Subgroups — C₃₁×#21248 #21478007

Order 7936.

1458 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2896	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#413	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1979	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1978	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1977	C₂	1	4	yes	maximal, complemented
9	C₂²×#41×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×#135	—	2	1	no	—
11	C₂×C₃₁×#135	—	2	1	no	—
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₂×C₃₁×#135	—	2	1	no	—
14	C₂×C₃₁×#135	—	2	4	no	—
15	C₂×C₃₁×#135	C₄	1	8	yes	complemented
16	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
18	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#108	—	2	4	no	—
24	C₂×C₃₁×#108	C₂²	1	2	yes	—
25	C₂×C₃₁×#108	C₂²	1	2	yes	—
26	C₃₁×#419	C₂²	1	4	yes	—
27	C₃₁×#418	C₂²	1	4	yes	—
28	C₃₁×#417	—	2	4	no	—
29	C₃₁×#416	—	2	4	no	—
30	C₃₁×#413	—	2	8	no	—
31	C₂×C₃₁×#101	C₂²	1	4	yes	—
32	C₂×C₃₁×#100	C₂²	1	4	yes	—
33	C₃₁×#405	—	2	1	no	—
34	C₃₁×#405	—	2	1	no	—
35	C₃₁×#405	C₂²	1	2	yes	—
36	C₃₁×#405	C₂²	1	2	yes	—
37	C₂×C₃₁×#97	—	2	1	no	—
38	C₂×C₄×C₈×C₃₁	—	2	1	no	—
39	C₂×#41×C₃₁	—	2	4	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₂×#41×C₃₁	—	2	4	no	—
45	C₂×#41×C₃₁	—	2	4	no	—
46	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
47	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×D₄×C₃₁	—	2	2	no	—
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂³×C₄×C₃₁	—	2	2	no	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#135	—	2	8	no	—
53	C₃₁×#135	—	2	8	no	—
54	C₃₁×#135	—	2	8	no	—
55	C₃₁×#135	—	2	8	no	—
56	C₂×Q₁₆×C₃₁	—	2	1	no	—
57	C₂×Q₁₆×C₃₁	—	2	1	no	—
58	C₂×Q₁₆×C₃₁	—	2	2	no	—
59	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
60	C₂×SD₁₆×C₃₁	—	2	2	no	—
61	C₂×SD₁₆×C₃₁	—	2	2	no	—
62	C₂×SD₁₆×C₃₁	—	2	4	no	—
63	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
64	C₂×D₈×C₃₁	—	2	1	no	—
65	C₂×D₈×C₃₁	—	2	1	no	—
66	C₂×D₈×C₃₁	—	2	2	no	—
67	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
68	C₂×M₁₆×C₃₁	—	2	1	no	—
69	C₂×M₁₆×C₃₁	—	2	2	no	—
70	C₂×M₁₆×C₃₁	—	2	2	no	—
71	C₂×M₁₆×C₃₁	—	2	4	no	—
72	C₂×M₁₆×C₃₁	—	2	4	no	—
73	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
74	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
75	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₃₁	—	2	1	no	—
77	C₂²×C₈×C₃₁	—	2	2	no	—
78	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
79	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
83	C₂²×C₈×C₃₁	D₄	1	4	yes	—
84	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#117	—	2	1	no	—
87	C₃₁×#117	—	2	1	no	—
88	C₃₁×#117	D₄	1	2	yes	—
89	C₃₁×#117	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
92	C₂×C₄²×C₃₁	—	2	1	no	—
93	C₃₁×#108	—	2	4	no	—
94	C₃₁×#108	—	2	4	no	—
95	C₃₁×#103	—	2	8	no	—
96	C₃₁×#102	—	2	8	no	—
97	C₃₁×#101	—	2	8	no	—
98	C₃₁×#100	—	2	8	no	—
99	C₃₁×#97	—	2	2	no	—
100	C₃₁×#97	—	2	2	no	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	2	no	—
103	C₂⁴×C₃₁	—	2	2	no	—
104	C₂⁴×C₃₁	#41	1	1	yes	characteristic
105	#41×C₃₁	—	2	8	no	—
106	#41×C₃₁	—	2	8	no	—
107	#41×C₃₁	—	2	8	no	—
108	#41×C₃₁	—	2	8	no	—
109	#41×C₃₁	—	4	4	no	—
110	#41×C₃₁	—	4	4	no	—
111	#41×C₃₁	—	4	4	no	—
112	#41×C₃₁	—	4	4	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	—	2	2	no	—
118	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	4	2	no	—
126	C₂×D₄×C₃₁	—	4	2	no	—
127	C₂×D₄×C₃₁	—	4	4	no	—
128	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	8	no	—
132	C₂²×C₄×C₃₁	—	4	2	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	—	4	4	no	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	#41	1	2	yes	—
143	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
145	Q₁₆×C₃₁	—	2	4	no	—
146	Q₁₆×C₃₁	—	2	4	no	—
147	SD₁₆×C₃₁	—	2	8	no	—
148	SD₁₆×C₃₁	—	2	8	no	—
149	D₈×C₃₁	—	2	4	no	—
150	D₈×C₃₁	—	2	4	no	—
151	M₁₆×C₃₁	—	2	4	no	—
152	M₁₆×C₃₁	—	2	2	no	—
153	M₁₆×C₃₁	—	2	4	no	—
154	M₁₆×C₃₁	—	2	4	no	—
155	M₁₆×C₃₁	—	4	4	no	—
156	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
157	C₂×C₈×C₃₁	—	2	4	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	4	no	—
160	C₂×C₈×C₃₁	—	4	4	no	—
161	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
162	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
163	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
164	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
165	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
166	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
167	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	2	2	no	—
177	C₄²×C₃₁	—	2	2	no	—
178	C₄²×C₃₁	—	2	2	no	—
179	#21248	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	2	4	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	8	4	no	—
185	C₂³×C₃₁	C₄×D₄	1	2	yes	—
186	C₂³×C₃₁	#121	1	1	yes	characteristic
187	C₂³×C₃₁	#121	1	1	yes	characteristic
188	C₂³×C₃₁	#124	1	2	yes	—
189	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
190	Q₈×C₃₁	—	2	4	no	—
191	Q₈×C₃₁	—	2	4	no	—
192	Q₈×C₃₁	—	4	2	no	—
193	Q₈×C₃₁	—	4	2	no	—
194	D₄×C₃₁	—	2	4	no	—
195	D₄×C₃₁	—	2	4	no	—
196	D₄×C₃₁	—	4	8	no	—
197	D₄×C₃₁	—	4	8	no	—
198	D₄×C₃₁	—	4	2	no	—
199	D₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	8	no	—
206	C₂×C₄×C₃₁	—	4	8	no	—
207	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
209	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
211	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
212	C₂×C₄×C₃₁	#121	1	2	yes	—
213	C₂×C₄×C₃₁	#121	1	2	yes	—
214	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
215	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
216	C₂×C₄×C₃₁	#124	1	2	yes	—
217	C₂×C₄×C₃₁	#127	1	2	yes	—
218	C₂×C₄×C₃₁	#127	1	2	yes	—
219	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
225	C₈×C₃₁	—	2	4	no	—
226	C₈×C₃₁	—	4	4	no	—
227	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
228	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
229	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
230	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
231	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic
232	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#2896	C₂×C₃₁	1	2	yes	complemented
234	C₂×#413	C₂×C₃₁	1	2	yes	complemented
235	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1979	C₂×C₃₁	1	2	yes	complemented
237	#1978	C₂×C₃₁	1	2	yes	complemented
238	#1977	C₂×C₃₁	1	4	yes	complemented
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	4	4	no	—
245	C₂²×C₃₁	—	8	4	no	—
246	C₂²×C₃₁	—	8	4	no	—
247	C₂²×C₃₁	—	8	4	no	—
248	C₂²×C₃₁	#390	1	2	yes	—
249	C₂²×C₃₁	#390	1	2	yes	—
250	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
252	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
253	C₄×C₃₁	—	4	4	no	—
254	C₄×C₃₁	—	4	4	no	—
255	C₄×C₃₁	#390	1	2	yes	—
256	C₄×C₃₁	#390	1	2	yes	—
257	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
258	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
259	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
260	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
261	C₂²×#41	C₂²×C₃₁	1	2	yes	—
262	C₂×#135	—	2	1	no	—
263	C₂×#135	—	2	1	no	—
264	C₂×#135	—	2	1	no	—
265	C₂×#135	—	2	1	no	—
266	C₂×#135	—	2	4	no	—
267	C₂×#135	C₄×C₃₁	1	8	yes	complemented
268	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
269	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
270	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
272	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
273	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#108	—	2	4	no	—
276	C₂×#108	C₂²×C₃₁	1	2	yes	—
277	C₂×#108	C₂²×C₃₁	1	2	yes	—
278	#419	C₂²×C₃₁	1	4	yes	—
279	#418	C₂²×C₃₁	1	4	yes	—
280	#417	—	2	4	no	—
281	#416	—	2	4	no	—
282	#413	—	2	8	no	—
283	C₂×#101	C₂²×C₃₁	1	4	yes	—
284	C₂×#100	C₂²×C₃₁	1	4	yes	—
285	#405	—	2	1	no	—
286	#405	—	2	1	no	—
287	#405	C₂²×C₃₁	1	2	yes	—
288	#405	C₂²×C₃₁	1	2	yes	—
289	C₂×#97	—	2	1	no	—
290	C₂×C₄×C₈	—	2	1	no	—
291	C₂×C₃₁	—	2	1	no	—
292	C₂×C₃₁	—	2	1	no	—
293	C₂×C₃₁	—	2	2	no	—
294	C₂×C₃₁	—	2	2	no	—
295	C₂×C₃₁	—	8	4	no	—
296	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
297	C₂×C₃₁	#3067	1	2	yes	—
298	C₂×#41	—	2	4	no	—
299	C₂×#41	—	2	2	no	—
300	C₂×#41	—	2	2	no	—
301	C₂×#41	—	2	4	no	—
302	C₂×#41	—	2	2	no	—
303	C₂×#41	—	2	2	no	—
304	C₂×#41	—	2	4	no	—
305	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
306	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
307	C₂²×D₄	—	2	2	no	—
308	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
309	C₂³×C₄	—	2	2	no	—
310	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
311	#135	—	2	8	no	—
312	#135	—	2	8	no	—
313	#135	—	2	8	no	—
314	#135	—	2	8	no	—
315	C₂×Q₁₆	—	2	1	no	—
316	C₂×Q₁₆	—	2	1	no	—
317	C₂×Q₁₆	—	2	2	no	—
318	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
319	C₂×SD₁₆	—	2	2	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	4	no	—
322	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
323	C₂×D₈	—	2	1	no	—
324	C₂×D₈	—	2	1	no	—
325	C₂×D₈	—	2	2	no	—
326	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
327	C₂×M₁₆	—	2	1	no	—
328	C₂×M₁₆	—	2	2	no	—
329	C₂×M₁₆	—	2	2	no	—
330	C₂×M₁₆	—	2	4	no	—
331	C₂×M₁₆	—	2	4	no	—
332	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
333	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
334	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
335	C₂²×C₈	—	2	1	no	—
336	C₂²×C₈	—	2	2	no	—
337	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
338	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
339	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
340	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₈	D₄×C₃₁	1	4	yes	—
343	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	#117	—	2	1	no	—
346	#117	—	2	1	no	—
347	#117	D₄×C₃₁	1	2	yes	—
348	#117	C₂³×C₃₁	1	2	yes	—
349	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₄²	—	2	1	no	—
352	#108	—	2	4	no	—
353	#108	—	2	4	no	—
354	#103	—	2	8	no	—
355	#102	—	2	8	no	—
356	#101	—	2	8	no	—
357	#100	—	2	8	no	—
358	#97	—	2	2	no	—
359	#97	—	2	2	no	—
360	C₄×C₈	—	2	2	no	—
361	C₄×C₈	—	2	2	no	—
362	C₃₁	#21248	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	2	2	no	—
364	C₂⁴	#41×C₃₁	1	1	yes	characteristic
365	#41	—	2	8	no	—
366	#41	—	2	8	no	—
367	#41	—	2	8	no	—
368	#41	—	2	8	no	—
369	#41	—	4	4	no	—
370	#41	—	4	4	no	—
371	#41	—	4	4	no	—
372	#41	—	4	4	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	4	2	no	—
386	C₂×D₄	—	4	2	no	—
387	C₂×D₄	—	4	4	no	—
388	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	8	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	2	no	—
394	C₂²×C₄	—	4	4	no	—
395	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
402	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	#41×C₃₁	1	2	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
405	Q₁₆	—	2	4	no	—
406	Q₁₆	—	2	4	no	—
407	SD₁₆	—	2	8	no	—
408	SD₁₆	—	2	8	no	—
409	D₈	—	2	4	no	—
410	D₈	—	2	4	no	—
411	M₁₆	—	2	4	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	2	4	no	—
414	M₁₆	—	2	4	no	—
415	M₁₆	—	4	4	no	—
416	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
417	C₂×C₈	—	2	4	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	4	no	—
420	C₂×C₈	—	4	4	no	—
421	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
422	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
423	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
424	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
425	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
426	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
427	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
432	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₄²	—	2	2	no	—
438	C₄²	—	2	2	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	8	4	no	—
444	C₂³	C₄×D₄×C₃₁	1	2	yes	—
445	C₂³	C₃₁×#121	1	1	yes	characteristic
446	C₂³	C₃₁×#121	1	1	yes	characteristic
447	C₂³	C₃₁×#124	1	2	yes	—
448	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
449	Q₈	—	2	4	no	—
450	Q₈	—	2	4	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	D₄	—	2	4	no	—
454	D₄	—	2	4	no	—
455	D₄	—	4	8	no	—
456	D₄	—	4	8	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	8	no	—
465	C₂×C₄	—	4	8	no	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
467	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
468	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
469	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
470	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
471	C₂×C₄	C₃₁×#121	1	2	yes	—
472	C₂×C₄	C₃₁×#121	1	2	yes	—
473	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
474	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
475	C₂×C₄	C₃₁×#124	1	2	yes	—
476	C₂×C₄	C₃₁×#127	1	2	yes	—
477	C₂×C₄	C₃₁×#127	1	2	yes	—
478	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
484	C₈	—	2	4	no	—
485	C₈	—	4	4	no	—
486	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
487	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
488	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
489	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	8	4	no	—
497	C₂²	—	8	4	no	—
498	C₂²	—	8	4	no	—
499	C₂²	C₃₁×#390	1	2	yes	—
500	C₂²	C₃₁×#390	1	2	yes	—
501	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
502	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
503	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
504	C₄	—	4	4	no	—
505	C₄	—	4	4	no	—
506	C₄	C₃₁×#390	1	2	yes	—
507	C₄	C₃₁×#390	1	2	yes	—
508	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
509	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
510	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
511	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
512	C₂	—	2	1	no	—
513	C₂	—	2	1	no	—
514	C₂	—	2	2	no	—
515	C₂	—	2	2	no	—
516	C₂	—	8	4	no	—
517	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
518	C₂	C₃₁×#3067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
prime radical	C₂²×C₃₁×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#108	C₂	semidirect
3. C₃₁×#2896	C₂	semidirect
4. C₂×C₃₁×#413	C₂	semidirect
5. C₂×C₃₁×#405	C₂	semidirect
6. C₃₁×#1979	C₂	semidirect
7. C₃₁×#1978	C₂	semidirect
8. C₃₁×#1977	C₂	semidirect
15. C₂×C₃₁×#135	C₄	semidirect
59. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
63. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
67. C₂×D₈×C₃₁	C₂×C₄	semidirect
179. #21248	C₃₁	direct (N × H)
232. C₂²×#108	C₂×C₃₁	semidirect
233. #2896	C₂×C₃₁	semidirect
234. C₂×#413	C₂×C₃₁	semidirect
235. C₂×#405	C₂×C₃₁	semidirect
236. #1979	C₂×C₃₁	semidirect
237. #1978	C₂×C₃₁	semidirect
238. #1977	C₂×C₃₁	semidirect
267. C₂×#135	C₄×C₃₁	semidirect
318. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
322. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
326. C₂×D₈	C₂×C₄×C₃₁	semidirect
362. C₃₁	#21248	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×C₃₁×#135	2
C₂²×C₃₁×#108	2
C₃₁×#2896	2
C₂×C₃₁×#413	2
C₂×C₃₁×#405	2
C₃₁×#1979	2
C₃₁×#1978	2
C₃₁×#1977	2
#21248	31

Subgroups — C31×#21248 #21478007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21248 #21478007