Subgroups — C₂×C₃₁×#1985 #21478058

Order 7936.

1438 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1985	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#336	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₃₁	—	2	4	no	—
10	C₂×C₄×Q₈×C₃₁	—	2	4	no	—
11	C₂×C₄×Q₈×C₃₁	C₄	1	4	yes	—
12	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
19	C₂×C₃₁×#97	—	2	4	no	—
20	C₂×C₃₁×#97	C₂²	1	4	yes	—
21	C₃₁×#336	C₂²	1	16	yes	—
22	C₂²×Q₈×C₃₁	—	2	4	no	—
23	C₂²×Q₈×C₃₁	—	2	8	no	—
24	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	—	2	4	no	—
31	C₂²×C₈×C₃₁	—	2	4	no	—
32	C₂²×C₈×C₃₁	—	2	4	no	—
33	C₂²×C₈×C₃₁	—	2	8	no	—
34	C₂²×C₈×C₃₁	Q₈	1	8	yes	complemented
35	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
37	C₄×Q₈×C₃₁	—	2	16	no	—
38	C₄×Q₈×C₃₁	—	2	8	no	—
39	C₄×Q₈×C₃₁	—	2	8	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
44	C₂×C₄²×C₃₁	—	2	4	no	—
45	C₂×C₄²×C₃₁	—	2	4	no	—
46	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₃₁	D₄	1	4	yes	—
49	C₂×C₄²×C₃₁	D₄	1	4	yes	—
50	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
51	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
52	C₃₁×C₄⋊C₈	—	2	8	no	—
53	C₃₁×#97	—	2	8	no	—
54	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₃₁	—	2	32	no	—
56	C₂×Q₈×C₃₁	—	2	8	no	—
57	C₂×Q₈×C₃₁	—	2	16	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	2	8	no	—
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₃₁	M₁₆	1	8	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₃₁	#41	1	4	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₃₁	—	2	16	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	2	8	no	—
82	C₂×C₈×C₃₁	—	4	8	no	—
83	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	complemented
84	C₄⋊C₄×C₃₁	—	2	16	no	—
85	C₄⋊C₄×C₃₁	—	2	8	no	—
86	C₄²×C₃₁	—	2	8	no	—
87	C₄²×C₃₁	—	2	4	no	—
88	C₄²×C₃₁	—	2	4	no	—
89	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
90	C₄²×C₃₁	C₂×D₄	1	4	yes	—
91	C₄²×C₃₁	C₂×D₄	1	4	yes	—
92	C₂×#1985	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂³×C₃₁	C₄×Q₈	1	4	yes	—
95	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
96	C₂³×C₃₁	#131	1	4	yes	—
97	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
98	Q₈×C₃₁	—	2	32	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	8	no	—
104	C₂×C₄×C₃₁	—	2	16	no	—
105	C₂×C₄×C₃₁	—	2	8	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	8	no	—
108	C₂×C₄×C₃₁	—	4	8	no	—
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
114	C₂×C₄×C₃₁	#124	1	4	yes	—
115	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
116	C₂×C₄×C₃₁	C₂×M₁₆	1	8	yes	—
117	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
121	C₈×C₃₁	—	2	8	no	—
122	C₈×C₃₁	—	4	4	no	—
123	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
124	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
125	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic
126	#1985	C₂×C₃₁	1	8	yes	direct product factor
127	C₂×#336	C₂×C₃₁	1	4	yes	—
128	C₂²×C₃₁	#391	1	4	yes	—
129	C₂²×C₃₁	#407	1	4	yes	—
130	C₂²×C₃₁	#408	1	4	yes	—
131	C₂²×C₃₁	#446	1	16	yes	—
132	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×C₄×Q₈	1	2	yes	—
134	C₂²×C₃₁	C₂²×M₁₆	1	2	yes	—
135	C₂²×C₃₁	C₂×#131	1	2	yes	—
136	C₄×C₃₁	—	2	4	no	—
137	C₄×C₃₁	—	2	8	no	—
138	C₄×C₃₁	—	2	4	no	—
139	C₄×C₃₁	—	4	4	no	—
140	C₄×C₃₁	#391	1	4	yes	—
141	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
142	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
143	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₂³×Q₈	C₄×C₃₁	1	1	yes	characteristic
145	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
146	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
147	C₂×C₄×Q₈	—	2	4	no	—
148	C₂×C₄×Q₈	—	2	4	no	—
149	C₂×C₄×Q₈	C₄×C₃₁	1	4	yes	—
150	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
151	C₂²×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
152	C₂²×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
153	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
154	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₄⋊C₈	—	2	4	no	—
156	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
157	C₂×#97	—	2	4	no	—
158	C₂×#97	C₂²×C₃₁	1	4	yes	—
159	#336	C₂²×C₃₁	1	16	yes	—
160	C₂×C₃₁	#1985	1	8	yes	direct product factor
161	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
162	C₂×C₃₁	C₂×#407	1	1	yes	characteristic
163	C₂×C₃₁	C₂×#408	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#446	1	4	yes	—
165	C₂²×Q₈	—	2	4	no	—
166	C₂²×Q₈	—	2	8	no	—
167	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
168	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
169	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
172	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	8	no	—
177	C₂²×C₈	Q₈×C₃₁	1	8	yes	complemented
178	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
179	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
180	C₄×Q₈	—	2	16	no	—
181	C₄×Q₈	—	2	8	no	—
182	C₄×Q₈	—	2	8	no	—
183	C₂×C₄⋊C₄	—	2	8	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
187	C₂×C₄²	—	2	4	no	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
191	C₂×C₄²	D₄×C₃₁	1	4	yes	—
192	C₂×C₄²	D₄×C₃₁	1	4	yes	—
193	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
194	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
195	C₄⋊C₈	—	2	8	no	—
196	#97	—	2	8	no	—
197	C₃₁	C₂×#1985	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	C₂×Q₈	—	2	32	no	—
200	C₂×Q₈	—	2	8	no	—
201	C₂×Q₈	—	2	16	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
209	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
210	C₂²×C₄	M₁₆×C₃₁	1	8	yes	—
211	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
212	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
215	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
219	C₂²×C₄	#41×C₃₁	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
221	C₂×C₈	—	2	16	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	4	8	no	—
227	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	complemented
228	C₄⋊C₄	—	2	16	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄²	—	2	8	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
234	C₄²	C₂×D₄×C₃₁	1	4	yes	—
235	C₄²	C₂×D₄×C₃₁	1	4	yes	—
236	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
237	C₂³	C₄×Q₈×C₃₁	1	4	yes	—
238	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
239	C₂³	C₃₁×#131	1	4	yes	—
240	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
241	Q₈	—	2	32	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	16	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
255	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
256	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
257	C₂×C₄	C₃₁×#124	1	4	yes	—
258	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
259	C₂×C₄	C₂×M₁₆×C₃₁	1	8	yes	—
260	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
263	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
264	C₈	—	2	8	no	—
265	C₈	—	4	4	no	—
266	C₂²	C₃₁×#391	1	4	yes	—
267	C₂²	C₃₁×#407	1	4	yes	—
268	C₂²	C₃₁×#408	1	4	yes	—
269	C₂²	C₃₁×#446	1	16	yes	—
270	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂²	C₂×C₄×Q₈×C₃₁	1	2	yes	—
272	C₂²	C₂²×M₁₆×C₃₁	1	2	yes	—
273	C₂²	C₂×C₃₁×#131	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	8	no	—
276	C₄	—	2	4	no	—
277	C₄	—	4	4	no	—
278	C₄	C₃₁×#391	1	4	yes	—
279	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
280	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
281	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
282	C₂	C₃₁×#1985	1	8	yes	direct product factor
283	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
284	C₂	C₂×C₃₁×#407	1	1	yes	characteristic
285	C₂	C₂×C₃₁×#408	1	1	yes	characteristic
286	C₂	C₂×C₃₁×#446	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₃₁	C₂³
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle, prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1985	C₂	direct (N × H)
34. C₂²×C₈×C₃₁	Q₈	semidirect
83. C₂×C₈×C₃₁	C₂×Q₈	semidirect
92. C₂×#1985	C₃₁	direct (N × H)
126. #1985	C₂×C₃₁	direct (N × H)
160. C₂×C₃₁	#1985	direct (N × H)
177. C₂²×C₈	Q₈×C₃₁	semidirect
197. C₃₁	C₂×#1985	direct (N × H)
227. C₂×C₈	C₂×Q₈×C₃₁	semidirect
282. C₂	C₃₁×#1985	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₃₁×C₄⋊C₈	2
C₂²×C₃₁×#97	2
C₃₁×#1985	2
C₂×C₃₁×#336	2
C₂×#1985	31

Subgroups — C2×C31×#1985 #21478058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1985 #21478058