no	—
80	Q₁₆×C₃₁	—	4	8	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	4	no	—
84	C₂×C₈×C₃₁	—	4	2	no	—
85	C₂×C₈×C₃₁	—	4	4	no	—
86	C₂×C₈×C₃₁	—	4	8	no	—
87	C₂×C₈×C₃₁	D₈	1	4	yes	—
88	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₃₁	—	2	1	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂²⋊C₄×C₃₁	—	2	2	no	—
97	C₂²⋊C₄×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₃₁	#41	1	1	yes	characteristic
101	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₂×#1669	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	2	2	no	—
104	C₂³×C₃₁	—	2	4	no	—
105	C₂³×C₃₁	—	4	2	no	—
106	C₂³×C₃₁	—	4	2	no	—
107	C₂³×C₃₁	#121	1	1	yes	characteristic
108	Q₈×C₃₁	—	4	4	no	—
109	Q₈×C₃₁	—	4	4	no	—
110	Q₈×C₃₁	—	8	4	no	—
111	D₄×C₃₁	—	2	2	no	—
112	D₄×C₃₁	—	2	4	no	—
113	D₄×C₃₁	C₂×Q₁₆	1	4	yes	complemented
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	8	4	no	—
122	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
127	C₂×C₄×C₃₁	#135	1	2	yes	—
128	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₃₁	—	2	2	no	—
133	C₈×C₃₁	—	4	2	no	—
134	C₈×C₃₁	—	4	4	no	—
135	C₈×C₃₁	C₂×D₈	1	4	yes	—
136	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
137	C₂×#474	C₂×C₃₁	1	2	yes	—
138	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
139	#1669	C₂×C₃₁	1	8	yes	direct product factor
140	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
141	C₂×#332	C₂×C₃₁	1	2	yes	complemented
142	C₂²×C₃₁	—	2	4	no	—
143	C₂²×C₃₁	—	4	8	no	—
144	C₂²×C₃₁	—	4	2	no	—
145	C₂²×C₃₁	—	4	2	no	—
146	C₂²×C₃₁	#459	1	2	yes	—
147	C₂²×C₃₁	#467	1	2	yes	—
148	C₂²×C₃₁	#473	1	2	yes	—
149	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	2	1	no	—
152	C₄×C₃₁	—	4	2	no	—
153	C₄×C₃₁	—	8	4	no	—
154	C₄×C₃₁	#459	1	2	yes	—
155	C₄×C₃₁	#467	1	2	yes	—
156	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
158	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
159	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂²×Q₁₆	—	2	1	no	—
161	C₂²×Q₁₆	—	2	1	no	—
162	C₂³×C₈	—	2	1	no	—
163	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
164	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
165	#494	—	2	2	no	—
166	#494	C₂²×C₃₁	1	4	yes	complemented
167	#474	—	2	4	no	—
168	#474	C₂²×C₃₁	1	8	yes	—
169	C₈×D₄	—	2	2	no	—
170	C₈×D₄	C₂²×C₃₁	1	4	yes	—
171	C₂×#107	—	2	2	no	—
172	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
174	C₂×#102	—	2	2	no	—
175	C₂×C₂²⋊C₈	—	2	1	no	—
176	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
177	#335	C₂²×C₃₁	1	4	yes	complemented
178	#332	C₂²×C₃₁	1	8	yes	complemented
179	C₂×C₃₁	—	4	4	no	—
180	C₂×C₃₁	#1669	1	4	yes	direct product factor
181	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
182	C₂×C₃₁	C₂×#467	1	1	yes	characteristic
183	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
184	C₂²×Q₈	—	2	2	no	—
185	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
186	C₂³×C₄	—	2	1	no	—
187	C₂×Q₁₆	—	2	4	no	—
188	C₂×Q₁₆	—	2	4	no	—
189	C₂×Q₁₆	—	4	4	no	—
190	C₂×Q₁₆	—	4	4	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	4	4	no	—
195	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	D₄×C₃₁	1	2	yes	—
204	C₄×D₄	C₂³×C₃₁	1	2	yes	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	—	2	1	no	—
209	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
210	#107	—	2	8	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
214	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
215	#102	—	2	8	no	—
216	C₂²⋊C₈	—	2	4	no	—
217	C₄×C₈	D₄×C₃₁	1	2	yes	—
218	C₄×C₈	C₂³×C₃₁	1	2	yes	—
219	C₃₁	C₂×#1669	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	—	2	1	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	Q₁₆×C₃₁	1	4	yes	complemented
228	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
238	Q₁₆	—	4	8	no	—
239	Q₁₆	—	4	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	—	4	8	no	—
246	C₂×C₈	D₈×C₃₁	1	4	yes	—
247	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₄²	#41×C₃₁	1	1	yes	characteristic
260	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
261	C₂³	—	2	4	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	C₃₁×#121	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	8	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	D₄	C₂×Q₁₆×C₃₁	1	4	yes	complemented
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
281	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
284	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
285	C₂×C₄	C₃₁×#135	1	2	yes	—
286	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
287	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	4	no	—
293	C₈	C₂×D₈×C₃₁	1	4	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₃₁×#459	1	2	yes	—
299	C₂²	C₃₁×#467	1	2	yes	—
300	C₂²	C₃₁×#473	1	2	yes	—
301	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	8	4	no	—
306	C₄	C₃₁×#459	1	2	yes	—
307	C₄	C₃₁×#467	1	2	yes	—
308	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
309	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
310	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
311	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₂	—	4	4	no	—
313	C₂	C₃₁×#1669	1	4	yes	direct product factor
314	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
315	C₂	C₂×C₃₁×#467	1	1	yes	characteristic
316	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#494	C₂	semidirect
4. C₃₁×#1669	C₂	direct (N × H)
5. C₂×C₃₁×#335	C₂	semidirect
6. C₂×C₃₁×#332	C₂	semidirect
13. C₃₁×#494	C₂²	semidirect
24. C₃₁×#335	C₂²	semidirect
25. C₃₁×#332	C₂²	semidirect
68. C₂×D₄×C₃₁	Q₁₆	semidirect
102. C₂×#1669	C₃₁	direct (N × H)
113. D₄×C₃₁	C₂×Q₁₆	semidirect
136. C₂×#494	C₂×C₃₁	semidirect
139. #1669	C₂×C₃₁	direct (N × H)
140. C₂×#335	C₂×C₃₁	semidirect
141. C₂×#332	C₂×C₃₁	semidirect
166. #494	C₂²×C₃₁	semidirect
177. #335	C₂²×C₃₁	semidirect
178. #332	C₂²×C₃₁	semidirect
180. C₂×C₃₁	#1669	direct (N × H)
219. C₃₁	C₂×#1669	direct (N × H)
227. C₂×D₄	Q₁₆×C₃₁	semidirect
271. D₄	C₂×Q₁₆×C₃₁	semidirect
313. C₂	C₃₁×#1669	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#494	2
C₂×C₃₁×#474	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#1669	2
C₂×C₃₁×#335	2
C₂×C₃₁×#332	2
C₂×#1669	31

Subgroups — C2×C31×#1669 #21475859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1669 #21475859