Subgroups — C₃₁×#60126 #21516885

Order 7936.

2042 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2925	C₂	1	8	yes	maximal, complemented
3	C₃₁×#2924	C₂	1	8	yes	maximal, complemented
4	C₃₁×#2923	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2922	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2879	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#520	C₄	1	4	yes	—
18	C₃₁×#520	C₄	1	4	yes	—
19	C₃₁×#519	C₄	1	2	yes	—
20	C₃₁×#518	C₄	1	2	yes	—
21	C₃₁×#518	C₄	1	4	yes	—
22	C₄×#41×C₃₁	C₂²	1	4	yes	—
23	C₄×#41×C₃₁	C₂²	1	4	yes	—
24	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#117	C₂²	1	4	yes	—
27	C₃₁×#433	C₂²	1	16	yes	complemented
28	C₃₁×#433	C₂²	1	16	yes	complemented
29	C₃₁×#432	C₂²	1	16	yes	complemented
30	C₃₁×#431	C₂²	1	16	yes	complemented
31	C₃₁×#424	C₂²	1	4	yes	—
32	C₃₁×#424	C₂²	1	4	yes	—
33	C₃₁×#424	C₂²	1	2	yes	complemented
34	C₃₁×#424	C₂²	1	2	yes	complemented
35	C₃₁×#424	C₂²	1	4	yes	complemented
36	C₃₁×#423	C₂²	1	4	yes	complemented
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
38	C₃₁×#408	C₂²	1	2	yes	—
39	C₃₁×#408	C₂²	1	2	yes	—
40	C₃₁×#408	C₂²	1	4	yes	—
41	C₃₁×#408	C₂²	1	4	yes	—
42	C₃₁×#408	C₂²	1	4	yes	—
43	C₃₁×#407	C₂²	1	4	yes	—
44	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
45	C₃₁×#405	C₂²	1	4	yes	—
46	C₃₁×#405	C₂²	1	4	yes	—
47	C₃₁×#405	C₂²	1	4	yes	complemented
48	C₃₁×#405	C₂²	1	4	yes	complemented
49	C₄×M₁₆×C₃₁	C₂²	1	4	yes	complemented
50	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
51	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
52	C₂×#41×C₃₁	—	2	4	no	—
53	C₂×#41×C₃₁	—	2	4	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×#41×C₃₁	C₂³	1	2	yes	—
57	C₂×#41×C₃₁	C₂³	1	2	yes	—
58	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₃₁	—	2	4	no	—
64	C₂×M₁₆×C₃₁	—	2	4	no	—
65	C₂×M₁₆×C₃₁	—	2	4	no	—
66	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
68	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
72	C₂²×C₈×C₃₁	—	2	4	no	—
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
77	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
78	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
79	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
80	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	—
81	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	—
82	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	—
83	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
84	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	—
85	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
86	C₃₁×#117	C₂×C₄	1	8	yes	—
87	C₃₁×#117	C₂×C₄	1	2	yes	—
88	C₃₁×#117	C₂×C₄	1	2	yes	—
89	C₃₁×#117	C₂×C₄	1	8	yes	—
90	C₃₁×#117	C₂×C₄	1	2	yes	—
91	C₃₁×#117	C₂×C₄	1	2	yes	—
92	C₃₁×#117	C₂³	1	2	yes	—
93	C₃₁×#117	C₂³	1	2	yes	—
94	C₃₁×#117	C₂³	1	2	yes	—
95	C₃₁×#117	C₂³	1	2	yes	—
96	C₃₁×#117	C₂³	1	8	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
105	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
106	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
107	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
108	C₃₁×C₄⋊C₈	C₂³	1	16	yes	—
109	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
110	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
111	C₃₁×C₂²⋊C₈	C₂³	1	16	yes	—
112	C₃₁×#97	C₂³	1	8	yes	—
113	C₃₁×#97	C₂³	1	8	yes	—
114	C₄×C₈×C₃₁	C₂³	1	8	yes	—
115	C₄×C₈×C₃₁	C₂³	1	8	yes	—
116	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	#41×C₃₁	—	2	8	no	—
119	#41×C₃₁	—	2	8	no	—
120	#41×C₃₁	—	2	4	no	—
121	#41×C₃₁	—	2	4	no	—
122	#41×C₃₁	—	2	4	no	—
123	#41×C₃₁	—	2	4	no	—
124	C₂×Q₈×C₃₁	—	2	4	no	—
125	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
127	C₂×D₄×C₃₁	—	2	8	no	—
128	C₂×D₄×C₃₁	—	2	4	no	—
129	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
131	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
132	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
138	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
139	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	#41	1	4	yes	—
144	C₂²×C₄×C₃₁	#41	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
149	M₁₆×C₃₁	—	2	4	no	—
150	M₁₆×C₃₁	—	2	4	no	—
151	M₁₆×C₃₁	—	2	4	no	—
152	M₁₆×C₃₁	—	2	4	no	—
153	M₁₆×C₃₁	—	2	4	no	—
154	M₁₆×C₃₁	—	2	4	no	—
155	C₂×C₈×C₃₁	—	2	4	no	—
156	C₂×C₈×C₃₁	—	2	4	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	4	no	—
160	C₂×C₈×C₃₁	—	2	4	no	—
161	C₂×C₈×C₃₁	—	2	4	no	—
162	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
163	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
164	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
165	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
166	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
167	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
170	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
171	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
172	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
173	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
174	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
175	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
176	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
177	C₄²×C₃₁	C₂²×C₄	1	16	yes	—
178	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
179	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
180	C₄²×C₃₁	C₂⁴	1	4	yes	—
181	C₄²×C₃₁	C₂⁴	1	4	yes	—
182	#60126	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	4	no	—
184	C₂³×C₃₁	—	4	2	no	—
185	C₂³×C₃₁	—	4	4	no	—
186	C₂³×C₃₁	#117	1	4	yes	—
187	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
188	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
189	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
190	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
191	Q₈×C₃₁	—	2	4	no	—
192	Q₈×C₃₁	—	2	4	no	—
193	D₄×C₃₁	—	2	8	no	—
194	D₄×C₃₁	—	2	8	no	—
195	D₄×C₃₁	—	2	4	no	—
196	D₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	8	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	4	4	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	4	16	no	—
204	C₂×C₄×C₃₁	#117	1	4	yes	—
205	C₂×C₄×C₃₁	#117	1	4	yes	—
206	C₂×C₄×C₃₁	#117	1	4	yes	—
207	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
208	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
209	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
210	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
214	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
215	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
216	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
217	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
218	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
219	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	2	2	no	—
222	C₈×C₃₁	—	2	4	no	—
223	C₈×C₃₁	—	2	4	no	—
224	C₈×C₃₁	—	2	4	no	—
225	#3428	C₂×C₃₁	1	1	yes	characteristic
226	#2925	C₂×C₃₁	1	8	yes	complemented
227	#2924	C₂×C₃₁	1	8	yes	complemented
228	#2923	C₂×C₃₁	1	4	yes	complemented
229	#2922	C₂×C₃₁	1	4	yes	complemented
230	#2905	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂×#424	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2879	C₂×C₃₁	1	1	yes	characteristic
233	C₂×#408	C₂×C₃₁	1	1	yes	characteristic
234	#2873	C₂×C₃₁	1	1	yes	characteristic, complemented
235	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂²×C₃₁	—	2	2	no	—
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	2	2	no	—
239	C₂²×C₃₁	—	2	4	no	—
240	C₂²×C₃₁	—	2	8	no	—
241	C₂²×C₃₁	—	4	4	no	—
242	C₂²×C₃₁	—	8	4	no	—
243	C₂²×C₃₁	C₂×#117	1	2	yes	—
244	C₂²×C₃₁	C₂×#117	1	2	yes	—
245	C₂²×C₃₁	C₂×#117	1	2	yes	—
246	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
247	C₄×C₃₁	—	2	4	no	—
248	C₄×C₃₁	—	2	8	no	—
249	C₄×C₃₁	—	2	8	no	—
250	C₄×C₃₁	C₂×#117	1	2	yes	—
251	C₄×C₃₁	C₂×#117	1	2	yes	—
252	C₄×C₃₁	C₂×#117	1	2	yes	—
253	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
254	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
255	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
256	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
257	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
258	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
259	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	#520	C₄×C₃₁	1	4	yes	—
261	#520	C₄×C₃₁	1	4	yes	—
262	#519	C₄×C₃₁	1	2	yes	—
263	#518	C₄×C₃₁	1	2	yes	—
264	#518	C₄×C₃₁	1	4	yes	—
265	C₄×#41	C₂²×C₃₁	1	4	yes	—
266	C₄×#41	C₂²×C₃₁	1	4	yes	—
267	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#117	C₂²×C₃₁	1	4	yes	—
270	#433	C₂²×C₃₁	1	16	yes	complemented
271	#433	C₂²×C₃₁	1	16	yes	complemented
272	#432	C₂²×C₃₁	1	16	yes	complemented
273	#431	C₂²×C₃₁	1	16	yes	complemented
274	#424	C₂²×C₃₁	1	4	yes	—
275	#424	C₂²×C₃₁	1	4	yes	—
276	#424	C₂²×C₃₁	1	2	yes	complemented
277	#424	C₂²×C₃₁	1	2	yes	complemented
278	#424	C₂²×C₃₁	1	4	yes	complemented
279	#423	C₂²×C₃₁	1	4	yes	complemented
280	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
281	#408	C₂²×C₃₁	1	2	yes	—
282	#408	C₂²×C₃₁	1	2	yes	—
283	#408	C₂²×C₃₁	1	4	yes	—
284	#408	C₂²×C₃₁	1	4	yes	—
285	#408	C₂²×C₃₁	1	4	yes	—
286	#407	C₂²×C₃₁	1	4	yes	—
287	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
288	#405	C₂²×C₃₁	1	4	yes	—
289	#405	C₂²×C₃₁	1	4	yes	—
290	#405	C₂²×C₃₁	1	4	yes	complemented
291	#405	C₂²×C₃₁	1	4	yes	complemented
292	C₄×M₁₆	C₂²×C₃₁	1	4	yes	complemented
293	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
294	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
295	C₂×C₃₁	—	2	2	no	—
296	C₂×C₃₁	—	2	2	no	—
297	C₂×C₃₁	—	2	2	no	—
298	C₂×C₃₁	—	4	4	no	—
299	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
300	C₂×C₃₁	#3573	1	2	yes	—
301	C₂×#41	—	2	4	no	—
302	C₂×#41	—	2	4	no	—
303	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
304	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
305	C₂×#41	C₂³×C₃₁	1	2	yes	—
306	C₂×#41	C₂³×C₃₁	1	2	yes	—
307	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
308	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
309	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
311	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×M₁₆	—	2	4	no	—
313	C₂×M₁₆	—	2	4	no	—
314	C₂×M₁₆	—	2	4	no	—
315	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
316	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
317	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
318	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
319	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
320	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
321	C₂²×C₈	—	2	4	no	—
322	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
324	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
325	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
326	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
327	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
328	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
329	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	—
330	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	—
331	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	—
332	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
333	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	—
334	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
335	#117	C₂×C₄×C₃₁	1	8	yes	—
336	#117	C₂×C₄×C₃₁	1	2	yes	—
337	#117	C₂×C₄×C₃₁	1	2	yes	—
338	#117	C₂×C₄×C₃₁	1	8	yes	—
339	#117	C₂×C₄×C₃₁	1	2	yes	—
340	#117	C₂×C₄×C₃₁	1	2	yes	—
341	#117	C₂³×C₃₁	1	2	yes	—
342	#117	C₂³×C₃₁	1	2	yes	—
343	#117	C₂³×C₃₁	1	2	yes	—
344	#117	C₂³×C₃₁	1	2	yes	—
345	#117	C₂³×C₃₁	1	8	yes	—
346	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
347	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
348	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
349	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
351	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
352	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
353	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
354	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
355	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
356	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
357	C₄⋊C₈	C₂³×C₃₁	1	16	yes	—
358	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
359	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
360	C₂²⋊C₈	C₂³×C₃₁	1	16	yes	—
361	#97	C₂³×C₃₁	1	8	yes	—
362	#97	C₂³×C₃₁	1	8	yes	—
363	C₄×C₈	C₂³×C₃₁	1	8	yes	—
364	C₄×C₈	C₂³×C₃₁	1	8	yes	—
365	C₃₁	#60126	1	1	yes	characteristic, direct product factor, Sylow
366	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
367	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
368	#41	—	2	8	no	—
369	#41	—	2	8	no	—
370	#41	—	2	4	no	—
371	#41	—	2	4	no	—
372	#41	—	2	4	no	—
373	#41	—	2	4	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
376	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
380	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
381	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
382	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
386	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
388	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
389	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
390	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	#41×C₃₁	1	4	yes	—
394	C₂²×C₄	#41×C₃₁	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
399	M₁₆	—	2	4	no	—
400	M₁₆	—	2	4	no	—
401	M₁₆	—	2	4	no	—
402	M₁₆	—	2	4	no	—
403	M₁₆	—	2	4	no	—
404	M₁₆	—	2	4	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	—	2	4	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	4	no	—
410	C₂×C₈	—	2	4	no	—
411	C₂×C₈	—	2	4	no	—
412	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
413	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
414	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
415	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
416	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
417	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
418	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
419	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
420	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
421	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
422	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
423	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
424	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
425	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
426	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
427	C₄²	C₂²×C₄×C₃₁	1	16	yes	—
428	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
429	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
430	C₄²	C₂⁴×C₃₁	1	4	yes	—
431	C₄²	C₂⁴×C₃₁	1	4	yes	—
432	C₂³	—	2	4	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	4	no	—
435	C₂³	C₃₁×#117	1	4	yes	—
436	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
437	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
438	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
439	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
440	Q₈	—	2	4	no	—
441	Q₈	—	2	4	no	—
442	D₄	—	2	8	no	—
443	D₄	—	2	8	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	16	no	—
453	C₂×C₄	C₃₁×#117	1	4	yes	—
454	C₂×C₄	C₃₁×#117	1	4	yes	—
455	C₂×C₄	C₃₁×#117	1	4	yes	—
456	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
457	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
458	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
459	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
463	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
464	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
465	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
466	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
467	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
468	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	2	2	no	—
471	C₈	—	2	4	no	—
472	C₈	—	2	4	no	—
473	C₈	—	2	4	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	8	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	8	4	no	—
481	C₂²	C₂×C₃₁×#117	1	2	yes	—
482	C₂²	C₂×C₃₁×#117	1	2	yes	—
483	C₂²	C₂×C₃₁×#117	1	2	yes	—
484	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
485	C₄	—	2	4	no	—
486	C₄	—	2	8	no	—
487	C₄	—	2	8	no	—
488	C₄	C₂×C₃₁×#117	1	2	yes	—
489	C₄	C₂×C₃₁×#117	1	2	yes	—
490	C₄	C₂×C₃₁×#117	1	2	yes	—
491	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
492	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
493	C₂	—	2	2	no	—
494	C₂	—	2	2	no	—
495	C₂	—	2	2	no	—
496	C₂	—	4	4	no	—
497	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
498	C₂	C₃₁×#3573	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2925	C₂	semidirect
3. C₃₁×#2924	C₂	semidirect
4. C₃₁×#2923	C₂	semidirect
5. C₃₁×#2922	C₂	semidirect
6. C₃₁×#2905	C₂	semidirect
7. C₂×C₃₁×#424	C₂	semidirect
10. C₃₁×#2873	C₂	semidirect
11. C₂×C₃₁×#405	C₂	semidirect
27. C₃₁×#433	C₂²	semidirect
28. C₃₁×#433	C₂²	semidirect
29. C₃₁×#432	C₂²	semidirect
30. C₃₁×#431	C₂²	semidirect
33. C₃₁×#424	C₂²	semidirect
34. C₃₁×#424	C₂²	semidirect
35. C₃₁×#424	C₂²	semidirect
36. C₃₁×#423	C₂²	semidirect
37. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
47. C₃₁×#405	C₂²	semidirect
48. C₃₁×#405	C₂²	semidirect
49. C₄×M₁₆×C₃₁	C₂²	semidirect
50. C₂×C₃₁×#97	C₂²	semidirect
51. C₂×C₄×C₈×C₃₁	C₂²	semidirect
182. #60126	C₃₁	direct (N × H)
226. #2925	C₂×C₃₁	semidirect
227. #2924	C₂×C₃₁	semidirect
228. #2923	C₂×C₃₁	semidirect
229. #2922	C₂×C₃₁	semidirect
230. #2905	C₂×C₃₁	semidirect
231. C₂×#424	C₂×C₃₁	semidirect
234. #2873	C₂×C₃₁	semidirect
235. C₂×#405	C₂×C₃₁	semidirect
270. #433	C₂²×C₃₁	semidirect
271. #433	C₂²×C₃₁	semidirect
272. #432	C₂²×C₃₁	semidirect
273. #431	C₂²×C₃₁	semidirect
276. #424	C₂²×C₃₁	semidirect
277. #424	C₂²×C₃₁	semidirect
278. #424	C₂²×C₃₁	semidirect
279. #423	C₂²×C₃₁	semidirect
280. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
290. #405	C₂²×C₃₁	semidirect
291. #405	C₂²×C₃₁	semidirect
292. C₄×M₁₆	C₂²×C₃₁	semidirect
293. C₂×#97	C₂²×C₃₁	semidirect
294. C₂×C₄×C₈	C₂²×C₃₁	semidirect
365. C₃₁	#60126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3428	2
C₃₁×#2925	2
C₃₁×#2924	2
C₃₁×#2923	2
C₃₁×#2922	2
C₃₁×#2905	2
C₂×C₃₁×#424	2
C₃₁×#2879	2
C₂×C₃₁×#408	2
C₃₁×#2873	2
C₂×C₃₁×#405	2
#60126	31

Subgroups — C31×#60126 #21516885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60126 #21516885