Subgroups — C₃₁×#61948 #21518707

Order 7936.

2114 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3392	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3359	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3298	C₂	1	6	yes	maximal, complemented
6	D₄×SD₁₆×C₃₁	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3246	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3206	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3101	C₂	1	3	yes	maximal, complemented
10	C₂×C₃₁×#461	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	—	2	1	no	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	6	yes	—
14	Q₈²×C₃₁	—	2	1	no	—
15	C₃₁×#557	C₂²	1	2	yes	complemented
16	C₃₁×#554	—	2	3	no	—
17	C₃₁×#550	C₂²	1	1	yes	characteristic
18	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₃₁	C₂²	1	6	yes	complemented
20	C₃₁×#548	—	2	1	no	—
21	C₃₁×#547	C₂²	1	3	yes	—
22	C₃₁×#546	—	2	3	no	—
23	D₄²×C₃₁	C₂²	1	3	yes	—
24	C₃₁×#544	—	2	3	no	—
25	C₃₁×#533	C₂²	1	3	yes	complemented
26	C₃₁×#532	C₂²	1	3	yes	—
27	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	complemented
28	C₃₁×#525	—	2	3	no	—
29	C₃₁×#520	—	2	1	no	—
30	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
32	C₃₁×#492	C₂²	1	6	yes	complemented
33	C₃₁×#487	C₂²	1	6	yes	complemented
34	C₃₁×#483	C₂²	1	6	yes	complemented
35	C₃₁×#476	C₂²	1	1	yes	characteristic
36	C₃₁×#476	C₂²	1	3	yes	complemented
37	C₃₁×#475	C₂²	1	6	yes	complemented
38	C₃₁×#465	C₂²	1	12	yes	complemented
39	C₃₁×#461	C₂²	1	3	yes	—
40	C₃₁×#461	C₂²	1	6	yes	—
41	C₃₁×#461	C₂²	1	3	yes	complemented
42	C₃₁×#461	C₂²	1	6	yes	complemented
43	C₃₁×#461	C₂²	1	6	yes	complemented
44	C₃₁×#460	C₂²	1	6	yes	complemented
45	C₃₁×#450	C₂²	1	6	yes	—
46	C₃₁×#448	C₂²	1	12	yes	complemented
47	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
48	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	—
49	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	complemented
50	C₈×D₄×C₃₁	C₂²	1	6	yes	complemented
51	C₃₁×#431	C₂²	1	3	yes	complemented
52	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	complemented
53	C₂×C₃₁×#103	C₂²	1	6	yes	complemented
54	C₂×#41×C₃₁	—	2	1	no	—
55	C₂×#41×C₃₁	—	2	3	no	—
56	C₂×#41×C₃₁	—	2	3	no	—
57	C₂×#41×C₃₁	—	4	8	no	—
58	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
59	C₂²×Q₈×C₃₁	—	2	1	no	—
60	C₂²×Q₈×C₃₁	C₂³	1	6	yes	—
61	C₂²×D₄×C₃₁	—	2	3	no	—
62	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
63	C₂³×C₄×C₃₁	—	2	3	no	—
64	C₂×SD₁₆×C₃₁	—	4	12	no	—
65	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
66	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
67	C₂×SD₁₆×C₃₁	C₂³	1	12	yes	—
68	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
69	C₃₁×C₄⋊Q₈	—	2	6	no	—
70	C₃₁×C₄⋊Q₈	—	2	3	no	—
71	C₃₁×C₄⋊Q₈	—	2	3	no	—
72	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
73	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
74	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₃₁×#127	C₂³	1	3	yes	—
77	C₃₁×#125	—	2	6	no	—
78	C₃₁×#125	—	2	3	no	—
79	C₃₁×#124	—	2	3	no	—
80	C₃₁×#123	—	2	6	no	—
81	C₃₁×C₂²⋊Q₈	—	2	3	no	—
82	C₃₁×C₂²⋊Q₈	—	2	3	no	—
83	C₃₁×C₂²⋊Q₈	—	2	3	no	—
84	C₃₁×C₂²⋊Q₈	—	2	12	no	—
85	C₃₁×C₂²⋊Q₈	C₂³	1	6	yes	—
86	C₃₁×#121	—	2	3	no	—
87	C₃₁×#121	—	2	3	no	—
88	C₃₁×#121	—	2	3	no	—
89	C₃₁×#121	C₂³	1	6	yes	—
90	C₃₁×#120	—	2	6	no	—
91	C₄×Q₈×C₃₁	—	2	2	no	—
92	C₄×Q₈×C₃₁	—	2	3	no	—
93	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂³	1	6	yes	—
95	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₃₁	—	2	6	no	—
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	—	2	3	no	—
99	C₄×D₄×C₃₁	—	2	3	no	—
100	C₄×D₄×C₃₁	D₄	1	3	yes	—
101	C₄×D₄×C₃₁	C₂³	1	3	yes	—
102	C₄×D₄×C₃₁	C₂³	1	3	yes	—
103	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₄×D₄×C₃₁	C₂³	1	3	yes	—
105	C₄×D₄×C₃₁	C₂³	1	3	yes	—
106	C₃₁×#117	—	2	3	no	—
107	C₃₁×#117	D₄	1	3	yes	—
108	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
109	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
112	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
113	C₃₁×#106	C₂³	1	6	yes	—
114	C₃₁×#106	C₂³	1	2	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₃₁×#103	C₂³	1	6	yes	—
119	C₃₁×#103	C₂³	1	12	yes	—
120	C₃₁×#103	C₂³	1	6	yes	—
121	C₃₁×#102	C₂³	1	2	yes	—
122	C₃₁×#102	C₂³	1	6	yes	—
123	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
124	C₄×C₈×C₃₁	C₂³	1	6	yes	—
125	C₂⁴×C₃₁	—	2	3	no	—
126	#41×C₃₁	—	4	4	no	—
127	#41×C₃₁	—	4	12	no	—
128	#41×C₃₁	—	4	12	no	—
129	#41×C₃₁	SD₁₆	1	8	yes	complemented
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	3	no	—
132	C₂×Q₈×C₃₁	—	2	6	no	—
133	C₂×Q₈×C₃₁	—	4	6	no	—
134	C₂×Q₈×C₃₁	—	4	6	no	—
135	C₂×Q₈×C₃₁	—	4	4	no	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×Q₈×C₃₁	C₂⁴	1	6	yes	—
138	C₂×D₄×C₃₁	—	2	6	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	3	no	—
141	C₂×D₄×C₃₁	—	2	3	no	—
142	C₂×D₄×C₃₁	—	2	3	no	—
143	C₂×D₄×C₃₁	—	4	3	no	—
144	C₂×D₄×C₃₁	—	4	12	no	—
145	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
146	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
147	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	—	2	3	no	—
149	C₂²×C₄×C₃₁	—	2	6	no	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	3	no	—
152	C₂²×C₄×C₃₁	—	2	3	no	—
153	C₂²×C₄×C₃₁	—	2	3	no	—
154	C₂²×C₄×C₃₁	—	4	12	no	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
158	SD₁₆×C₃₁	—	2	12	no	—
159	SD₁₆×C₃₁	—	4	12	no	—
160	SD₁₆×C₃₁	—	4	6	no	—
161	C₂×C₈×C₃₁	—	4	6	no	—
162	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
163	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
164	C₄⋊C₄×C₃₁	—	2	3	no	—
165	C₄⋊C₄×C₃₁	—	2	3	no	—
166	C₄⋊C₄×C₃₁	—	2	6	no	—
167	C₄⋊C₄×C₃₁	—	2	6	no	—
168	C₄⋊C₄×C₃₁	—	2	3	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	3	no	—
171	C₄⋊C₄×C₃₁	—	2	6	no	—
172	C₄⋊C₄×C₃₁	—	2	6	no	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
175	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
178	C₂²⋊C₄×C₃₁	—	2	6	no	—
179	C₂²⋊C₄×C₃₁	—	2	1	no	—
180	C₂²⋊C₄×C₃₁	—	2	3	no	—
181	C₂²⋊C₄×C₃₁	—	2	3	no	—
182	C₂²⋊C₄×C₃₁	—	2	3	no	—
183	C₂²⋊C₄×C₃₁	—	2	6	no	—
184	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
185	C₄²×C₃₁	—	2	6	no	—
186	C₄²×C₃₁	C₂×D₄	1	3	yes	—
187	C₄²×C₃₁	C₂×D₄	1	3	yes	—
188	C₄²×C₃₁	C₂⁴	1	3	yes	—
189	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
190	#61948	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	1	no	—
192	C₂³×C₃₁	—	2	3	no	—
193	C₂³×C₃₁	—	4	6	no	—
194	C₂³×C₃₁	—	4	6	no	—
195	C₂³×C₃₁	—	8	3	no	—
196	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
197	Q₈×C₃₁	—	2	6	no	—
198	Q₈×C₃₁	—	2	6	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	4	6	no	—
201	Q₈×C₃₁	—	8	6	no	—
202	Q₈×C₃₁	C₂×SD₁₆	1	4	yes	complemented
203	D₄×C₃₁	—	2	3	no	—
204	D₄×C₃₁	—	2	1	no	—
205	D₄×C₃₁	—	2	3	no	—
206	D₄×C₃₁	—	4	6	no	—
207	D₄×C₃₁	—	4	12	no	—
208	D₄×C₃₁	—	4	6	no	—
209	D₄×C₃₁	—	8	3	no	—
210	D₄×C₃₁	C₂×SD₁₆	1	12	yes	complemented
211	D₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
212	C₂×C₄×C₃₁	—	2	3	no	—
213	C₂×C₄×C₃₁	—	2	3	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	6	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	3	no	—
218	C₂×C₄×C₃₁	—	4	3	no	—
219	C₂×C₄×C₃₁	—	4	6	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	6	no	—
222	C₂×C₄×C₃₁	—	4	6	no	—
223	C₂×C₄×C₃₁	—	4	6	no	—
224	C₂×C₄×C₃₁	—	8	6	no	—
225	C₂×C₄×C₃₁	C₂×SD₁₆	1	12	yes	complemented
226	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
229	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
232	C₈×C₃₁	—	2	6	no	—
233	C₈×C₃₁	—	2	2	no	—
234	#3503	C₂×C₃₁	1	1	yes	characteristic, complemented
235	D₄×#41	C₂×C₃₁	1	1	yes	characteristic
236	#3392	C₂×C₃₁	1	2	yes	complemented
237	#3359	C₂×C₃₁	1	2	yes	complemented
238	#3298	C₂×C₃₁	1	6	yes	complemented
239	D₄×SD₁₆	C₂×C₃₁	1	6	yes	complemented
240	#3246	C₂×C₃₁	1	3	yes	complemented
241	#3206	C₂×C₃₁	1	3	yes	complemented
242	#3101	C₂×C₃₁	1	3	yes	complemented
243	C₂×#461	C₂×C₃₁	1	3	yes	complemented
244	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₂²×C₃₁	—	2	3	no	—
246	C₂²×C₃₁	—	2	3	no	—
247	C₂²×C₃₁	—	2	2	no	—
248	C₂²×C₃₁	—	4	1	no	—
249	C₂²×C₃₁	—	4	1	no	—
250	C₂²×C₃₁	—	4	3	no	—
251	C₂²×C₃₁	—	4	3	no	—
252	C₂²×C₃₁	—	4	3	no	—
253	C₂²×C₃₁	—	8	12	no	—
254	C₂²×C₃₁	C₂²×SD₁₆	1	6	yes	—
255	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
256	C₄×C₃₁	—	2	1	no	—
257	C₄×C₃₁	—	2	3	no	—
258	C₄×C₃₁	—	2	3	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	6	no	—
261	C₄×C₃₁	—	4	1	no	—
262	C₄×C₃₁	—	4	3	no	—
263	C₄×C₃₁	C₂²×SD₁₆	1	2	yes	—
264	C₄×C₃₁	C₂²×SD₁₆	1	6	yes	—
265	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
266	C₄×C₃₁	#585	1	1	yes	characteristic
267	C₂²×#41	—	2	1	no	—
268	C₂²×SD₁₆	C₂²×C₃₁	1	6	yes	—
269	Q₈²	—	2	1	no	—
270	#557	C₂²×C₃₁	1	2	yes	complemented
271	#554	—	2	3	no	—
272	#550	C₂²×C₃₁	1	1	yes	characteristic
273	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
274	D₄×Q₈	C₂²×C₃₁	1	6	yes	complemented
275	#548	—	2	1	no	—
276	#547	C₂²×C₃₁	1	3	yes	—
277	#546	—	2	3	no	—
278	D₄²	C₂²×C₃₁	1	3	yes	—
279	#544	—	2	3	no	—
280	#533	C₂²×C₃₁	1	3	yes	complemented
281	#532	C₂²×C₃₁	1	3	yes	—
282	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	complemented
283	#525	—	2	3	no	—
284	#520	—	2	1	no	—
285	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
287	#492	C₂²×C₃₁	1	6	yes	complemented
288	#487	C₂²×C₃₁	1	6	yes	complemented
289	#483	C₂²×C₃₁	1	6	yes	complemented
290	#476	C₂²×C₃₁	1	1	yes	characteristic
291	#476	C₂²×C₃₁	1	3	yes	complemented
292	#475	C₂²×C₃₁	1	6	yes	complemented
293	#465	C₂²×C₃₁	1	12	yes	complemented
294	#461	C₂²×C₃₁	1	3	yes	—
295	#461	C₂²×C₃₁	1	6	yes	—
296	#461	C₂²×C₃₁	1	3	yes	complemented
297	#461	C₂²×C₃₁	1	6	yes	complemented
298	#461	C₂²×C₃₁	1	6	yes	complemented
299	#460	C₂²×C₃₁	1	6	yes	complemented
300	#450	C₂²×C₃₁	1	6	yes	—
301	#448	C₂²×C₃₁	1	12	yes	complemented
302	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
303	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	—
304	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	complemented
305	C₈×D₄	C₂²×C₃₁	1	6	yes	complemented
306	#431	C₂²×C₃₁	1	3	yes	complemented
307	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
308	C₂×#103	C₂²×C₃₁	1	6	yes	complemented
309	C₂×C₃₁	—	2	6	no	—
310	C₂×C₃₁	—	4	2	no	—
311	C₂×C₃₁	—	8	3	no	—
312	C₂×C₃₁	#3488	1	1	yes	characteristic
313	C₂×C₃₁	C₂³×SD₁₆	1	1	yes	characteristic
314	C₂×C₃₁	#3586	1	1	yes	characteristic
315	C₂×#41	—	2	1	no	—
316	C₂×#41	—	2	3	no	—
317	C₂×#41	—	2	3	no	—
318	C₂×#41	—	4	8	no	—
319	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
320	C₂²×Q₈	—	2	1	no	—
321	C₂²×Q₈	C₂³×C₃₁	1	6	yes	—
322	C₂²×D₄	—	2	3	no	—
323	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
324	C₂³×C₄	—	2	3	no	—
325	C₂×SD₁₆	—	4	12	no	—
326	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
327	C₂×SD₁₆	C₂³×C₃₁	1	12	yes	—
328	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
329	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
330	C₄⋊Q₈	—	2	6	no	—
331	C₄⋊Q₈	—	2	3	no	—
332	C₄⋊Q₈	—	2	3	no	—
333	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
334	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
335	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
336	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
337	#127	C₂³×C₃₁	1	3	yes	—
338	#125	—	2	6	no	—
339	#125	—	2	3	no	—
340	#124	—	2	3	no	—
341	#123	—	2	6	no	—
342	C₂²⋊Q₈	—	2	3	no	—
343	C₂²⋊Q₈	—	2	3	no	—
344	C₂²⋊Q₈	—	2	3	no	—
345	C₂²⋊Q₈	—	2	12	no	—
346	C₂²⋊Q₈	C₂³×C₃₁	1	6	yes	—
347	#121	—	2	3	no	—
348	#121	—	2	3	no	—
349	#121	—	2	3	no	—
350	#121	C₂³×C₃₁	1	6	yes	—
351	#120	—	2	6	no	—
352	C₄×Q₈	—	2	2	no	—
353	C₄×Q₈	—	2	3	no	—
354	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
355	C₄×Q₈	C₂³×C₃₁	1	6	yes	—
356	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₄×D₄	—	2	6	no	—
358	C₄×D₄	—	2	3	no	—
359	C₄×D₄	—	2	3	no	—
360	C₄×D₄	—	2	3	no	—
361	C₄×D₄	D₄×C₃₁	1	3	yes	—
362	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₄×D₄	C₂³×C₃₁	1	3	yes	—
364	C₄×D₄	C₂³×C₃₁	1	3	yes	—
365	C₄×D₄	C₂³×C₃₁	1	3	yes	—
366	C₄×D₄	C₂³×C₃₁	1	3	yes	—
367	#117	—	2	3	no	—
368	#117	D₄×C₃₁	1	3	yes	—
369	C₂×C₄⋊C₄	—	2	3	no	—
370	C₂×C₄⋊C₄	—	2	3	no	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
372	C₂×C₂²⋊C₄	—	2	3	no	—
373	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
374	#106	C₂³×C₃₁	1	6	yes	—
375	#106	C₂³×C₃₁	1	2	yes	—
376	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
377	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
378	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
379	#103	C₂³×C₃₁	1	6	yes	—
380	#103	C₂³×C₃₁	1	12	yes	—
381	#103	C₂³×C₃₁	1	6	yes	—
382	#102	C₂³×C₃₁	1	2	yes	—
383	#102	C₂³×C₃₁	1	6	yes	—
384	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
385	C₄×C₈	C₂³×C₃₁	1	6	yes	—
386	C₃₁	#61948	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	—	2	3	no	—
388	#41	—	4	4	no	—
389	#41	—	4	12	no	—
390	#41	—	4	12	no	—
391	#41	SD₁₆×C₃₁	1	8	yes	complemented
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	3	no	—
394	C₂×Q₈	—	2	6	no	—
395	C₂×Q₈	—	4	6	no	—
396	C₂×Q₈	—	4	6	no	—
397	C₂×Q₈	—	4	4	no	—
398	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂×Q₈	C₂⁴×C₃₁	1	6	yes	—
400	C₂×D₄	—	2	6	no	—
401	C₂×D₄	—	2	6	no	—
402	C₂×D₄	—	2	3	no	—
403	C₂×D₄	—	2	3	no	—
404	C₂×D₄	—	2	3	no	—
405	C₂×D₄	—	4	3	no	—
406	C₂×D₄	—	4	12	no	—
407	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
408	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
409	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
410	C₂²×C₄	—	2	3	no	—
411	C₂²×C₄	—	2	6	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	3	no	—
414	C₂²×C₄	—	2	3	no	—
415	C₂²×C₄	—	2	3	no	—
416	C₂²×C₄	—	4	12	no	—
417	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
419	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
420	SD₁₆	—	2	12	no	—
421	SD₁₆	—	4	12	no	—
422	SD₁₆	—	4	6	no	—
423	C₂×C₈	—	4	6	no	—
424	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
425	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
426	C₄⋊C₄	—	2	3	no	—
427	C₄⋊C₄	—	2	3	no	—
428	C₄⋊C₄	—	2	6	no	—
429	C₄⋊C₄	—	2	6	no	—
430	C₄⋊C₄	—	2	3	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	3	no	—
433	C₄⋊C₄	—	2	6	no	—
434	C₄⋊C₄	—	2	6	no	—
435	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
436	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
437	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
438	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
439	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
440	C₂²⋊C₄	—	2	6	no	—
441	C₂²⋊C₄	—	2	1	no	—
442	C₂²⋊C₄	—	2	3	no	—
443	C₂²⋊C₄	—	2	3	no	—
444	C₂²⋊C₄	—	2	3	no	—
445	C₂²⋊C₄	—	2	6	no	—
446	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
447	C₄²	—	2	6	no	—
448	C₄²	C₂×D₄×C₃₁	1	3	yes	—
449	C₄²	C₂×D₄×C₃₁	1	3	yes	—
450	C₄²	C₂⁴×C₃₁	1	3	yes	—
451	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
452	C₂³	—	2	1	no	—
453	C₂³	—	2	3	no	—
454	C₂³	—	4	6	no	—
455	C₂³	—	4	6	no	—
456	C₂³	—	8	3	no	—
457	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
458	Q₈	—	2	6	no	—
459	Q₈	—	2	6	no	—
460	Q₈	—	4	2	no	—
461	Q₈	—	4	6	no	—
462	Q₈	—	8	6	no	—
463	Q₈	C₂×SD₁₆×C₃₁	1	4	yes	complemented
464	D₄	—	2	3	no	—
465	D₄	—	2	1	no	—
466	D₄	—	2	3	no	—
467	D₄	—	4	6	no	—
468	D₄	—	4	12	no	—
469	D₄	—	4	6	no	—
470	D₄	—	8	3	no	—
471	D₄	C₂×SD₁₆×C₃₁	1	12	yes	complemented
472	D₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
473	C₂×C₄	—	2	3	no	—
474	C₂×C₄	—	2	3	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	6	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	3	no	—
479	C₂×C₄	—	4	3	no	—
480	C₂×C₄	—	4	6	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	6	no	—
483	C₂×C₄	—	4	6	no	—
484	C₂×C₄	—	4	6	no	—
485	C₂×C₄	—	8	6	no	—
486	C₂×C₄	C₂×SD₁₆×C₃₁	1	12	yes	complemented
487	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
489	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
490	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
493	C₈	—	2	6	no	—
494	C₈	—	2	2	no	—
495	C₂²	—	2	3	no	—
496	C₂²	—	2	3	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	3	no	—
501	C₂²	—	4	3	no	—
502	C₂²	—	4	3	no	—
503	C₂²	—	8	12	no	—
504	C₂²	C₂²×SD₁₆×C₃₁	1	6	yes	—
505	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
506	C₄	—	2	3	no	—
507	C₄	—	2	1	no	—
508	C₄	—	2	3	no	—
509	C₄	—	4	6	no	—
510	C₄	—	4	1	no	—
511	C₄	—	4	3	no	—
512	C₄	—	4	2	no	—
513	C₄	C₂²×SD₁₆×C₃₁	1	2	yes	—
514	C₄	C₂²×SD₁₆×C₃₁	1	6	yes	—
515	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
516	C₄	C₃₁×#585	1	1	yes	characteristic
517	C₂	—	2	6	no	—
518	C₂	—	4	2	no	—
519	C₂	—	8	3	no	—
520	C₂	C₃₁×#3488	1	1	yes	characteristic
521	C₂	C₂³×SD₁₆×C₃₁	1	1	yes	characteristic
522	C₂	C₃₁×#3586	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3503	C₂	semidirect
3. C₃₁×#3392	C₂	semidirect
4. C₃₁×#3359	C₂	semidirect
5. C₃₁×#3298	C₂	semidirect
6. D₄×SD₁₆×C₃₁	C₂	semidirect
7. C₃₁×#3246	C₂	semidirect
8. C₃₁×#3206	C₂	semidirect
9. C₃₁×#3101	C₂	semidirect
10. C₂×C₃₁×#461	C₂	semidirect
11. C₃₁×#2988	C₂	semidirect
15. C₃₁×#557	C₂²	semidirect
19. D₄×Q₈×C₃₁	C₂²	semidirect
25. C₃₁×#533	C₂²	semidirect
27. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
32. C₃₁×#492	C₂²	semidirect
33. C₃₁×#487	C₂²	semidirect
34. C₃₁×#483	C₂²	semidirect
36. C₃₁×#476	C₂²	semidirect
37. C₃₁×#475	C₂²	semidirect
38. C₃₁×#465	C₂²	semidirect
41. C₃₁×#461	C₂²	semidirect
42. C₃₁×#461	C₂²	semidirect
43. C₃₁×#461	C₂²	semidirect
44. C₃₁×#460	C₂²	semidirect
46. C₃₁×#448	C₂²	semidirect
47. C₈×Q₈×C₃₁	C₂²	semidirect
49. C₄×SD₁₆×C₃₁	C₂²	semidirect
50. C₈×D₄×C₃₁	C₂²	semidirect
51. C₃₁×#431	C₂²	semidirect
52. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
53. C₂×C₃₁×#103	C₂²	semidirect
129. #41×C₃₁	SD₁₆	semidirect
190. #61948	C₃₁	direct (N × H)
202. Q₈×C₃₁	C₂×SD₁₆	semidirect
210. D₄×C₃₁	C₂×SD₁₆	semidirect
225. C₂×C₄×C₃₁	C₂×SD₁₆	semidirect
234. #3503	C₂×C₃₁	semidirect
236. #3392	C₂×C₃₁	semidirect
237. #3359	C₂×C₃₁	semidirect
238. #3298	C₂×C₃₁	semidirect
239. D₄×SD₁₆	C₂×C₃₁	semidirect
240. #3246	C₂×C₃₁	semidirect
241. #3206	C₂×C₃₁	semidirect
242. #3101	C₂×C₃₁	semidirect
243. C₂×#461	C₂×C₃₁	semidirect
244. #2988	C₂×C₃₁	semidirect
270. #557	C₂²×C₃₁	semidirect
274. D₄×Q₈	C₂²×C₃₁	semidirect
280. #533	C₂²×C₃₁	semidirect
282. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
287. #492	C₂²×C₃₁	semidirect
288. #487	C₂²×C₃₁	semidirect
289. #483	C₂²×C₃₁	semidirect
291. #476	C₂²×C₃₁	semidirect
292. #475	C₂²×C₃₁	semidirect
293. #465	C₂²×C₃₁	semidirect
296. #461	C₂²×C₃₁	semidirect
297. #461	C₂²×C₃₁	semidirect
298. #461	C₂²×C₃₁	semidirect
299. #460	C₂²×C₃₁	semidirect
301. #448	C₂²×C₃₁	semidirect
302. C₈×Q₈	C₂²×C₃₁	semidirect
304. C₄×SD₁₆	C₂²×C₃₁	semidirect
305. C₈×D₄	C₂²×C₃₁	semidirect
306. #431	C₂²×C₃₁	semidirect
307. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
308. C₂×#103	C₂²×C₃₁	semidirect
386. C₃₁	#61948	direct (N × H)
391. #41	SD₁₆×C₃₁	semidirect
463. Q₈	C₂×SD₁₆×C₃₁	semidirect
471. D₄	C₂×SD₁₆×C₃₁	semidirect
486. C₂×C₄	C₂×SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3503	2
D₄×#41×C₃₁	2
C₃₁×#3392	2
C₃₁×#3359	2
C₃₁×#3298	2
D₄×SD₁₆×C₃₁	2
C₃₁×#3246	2
C₃₁×#3206	2
C₃₁×#3101	2
C₂×C₃₁×#461	2
C₃₁×#2988	2
#61948	31

Subgroups — C31×#61948 #21518707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61948 #21518707