Subgroups — C₃₁×#61111 #21517870

Order 7936.

2600 conjugacy classes of subgroups.

The subgroup lattice has 474 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	4	yes	maximal, complemented
2	C₃₁×#3293	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3291	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3289	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3069	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3018	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#456	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2956	C₂	1	2	yes	maximal, complemented
10	C₃₁×#585	C₂²	1	2	yes	complemented
11	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
12	C₃₁×#578	—	2	8	no	—
13	C₃₁×#578	C₂²	1	16	yes	complemented
14	C₃₁×#577	—	2	8	no	—
15	C₃₁×#577	C₂²	1	16	yes	complemented
16	C₃₁×#575	—	2	8	no	—
17	C₃₁×#575	C₂²	1	4	yes	—
18	C₃₁×#575	C₂²	1	2	yes	—
19	C₂×C₃₁×#137	—	2	8	no	—
20	C₂×C₃₁×#137	C₂²	1	4	yes	complemented
21	C₂×C₃₁×#137	C₂²	1	4	yes	complemented
22	C₃₁×#568	C₂²	1	4	yes	complemented
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#497	C₂²	1	4	yes	complemented
26	C₃₁×#494	C₂²	1	2	yes	complemented
27	C₃₁×#492	C₂²	1	2	yes	complemented
28	C₃₁×#473	C₂²	1	16	yes	complemented
29	C₃₁×#471	C₂²	1	16	yes	complemented
30	C₃₁×#456	C₂²	1	8	yes	complemented
31	C₃₁×#456	C₂²	1	8	yes	complemented
32	C₃₁×#456	C₂²	1	4	yes	complemented
33	C₃₁×#455	C₂²	1	4	yes	complemented
34	C₃₁×#454	C₂²	1	4	yes	complemented
35	C₃₁×#453	C₂²	1	4	yes	complemented
36	C₃₁×#444	C₂²	1	8	yes	complemented
37	C₃₁×#443	C₂²	1	8	yes	complemented
38	C₃₁×#430	C₂²	1	2	yes	complemented
39	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
40	C₃₁×#413	C₂²	1	2	yes	complemented
41	C₂×C₃₁×#101	C₂²	1	2	yes	complemented
42	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×2₋¹⁺⁴	—	2	8	no	—
44	C₃₁×2₋¹⁺⁴	—	2	4	no	—
45	C₃₁×2₋¹⁺⁴	—	4	2	no	—
46	C₃₁×2₋¹⁺⁴	C₂³	1	4	yes	—
47	C₃₁×2₋¹⁺⁴	C₂³	1	8	yes	—
48	C₃₁×H(5,𝔽₂)	—	2	4	no	—
49	C₃₁×H(5,𝔽₂)	—	4	2	no	—
50	C₃₁×H(5,𝔽₂)	C₂³	1	4	yes	—
51	C₂×#41×C₃₁	—	2	8	no	—
52	C₂×#41×C₃₁	—	2	4	no	—
53	C₂×#41×C₃₁	—	4	2	no	—
54	C₂×#41×C₃₁	—	4	2	no	—
55	C₂×#41×C₃₁	C₂³	1	4	yes	—
56	C₂×#41×C₃₁	C₂³	1	2	yes	—
57	C₂²×Q₈×C₃₁	—	2	4	no	—
58	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₃₁×#137	—	2	8	no	—
60	C₃₁×#137	—	2	4	no	—
61	C₃₁×#137	—	2	16	no	—
62	C₃₁×#137	—	2	16	no	—
63	C₃₁×#137	—	2	16	no	—
64	C₃₁×#137	—	2	8	no	—
65	C₃₁×#137	—	2	8	no	—
66	C₃₁×#137	D₄	1	8	yes	complemented
67	C₃₁×#137	C₂³	1	8	yes	—
68	C₃₁×#137	C₂³	1	16	yes	—
69	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
70	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
71	C₃₁×AGL(1,ℤ/8)	D₄	1	8	yes	complemented
72	C₃₁×AGL(1,ℤ/8)	C₂³	1	8	yes	—
73	C₃₁×#135	—	2	8	no	—
74	C₃₁×#135	—	2	16	no	—
75	C₃₁×#135	—	2	16	no	—
76	C₃₁×#135	C₂³	1	8	yes	—
77	C₂×Q₁₆×C₃₁	—	2	4	no	—
78	C₂×Q₁₆×C₃₁	—	2	16	no	—
79	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
80	C₂×SD₁₆×C₃₁	—	2	4	no	—
81	C₂×SD₁₆×C₃₁	—	2	16	no	—
82	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
83	C₃₁×#131	—	2	8	no	—
84	C₃₁×#131	C₂³	1	16	yes	—
85	C₂×M₁₆×C₃₁	—	2	8	no	—
86	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
87	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
88	C₃₁×C₄⋊Q₈	—	4	1	no	—
89	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
90	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
92	C₃₁×#127	C₂³	1	1	yes	characteristic
93	C₃₁×#124	C₂³	1	2	yes	—
94	C₃₁×#121	—	4	1	no	—
95	C₄×D₄×C₃₁	—	4	1	no	—
96	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
97	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
98	C₃₁×#108	C₂³	1	8	yes	—
99	C₃₁×#104	C₂³	1	8	yes	—
100	C₃₁×#104	C₂³	1	8	yes	—
101	C₃₁×#101	C₂³	1	8	yes	—
102	C₃₁×#101	C₂³	1	4	yes	—
103	C₃₁×#100	C₂³	1	4	yes	—
104	C₃₁×#97	C₂³	1	2	yes	—
105	C₄×C₈×C₃₁	C₂³	1	2	yes	—
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	4	no	—
108	#41×C₃₁	—	2	8	no	—
109	#41×C₃₁	—	2	16	no	—
110	#41×C₃₁	—	2	8	no	—
111	#41×C₃₁	—	2	4	no	—
112	#41×C₃₁	—	2	4	no	—
113	#41×C₃₁	—	4	4	no	—
114	#41×C₃₁	—	4	4	no	—
115	#41×C₃₁	—	4	8	no	—
116	#41×C₃₁	—	4	8	no	—
117	#41×C₃₁	—	4	8	no	—
118	#41×C₃₁	—	4	8	no	—
119	#41×C₃₁	—	4	4	no	—
120	#41×C₃₁	C₂×D₄	1	8	yes	—
121	#41×C₃₁	C₂⁴	1	8	yes	—
122	C₂×Q₈×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	—	2	16	no	—
124	C₂×Q₈×C₃₁	—	2	8	no	—
125	C₂×Q₈×C₃₁	—	2	4	no	—
126	C₂×Q₈×C₃₁	—	2	4	no	—
127	C₂×Q₈×C₃₁	—	4	2	no	—
128	C₂×Q₈×C₃₁	—	4	8	no	—
129	C₂×Q₈×C₃₁	—	4	2	no	—
130	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
131	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
132	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
133	C₂×D₄×C₃₁	—	2	4	no	—
134	C₂×D₄×C₃₁	—	2	8	no	—
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	4	1	no	—
137	C₂×D₄×C₃₁	—	4	4	no	—
138	C₂×D₄×C₃₁	—	4	2	no	—
139	C₂×D₄×C₃₁	—	4	8	no	—
140	C₂×D₄×C₃₁	—	4	2	no	—
141	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
142	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₃₁	—	2	4	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	4	4	no	—
148	C₂²×C₄×C₃₁	—	4	4	no	—
149	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	Q₁₆×C₃₁	—	2	16	no	—
151	Q₁₆×C₃₁	—	2	8	no	—
152	Q₁₆×C₃₁	—	2	16	no	—
153	Q₁₆×C₃₁	C₂×D₄	1	8	yes	—
154	SD₁₆×C₃₁	—	2	8	no	—
155	SD₁₆×C₃₁	—	2	16	no	—
156	SD₁₆×C₃₁	—	2	8	no	—
157	SD₁₆×C₃₁	—	2	16	no	—
158	SD₁₆×C₃₁	C₂×D₄	1	16	yes	—
159	D₈×C₃₁	—	2	8	no	—
160	D₈×C₃₁	C₂×D₄	1	8	yes	—
161	M₁₆×C₃₁	—	2	4	no	—
162	M₁₆×C₃₁	—	2	16	no	—
163	M₁₆×C₃₁	C₂×D₄	1	8	yes	—
164	M₁₆×C₃₁	C₂⁴	1	8	yes	—
165	C₂×C₈×C₃₁	—	2	4	no	—
166	C₂×C₈×C₃₁	—	2	16	no	—
167	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
168	C₄⋊C₄×C₃₁	—	4	2	no	—
169	C₄⋊C₄×C₃₁	—	4	2	no	—
170	C₄⋊C₄×C₃₁	—	4	1	no	—
171	C₂²⋊C₄×C₃₁	—	4	2	no	—
172	C₄²×C₃₁	—	2	1	no	—
173	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
175	#61111	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	4	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	4	4	no	—
179	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
180	Q₈×C₃₁	—	2	4	no	—
181	Q₈×C₃₁	—	2	8	no	—
182	Q₈×C₃₁	—	2	4	no	—
183	Q₈×C₃₁	—	4	8	no	—
184	Q₈×C₃₁	—	4	4	no	—
185	Q₈×C₃₁	—	4	4	no	—
186	Q₈×C₃₁	C₂²×D₄	1	8	yes	—
187	D₄×C₃₁	—	2	8	no	—
188	D₄×C₃₁	—	2	8	no	—
189	D₄×C₃₁	—	2	4	no	—
190	D₄×C₃₁	—	2	4	no	—
191	D₄×C₃₁	—	4	8	no	—
192	D₄×C₃₁	—	4	4	no	—
193	D₄×C₃₁	—	4	8	no	—
194	D₄×C₃₁	—	4	4	no	—
195	D₄×C₃₁	—	8	2	no	—
196	D₄×C₃₁	C₂²×D₄	1	8	yes	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	8	no	—
200	C₂×C₄×C₃₁	—	2	8	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	4	4	no	—
205	C₂×C₄×C₃₁	—	4	8	no	—
206	C₂×C₄×C₃₁	—	4	16	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
213	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
215	C₈×C₃₁	—	2	8	no	—
216	C₈×C₃₁	C₂²×D₄	1	8	yes	—
217	#3586	C₂×C₃₁	1	4	yes	complemented
218	#3293	C₂×C₃₁	1	4	yes	complemented
219	#3291	C₂×C₃₁	1	4	yes	complemented
220	#3289	C₂×C₃₁	1	8	yes	complemented
221	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#3069	C₂×C₃₁	1	4	yes	complemented
223	#3018	C₂×C₃₁	1	2	yes	complemented
224	C₂×#456	C₂×C₃₁	1	2	yes	complemented
225	#2956	C₂×C₃₁	1	2	yes	complemented
226	C₂²×C₃₁	—	2	8	no	—
227	C₂²×C₃₁	—	4	1	no	—
228	C₂²×C₃₁	—	4	2	no	—
229	C₂²×C₃₁	—	8	4	no	—
230	C₂²×C₃₁	—	8	2	no	—
231	C₂²×C₃₁	—	16	4	no	—
232	C₂²×C₃₁	D₄²	1	2	yes	—
233	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	2	8	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	4	no	—
239	C₄×C₃₁	D₄²	1	2	yes	—
240	C₄×C₃₁	C₂³×D₄	1	2	yes	—
241	#585	C₂²×C₃₁	1	2	yes	complemented
242	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
243	#578	—	2	8	no	—
244	#578	C₂²×C₃₁	1	16	yes	complemented
245	#577	—	2	8	no	—
246	#577	C₂²×C₃₁	1	16	yes	complemented
247	#575	—	2	8	no	—
248	#575	C₂²×C₃₁	1	4	yes	—
249	#575	C₂²×C₃₁	1	2	yes	—
250	C₂×#137	—	2	8	no	—
251	C₂×#137	C₂²×C₃₁	1	4	yes	complemented
252	C₂×#137	C₂²×C₃₁	1	4	yes	complemented
253	#568	C₂²×C₃₁	1	4	yes	complemented
254	#532	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
256	#497	C₂²×C₃₁	1	4	yes	complemented
257	#494	C₂²×C₃₁	1	2	yes	complemented
258	#492	C₂²×C₃₁	1	2	yes	complemented
259	#473	C₂²×C₃₁	1	16	yes	complemented
260	#471	C₂²×C₃₁	1	16	yes	complemented
261	#456	C₂²×C₃₁	1	8	yes	complemented
262	#456	C₂²×C₃₁	1	8	yes	complemented
263	#456	C₂²×C₃₁	1	4	yes	complemented
264	#455	C₂²×C₃₁	1	4	yes	complemented
265	#454	C₂²×C₃₁	1	4	yes	complemented
266	#453	C₂²×C₃₁	1	4	yes	complemented
267	#444	C₂²×C₃₁	1	8	yes	complemented
268	#443	C₂²×C₃₁	1	8	yes	complemented
269	#430	C₂²×C₃₁	1	2	yes	complemented
270	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
271	#413	C₂²×C₃₁	1	2	yes	complemented
272	C₂×#101	C₂²×C₃₁	1	2	yes	complemented
273	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	C₂×C₃₁	—	2	1	no	—
275	C₂×C₃₁	—	2	2	no	—
276	C₂×C₃₁	—	4	8	no	—
277	C₂×C₃₁	—	8	2	no	—
278	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
279	2₋¹⁺⁴	—	2	8	no	—
280	2₋¹⁺⁴	—	2	4	no	—
281	2₋¹⁺⁴	—	4	2	no	—
282	2₋¹⁺⁴	C₂³×C₃₁	1	8	yes	—
283	2₋¹⁺⁴	C₂³×C₃₁	1	4	yes	—
284	H(5,𝔽₂)	—	2	4	no	—
285	H(5,𝔽₂)	—	4	2	no	—
286	H(5,𝔽₂)	C₂³×C₃₁	1	4	yes	—
287	C₂×#41	—	2	8	no	—
288	C₂×#41	—	2	4	no	—
289	C₂×#41	—	4	2	no	—
290	C₂×#41	—	4	2	no	—
291	C₂×#41	C₂³×C₃₁	1	4	yes	—
292	C₂×#41	C₂³×C₃₁	1	2	yes	—
293	C₂²×Q₈	—	2	4	no	—
294	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
295	#137	—	2	8	no	—
296	#137	—	2	4	no	—
297	#137	—	2	16	no	—
298	#137	—	2	16	no	—
299	#137	—	2	16	no	—
300	#137	—	2	8	no	—
301	#137	—	2	8	no	—
302	#137	D₄×C₃₁	1	8	yes	complemented
303	#137	C₂³×C₃₁	1	16	yes	—
304	#137	C₂³×C₃₁	1	8	yes	—
305	AGL(1,ℤ/8)	—	2	4	no	—
306	AGL(1,ℤ/8)	—	2	16	no	—
307	AGL(1,ℤ/8)	D₄×C₃₁	1	8	yes	complemented
308	AGL(1,ℤ/8)	C₂³×C₃₁	1	8	yes	—
309	#135	—	2	8	no	—
310	#135	—	2	16	no	—
311	#135	—	2	16	no	—
312	#135	C₂³×C₃₁	1	8	yes	—
313	C₂×Q₁₆	—	2	4	no	—
314	C₂×Q₁₆	—	2	16	no	—
315	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
316	C₂×SD₁₆	—	2	4	no	—
317	C₂×SD₁₆	—	2	16	no	—
318	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
319	#131	—	2	8	no	—
320	#131	C₂³×C₃₁	1	16	yes	—
321	C₂×M₁₆	—	2	8	no	—
322	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
323	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
324	C₄⋊Q₈	—	4	1	no	—
325	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
326	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
327	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
328	#127	C₂³×C₃₁	1	1	yes	characteristic
329	#124	C₂³×C₃₁	1	2	yes	—
330	#121	—	4	1	no	—
331	C₄×D₄	—	4	1	no	—
332	C₂×C₄⋊C₄	—	4	1	no	—
333	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
334	#108	C₂³×C₃₁	1	8	yes	—
335	#104	C₂³×C₃₁	1	8	yes	—
336	#104	C₂³×C₃₁	1	8	yes	—
337	#101	C₂³×C₃₁	1	8	yes	—
338	#101	C₂³×C₃₁	1	4	yes	—
339	#100	C₂³×C₃₁	1	4	yes	—
340	#97	C₂³×C₃₁	1	2	yes	—
341	C₄×C₈	C₂³×C₃₁	1	2	yes	—
342	C₃₁	#61111	1	1	yes	characteristic, direct product factor, Sylow
343	#41	—	2	4	no	—
344	#41	—	2	4	no	—
345	#41	—	2	8	no	—
346	#41	—	2	16	no	—
347	#41	—	2	8	no	—
348	#41	—	2	4	no	—
349	#41	—	2	4	no	—
350	#41	—	4	4	no	—
351	#41	—	4	4	no	—
352	#41	—	4	8	no	—
353	#41	—	4	8	no	—
354	#41	—	4	8	no	—
355	#41	—	4	8	no	—
356	#41	—	4	4	no	—
357	#41	C₂×D₄×C₃₁	1	8	yes	—
358	#41	C₂⁴×C₃₁	1	8	yes	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	16	no	—
361	C₂×Q₈	—	2	8	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	4	2	no	—
365	C₂×Q₈	—	4	8	no	—
366	C₂×Q₈	—	4	2	no	—
367	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
368	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
369	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	8	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	4	1	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	—	4	2	no	—
376	C₂×D₄	—	4	8	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
379	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
387	Q₁₆	—	2	16	no	—
388	Q₁₆	—	2	8	no	—
389	Q₁₆	—	2	16	no	—
390	Q₁₆	C₂×D₄×C₃₁	1	8	yes	—
391	SD₁₆	—	2	8	no	—
392	SD₁₆	—	2	16	no	—
393	SD₁₆	—	2	8	no	—
394	SD₁₆	—	2	16	no	—
395	SD₁₆	C₂×D₄×C₃₁	1	16	yes	—
396	D₈	—	2	8	no	—
397	D₈	C₂×D₄×C₃₁	1	8	yes	—
398	M₁₆	—	2	4	no	—
399	M₁₆	—	2	16	no	—
400	M₁₆	C₂×D₄×C₃₁	1	8	yes	—
401	M₁₆	C₂⁴×C₃₁	1	8	yes	—
402	C₂×C₈	—	2	4	no	—
403	C₂×C₈	—	2	16	no	—
404	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	—	4	2	no	—
407	C₄⋊C₄	—	4	1	no	—
408	C₂²⋊C₄	—	4	2	no	—
409	C₄²	—	2	1	no	—
410	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
411	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
412	C₂³	—	2	4	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	4	4	no	—
415	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
416	Q₈	—	2	4	no	—
417	Q₈	—	2	8	no	—
418	Q₈	—	2	4	no	—
419	Q₈	—	4	8	no	—
420	Q₈	—	4	4	no	—
421	Q₈	—	4	4	no	—
422	Q₈	C₂²×D₄×C₃₁	1	8	yes	—
423	D₄	—	2	8	no	—
424	D₄	—	2	8	no	—
425	D₄	—	2	4	no	—
426	D₄	—	2	4	no	—
427	D₄	—	4	8	no	—
428	D₄	—	4	4	no	—
429	D₄	—	4	8	no	—
430	D₄	—	4	4	no	—
431	D₄	—	8	2	no	—
432	D₄	C₂²×D₄×C₃₁	1	8	yes	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	8	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	8	no	—
442	C₂×C₄	—	4	16	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
449	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
450	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
451	C₈	—	2	8	no	—
452	C₈	C₂²×D₄×C₃₁	1	8	yes	—
453	C₂²	—	2	8	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	8	4	no	—
457	C₂²	—	8	2	no	—
458	C₂²	—	16	4	no	—
459	C₂²	D₄²×C₃₁	1	2	yes	—
460	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
461	C₄	—	2	2	no	—
462	C₄	—	2	8	no	—
463	C₄	—	4	1	no	—
464	C₄	—	4	2	no	—
465	C₄	—	4	4	no	—
466	C₄	D₄²×C₃₁	1	2	yes	—
467	C₄	C₂³×D₄×C₃₁	1	2	yes	—
468	C₂	—	2	1	no	—
469	C₂	—	2	2	no	—
470	C₂	—	4	8	no	—
471	C₂	—	8	2	no	—
472	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂×D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3586	C₂	semidirect
2. C₃₁×#3293	C₂	semidirect
3. C₃₁×#3291	C₂	semidirect
4. C₃₁×#3289	C₂	semidirect
5. C₃₁×#3152	C₂	semidirect
6. C₃₁×#3069	C₂	semidirect
7. C₃₁×#3018	C₂	semidirect
8. C₂×C₃₁×#456	C₂	semidirect
9. C₃₁×#2956	C₂	semidirect
10. C₃₁×#585	C₂²	semidirect
13. C₃₁×#578	C₂²	semidirect
15. C₃₁×#577	C₂²	semidirect
20. C₂×C₃₁×#137	C₂²	semidirect
21. C₂×C₃₁×#137	C₂²	semidirect
22. C₃₁×#568	C₂²	semidirect
24. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
25. C₃₁×#497	C₂²	semidirect
26. C₃₁×#494	C₂²	semidirect
27. C₃₁×#492	C₂²	semidirect
28. C₃₁×#473	C₂²	semidirect
29. C₃₁×#471	C₂²	semidirect
30. C₃₁×#456	C₂²	semidirect
31. C₃₁×#456	C₂²	semidirect
32. C₃₁×#456	C₂²	semidirect
33. C₃₁×#455	C₂²	semidirect
34. C₃₁×#454	C₂²	semidirect
35. C₃₁×#453	C₂²	semidirect
36. C₃₁×#444	C₂²	semidirect
37. C₃₁×#443	C₂²	semidirect
38. C₃₁×#430	C₂²	semidirect
39. C₂×C₃₁×#104	C₂²	semidirect
40. C₃₁×#413	C₂²	semidirect
41. C₂×C₃₁×#101	C₂²	semidirect
42. C₄×M₁₆×C₃₁	C₂²	semidirect
66. C₃₁×#137	D₄	semidirect
71. C₃₁×AGL(1,ℤ/8)	D₄	semidirect
175. #61111	C₃₁	direct (N × H)
217. #3586	C₂×C₃₁	semidirect
218. #3293	C₂×C₃₁	semidirect
219. #3291	C₂×C₃₁	semidirect
220. #3289	C₂×C₃₁	semidirect
221. #3152	C₂×C₃₁	semidirect
222. #3069	C₂×C₃₁	semidirect
223. #3018	C₂×C₃₁	semidirect
224. C₂×#456	C₂×C₃₁	semidirect
225. #2956	C₂×C₃₁	semidirect
241. #585	C₂²×C₃₁	semidirect
244. #578	C₂²×C₃₁	semidirect
246. #577	C₂²×C₃₁	semidirect
251. C₂×#137	C₂²×C₃₁	semidirect
252. C₂×#137	C₂²×C₃₁	semidirect
253. #568	C₂²×C₃₁	semidirect
255. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
256. #497	C₂²×C₃₁	semidirect
257. #494	C₂²×C₃₁	semidirect
258. #492	C₂²×C₃₁	semidirect
259. #473	C₂²×C₃₁	semidirect
260. #471	C₂²×C₃₁	semidirect
261. #456	C₂²×C₃₁	semidirect
262. #456	C₂²×C₃₁	semidirect
263. #456	C₂²×C₃₁	semidirect
264. #455	C₂²×C₃₁	semidirect
265. #454	C₂²×C₃₁	semidirect
266. #453	C₂²×C₃₁	semidirect
267. #444	C₂²×C₃₁	semidirect
268. #443	C₂²×C₃₁	semidirect
269. #430	C₂²×C₃₁	semidirect
270. C₂×#104	C₂²×C₃₁	semidirect
271. #413	C₂²×C₃₁	semidirect
272. C₂×#101	C₂²×C₃₁	semidirect
273. C₄×M₁₆	C₂²×C₃₁	semidirect
302. #137	D₄×C₃₁	semidirect
307. AGL(1,ℤ/8)	D₄×C₃₁	semidirect
342. C₃₁	#61111	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
C₃₁×#3293	2
C₃₁×#3291	2
C₃₁×#3289	2
C₃₁×#3152	2
C₃₁×#3069	2
C₃₁×#3018	2
C₂×C₃₁×#456	2
C₃₁×#2956	2
#61111	31

Subgroups — C31×#61111 #21517870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61111 #21517870