Subgroups — C₃₁×#36576 #21493335

Order 7936.

3244 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#548	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2856	C₂	1	3	yes	maximal, complemented
5	C₃₁×#2548	C₂	1	3	yes	maximal, complemented
6	C₃₁×#2538	C₂	1	3	yes	maximal, complemented
7	C₃₁×#2426	C₂	1	6	yes	maximal, complemented
8	C₃₁×#2416	C₂	1	6	yes	maximal, complemented
9	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2341	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#121	C₂	1	3	yes	maximal, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#550	—	2	8	no	—
15	C₃₁×#548	—	2	16	no	—
16	C₂×C₃₁×#127	C₂²	1	3	yes	—
17	C₂×C₃₁×#124	C₂²	1	6	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	6	yes	—
19	C₂×C₃₁×#121	C₂²	1	3	yes	—
20	C₂×C₃₁×#121	C₂²	1	6	yes	—
21	C₂×C₃₁×#121	C₂²	1	6	yes	—
22	C₂×C₃₁×#121	C₂²	1	6	yes	—
23	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
27	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
32	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
33	C₃₁×#396	C₂²	1	12	yes	—
34	C₃₁×#394	C₂²	1	12	yes	—
35	C₃₁×#390	C₂²	1	6	yes	—
36	C₃₁×#390	C₂²	1	6	yes	—
37	C₃₁×#390	C₂²	1	6	yes	—
38	C₃₁×#390	C₂²	1	6	yes	—
39	C₃₁×#390	C₂²	1	3	yes	—
40	C₃₁×#389	C₂²	1	6	yes	—
41	C₃₁×#389	C₂²	1	3	yes	—
42	C₃₁×#388	C₂²	1	6	yes	—
43	C₃₁×#386	C₂²	1	6	yes	—
44	C₃₁×#384	C₂²	1	3	yes	—
45	C₃₁×#380	C₂²	1	6	yes	—
46	C₃₁×#380	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
48	C₄×C₂²⋊C₄×C₃₁	C₂²	1	6	yes	—
49	C₂×#41×C₃₁	—	2	8	no	—
50	C₂×#41×C₃₁	—	2	8	no	—
51	C₂×#41×C₃₁	—	2	8	no	—
52	C₂²×Q₈×C₃₁	—	2	4	no	—
53	C₂²×Q₈×C₃₁	—	2	4	no	—
54	C₂²×Q₈×C₃₁	D₄	1	8	yes	complemented
55	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
56	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
58	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
59	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
60	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#127	—	2	12	no	—
66	C₃₁×#124	—	2	24	no	—
67	C₃₁×C₂²⋊Q₈	—	2	24	no	—
68	C₃₁×#121	—	2	24	no	—
69	C₃₁×#121	—	2	12	no	—
70	C₃₁×#121	—	2	24	no	—
71	C₃₁×#121	—	4	12	no	—
72	C₄×Q₈×C₃₁	—	2	4	no	—
73	C₄×Q₈×C₃₁	—	2	8	no	—
74	C₄×Q₈×C₃₁	—	2	12	no	—
75	C₄×Q₈×C₃₁	—	2	12	no	—
76	C₄×Q₈×C₃₁	D₄	1	8	yes	complemented
77	C₄×D₄×C₃₁	—	2	24	no	—
78	C₄×D₄×C₃₁	—	2	12	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
96	C₂×C₄²×C₃₁	—	2	12	no	—
97	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
98	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
99	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
102	C₃₁×#95	C₂³	1	6	yes	—
103	C₃₁×#95	C₂³	1	2	yes	—
104	C₃₁×#95	C₂³	1	6	yes	—
105	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
106	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
107	#41×C₃₁	—	4	32	no	—
108	C₂×Q₈×C₃₁	—	2	32	no	—
109	C₂×Q₈×C₃₁	—	2	6	no	—
110	C₂×Q₈×C₃₁	—	2	6	no	—
111	C₂×Q₈×C₃₁	C₂×D₄	1	12	yes	—
112	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
113	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
114	C₂×Q₈×C₃₁	#41	1	4	yes	—
115	C₂×D₄×C₃₁	—	2	12	no	—
116	C₂×D₄×C₃₁	—	2	12	no	—
117	C₂×D₄×C₃₁	—	2	12	no	—
118	C₂×D₄×C₃₁	—	2	6	no	—
119	C₂×D₄×C₃₁	—	2	12	no	—
120	C₂×D₄×C₃₁	—	2	12	no	—
121	C₂×D₄×C₃₁	—	2	12	no	—
122	C₂×D₄×C₃₁	—	2	6	no	—
123	C₂×D₄×C₃₁	—	2	12	no	—
124	C₂×D₄×C₃₁	—	2	6	no	—
125	C₂×D₄×C₃₁	—	4	6	no	—
126	C₂×D₄×C₃₁	—	4	6	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	6	no	—
129	C₂²×C₄×C₃₁	—	2	6	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	6	no	—
132	C₂²×C₄×C₃₁	—	2	6	no	—
133	C₂²×C₄×C₃₁	—	2	12	no	—
134	C₂²×C₄×C₃₁	—	2	12	no	—
135	C₂²×C₄×C₃₁	—	2	12	no	—
136	C₂²×C₄×C₃₁	—	4	3	no	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
138	C₂²×C₄×C₃₁	#41	1	4	yes	complemented
139	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	C₄⋊C₄×C₃₁	—	2	12	no	—
149	C₄⋊C₄×C₃₁	—	2	6	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	12	no	—
152	C₄⋊C₄×C₃₁	—	2	12	no	—
153	C₄⋊C₄×C₃₁	—	2	6	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	4	6	no	—
156	C₄⋊C₄×C₃₁	C₂×D₄	1	12	yes	—
157	C₂²⋊C₄×C₃₁	—	2	4	no	—
158	C₂²⋊C₄×C₃₁	—	2	12	no	—
159	C₂²⋊C₄×C₃₁	—	2	12	no	—
160	C₂²⋊C₄×C₃₁	—	2	24	no	—
161	C₂²⋊C₄×C₃₁	—	2	12	no	—
162	C₂²⋊C₄×C₃₁	—	4	6	no	—
163	C₂²⋊C₄×C₃₁	—	4	12	no	—
164	C₄²×C₃₁	—	2	6	no	—
165	C₄²×C₃₁	—	2	6	no	—
166	C₄²×C₃₁	—	2	6	no	—
167	C₄²×C₃₁	—	2	12	no	—
168	C₄²×C₃₁	C₂×D₄	1	12	yes	—
169	#36576	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	—	2	1	no	—
174	C₂³×C₃₁	—	2	6	no	—
175	C₂³×C₃₁	—	2	6	no	—
176	C₂³×C₃₁	—	2	6	no	—
177	C₂³×C₃₁	—	2	6	no	—
178	C₂³×C₃₁	—	2	6	no	—
179	C₂³×C₃₁	—	2	6	no	—
180	C₂³×C₃₁	—	2	6	no	—
181	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
182	C₂³×C₃₁	C₂×#41	1	2	yes	—
183	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
184	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
185	Q₈×C₃₁	—	2	24	no	—
186	Q₈×C₃₁	#121	1	16	yes	complemented
187	D₄×C₃₁	—	4	24	no	—
188	D₄×C₃₁	—	4	24	no	—
189	D₄×C₃₁	—	4	12	no	—
190	D₄×C₃₁	—	8	12	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	3	no	—
193	C₂×C₄×C₃₁	—	2	3	no	—
194	C₂×C₄×C₃₁	—	2	3	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	3	no	—
197	C₂×C₄×C₃₁	—	2	24	no	—
198	C₂×C₄×C₃₁	—	2	8	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	3	no	—
201	C₂×C₄×C₃₁	—	2	6	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	6	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	3	no	—
207	C₂×C₄×C₃₁	—	2	6	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	3	no	—
211	C₂×C₄×C₃₁	—	4	12	no	—
212	C₂×C₄×C₃₁	—	4	24	no	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
215	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
219	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
220	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
221	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
222	C₂×#550	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₂×#548	C₂×C₃₁	1	2	yes	complemented
224	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#2856	C₂×C₃₁	1	3	yes	complemented
226	#2548	C₂×C₃₁	1	3	yes	complemented
227	#2538	C₂×C₃₁	1	3	yes	complemented
228	#2426	C₂×C₃₁	1	6	yes	complemented
229	#2416	C₂×C₃₁	1	6	yes	complemented
230	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2341	C₂×C₃₁	1	2	yes	complemented
232	C₄×#121	C₂×C₃₁	1	3	yes	complemented
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	2	2	no	—
236	C₂²×C₃₁	—	2	2	no	—
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	2	2	no	—
239	C₂²×C₃₁	—	4	6	no	—
240	C₂²×C₃₁	—	4	6	no	—
241	C₂²×C₃₁	—	4	6	no	—
242	C₂²×C₃₁	—	4	6	no	—
243	C₂²×C₃₁	—	4	6	no	—
244	C₂²×C₃₁	—	4	6	no	—
245	C₂²×C₃₁	—	4	6	no	—
246	C₂²×C₃₁	#550	1	4	yes	complemented
247	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
248	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
249	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
250	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
252	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
253	C₂²×C₃₁	#585	1	1	yes	characteristic
254	C₄×C₃₁	—	2	6	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	2	6	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	6	no	—
259	C₄×C₃₁	—	4	3	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	C₂×#121	1	12	yes	—
262	C₄×C₃₁	#548	1	4	yes	complemented
263	C₂²×#41	C₂²×C₃₁	1	2	yes	—
264	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
265	#550	—	2	8	no	—
266	#548	—	2	16	no	—
267	C₂×#127	C₂²×C₃₁	1	3	yes	—
268	C₂×#124	C₂²×C₃₁	1	6	yes	—
269	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	6	yes	—
270	C₂×#121	C₂²×C₃₁	1	3	yes	—
271	C₂×#121	C₂²×C₃₁	1	6	yes	—
272	C₂×#121	C₂²×C₃₁	1	6	yes	—
273	C₂×#121	C₂²×C₃₁	1	6	yes	—
274	C₂×C₄×Q₈	—	2	8	no	—
275	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
277	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
278	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
279	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
281	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
282	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
283	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
284	#396	C₂²×C₃₁	1	12	yes	—
285	#394	C₂²×C₃₁	1	12	yes	—
286	#390	C₂²×C₃₁	1	6	yes	—
287	#390	C₂²×C₃₁	1	6	yes	—
288	#390	C₂²×C₃₁	1	6	yes	—
289	#390	C₂²×C₃₁	1	6	yes	—
290	#390	C₂²×C₃₁	1	3	yes	—
291	#389	C₂²×C₃₁	1	6	yes	—
292	#389	C₂²×C₃₁	1	3	yes	—
293	#388	C₂²×C₃₁	1	6	yes	—
294	#386	C₂²×C₃₁	1	6	yes	—
295	#384	C₂²×C₃₁	1	3	yes	—
296	#380	C₂²×C₃₁	1	6	yes	—
297	#380	C₂²×C₃₁	1	1	yes	characteristic
298	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
299	C₄×C₂²⋊C₄	C₂²×C₃₁	1	6	yes	—
300	C₂×C₃₁	—	2	4	no	—
301	C₂×C₃₁	—	8	6	no	—
302	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
303	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
304	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
305	C₂×C₃₁	#3480	1	1	yes	characteristic
306	C₂×C₃₁	#3488	1	1	yes	characteristic
307	C₂×C₃₁	#3497	1	1	yes	characteristic
308	C₂×C₃₁	#3522	1	1	yes	characteristic
309	C₂×#41	—	2	8	no	—
310	C₂×#41	—	2	8	no	—
311	C₂×#41	—	2	8	no	—
312	C₂²×Q₈	—	2	4	no	—
313	C₂²×Q₈	—	2	4	no	—
314	C₂²×Q₈	D₄×C₃₁	1	8	yes	complemented
315	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
316	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
318	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
319	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
320	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
321	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
322	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
323	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
324	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
325	#127	—	2	12	no	—
326	#124	—	2	24	no	—
327	C₂²⋊Q₈	—	2	24	no	—
328	#121	—	2	24	no	—
329	#121	—	2	12	no	—
330	#121	—	2	24	no	—
331	#121	—	4	12	no	—
332	C₄×Q₈	—	2	4	no	—
333	C₄×Q₈	—	2	8	no	—
334	C₄×Q₈	—	2	12	no	—
335	C₄×Q₈	—	2	12	no	—
336	C₄×Q₈	D₄×C₃₁	1	8	yes	complemented
337	C₄×D₄	—	2	24	no	—
338	C₄×D₄	—	2	12	no	—
339	C₂×C₄⋊C₄	—	2	12	no	—
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
346	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
347	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
348	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
356	C₂×C₄²	—	2	12	no	—
357	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
358	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
359	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
360	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
361	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
362	#95	C₂³×C₃₁	1	2	yes	—
363	#95	C₂³×C₃₁	1	6	yes	—
364	#95	C₂³×C₃₁	1	6	yes	—
365	C₃₁	#36576	1	1	yes	characteristic, direct product factor, Sylow
366	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
367	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
368	#41	—	4	32	no	—
369	C₂×Q₈	—	2	32	no	—
370	C₂×Q₈	—	2	6	no	—
371	C₂×Q₈	—	2	6	no	—
372	C₂×Q₈	C₂×D₄×C₃₁	1	12	yes	—
373	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
374	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
375	C₂×Q₈	#41×C₃₁	1	4	yes	—
376	C₂×D₄	—	2	12	no	—
377	C₂×D₄	—	2	12	no	—
378	C₂×D₄	—	2	12	no	—
379	C₂×D₄	—	2	6	no	—
380	C₂×D₄	—	2	12	no	—
381	C₂×D₄	—	2	12	no	—
382	C₂×D₄	—	2	12	no	—
383	C₂×D₄	—	2	6	no	—
384	C₂×D₄	—	2	12	no	—
385	C₂×D₄	—	2	6	no	—
386	C₂×D₄	—	4	6	no	—
387	C₂×D₄	—	4	6	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	6	no	—
390	C₂²×C₄	—	2	6	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	6	no	—
393	C₂²×C₄	—	2	6	no	—
394	C₂²×C₄	—	2	12	no	—
395	C₂²×C₄	—	2	12	no	—
396	C₂²×C₄	—	2	12	no	—
397	C₂²×C₄	—	4	3	no	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
399	C₂²×C₄	#41×C₃₁	1	4	yes	complemented
400	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
402	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
403	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
405	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
406	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
407	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
408	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
409	C₄⋊C₄	—	2	12	no	—
410	C₄⋊C₄	—	2	6	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	12	no	—
413	C₄⋊C₄	—	2	12	no	—
414	C₄⋊C₄	—	2	6	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	4	6	no	—
417	C₄⋊C₄	C₂×D₄×C₃₁	1	12	yes	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	2	12	no	—
420	C₂²⋊C₄	—	2	12	no	—
421	C₂²⋊C₄	—	2	24	no	—
422	C₂²⋊C₄	—	2	12	no	—
423	C₂²⋊C₄	—	4	6	no	—
424	C₂²⋊C₄	—	4	12	no	—
425	C₄²	—	2	6	no	—
426	C₄²	—	2	6	no	—
427	C₄²	—	2	6	no	—
428	C₄²	—	2	12	no	—
429	C₄²	C₂×D₄×C₃₁	1	12	yes	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	6	no	—
435	C₂³	—	2	6	no	—
436	C₂³	—	2	6	no	—
437	C₂³	—	2	6	no	—
438	C₂³	—	2	6	no	—
439	C₂³	—	2	6	no	—
440	C₂³	—	2	6	no	—
441	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
442	C₂³	C₂×#41×C₃₁	1	2	yes	—
443	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
444	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
445	Q₈	—	2	24	no	—
446	Q₈	C₃₁×#121	1	16	yes	complemented
447	D₄	—	4	24	no	—
448	D₄	—	4	24	no	—
449	D₄	—	4	12	no	—
450	D₄	—	8	12	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	3	no	—
453	C₂×C₄	—	2	3	no	—
454	C₂×C₄	—	2	3	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	3	no	—
457	C₂×C₄	—	2	24	no	—
458	C₂×C₄	—	2	8	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	3	no	—
461	C₂×C₄	—	2	6	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	6	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	3	no	—
467	C₂×C₄	—	2	6	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	3	no	—
471	C₂×C₄	—	4	12	no	—
472	C₂×C₄	—	4	24	no	—
473	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
474	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
475	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
476	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
477	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
478	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
479	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
480	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
481	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	2	no	—
488	C₂²	—	4	6	no	—
489	C₂²	—	4	6	no	—
490	C₂²	—	4	6	no	—
491	C₂²	—	4	6	no	—
492	C₂²	—	4	6	no	—
493	C₂²	—	4	6	no	—
494	C₂²	—	4	6	no	—
495	C₂²	C₃₁×#550	1	4	yes	complemented
496	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
497	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
498	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
499	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
500	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
501	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
502	C₂²	C₃₁×#585	1	1	yes	characteristic
503	C₄	—	2	6	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	6	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	6	no	—
508	C₄	—	4	3	no	—
509	C₄	—	4	2	no	—
510	C₄	C₂×C₃₁×#121	1	12	yes	—
511	C₄	C₃₁×#548	1	4	yes	complemented
512	C₂	—	2	4	no	—
513	C₂	—	8	6	no	—
514	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
515	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
516	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
517	C₂	C₃₁×#3480	1	1	yes	characteristic
518	C₂	C₃₁×#3488	1	1	yes	characteristic
519	C₂	C₃₁×#3497	1	1	yes	characteristic
520	C₂	C₃₁×#3522	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#550	C₂	semidirect
2. C₂×C₃₁×#548	C₂	semidirect
3. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#2856	C₂	semidirect
5. C₃₁×#2548	C₂	semidirect
6. C₃₁×#2538	C₂	semidirect
7. C₃₁×#2426	C₂	semidirect
8. C₃₁×#2416	C₂	semidirect
9. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
10. C₃₁×#2341	C₂	semidirect
11. C₄×C₃₁×#121	C₂	semidirect
54. C₂²×Q₈×C₃₁	D₄	semidirect
76. C₄×Q₈×C₃₁	D₄	semidirect
138. C₂²×C₄×C₃₁	#41	semidirect
169. #36576	C₃₁	direct (N × H)
186. Q₈×C₃₁	#121	semidirect
222. C₂×#550	C₂×C₃₁	semidirect
223. C₂×#548	C₂×C₃₁	semidirect
224. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
225. #2856	C₂×C₃₁	semidirect
226. #2548	C₂×C₃₁	semidirect
227. #2538	C₂×C₃₁	semidirect
228. #2426	C₂×C₃₁	semidirect
229. #2416	C₂×C₃₁	semidirect
230. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
231. #2341	C₂×C₃₁	semidirect
232. C₄×#121	C₂×C₃₁	semidirect
246. C₂²×C₃₁	#550	semidirect
262. C₄×C₃₁	#548	semidirect
314. C₂²×Q₈	D₄×C₃₁	semidirect
336. C₄×Q₈	D₄×C₃₁	semidirect
365. C₃₁	#36576	direct (N × H)
399. C₂²×C₄	#41×C₃₁	semidirect
446. Q₈	C₃₁×#121	semidirect
495. C₂²	C₃₁×#550	semidirect
511. C₄	C₃₁×#548	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#550	2
C₂×C₃₁×#548	2
C₂²×C₄×Q₈×C₃₁	2
C₃₁×#2856	2
C₃₁×#2548	2
C₃₁×#2538	2
C₃₁×#2426	2
C₃₁×#2416	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#2341	2
C₄×C₃₁×#121	2
#36576	31

Subgroups — C31×#36576 #21493335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36576 #21493335