Subgroups — C₃₁×#60472 #21517231

Order 7936.

2578 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3586	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#578	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3573	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3069	C₂	1	12	yes	maximal, complemented
5	C₃₁×#3067	C₂	1	6	yes	maximal, complemented
6	C₂×C₃₁×#473	C₂	1	6	yes	maximal, complemented
7	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2889	C₂	1	2	yes	maximal, complemented
9	C₃₁×#585	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
11	C₃₁×#578	—	2	8	no	—
12	C₃₁×#578	—	2	4	no	—
13	C₃₁×#578	—	2	4	no	—
14	C₃₁×#577	—	2	8	no	—
15	C₃₁×#575	C₂²	1	12	yes	—
16	C₂×C₃₁×#137	C₂²	1	12	yes	—
17	C₂×C₃₁×#137	C₂²	1	6	yes	complemented
18	C₂×C₃₁×#135	C₂²	1	3	yes	complemented
19	C₂²×Q₁₆×C₃₁	C₂²	1	3	yes	complemented
20	C₃₁×#568	—	2	4	no	—
21	C₃₁×#568	C₂²	1	6	yes	—
22	C₃₁×#568	C₂²	1	1	yes	characteristic
23	C₃₁×#568	C₂²	1	2	yes	—
24	C₂×C₃₁×#131	—	2	4	no	—
25	C₂×C₃₁×#131	C₂²	1	6	yes	—
26	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
27	C₃₁×#473	C₂²	1	12	yes	complemented
28	C₃₁×#473	C₂²	1	24	yes	complemented
29	C₃₁×#472	C₂²	1	4	yes	complemented
30	C₃₁×#471	C₂²	1	24	yes	complemented
31	C₃₁×#430	C₂²	1	6	yes	complemented
32	C₂×C₃₁×#108	C₂²	1	6	yes	complemented
33	C₃₁×#413	C₂²	1	12	yes	complemented
34	C₂×C₃₁×#101	C₂²	1	12	yes	complemented
35	C₃₁×2₋¹⁺⁴	—	2	1	no	—
36	C₃₁×2₋¹⁺⁴	—	2	8	no	—
37	C₃₁×2₋¹⁺⁴	—	2	8	no	—
38	C₃₁×2₋¹⁺⁴	D₄	1	4	yes	—
39	C₃₁×H(5,𝔽₂)	—	2	3	no	—
40	C₃₁×H(5,𝔽₂)	D₄	1	4	yes	—
41	C₂×#41×C₃₁	—	2	4	no	—
42	C₂×#41×C₃₁	—	2	6	no	—
43	C₂×#41×C₃₁	C₂³	1	6	yes	—
44	C₂×#41×C₃₁	C₂³	1	6	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	—	2	2	no	—
47	C₂²×Q₈×C₃₁	C₂³	1	6	yes	—
48	C₃₁×#137	—	2	24	no	—
49	C₃₁×#137	—	2	12	no	—
50	C₃₁×#137	—	2	24	no	—
51	C₃₁×#137	C₂³	1	24	yes	—
52	C₃₁×AGL(1,ℤ/8)	—	2	12	no	—
53	C₃₁×AGL(1,ℤ/8)	C₂³	1	8	yes	—
54	C₃₁×#135	—	2	24	no	—
55	C₃₁×#135	—	2	6	no	—
56	C₃₁×#135	—	2	6	no	—
57	C₂×Q₁₆×C₃₁	—	2	12	no	—
58	C₂×Q₁₆×C₃₁	—	2	6	no	—
59	C₂×Q₁₆×C₃₁	—	2	6	no	—
60	C₂×Q₁₆×C₃₁	C₂³	1	3	yes	—
61	C₂×Q₁₆×C₃₁	C₂³	1	6	yes	—
62	C₂×SD₁₆×C₃₁	—	2	12	no	—
63	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
64	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#131	—	2	4	no	—
66	C₃₁×#131	—	2	8	no	—
67	C₃₁×#131	—	2	12	no	—
68	C₃₁×#131	—	2	6	no	—
69	C₃₁×#131	—	2	6	no	—
70	C₃₁×#131	D₄	1	4	yes	—
71	C₃₁×#131	D₄	1	4	yes	—
72	C₃₁×#131	C₂³	1	12	yes	—
73	C₃₁×#131	C₂³	1	4	yes	—
74	C₂×M₁₆×C₃₁	—	2	12	no	—
75	C₂×M₁₆×C₃₁	—	2	6	no	—
76	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
77	C₂×M₁₆×C₃₁	C₂³	1	12	yes	—
78	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
79	C₂²×C₈×C₃₁	—	2	6	no	—
80	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
81	C₃₁×#108	C₂³	1	4	yes	—
82	C₃₁×#108	C₂³	1	12	yes	—
83	C₃₁×#101	C₂³	1	24	yes	—
84	C₃₁×#100	C₂³	1	8	yes	—
85	#41×C₃₁	—	2	4	no	—
86	#41×C₃₁	—	2	12	no	—
87	#41×C₃₁	—	2	6	no	—
88	#41×C₃₁	—	2	6	no	—
89	#41×C₃₁	—	2	12	no	—
90	#41×C₃₁	—	2	12	no	—
91	#41×C₃₁	—	4	24	no	—
92	#41×C₃₁	C₂×D₄	1	12	yes	—
93	#41×C₃₁	C₂×D₄	1	4	yes	—
94	#41×C₃₁	C₂×D₄	1	4	yes	—
95	#41×C₃₁	C₂⁴	1	4	yes	—
96	C₂×Q₈×C₃₁	—	2	4	no	—
97	C₂×Q₈×C₃₁	—	2	6	no	—
98	C₂×Q₈×C₃₁	—	2	12	no	—
99	C₂×Q₈×C₃₁	—	2	12	no	—
100	C₂×Q₈×C₃₁	—	4	8	no	—
101	C₂×Q₈×C₃₁	C₂×D₄	1	6	yes	—
102	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
103	C₂×Q₈×C₃₁	C₂⁴	1	6	yes	—
104	C₂×D₄×C₃₁	—	2	6	no	—
105	C₂×D₄×C₃₁	—	2	12	no	—
106	C₂×D₄×C₃₁	—	2	6	no	—
107	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
108	C₂×D₄×C₃₁	#41	1	3	yes	—
109	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₃₁	—	2	6	no	—
111	C₂²×C₄×C₃₁	—	2	6	no	—
112	C₂²×C₄×C₃₁	—	2	6	no	—
113	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
114	Q₁₆×C₃₁	—	2	24	no	—
115	Q₁₆×C₃₁	—	2	3	no	—
116	Q₁₆×C₃₁	—	2	3	no	—
117	Q₁₆×C₃₁	—	2	12	no	—
118	SD₁₆×C₃₁	—	2	8	no	—
119	SD₁₆×C₃₁	—	2	24	no	—
120	SD₁₆×C₃₁	—	2	12	no	—
121	D₈×C₃₁	—	2	8	no	—
122	D₈×C₃₁	—	2	1	no	—
123	D₈×C₃₁	—	2	1	no	—
124	M₁₆×C₃₁	—	2	6	no	—
125	M₁₆×C₃₁	—	2	12	no	—
126	M₁₆×C₃₁	—	2	12	no	—
127	M₁₆×C₃₁	—	2	4	no	—
128	M₁₆×C₃₁	C₂×D₄	1	12	yes	—
129	M₁₆×C₃₁	C₂⁴	1	4	yes	—
130	M₁₆×C₃₁	C₂⁴	1	8	yes	—
131	C₂×C₈×C₃₁	—	2	6	no	—
132	C₂×C₈×C₃₁	—	2	12	no	—
133	C₂×C₈×C₃₁	—	2	4	no	—
134	C₂×C₈×C₃₁	—	2	12	no	—
135	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
136	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
137	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
138	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
139	#60472	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂³×C₃₁	—	2	6	no	—
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	C₂×#41	1	3	yes	—
143	Q₈×C₃₁	—	2	6	no	—
144	Q₈×C₃₁	—	2	6	no	—
145	Q₈×C₃₁	—	2	6	no	—
146	Q₈×C₃₁	—	4	12	no	—
147	Q₈×C₃₁	#121	1	4	yes	—
148	Q₈×C₃₁	C₂²×D₄	1	4	yes	—
149	D₄×C₃₁	—	2	12	no	—
150	D₄×C₃₁	—	2	4	no	—
151	D₄×C₃₁	—	2	6	no	—
152	D₄×C₃₁	—	2	2	no	—
153	D₄×C₃₁	—	2	2	no	—
154	D₄×C₃₁	—	4	12	no	—
155	D₄×C₃₁	#121	1	12	yes	—
156	D₄×C₃₁	C₂²×D₄	1	4	yes	—
157	C₂×C₄×C₃₁	—	2	6	no	—
158	C₂×C₄×C₃₁	—	2	12	no	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	6	no	—
161	C₂×C₄×C₃₁	—	4	12	no	—
162	C₂×C₄×C₃₁	—	4	12	no	—
163	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
164	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
165	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
166	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
167	C₈×C₃₁	—	2	4	no	—
168	C₈×C₃₁	—	2	8	no	—
169	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	C₈×C₃₁	C₂²×D₄	1	6	yes	—
171	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	#3586	C₂×C₃₁	1	2	yes	complemented
173	C₂×#578	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#3573	C₂×C₃₁	1	1	yes	characteristic, complemented
175	#3069	C₂×C₃₁	1	12	yes	complemented
176	#3067	C₂×C₃₁	1	6	yes	complemented
177	C₂×#473	C₂×C₃₁	1	6	yes	complemented
178	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#2889	C₂×C₃₁	1	2	yes	complemented
180	C₂²×C₃₁	—	2	4	no	—
181	C₂²×C₃₁	—	4	3	no	—
182	C₂²×C₃₁	—	4	2	no	—
183	C₂²×C₃₁	—	8	6	no	—
184	C₂²×C₃₁	—	8	2	no	—
185	C₂²×C₃₁	C₂×#121	1	6	yes	—
186	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
187	C₄×C₃₁	—	2	4	no	—
188	C₄×C₃₁	—	4	6	no	—
189	C₄×C₃₁	C₂×#121	1	6	yes	—
190	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
191	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
192	#585	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
194	#578	—	2	8	no	—
195	#578	—	2	4	no	—
196	#578	—	2	4	no	—
197	#577	—	2	8	no	—
198	#575	C₂²×C₃₁	1	12	yes	—
199	C₂×#137	C₂²×C₃₁	1	12	yes	—
200	C₂×#137	C₂²×C₃₁	1	6	yes	complemented
201	C₂×#135	C₂²×C₃₁	1	3	yes	complemented
202	C₂²×Q₁₆	C₂²×C₃₁	1	3	yes	complemented
203	#568	—	2	4	no	—
204	#568	C₂²×C₃₁	1	6	yes	—
205	#568	C₂²×C₃₁	1	2	yes	—
206	#568	C₂²×C₃₁	1	1	yes	characteristic
207	C₂×#131	—	2	4	no	—
208	C₂×#131	C₂²×C₃₁	1	6	yes	—
209	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
210	#473	C₂²×C₃₁	1	12	yes	complemented
211	#473	C₂²×C₃₁	1	24	yes	complemented
212	#472	C₂²×C₃₁	1	4	yes	complemented
213	#471	C₂²×C₃₁	1	24	yes	complemented
214	#430	C₂²×C₃₁	1	6	yes	complemented
215	C₂×#108	C₂²×C₃₁	1	6	yes	complemented
216	#413	C₂²×C₃₁	1	12	yes	complemented
217	C₂×#101	C₂²×C₃₁	1	12	yes	complemented
218	C₂×C₃₁	—	2	1	no	—
219	C₂×C₃₁	—	2	6	no	—
220	C₂×C₃₁	—	4	4	no	—
221	C₂×C₃₁	—	8	2	no	—
222	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
223	2₋¹⁺⁴	—	2	1	no	—
224	2₋¹⁺⁴	—	2	8	no	—
225	2₋¹⁺⁴	—	2	8	no	—
226	2₋¹⁺⁴	D₄×C₃₁	1	4	yes	—
227	H(5,𝔽₂)	—	2	3	no	—
228	H(5,𝔽₂)	D₄×C₃₁	1	4	yes	—
229	C₂×#41	—	2	4	no	—
230	C₂×#41	—	2	6	no	—
231	C₂×#41	C₂³×C₃₁	1	6	yes	—
232	C₂×#41	C₂³×C₃₁	1	6	yes	—
233	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
234	C₂²×Q₈	—	2	2	no	—
235	C₂²×Q₈	C₂³×C₃₁	1	6	yes	—
236	#137	—	2	24	no	—
237	#137	—	2	12	no	—
238	#137	—	2	24	no	—
239	#137	C₂³×C₃₁	1	24	yes	—
240	AGL(1,ℤ/8)	—	2	12	no	—
241	AGL(1,ℤ/8)	C₂³×C₃₁	1	8	yes	—
242	#135	—	2	24	no	—
243	#135	—	2	6	no	—
244	#135	—	2	6	no	—
245	C₂×Q₁₆	—	2	12	no	—
246	C₂×Q₁₆	—	2	6	no	—
247	C₂×Q₁₆	—	2	6	no	—
248	C₂×Q₁₆	C₂³×C₃₁	1	3	yes	—
249	C₂×Q₁₆	C₂³×C₃₁	1	6	yes	—
250	C₂×SD₁₆	—	2	12	no	—
251	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
252	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
253	#131	—	2	4	no	—
254	#131	—	2	8	no	—
255	#131	—	2	12	no	—
256	#131	—	2	6	no	—
257	#131	—	2	6	no	—
258	#131	D₄×C₃₁	1	4	yes	—
259	#131	D₄×C₃₁	1	4	yes	—
260	#131	C₂³×C₃₁	1	4	yes	—
261	#131	C₂³×C₃₁	1	12	yes	—
262	C₂×M₁₆	—	2	12	no	—
263	C₂×M₁₆	—	2	6	no	—
264	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
265	C₂×M₁₆	C₂³×C₃₁	1	12	yes	—
266	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
267	C₂²×C₈	—	2	6	no	—
268	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
269	#108	C₂³×C₃₁	1	4	yes	—
270	#108	C₂³×C₃₁	1	12	yes	—
271	#101	C₂³×C₃₁	1	24	yes	—
272	#100	C₂³×C₃₁	1	8	yes	—
273	C₃₁	#60472	1	1	yes	characteristic, direct product factor, Sylow
274	#41	—	2	4	no	—
275	#41	—	2	12	no	—
276	#41	—	2	6	no	—
277	#41	—	2	6	no	—
278	#41	—	2	12	no	—
279	#41	—	2	12	no	—
280	#41	—	4	24	no	—
281	#41	C₂×D₄×C₃₁	1	12	yes	—
282	#41	C₂×D₄×C₃₁	1	4	yes	—
283	#41	C₂×D₄×C₃₁	1	4	yes	—
284	#41	C₂⁴×C₃₁	1	4	yes	—
285	C₂×Q₈	—	2	4	no	—
286	C₂×Q₈	—	2	6	no	—
287	C₂×Q₈	—	2	12	no	—
288	C₂×Q₈	—	2	12	no	—
289	C₂×Q₈	—	4	8	no	—
290	C₂×Q₈	C₂×D₄×C₃₁	1	6	yes	—
291	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
292	C₂×Q₈	C₂⁴×C₃₁	1	6	yes	—
293	C₂×D₄	—	2	6	no	—
294	C₂×D₄	—	2	12	no	—
295	C₂×D₄	—	2	6	no	—
296	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
297	C₂×D₄	#41×C₃₁	1	3	yes	—
298	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
299	C₂²×C₄	—	2	6	no	—
300	C₂²×C₄	—	2	6	no	—
301	C₂²×C₄	—	2	6	no	—
302	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
303	Q₁₆	—	2	24	no	—
304	Q₁₆	—	2	3	no	—
305	Q₁₆	—	2	3	no	—
306	Q₁₆	—	2	12	no	—
307	SD₁₆	—	2	8	no	—
308	SD₁₆	—	2	24	no	—
309	SD₁₆	—	2	12	no	—
310	D₈	—	2	8	no	—
311	D₈	—	2	1	no	—
312	D₈	—	2	1	no	—
313	M₁₆	—	2	6	no	—
314	M₁₆	—	2	12	no	—
315	M₁₆	—	2	12	no	—
316	M₁₆	—	2	4	no	—
317	M₁₆	C₂×D₄×C₃₁	1	12	yes	—
318	M₁₆	C₂⁴×C₃₁	1	4	yes	—
319	M₁₆	C₂⁴×C₃₁	1	8	yes	—
320	C₂×C₈	—	2	6	no	—
321	C₂×C₈	—	2	12	no	—
322	C₂×C₈	—	2	4	no	—
323	C₂×C₈	—	2	12	no	—
324	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
325	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
326	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
327	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
328	C₂³	—	2	6	no	—
329	C₂³	—	2	2	no	—
330	C₂³	C₂×#41×C₃₁	1	3	yes	—
331	Q₈	—	2	6	no	—
332	Q₈	—	2	6	no	—
333	Q₈	—	2	6	no	—
334	Q₈	—	4	12	no	—
335	Q₈	C₃₁×#121	1	4	yes	—
336	Q₈	C₂²×D₄×C₃₁	1	4	yes	—
337	D₄	—	2	12	no	—
338	D₄	—	2	4	no	—
339	D₄	—	2	6	no	—
340	D₄	—	2	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	4	12	no	—
343	D₄	C₃₁×#121	1	12	yes	—
344	D₄	C₂²×D₄×C₃₁	1	4	yes	—
345	C₂×C₄	—	2	6	no	—
346	C₂×C₄	—	2	12	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	6	no	—
349	C₂×C₄	—	4	12	no	—
350	C₂×C₄	—	4	12	no	—
351	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
352	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
353	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
354	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
355	C₈	—	2	4	no	—
356	C₈	—	2	8	no	—
357	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
358	C₈	C₂²×D₄×C₃₁	1	6	yes	—
359	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
360	C₂²	—	2	4	no	—
361	C₂²	—	4	3	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	6	no	—
364	C₂²	—	8	2	no	—
365	C₂²	C₂×C₃₁×#121	1	6	yes	—
366	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
367	C₄	—	2	4	no	—
368	C₄	—	4	6	no	—
369	C₄	C₂×C₃₁×#121	1	6	yes	—
370	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
371	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
372	C₂	—	2	1	no	—
373	C₂	—	2	6	no	—
374	C₂	—	4	4	no	—
375	C₂	—	8	2	no	—
376	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂²×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3586	C₂	semidirect
2. C₂×C₃₁×#578	C₂	semidirect
3. C₃₁×#3573	C₂	semidirect
4. C₃₁×#3069	C₂	semidirect
5. C₃₁×#3067	C₂	semidirect
6. C₂×C₃₁×#473	C₂	semidirect
7. C₃₁×#2916	C₂	semidirect
8. C₃₁×#2889	C₂	semidirect
17. C₂×C₃₁×#137	C₂²	semidirect
18. C₂×C₃₁×#135	C₂²	semidirect
19. C₂²×Q₁₆×C₃₁	C₂²	semidirect
27. C₃₁×#473	C₂²	semidirect
28. C₃₁×#473	C₂²	semidirect
29. C₃₁×#472	C₂²	semidirect
30. C₃₁×#471	C₂²	semidirect
31. C₃₁×#430	C₂²	semidirect
32. C₂×C₃₁×#108	C₂²	semidirect
33. C₃₁×#413	C₂²	semidirect
34. C₂×C₃₁×#101	C₂²	semidirect
139. #60472	C₃₁	direct (N × H)
172. #3586	C₂×C₃₁	semidirect
173. C₂×#578	C₂×C₃₁	semidirect
174. #3573	C₂×C₃₁	semidirect
175. #3069	C₂×C₃₁	semidirect
176. #3067	C₂×C₃₁	semidirect
177. C₂×#473	C₂×C₃₁	semidirect
178. #2916	C₂×C₃₁	semidirect
179. #2889	C₂×C₃₁	semidirect
200. C₂×#137	C₂²×C₃₁	semidirect
201. C₂×#135	C₂²×C₃₁	semidirect
202. C₂²×Q₁₆	C₂²×C₃₁	semidirect
210. #473	C₂²×C₃₁	semidirect
211. #473	C₂²×C₃₁	semidirect
212. #472	C₂²×C₃₁	semidirect
213. #471	C₂²×C₃₁	semidirect
214. #430	C₂²×C₃₁	semidirect
215. C₂×#108	C₂²×C₃₁	semidirect
216. #413	C₂²×C₃₁	semidirect
217. C₂×#101	C₂²×C₃₁	semidirect
273. C₃₁	#60472	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×#3586	2
C₂×C₃₁×#578	2
C₃₁×#3573	2
C₃₁×#3069	2
C₃₁×#3067	2
C₂×C₃₁×#473	2
C₃₁×#2916	2
C₃₁×#2889	2
#60472	31

Subgroups — C31×#60472 #21517231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60472 #21517231