Subgroups — C₃₁×#60093 #21516852

Order 7936.

4852 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#585	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2901	C₂	1	16	yes	maximal, complemented
5	C₂×C₃₁×#421	C₂	1	12	yes	maximal, complemented
6	C₃₁×#585	—	2	4	no	—
7	C₃₁×#585	—	2	4	no	—
8	C₃₁×#585	C₄	1	8	yes	complemented
9	C₃₁×#585	C₂²	1	8	yes	—
10	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
11	C₂×C₃₁×2₋¹⁺⁴	C₄	1	4	yes	complemented
12	C₂×C₃₁×H(5,𝔽₂)	—	2	3	no	—
13	C₂×C₃₁×H(5,𝔽₂)	C₄	1	4	yes	complemented
14	C₂²×#41×C₃₁	—	2	4	no	—
15	C₂²×#41×C₃₁	C₂²	1	6	yes	—
16	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#568	—	2	4	no	—
18	C₃₁×#568	C₂²	1	8	yes	complemented
19	C₂×C₃₁×#131	—	2	4	no	—
20	C₂²×M₁₆×C₃₁	C₂²	1	6	yes	complemented
21	C₃₁×#520	—	2	4	no	—
22	C₃₁×#519	—	2	1	no	—
23	C₃₁×#518	—	2	3	no	—
24	C₄×#41×C₃₁	C₂²	1	8	yes	complemented
25	C₂×C₃₁×#117	C₂²	1	6	yes	complemented
26	C₃₁×#421	C₂²	1	16	yes	complemented
27	C₃₁×#421	C₂²	1	48	yes	complemented
28	C₂×C₃₁×#104	C₂²	1	48	yes	complemented
29	C₃₁×2₋¹⁺⁴	—	2	16	no	—
30	C₃₁×2₋¹⁺⁴	—	2	8	no	—
31	C₃₁×2₋¹⁺⁴	—	2	8	no	—
32	C₃₁×2₋¹⁺⁴	C₂×C₄	1	32	yes	complemented
33	C₃₁×H(5,𝔽₂)	—	2	16	no	—
34	C₃₁×H(5,𝔽₂)	—	2	24	no	—
35	C₃₁×H(5,𝔽₂)	—	2	24	no	—
36	C₃₁×H(5,𝔽₂)	C₂×C₄	1	32	yes	complemented
37	C₂×#41×C₃₁	—	2	32	no	—
38	C₂×#41×C₃₁	—	2	32	no	—
39	C₂×#41×C₃₁	—	2	4	no	—
40	C₂×#41×C₃₁	—	2	12	no	—
41	C₂×#41×C₃₁	—	2	6	no	—
42	C₂×#41×C₃₁	—	2	6	no	—
43	C₂×#41×C₃₁	—	2	12	no	—
44	C₂×#41×C₃₁	—	2	12	no	—
45	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
46	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
47	C₂×#41×C₃₁	C₂×C₄	1	12	yes	complemented
48	C₂×#41×C₃₁	C₂×C₄	1	24	yes	complemented
49	C₂×#41×C₃₁	D₄	1	4	yes	—
50	C₂×#41×C₃₁	D₄	1	4	yes	—
51	C₂×#41×C₃₁	D₄	1	4	yes	—
52	C₂×#41×C₃₁	C₂³	1	24	yes	—
53	C₂×#41×C₃₁	C₂³	1	4	yes	—
54	C₂×#41×C₃₁	C₂³	1	4	yes	—
55	C₂²×Q₈×C₃₁	—	2	4	no	—
56	C₂²×Q₈×C₃₁	C₂×C₄	1	6	yes	complemented
57	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₃₁	—	2	6	no	—
59	C₂²×D₄×C₃₁	—	2	12	no	—
60	C₂²×D₄×C₃₁	C₂×C₄	1	6	yes	complemented
61	C₂²×D₄×C₃₁	D₄	1	3	yes	—
62	C₂³×C₄×C₃₁	—	2	6	no	—
63	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
64	C₃₁×#131	—	2	32	no	—
65	C₂×M₁₆×C₃₁	—	2	6	no	—
66	C₂×M₁₆×C₃₁	—	2	12	no	—
67	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
68	C₂×M₁₆×C₃₁	C₂³	1	24	yes	complemented
69	C₂²×C₈×C₃₁	—	2	6	no	—
70	C₄×Q₈×C₃₁	—	2	8	no	—
71	C₄×D₄×C₃₁	—	2	24	no	—
72	C₃₁×#117	—	2	6	no	—
73	C₃₁×#117	—	2	6	no	—
74	C₃₁×#117	C₂³	1	4	yes	—
75	C₃₁×#117	C₂³	1	12	yes	complemented
76	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
78	C₂×C₄²×C₃₁	C₂³	1	12	yes	complemented
79	C₃₁×#104	C₂³	1	64	yes	complemented
80	C₂⁴×C₃₁	—	2	6	no	—
81	C₂⁴×C₃₁	C₂×D₄	1	3	yes	—
82	#41×C₃₁	—	2	32	no	—
83	#41×C₃₁	—	2	96	no	—
84	#41×C₃₁	—	2	24	no	—
85	#41×C₃₁	—	2	24	no	—
86	#41×C₃₁	—	2	24	no	—
87	#41×C₃₁	—	2	4	no	—
88	#41×C₃₁	—	2	4	no	—
89	#41×C₃₁	—	2	24	no	—
90	#41×C₃₁	—	2	24	no	—
91	#41×C₃₁	C₂²⋊C₄	1	16	yes	—
92	#41×C₃₁	C₂²⋊C₄	1	16	yes	—
93	#41×C₃₁	C₂²×C₄	1	8	yes	—
94	#41×C₃₁	C₂²×C₄	1	48	yes	—
95	#41×C₃₁	C₂²×C₄	1	16	yes	—
96	#41×C₃₁	C₂×D₄	1	16	yes	—
97	#41×C₃₁	C₂⁴	1	8	yes	—
98	C₂×Q₈×C₃₁	—	2	32	no	—
99	C₂×Q₈×C₃₁	—	2	6	no	—
100	C₂×Q₈×C₃₁	—	2	12	no	—
101	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
102	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
103	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
104	C₂×Q₈×C₃₁	C₂²×C₄	1	24	yes	—
105	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
106	C₂×D₄×C₃₁	—	2	12	no	—
107	C₂×D₄×C₃₁	—	2	24	no	—
108	C₂×D₄×C₃₁	—	2	4	no	—
109	C₂×D₄×C₃₁	—	2	24	no	—
110	C₂×D₄×C₃₁	—	2	96	no	—
111	C₂×D₄×C₃₁	—	2	6	no	—
112	C₂×D₄×C₃₁	—	2	12	no	—
113	C₂×D₄×C₃₁	C₂²⋊C₄	1	12	yes	—
114	C₂×D₄×C₃₁	C₂²⋊C₄	1	12	yes	—
115	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
116	C₂×D₄×C₃₁	C₂²×C₄	1	24	yes	—
117	C₂×D₄×C₃₁	C₂×D₄	1	12	yes	—
118	C₂²×C₄×C₃₁	—	2	6	no	—
119	C₂²×C₄×C₃₁	—	2	12	no	—
120	C₂²×C₄×C₃₁	—	2	24	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	24	no	—
123	C₂²×C₄×C₃₁	—	2	12	no	—
124	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
125	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	M₁₆×C₃₁	—	2	4	no	—
133	M₁₆×C₃₁	—	2	24	no	—
134	M₁₆×C₃₁	C₂⁴	1	8	yes	—
135	C₂×C₈×C₃₁	—	2	4	no	—
136	C₂×C₈×C₃₁	—	2	24	no	—
137	C₄⋊C₄×C₃₁	—	2	4	no	—
138	C₄⋊C₄×C₃₁	—	2	12	no	—
139	C₂²⋊C₄×C₃₁	—	2	4	no	—
140	C₂²⋊C₄×C₃₁	—	2	12	no	—
141	C₄²×C₃₁	—	2	12	no	—
142	C₄²×C₃₁	C₂⁴	1	8	yes	—
143	#60093	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	4	no	—
145	C₂³×C₃₁	—	2	24	no	—
146	C₂³×C₃₁	—	8	3	no	—
147	C₂³×C₃₁	—	8	12	no	—
148	C₂³×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
149	C₂³×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
150	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	C₂³×C₃₁	C₂²×D₄	1	6	yes	—
152	Q₈×C₃₁	—	2	24	no	—
153	Q₈×C₃₁	—	2	4	no	—
154	Q₈×C₃₁	C₂×C₂²⋊C₄	1	16	yes	—
155	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
156	D₄×C₃₁	—	2	48	no	—
157	D₄×C₃₁	—	2	8	no	—
158	D₄×C₃₁	—	2	8	no	—
159	D₄×C₃₁	—	2	24	no	—
160	D₄×C₃₁	—	2	4	no	—
161	D₄×C₃₁	C₂×C₂²⋊C₄	1	48	yes	—
162	D₄×C₃₁	C₂³×C₄	1	8	yes	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	24	no	—
165	C₂×C₄×C₃₁	—	2	8	no	—
166	C₂×C₄×C₃₁	—	2	48	no	—
167	C₂×C₄×C₃₁	—	2	8	no	—
168	C₂×C₄×C₃₁	—	2	8	no	—
169	C₂×C₄×C₃₁	—	8	12	no	—
170	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
171	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
172	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
173	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
174	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂³×C₄	1	12	yes	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂²×D₄	1	12	yes	—
181	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
183	C₈×C₃₁	—	2	8	no	—
184	C₂×#585	C₂×C₃₁	1	1	yes	characteristic
185	C₂×#568	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#3428	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#2901	C₂×C₃₁	1	16	yes	complemented
188	C₂×#421	C₂×C₃₁	1	12	yes	complemented
189	C₂²×C₃₁	—	2	8	no	—
190	C₂²×C₃₁	—	4	6	no	—
191	C₂²×C₃₁	—	4	6	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	—	4	6	no	—
194	C₂²×C₃₁	—	4	8	no	—
195	C₂²×C₃₁	—	8	24	no	—
196	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	12	yes	—
198	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
200	C₄×C₃₁	—	2	8	no	—
201	C₄×C₃₁	—	4	8	no	—
202	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
203	C₄×C₃₁	C₂²×C₂²⋊C₄	1	12	yes	—
204	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
205	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
206	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
207	#585	—	2	4	no	—
208	#585	—	2	4	no	—
209	#585	C₄×C₃₁	1	8	yes	complemented
210	#585	C₂²×C₃₁	1	8	yes	—
211	C₂×2₋¹⁺⁴	—	2	1	no	—
212	C₂×2₋¹⁺⁴	C₄×C₃₁	1	4	yes	complemented
213	C₂×H(5,𝔽₂)	—	2	3	no	—
214	C₂×H(5,𝔽₂)	C₄×C₃₁	1	4	yes	complemented
215	C₂²×#41	—	2	4	no	—
216	C₂²×#41	C₂²×C₃₁	1	6	yes	—
217	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
218	#568	—	2	4	no	—
219	#568	C₂²×C₃₁	1	8	yes	complemented
220	C₂×#131	—	2	4	no	—
221	C₂²×M₁₆	C₂²×C₃₁	1	6	yes	complemented
222	#520	—	2	4	no	—
223	#519	—	2	1	no	—
224	#518	—	2	3	no	—
225	C₄×#41	C₂²×C₃₁	1	8	yes	complemented
226	C₂×#117	C₂²×C₃₁	1	6	yes	complemented
227	#421	C₂²×C₃₁	1	16	yes	complemented
228	#421	C₂²×C₃₁	1	48	yes	complemented
229	C₂×#104	C₂²×C₃₁	1	48	yes	complemented
230	C₂×C₃₁	—	2	1	no	—
231	C₂×C₃₁	—	2	1	no	—
232	C₂×C₃₁	—	2	1	no	—
233	C₂×C₃₁	—	2	12	no	—
234	C₂×C₃₁	—	4	8	no	—
235	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
236	2₋¹⁺⁴	—	2	16	no	—
237	2₋¹⁺⁴	—	2	8	no	—
238	2₋¹⁺⁴	—	2	8	no	—
239	2₋¹⁺⁴	C₂×C₄×C₃₁	1	32	yes	complemented
240	H(5,𝔽₂)	—	2	16	no	—
241	H(5,𝔽₂)	—	2	24	no	—
242	H(5,𝔽₂)	—	2	24	no	—
243	H(5,𝔽₂)	C₂×C₄×C₃₁	1	32	yes	complemented
244	C₂×#41	—	2	32	no	—
245	C₂×#41	—	2	32	no	—
246	C₂×#41	—	2	4	no	—
247	C₂×#41	—	2	12	no	—
248	C₂×#41	—	2	6	no	—
249	C₂×#41	—	2	6	no	—
250	C₂×#41	—	2	12	no	—
251	C₂×#41	—	2	12	no	—
252	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
253	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
254	C₂×#41	C₂×C₄×C₃₁	1	12	yes	complemented
255	C₂×#41	C₂×C₄×C₃₁	1	24	yes	complemented
256	C₂×#41	D₄×C₃₁	1	4	yes	—
257	C₂×#41	D₄×C₃₁	1	4	yes	—
258	C₂×#41	D₄×C₃₁	1	4	yes	—
259	C₂×#41	C₂³×C₃₁	1	4	yes	—
260	C₂×#41	C₂³×C₃₁	1	24	yes	—
261	C₂×#41	C₂³×C₃₁	1	4	yes	—
262	C₂²×Q₈	—	2	4	no	—
263	C₂²×Q₈	C₂×C₄×C₃₁	1	6	yes	complemented
264	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
265	C₂²×D₄	—	2	6	no	—
266	C₂²×D₄	—	2	12	no	—
267	C₂²×D₄	C₂×C₄×C₃₁	1	6	yes	complemented
268	C₂²×D₄	D₄×C₃₁	1	3	yes	—
269	C₂³×C₄	—	2	6	no	—
270	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
271	#131	—	2	32	no	—
272	C₂×M₁₆	—	2	6	no	—
273	C₂×M₁₆	—	2	12	no	—
274	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
275	C₂×M₁₆	C₂³×C₃₁	1	24	yes	complemented
276	C₂²×C₈	—	2	6	no	—
277	C₄×Q₈	—	2	8	no	—
278	C₄×D₄	—	2	24	no	—
279	#117	—	2	6	no	—
280	#117	—	2	6	no	—
281	#117	C₂³×C₃₁	1	4	yes	—
282	#117	C₂³×C₃₁	1	12	yes	complemented
283	C₂×C₄⋊C₄	—	2	6	no	—
284	C₂×C₂²⋊C₄	—	2	6	no	—
285	C₂×C₄²	C₂³×C₃₁	1	12	yes	complemented
286	#104	C₂³×C₃₁	1	64	yes	complemented
287	C₃₁	#60093	1	1	yes	characteristic, direct product factor, Sylow
288	C₂⁴	—	2	6	no	—
289	C₂⁴	C₂×D₄×C₃₁	1	3	yes	—
290	#41	—	2	32	no	—
291	#41	—	2	96	no	—
292	#41	—	2	24	no	—
293	#41	—	2	24	no	—
294	#41	—	2	24	no	—
295	#41	—	2	4	no	—
296	#41	—	2	4	no	—
297	#41	—	2	24	no	—
298	#41	—	2	24	no	—
299	#41	C₂²⋊C₄×C₃₁	1	16	yes	—
300	#41	C₂²⋊C₄×C₃₁	1	16	yes	—
301	#41	C₂²×C₄×C₃₁	1	8	yes	—
302	#41	C₂²×C₄×C₃₁	1	48	yes	—
303	#41	C₂²×C₄×C₃₁	1	16	yes	—
304	#41	C₂×D₄×C₃₁	1	16	yes	—
305	#41	C₂⁴×C₃₁	1	8	yes	—
306	C₂×Q₈	—	2	32	no	—
307	C₂×Q₈	—	2	6	no	—
308	C₂×Q₈	—	2	12	no	—
309	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
310	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
311	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
312	C₂×Q₈	C₂²×C₄×C₃₁	1	24	yes	—
313	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
314	C₂×D₄	—	2	12	no	—
315	C₂×D₄	—	2	24	no	—
316	C₂×D₄	—	2	4	no	—
317	C₂×D₄	—	2	24	no	—
318	C₂×D₄	—	2	96	no	—
319	C₂×D₄	—	2	6	no	—
320	C₂×D₄	—	2	12	no	—
321	C₂×D₄	C₂²⋊C₄×C₃₁	1	12	yes	—
322	C₂×D₄	C₂²⋊C₄×C₃₁	1	12	yes	—
323	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
324	C₂×D₄	C₂²×C₄×C₃₁	1	24	yes	—
325	C₂×D₄	C₂×D₄×C₃₁	1	12	yes	—
326	C₂²×C₄	—	2	6	no	—
327	C₂²×C₄	—	2	12	no	—
328	C₂²×C₄	—	2	24	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	24	no	—
331	C₂²×C₄	—	2	12	no	—
332	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
333	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
334	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
335	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
337	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
338	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
339	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
340	M₁₆	—	2	4	no	—
341	M₁₆	—	2	24	no	—
342	M₁₆	C₂⁴×C₃₁	1	8	yes	—
343	C₂×C₈	—	2	4	no	—
344	C₂×C₈	—	2	24	no	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	12	no	—
347	C₂²⋊C₄	—	2	4	no	—
348	C₂²⋊C₄	—	2	12	no	—
349	C₄²	—	2	12	no	—
350	C₄²	C₂⁴×C₃₁	1	8	yes	—
351	C₂³	—	2	4	no	—
352	C₂³	—	2	24	no	—
353	C₂³	—	8	3	no	—
354	C₂³	—	8	12	no	—
355	C₂³	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
356	C₂³	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
357	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
358	C₂³	C₂²×D₄×C₃₁	1	6	yes	—
359	Q₈	—	2	24	no	—
360	Q₈	—	2	4	no	—
361	Q₈	C₂×C₂²⋊C₄×C₃₁	1	16	yes	—
362	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
363	D₄	—	2	48	no	—
364	D₄	—	2	8	no	—
365	D₄	—	2	8	no	—
366	D₄	—	2	24	no	—
367	D₄	—	2	4	no	—
368	D₄	C₂×C₂²⋊C₄×C₃₁	1	48	yes	—
369	D₄	C₂³×C₄×C₃₁	1	8	yes	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	24	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	2	48	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	8	12	no	—
377	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
378	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
379	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
380	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
381	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄	C₂³×C₄×C₃₁	1	12	yes	—
384	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
385	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₂²×D₄×C₃₁	1	12	yes	—
388	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
390	C₈	—	2	8	no	—
391	C₂²	—	2	8	no	—
392	C₂²	—	4	6	no	—
393	C₂²	—	4	6	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	6	no	—
396	C₂²	—	4	8	no	—
397	C₂²	—	8	24	no	—
398	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
399	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	12	yes	—
400	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
401	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
402	C₄	—	2	8	no	—
403	C₄	—	4	8	no	—
404	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
405	C₄	C₂²×C₂²⋊C₄×C₃₁	1	12	yes	—
406	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
407	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
408	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	12	no	—
413	C₂	—	4	8	no	—
414	C₂	C₂³×C₂²⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂³×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂×C₃₁	C₂³×C₂²⋊C₄
prime radical	C₂×C₃₁×#585	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#568	C₂	semidirect
3. C₃₁×#3428	C₂	semidirect
4. C₃₁×#2901	C₂	semidirect
5. C₂×C₃₁×#421	C₂	semidirect
8. C₃₁×#585	C₄	semidirect
11. C₂×C₃₁×2₋¹⁺⁴	C₄	semidirect
13. C₂×C₃₁×H(5,𝔽₂)	C₄	semidirect
18. C₃₁×#568	C₂²	semidirect
20. C₂²×M₁₆×C₃₁	C₂²	semidirect
24. C₄×#41×C₃₁	C₂²	semidirect
25. C₂×C₃₁×#117	C₂²	semidirect
26. C₃₁×#421	C₂²	semidirect
27. C₃₁×#421	C₂²	semidirect
28. C₂×C₃₁×#104	C₂²	semidirect
32. C₃₁×2₋¹⁺⁴	C₂×C₄	semidirect
36. C₃₁×H(5,𝔽₂)	C₂×C₄	semidirect
47. C₂×#41×C₃₁	C₂×C₄	semidirect
48. C₂×#41×C₃₁	C₂×C₄	semidirect
56. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
60. C₂²×D₄×C₃₁	C₂×C₄	semidirect
68. C₂×M₁₆×C₃₁	C₂³	semidirect
75. C₃₁×#117	C₂³	semidirect
78. C₂×C₄²×C₃₁	C₂³	semidirect
79. C₃₁×#104	C₂³	semidirect
143. #60093	C₃₁	direct (N × H)
185. C₂×#568	C₂×C₃₁	semidirect
186. #3428	C₂×C₃₁	semidirect
187. #2901	C₂×C₃₁	semidirect
188. C₂×#421	C₂×C₃₁	semidirect
209. #585	C₄×C₃₁	semidirect
212. C₂×2₋¹⁺⁴	C₄×C₃₁	semidirect
214. C₂×H(5,𝔽₂)	C₄×C₃₁	semidirect
219. #568	C₂²×C₃₁	semidirect
221. C₂²×M₁₆	C₂²×C₃₁	semidirect
225. C₄×#41	C₂²×C₃₁	semidirect
226. C₂×#117	C₂²×C₃₁	semidirect
227. #421	C₂²×C₃₁	semidirect
228. #421	C₂²×C₃₁	semidirect
229. C₂×#104	C₂²×C₃₁	semidirect
239. 2₋¹⁺⁴	C₂×C₄×C₃₁	semidirect
243. H(5,𝔽₂)	C₂×C₄×C₃₁	semidirect
254. C₂×#41	C₂×C₄×C₃₁	semidirect
255. C₂×#41	C₂×C₄×C₃₁	semidirect
263. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
267. C₂²×D₄	C₂×C₄×C₃₁	semidirect
275. C₂×M₁₆	C₂³×C₃₁	semidirect
282. #117	C₂³×C₃₁	semidirect
285. C₂×C₄²	C₂³×C₃₁	semidirect
286. #104	C₂³×C₃₁	semidirect
287. C₃₁	#60093	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×#585	2
C₂×C₃₁×#568	2
C₃₁×#3428	2
C₃₁×#2901	2
C₂×C₃₁×#421	2
#60093	31

Subgroups — C31×#60093 #21516852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60093 #21516852