Subgroups — C₃₁×#60569 #21517328

Order 7936.

2010 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3127	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3123	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3120	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3095	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3080	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3039	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3007	C₂	1	2	yes	maximal
10	C₄×C₃₁×#137	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#137	C₂²	1	4	yes	—
14	C₃₁×#541	—	2	2	no	—
15	C₃₁×#541	—	2	2	no	—
16	C₃₁×#541	C₂²	1	4	yes	—
17	C₃₁×#541	C₂²	1	4	yes	complemented
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₃₁×#126	—	2	2	no	—
20	C₂×C₃₁×#125	—	2	2	no	—
21	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₃₁×#520	—	2	2	no	—
24	C₃₁×#520	C₂²	1	4	yes	—
25	C₃₁×#519	C₂²	1	2	yes	—
26	C₃₁×#518	—	2	2	no	—
27	C₃₁×#518	C₂²	1	2	yes	—
28	C₄×#41×C₃₁	—	2	2	no	—
29	C₄×#41×C₃₁	C₂²	1	2	yes	—
30	C₄×#41×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
32	C₂×C₃₁×#117	—	2	2	no	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₃₁×#485	C₂²	1	8	yes	—
36	C₃₁×#484	C₂²	1	4	yes	complemented
37	C₃₁×#483	C₂²	1	4	yes	complemented
38	C₃₁×#479	C₂²	1	8	yes	complemented
39	C₃₁×#478	C₂²	1	8	yes	complemented
40	C₃₁×#462	C₂²	1	8	yes	complemented
41	C₃₁×#461	C₂²	1	8	yes	complemented
42	C₃₁×#452	C₂²	1	8	yes	—
43	C₃₁×#451	C₂²	1	4	yes	—
44	C₃₁×#450	C₂²	1	4	yes	—
45	C₃₁×#441	C₂²	1	8	yes	complemented
46	C₃₁×#440	C₂²	1	8	yes	complemented
47	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
48	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
49	C₃₁×#433	C₂²	1	4	yes	—
50	C₃₁×#433	C₂²	1	4	yes	complemented
51	C₃₁×#428	C₂²	1	4	yes	complemented
52	C₃₁×#423	C₂²	1	2	yes	complemented
53	C₃₁×#419	C₂²	1	4	yes	complemented
54	C₃₁×#417	C₂²	1	4	yes	—
55	C₃₁×#407	C₂²	1	2	yes	—
56	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
57	C₂×#41×C₃₁	—	2	8	no	—
58	C₂×#41×C₃₁	—	2	1	no	—
59	C₂×#41×C₃₁	—	2	1	no	—
60	C₂×#41×C₃₁	C₂³	1	2	yes	—
61	C₂×#41×C₃₁	C₂³	1	2	yes	—
62	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
63	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₃₁	—	2	1	no	—
65	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	—	2	1	no	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#137	—	2	16	no	—
69	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
70	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
72	C₃₁×C₄⋊Q₈	—	2	2	no	—
73	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
74	C₃₁×#126	—	2	8	no	—
75	C₃₁×#126	—	2	8	no	—
76	C₃₁×#125	—	2	8	no	—
77	C₃₁×#125	—	2	8	no	—
78	C₃₁×#125	—	2	2	no	—
79	C₃₁×#125	C₂³	1	4	yes	—
80	C₃₁×C₂²⋊Q₈	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
83	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
84	C₄×Q₈×C₃₁	—	2	4	no	—
85	C₄×Q₈×C₃₁	—	2	4	no	—
86	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
87	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
88	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
89	C₄×D₄×C₃₁	—	2	4	no	—
90	C₄×D₄×C₃₁	—	2	4	no	—
91	C₄×D₄×C₃₁	—	2	4	no	—
92	C₄×D₄×C₃₁	—	2	4	no	—
93	C₄×D₄×C₃₁	—	2	4	no	—
94	C₄×D₄×C₃₁	C₂³	1	4	yes	—
95	C₄×D₄×C₃₁	C₂³	1	4	yes	—
96	C₃₁×#117	—	2	1	no	—
97	C₃₁×#117	—	2	1	no	—
98	C₃₁×#117	—	2	4	no	—
99	C₃₁×#117	—	2	4	no	—
100	C₃₁×#117	—	2	2	no	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	D₄	1	2	yes	—
103	C₃₁×#117	D₄	1	2	yes	—
104	C₃₁×#117	C₂³	1	2	yes	—
105	C₃₁×#117	C₂³	1	2	yes	—
106	C₃₁×#117	C₂³	1	2	yes	—
107	C₃₁×#117	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
119	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
120	C₂×C₄²×C₃₁	—	2	1	no	—
121	C₂×C₄²×C₃₁	—	2	2	no	—
122	C₂×C₄²×C₃₁	—	2	2	no	—
123	C₂×C₄²×C₃₁	—	2	2	no	—
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₃₁×#107	C₂³	1	8	yes	—
127	C₃₁×#106	C₂³	1	8	yes	—
128	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
129	C₃₁×#103	C₂³	1	8	yes	—
130	C₃₁×#103	C₂³	1	8	yes	—
131	C₃₁×#103	C₂³	1	8	yes	—
132	C₃₁×#102	C₂³	1	8	yes	—
133	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
134	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
135	C₃₁×#97	C₂³	1	4	yes	—
136	C₄×C₈×C₃₁	C₂³	1	4	yes	—
137	C₂⁴×C₃₁	—	2	1	no	—
138	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
139	#41×C₃₁	—	2	8	no	—
140	#41×C₃₁	—	2	8	no	—
141	#41×C₃₁	—	2	4	no	—
142	#41×C₃₁	—	2	4	no	—
143	#41×C₃₁	—	2	4	no	—
144	#41×C₃₁	#41	1	8	yes	—
145	C₂×Q₈×C₃₁	—	2	4	no	—
146	C₂×Q₈×C₃₁	—	2	1	no	—
147	C₂×Q₈×C₃₁	—	2	2	no	—
148	C₂×Q₈×C₃₁	—	2	8	no	—
149	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
150	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
151	C₂×D₄×C₃₁	—	2	8	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	—	2	2	no	—
154	C₂×D₄×C₃₁	—	2	4	no	—
155	C₂×D₄×C₃₁	—	2	1	no	—
156	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	2	1	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	4	no	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	2	4	no	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	Q₁₆×C₃₁	—	2	16	no	—
176	SD₁₆×C₃₁	—	2	16	no	—
177	M₁₆×C₃₁	—	2	8	no	—
178	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	8	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	2	4	no	—
185	C₄⋊C₄×C₃₁	—	2	8	no	—
186	C₄⋊C₄×C₃₁	—	2	8	no	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
190	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
191	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
192	C₂²⋊C₄×C₃₁	—	2	2	no	—
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	4	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	4	no	—
198	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
199	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
200	C₄²×C₃₁	—	2	2	no	—
201	C₄²×C₃₁	—	2	4	no	—
202	C₄²×C₃₁	—	2	4	no	—
203	C₄²×C₃₁	—	2	2	no	—
204	C₄²×C₃₁	—	2	8	no	—
205	C₄²×C₃₁	C₂×D₄	1	2	yes	—
206	C₄²×C₃₁	C₂×D₄	1	2	yes	—
207	C₄²×C₃₁	C₂⁴	1	2	yes	—
208	C₄²×C₃₁	C₂⁴	1	2	yes	—
209	#60569	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₃₁	—	2	4	no	—
211	C₂³×C₃₁	—	2	2	no	—
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	4	2	no	—
214	C₂³×C₃₁	—	8	2	no	—
215	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
217	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	Q₈×C₃₁	—	2	4	no	—
219	Q₈×C₃₁	—	2	2	no	—
220	Q₈×C₃₁	—	2	8	no	—
221	Q₈×C₃₁	—	4	4	no	—
222	Q₈×C₃₁	C₂×#41	1	4	yes	—
223	D₄×C₃₁	—	2	8	no	—
224	D₄×C₃₁	—	2	4	no	—
225	D₄×C₃₁	—	2	8	no	—
226	D₄×C₃₁	—	2	2	no	—
227	D₄×C₃₁	C₂×#41	1	4	yes	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	8	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	8	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	4	2	no	—
241	C₂×C₄×C₃₁	—	4	2	no	—
242	C₂×C₄×C₃₁	—	4	4	no	—
243	C₂×C₄×C₃₁	—	4	4	no	—
244	C₂×C₄×C₃₁	—	8	4	no	—
245	C₂×C₄×C₃₁	—	8	4	no	—
246	C₂×C₄×C₃₁	—	8	2	no	—
247	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
248	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
253	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
254	C₈×C₃₁	—	2	8	no	—
255	C₂×#541	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#3428	C₂×C₃₁	1	1	yes	characteristic
257	#3127	C₂×C₃₁	1	4	yes	complemented
258	#3123	C₂×C₃₁	1	4	yes	complemented
259	#3120	C₂×C₃₁	1	8	yes	complemented
260	#3095	C₂×C₃₁	1	2	yes	complemented
261	#3080	C₂×C₃₁	1	2	yes	complemented
262	#3039	C₂×C₃₁	1	2	yes	complemented
263	#3007	C₂×C₃₁	1	2	yes	—
264	C₄×#137	C₂×C₃₁	1	4	yes	complemented
265	#2921	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂²×C₃₁	—	2	2	no	—
267	C₂²×C₃₁	—	2	2	no	—
268	C₂²×C₃₁	—	2	2	no	—
269	C₂²×C₃₁	—	2	1	no	—
270	C₂²×C₃₁	—	2	1	no	—
271	C₂²×C₃₁	—	2	4	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	8	2	no	—
274	C₂²×C₃₁	—	8	4	no	—
275	C₂²×C₃₁	#525	1	2	yes	—
276	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
277	C₄×C₃₁	—	2	4	no	—
278	C₄×C₃₁	—	2	4	no	—
279	C₄×C₃₁	—	2	1	no	—
280	C₄×C₃₁	—	2	4	no	—
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	4	4	no	—
283	C₄×C₃₁	—	4	4	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	#525	1	2	yes	—
286	C₄×C₃₁	C₂²×#41	1	2	yes	—
287	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#137	C₂²×C₃₁	1	4	yes	—
289	#541	—	2	2	no	—
290	#541	—	2	2	no	—
291	#541	C₂²×C₃₁	1	4	yes	—
292	#541	C₂²×C₃₁	1	4	yes	complemented
293	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
294	C₂×#126	—	2	2	no	—
295	C₂×#125	—	2	2	no	—
296	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
297	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
298	#520	—	2	2	no	—
299	#520	C₂²×C₃₁	1	4	yes	—
300	#519	C₂²×C₃₁	1	2	yes	—
301	#518	—	2	2	no	—
302	#518	C₂²×C₃₁	1	2	yes	—
303	C₄×#41	—	2	2	no	—
304	C₄×#41	C₂²×C₃₁	1	2	yes	—
305	C₄×#41	C₂²×C₃₁	1	2	yes	—
306	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
307	C₂×#117	—	2	2	no	—
308	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
310	#485	C₂²×C₃₁	1	8	yes	—
311	#484	C₂²×C₃₁	1	4	yes	complemented
312	#483	C₂²×C₃₁	1	4	yes	complemented
313	#479	C₂²×C₃₁	1	8	yes	complemented
314	#478	C₂²×C₃₁	1	8	yes	complemented
315	#462	C₂²×C₃₁	1	8	yes	complemented
316	#461	C₂²×C₃₁	1	8	yes	complemented
317	#452	C₂²×C₃₁	1	8	yes	—
318	#451	C₂²×C₃₁	1	4	yes	—
319	#450	C₂²×C₃₁	1	4	yes	—
320	#441	C₂²×C₃₁	1	8	yes	complemented
321	#440	C₂²×C₃₁	1	8	yes	complemented
322	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
323	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
324	#433	C₂²×C₃₁	1	4	yes	—
325	#433	C₂²×C₃₁	1	4	yes	complemented
326	#428	C₂²×C₃₁	1	4	yes	complemented
327	#423	C₂²×C₃₁	1	2	yes	complemented
328	#419	C₂²×C₃₁	1	4	yes	complemented
329	#417	C₂²×C₃₁	1	4	yes	—
330	#407	C₂²×C₃₁	1	2	yes	—
331	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
332	C₂×C₃₁	—	2	2	no	—
333	C₂×C₃₁	—	4	1	no	—
334	C₂×C₃₁	—	4	1	no	—
335	C₂×C₃₁	—	4	4	no	—
336	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
337	C₂×C₃₁	#3586	1	2	yes	—
338	C₂×#41	—	2	8	no	—
339	C₂×#41	—	2	1	no	—
340	C₂×#41	—	2	1	no	—
341	C₂×#41	C₂³×C₃₁	1	2	yes	—
342	C₂×#41	C₂³×C₃₁	1	2	yes	—
343	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
344	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₂²×D₄	—	2	1	no	—
346	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
349	#137	—	2	16	no	—
350	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
351	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
352	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
353	C₄⋊Q₈	—	2	2	no	—
354	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
355	#126	—	2	8	no	—
356	#126	—	2	8	no	—
357	#125	—	2	8	no	—
358	#125	—	2	8	no	—
359	#125	—	2	2	no	—
360	#125	C₂³×C₃₁	1	4	yes	—
361	C₂²⋊Q₈	—	2	2	no	—
362	C₂²⋊Q₈	—	2	2	no	—
363	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
364	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×Q₈	—	2	4	no	—
367	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
368	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
369	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
370	C₄×D₄	—	2	4	no	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	2	4	no	—
373	C₄×D₄	—	2	4	no	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	C₂³×C₃₁	1	4	yes	—
376	C₄×D₄	C₂³×C₃₁	1	4	yes	—
377	#117	—	2	1	no	—
378	#117	—	2	1	no	—
379	#117	—	2	4	no	—
380	#117	—	2	4	no	—
381	#117	—	2	2	no	—
382	#117	—	2	2	no	—
383	#117	D₄×C₃₁	1	2	yes	—
384	#117	D₄×C₃₁	1	2	yes	—
385	#117	C₂³×C₃₁	1	2	yes	—
386	#117	C₂³×C₃₁	1	2	yes	—
387	#117	C₂³×C₃₁	1	2	yes	—
388	#117	C₂³×C₃₁	1	2	yes	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄⋊C₄	—	2	2	no	—
391	C₂×C₄⋊C₄	—	2	2	no	—
392	C₂×C₄⋊C₄	—	2	2	no	—
393	C₂×C₄⋊C₄	—	2	4	no	—
394	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	2	no	—
399	C₂×C₂²⋊C₄	—	2	4	no	—
400	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
401	C₂×C₄²	—	2	1	no	—
402	C₂×C₄²	—	2	2	no	—
403	C₂×C₄²	—	2	2	no	—
404	C₂×C₄²	—	2	2	no	—
405	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
407	#107	C₂³×C₃₁	1	8	yes	—
408	#106	C₂³×C₃₁	1	8	yes	—
409	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
410	#103	C₂³×C₃₁	1	8	yes	—
411	#103	C₂³×C₃₁	1	8	yes	—
412	#103	C₂³×C₃₁	1	8	yes	—
413	#102	C₂³×C₃₁	1	8	yes	—
414	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
415	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
416	#97	C₂³×C₃₁	1	4	yes	—
417	C₄×C₈	C₂³×C₃₁	1	4	yes	—
418	C₃₁	#60569	1	1	yes	characteristic, direct product factor, Sylow
419	C₂⁴	—	2	1	no	—
420	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
421	#41	—	2	8	no	—
422	#41	—	2	8	no	—
423	#41	—	2	4	no	—
424	#41	—	2	4	no	—
425	#41	—	2	4	no	—
426	#41	#41×C₃₁	1	8	yes	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	2	8	no	—
431	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
432	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
439	C₂²×C₄	—	2	8	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
457	Q₁₆	—	2	16	no	—
458	SD₁₆	—	2	16	no	—
459	M₁₆	—	2	8	no	—
460	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	2	8	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
471	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
472	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
473	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	2	4	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
481	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
482	C₄²	—	2	2	no	—
483	C₄²	—	2	4	no	—
484	C₄²	—	2	4	no	—
485	C₄²	—	2	2	no	—
486	C₄²	—	2	8	no	—
487	C₄²	C₂×D₄×C₃₁	1	2	yes	—
488	C₄²	C₂×D₄×C₃₁	1	2	yes	—
489	C₄²	C₂⁴×C₃₁	1	2	yes	—
490	C₄²	C₂⁴×C₃₁	1	2	yes	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	8	2	no	—
496	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
497	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
498	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
499	Q₈	—	2	4	no	—
500	Q₈	—	2	2	no	—
501	Q₈	—	2	8	no	—
502	Q₈	—	4	4	no	—
503	Q₈	C₂×#41×C₃₁	1	4	yes	—
504	D₄	—	2	8	no	—
505	D₄	—	2	4	no	—
506	D₄	—	2	8	no	—
507	D₄	—	2	2	no	—
508	D₄	C₂×#41×C₃₁	1	4	yes	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	8	4	no	—
526	C₂×C₄	—	8	4	no	—
527	C₂×C₄	—	8	2	no	—
528	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
529	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
531	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
533	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
534	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
535	C₈	—	2	8	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	8	2	no	—
544	C₂²	—	8	4	no	—
545	C₂²	C₃₁×#525	1	2	yes	—
546	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
547	C₄	—	2	1	no	—
548	C₄	—	2	4	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	4	no	—
551	C₄	—	2	4	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₄	C₃₁×#525	1	2	yes	—
556	C₄	C₂²×#41×C₃₁	1	2	yes	—
557	C₂	—	2	2	no	—
558	C₂	—	4	1	no	—
559	C₂	—	4	1	no	—
560	C₂	—	4	4	no	—
561	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
562	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#541	C₂	semidirect
3. C₃₁×#3127	C₂	semidirect
4. C₃₁×#3123	C₂	semidirect
5. C₃₁×#3120	C₂	semidirect
6. C₃₁×#3095	C₂	semidirect
7. C₃₁×#3080	C₂	semidirect
8. C₃₁×#3039	C₂	semidirect
10. C₄×C₃₁×#137	C₂	semidirect
11. C₃₁×#2921	C₂	semidirect
17. C₃₁×#541	C₂²	semidirect
18. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
21. C₂×C₃₁×#125	C₂²	semidirect
31. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
36. C₃₁×#484	C₂²	semidirect
37. C₃₁×#483	C₂²	semidirect
38. C₃₁×#479	C₂²	semidirect
39. C₃₁×#478	C₂²	semidirect
40. C₃₁×#462	C₂²	semidirect
41. C₃₁×#461	C₂²	semidirect
45. C₃₁×#441	C₂²	semidirect
46. C₃₁×#440	C₂²	semidirect
47. C₄×Q₁₆×C₃₁	C₂²	semidirect
48. C₄×SD₁₆×C₃₁	C₂²	semidirect
50. C₃₁×#433	C₂²	semidirect
51. C₃₁×#428	C₂²	semidirect
52. C₃₁×#423	C₂²	semidirect
53. C₃₁×#419	C₂²	semidirect
56. C₄×M₁₆×C₃₁	C₂²	semidirect
209. #60569	C₃₁	direct (N × H)
255. C₂×#541	C₂×C₃₁	semidirect
257. #3127	C₂×C₃₁	semidirect
258. #3123	C₂×C₃₁	semidirect
259. #3120	C₂×C₃₁	semidirect
260. #3095	C₂×C₃₁	semidirect
261. #3080	C₂×C₃₁	semidirect
262. #3039	C₂×C₃₁	semidirect
264. C₄×#137	C₂×C₃₁	semidirect
265. #2921	C₂×C₃₁	semidirect
292. #541	C₂²×C₃₁	semidirect
293. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
296. C₂×#125	C₂²×C₃₁	semidirect
306. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
311. #484	C₂²×C₃₁	semidirect
312. #483	C₂²×C₃₁	semidirect
313. #479	C₂²×C₃₁	semidirect
314. #478	C₂²×C₃₁	semidirect
315. #462	C₂²×C₃₁	semidirect
316. #461	C₂²×C₃₁	semidirect
320. #441	C₂²×C₃₁	semidirect
321. #440	C₂²×C₃₁	semidirect
322. C₄×Q₁₆	C₂²×C₃₁	semidirect
323. C₄×SD₁₆	C₂²×C₃₁	semidirect
325. #433	C₂²×C₃₁	semidirect
326. #428	C₂²×C₃₁	semidirect
327. #423	C₂²×C₃₁	semidirect
328. #419	C₂²×C₃₁	semidirect
331. C₄×M₁₆	C₂²×C₃₁	semidirect
418. C₃₁	#60569	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×#541	2
C₃₁×#3428	2
C₃₁×#3127	2
C₃₁×#3123	2
C₃₁×#3120	2
C₃₁×#3095	2
C₃₁×#3080	2
C₃₁×#3039	2
C₃₁×#3007	2
C₄×C₃₁×#137	2
C₃₁×#2921	2
#60569	31

Subgroups — C31×#60569 #21517328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60569 #21517328