Subgroups — C₃₁×#61105 #21517864

Order 7936.

2620 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3584	C₂	1	4	yes	maximal, complemented
2	C₃₁×#3293	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3292	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3291	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3290	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3067	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3018	C₂	1	4	yes	maximal, complemented
9	C₂×C₃₁×#443	C₂	1	2	yes	maximal, complemented
10	C₃₁×#585	C₂²	1	4	yes	complemented
11	C₃₁×#578	—	2	4	no	—
12	C₃₁×#578	C₂²	1	8	yes	complemented
13	C₃₁×#577	—	2	8	no	—
14	C₃₁×#577	C₂²	1	16	yes	complemented
15	C₃₁×#576	—	2	4	no	—
16	C₃₁×#576	C₂²	1	8	yes	complemented
17	C₃₁×#575	—	2	8	no	—
18	C₃₁×#575	C₂²	1	8	yes	complemented
19	C₂×C₃₁×#135	—	2	8	no	—
20	C₂×C₃₁×#135	C₂²	1	2	yes	—
21	C₂×C₃₁×#135	C₂²	1	4	yes	complemented
22	C₂×C₃₁×#131	C₂²	1	4	yes	complemented
23	C₃₁×#532	C₂²	1	2	yes	complemented
24	C₃₁×#495	C₂²	1	4	yes	complemented
25	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
26	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#492	C₂²	1	2	yes	complemented
28	C₃₁×#473	C₂²	1	8	yes	complemented
29	C₃₁×#472	C₂²	1	8	yes	complemented
30	C₃₁×#471	C₂²	1	16	yes	complemented
31	C₃₁×#456	C₂²	1	8	yes	complemented
32	C₃₁×#455	C₂²	1	8	yes	complemented
33	C₃₁×#454	C₂²	1	8	yes	complemented
34	C₃₁×#453	C₂²	1	8	yes	complemented
35	C₃₁×#443	C₂²	1	4	yes	complemented
36	C₃₁×#443	C₂²	1	4	yes	complemented
37	C₃₁×#443	C₂²	1	4	yes	complemented
38	C₃₁×#443	C₂²	1	4	yes	complemented
39	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
40	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
41	C₃₁×#413	C₂²	1	4	yes	complemented
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×2₋¹⁺⁴	—	2	8	no	—
44	C₃₁×2₋¹⁺⁴	C₂³	1	8	yes	—
45	C₃₁×H(5,𝔽₂)	—	2	8	no	—
46	C₃₁×H(5,𝔽₂)	—	4	4	no	—
47	C₃₁×H(5,𝔽₂)	C₂³	1	8	yes	—
48	C₂×#41×C₃₁	—	2	16	no	—
49	C₂×#41×C₃₁	—	4	4	no	—
50	C₂×#41×C₃₁	C₂³	1	4	yes	—
51	C₂×#41×C₃₁	C₂³	1	4	yes	—
52	C₃₁×#137	—	2	8	no	—
53	C₃₁×#137	—	2	16	no	—
54	C₃₁×#137	—	2	16	no	—
55	C₃₁×#137	C₂³	1	16	yes	—
56	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
57	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
58	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
59	C₃₁×AGL(1,ℤ/8)	C₂³	1	16	yes	—
60	C₃₁×#135	—	2	16	no	—
61	C₃₁×#135	—	2	8	no	—
62	C₃₁×#135	—	2	16	no	—
63	C₃₁×#135	—	2	8	no	—
64	C₃₁×#135	—	2	4	no	—
65	C₃₁×#135	—	2	4	no	—
66	C₃₁×#135	—	2	4	no	—
67	C₃₁×#135	—	2	4	no	—
68	C₃₁×#135	D₄	1	4	yes	complemented
69	C₃₁×#135	D₄	1	4	yes	complemented
70	C₃₁×#135	C₂³	1	4	yes	—
71	C₃₁×#135	C₂³	1	4	yes	—
72	C₂×Q₁₆×C₃₁	—	2	8	no	—
73	C₂×Q₁₆×C₃₁	D₄	1	2	yes	complemented
74	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₃₁	—	2	16	no	—
76	C₂×SD₁₆×C₃₁	D₄	1	4	yes	complemented
77	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
78	C₂×D₈×C₃₁	—	2	8	no	—
79	C₂×D₈×C₃₁	D₄	1	2	yes	complemented
80	C₂×D₈×C₃₁	C₂³	1	4	yes	—
81	C₃₁×#131	—	2	4	no	—
82	C₃₁×#131	—	2	4	no	—
83	C₃₁×#131	C₂³	1	8	yes	—
84	C₃₁×#131	C₂³	1	8	yes	—
85	C₂×M₁₆×C₃₁	—	2	4	no	—
86	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
87	C₂²×C₈×C₃₁	—	2	4	no	—
88	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
89	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
90	C₃₁×#127	C₂³	1	2	yes	—
91	C₃₁×#124	C₂³	1	4	yes	—
92	C₃₁×#121	—	4	2	no	—
93	C₄×D₄×C₃₁	—	4	2	no	—
94	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
95	C₃₁×#108	C₂³	1	4	yes	—
96	C₃₁×#108	C₂³	1	4	yes	—
97	C₃₁×#104	C₂³	1	8	yes	—
98	C₃₁×#104	C₂³	1	8	yes	—
99	C₃₁×#101	C₂³	1	8	yes	—
100	C₃₁×#100	C₂³	1	8	yes	—
101	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₄×C₈×C₃₁	C₂³	1	2	yes	—
104	#41×C₃₁	—	2	4	no	—
105	#41×C₃₁	—	2	4	no	—
106	#41×C₃₁	—	2	16	no	—
107	#41×C₃₁	—	2	16	no	—
108	#41×C₃₁	—	2	8	no	—
109	#41×C₃₁	—	2	8	no	—
110	#41×C₃₁	—	4	8	no	—
111	#41×C₃₁	—	4	8	no	—
112	#41×C₃₁	—	4	16	no	—
113	#41×C₃₁	—	4	8	no	—
114	#41×C₃₁	C₂×D₄	1	8	yes	—
115	#41×C₃₁	C₂⁴	1	8	yes	—
116	C₂×Q₈×C₃₁	—	2	16	no	—
117	C₂×Q₈×C₃₁	—	4	4	no	—
118	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
119	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
120	C₂×D₄×C₃₁	—	2	4	no	—
121	C₂×D₄×C₃₁	—	2	16	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	4	2	no	—
124	C₂×D₄×C₃₁	—	4	8	no	—
125	C₂×D₄×C₃₁	—	4	4	no	—
126	C₂×D₄×C₃₁	—	4	16	no	—
127	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
128	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	4	8	no	—
133	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
134	Q₁₆×C₃₁	—	2	16	no	—
135	Q₁₆×C₃₁	—	2	4	no	—
136	Q₁₆×C₃₁	—	2	4	no	—
137	Q₁₆×C₃₁	C₂×D₄	1	4	yes	—
138	Q₁₆×C₃₁	C₂×D₄	1	4	yes	—
139	SD₁₆×C₃₁	—	2	16	no	—
140	SD₁₆×C₃₁	—	2	16	no	—
141	SD₁₆×C₃₁	—	2	8	no	—
142	SD₁₆×C₃₁	—	2	8	no	—
143	SD₁₆×C₃₁	C₂×D₄	1	8	yes	—
144	SD₁₆×C₃₁	C₂×D₄	1	8	yes	—
145	D₈×C₃₁	—	2	16	no	—
146	D₈×C₃₁	—	2	4	no	—
147	D₈×C₃₁	—	2	4	no	—
148	D₈×C₃₁	C₂×D₄	1	4	yes	—
149	D₈×C₃₁	C₂×D₄	1	4	yes	—
150	M₁₆×C₃₁	—	2	4	no	—
151	M₁₆×C₃₁	—	2	8	no	—
152	M₁₆×C₃₁	—	2	8	no	—
153	M₁₆×C₃₁	C₂⁴	1	8	yes	—
154	C₂×C₈×C₃₁	—	2	4	no	—
155	C₂×C₈×C₃₁	—	2	8	no	—
156	C₂×C₈×C₃₁	—	2	8	no	—
157	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
162	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
163	C₄⋊C₄×C₃₁	—	4	2	no	—
164	C₂²⋊C₄×C₃₁	—	4	4	no	—
165	C₄²×C₃₁	—	2	1	no	—
166	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
167	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
168	#61105	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂³×C₃₁	—	2	4	no	—
170	C₂³×C₃₁	—	2	4	no	—
171	C₂³×C₃₁	—	4	8	no	—
172	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
173	Q₈×C₃₁	—	2	4	no	—
174	Q₈×C₃₁	—	2	8	no	—
175	Q₈×C₃₁	—	4	8	no	—
176	Q₈×C₃₁	C₂²×D₄	1	8	yes	—
177	D₄×C₃₁	—	2	8	no	—
178	D₄×C₃₁	—	2	8	no	—
179	D₄×C₃₁	—	2	4	no	—
180	D₄×C₃₁	—	2	8	no	—
181	D₄×C₃₁	—	4	16	no	—
182	D₄×C₃₁	—	4	8	no	—
183	D₄×C₃₁	—	4	8	no	—
184	D₄×C₃₁	—	8	4	no	—
185	D₄×C₃₁	C₂²×D₄	1	8	yes	—
186	C₂×C₄×C₃₁	—	2	4	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	8	no	—
189	C₂×C₄×C₃₁	—	2	8	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	4	no	—
192	C₂×C₄×C₃₁	—	4	8	no	—
193	C₂×C₄×C₃₁	—	4	16	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	4	no	—
196	C₂×C₄×C₃₁	—	4	4	no	—
197	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
198	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
201	C₈×C₃₁	—	2	8	no	—
202	C₈×C₃₁	C₂²×D₄	1	2	yes	—
203	C₈×C₃₁	C₂²×D₄	1	2	yes	—
204	C₈×C₃₁	C₂²×D₄	1	2	yes	—
205	C₈×C₃₁	C₂²×D₄	1	2	yes	—
206	#3584	C₂×C₃₁	1	4	yes	complemented
207	#3293	C₂×C₃₁	1	2	yes	complemented
208	#3292	C₂×C₃₁	1	2	yes	complemented
209	#3291	C₂×C₃₁	1	4	yes	complemented
210	#3290	C₂×C₃₁	1	8	yes	complemented
211	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#3067	C₂×C₃₁	1	4	yes	complemented
213	#3018	C₂×C₃₁	1	4	yes	complemented
214	C₂×#443	C₂×C₃₁	1	2	yes	complemented
215	C₂²×C₃₁	—	2	8	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	—	8	4	no	—
219	C₂²×C₃₁	—	8	4	no	—
220	C₂²×C₃₁	—	16	8	no	—
221	C₂²×C₃₁	D₄²	1	2	yes	—
222	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
223	C₄×C₃₁	—	2	2	no	—
224	C₄×C₃₁	—	2	8	no	—
225	C₄×C₃₁	—	4	1	no	—
226	C₄×C₃₁	—	4	4	no	—
227	C₄×C₃₁	D₄²	1	1	yes	characteristic
228	C₄×C₃₁	D₄²	1	1	yes	characteristic
229	C₄×C₃₁	C₂³×D₄	1	2	yes	—
230	#585	C₂²×C₃₁	1	4	yes	complemented
231	#578	—	2	4	no	—
232	#578	C₂²×C₃₁	1	8	yes	complemented
233	#577	—	2	8	no	—
234	#577	C₂²×C₃₁	1	16	yes	complemented
235	#576	—	2	4	no	—
236	#576	C₂²×C₃₁	1	8	yes	complemented
237	#575	—	2	8	no	—
238	#575	C₂²×C₃₁	1	8	yes	complemented
239	C₂×#135	—	2	8	no	—
240	C₂×#135	C₂²×C₃₁	1	2	yes	—
241	C₂×#135	C₂²×C₃₁	1	4	yes	complemented
242	C₂×#131	C₂²×C₃₁	1	4	yes	complemented
243	#532	C₂²×C₃₁	1	2	yes	complemented
244	#495	C₂²×C₃₁	1	4	yes	complemented
245	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
246	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
247	#492	C₂²×C₃₁	1	2	yes	complemented
248	#473	C₂²×C₃₁	1	8	yes	complemented
249	#472	C₂²×C₃₁	1	8	yes	complemented
250	#471	C₂²×C₃₁	1	16	yes	complemented
251	#456	C₂²×C₃₁	1	8	yes	complemented
252	#455	C₂²×C₃₁	1	8	yes	complemented
253	#454	C₂²×C₃₁	1	8	yes	complemented
254	#453	C₂²×C₃₁	1	8	yes	complemented
255	#443	C₂²×C₃₁	1	4	yes	complemented
256	#443	C₂²×C₃₁	1	4	yes	complemented
257	#443	C₂²×C₃₁	1	4	yes	complemented
258	#443	C₂²×C₃₁	1	4	yes	complemented
259	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
260	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
261	#413	C₂²×C₃₁	1	4	yes	complemented
262	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
263	C₂×C₃₁	—	2	1	no	—
264	C₂×C₃₁	—	2	2	no	—
265	C₂×C₃₁	—	4	8	no	—
266	C₂×C₃₁	—	8	4	no	—
267	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
268	2₋¹⁺⁴	—	2	8	no	—
269	2₋¹⁺⁴	C₂³×C₃₁	1	8	yes	—
270	H(5,𝔽₂)	—	2	8	no	—
271	H(5,𝔽₂)	—	4	4	no	—
272	H(5,𝔽₂)	C₂³×C₃₁	1	8	yes	—
273	C₂×#41	—	2	16	no	—
274	C₂×#41	—	4	4	no	—
275	C₂×#41	C₂³×C₃₁	1	4	yes	—
276	C₂×#41	C₂³×C₃₁	1	4	yes	—
277	#137	—	2	8	no	—
278	#137	—	2	16	no	—
279	#137	—	2	16	no	—
280	#137	C₂³×C₃₁	1	16	yes	—
281	AGL(1,ℤ/8)	—	2	8	no	—
282	AGL(1,ℤ/8)	—	2	16	no	—
283	AGL(1,ℤ/8)	—	2	16	no	—
284	AGL(1,ℤ/8)	C₂³×C₃₁	1	16	yes	—
285	#135	—	2	16	no	—
286	#135	—	2	8	no	—
287	#135	—	2	16	no	—
288	#135	—	2	8	no	—
289	#135	—	2	4	no	—
290	#135	—	2	4	no	—
291	#135	—	2	4	no	—
292	#135	—	2	4	no	—
293	#135	D₄×C₃₁	1	4	yes	complemented
294	#135	D₄×C₃₁	1	4	yes	complemented
295	#135	C₂³×C₃₁	1	4	yes	—
296	#135	C₂³×C₃₁	1	4	yes	—
297	C₂×Q₁₆	—	2	8	no	—
298	C₂×Q₁₆	D₄×C₃₁	1	2	yes	complemented
299	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
300	C₂×SD₁₆	—	2	16	no	—
301	C₂×SD₁₆	D₄×C₃₁	1	4	yes	complemented
302	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
303	C₂×D₈	—	2	8	no	—
304	C₂×D₈	D₄×C₃₁	1	2	yes	complemented
305	C₂×D₈	C₂³×C₃₁	1	4	yes	—
306	#131	—	2	4	no	—
307	#131	—	2	4	no	—
308	#131	C₂³×C₃₁	1	8	yes	—
309	#131	C₂³×C₃₁	1	8	yes	—
310	C₂×M₁₆	—	2	4	no	—
311	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
312	C₂²×C₈	—	2	4	no	—
313	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
314	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
315	#127	C₂³×C₃₁	1	2	yes	—
316	#124	C₂³×C₃₁	1	4	yes	—
317	#121	—	4	2	no	—
318	C₄×D₄	—	4	2	no	—
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	#108	C₂³×C₃₁	1	4	yes	—
321	#108	C₂³×C₃₁	1	4	yes	—
322	#104	C₂³×C₃₁	1	8	yes	—
323	#104	C₂³×C₃₁	1	8	yes	—
324	#101	C₂³×C₃₁	1	8	yes	—
325	#100	C₂³×C₃₁	1	8	yes	—
326	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
327	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
328	C₄×C₈	C₂³×C₃₁	1	2	yes	—
329	C₃₁	#61105	1	1	yes	characteristic, direct product factor, Sylow
330	#41	—	2	4	no	—
331	#41	—	2	4	no	—
332	#41	—	2	16	no	—
333	#41	—	2	16	no	—
334	#41	—	2	8	no	—
335	#41	—	2	8	no	—
336	#41	—	4	8	no	—
337	#41	—	4	8	no	—
338	#41	—	4	16	no	—
339	#41	—	4	8	no	—
340	#41	C₂×D₄×C₃₁	1	8	yes	—
341	#41	C₂⁴×C₃₁	1	8	yes	—
342	C₂×Q₈	—	2	16	no	—
343	C₂×Q₈	—	4	4	no	—
344	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
345	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	16	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	4	2	no	—
350	C₂×D₄	—	4	8	no	—
351	C₂×D₄	—	4	4	no	—
352	C₂×D₄	—	4	16	no	—
353	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
354	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	4	8	no	—
359	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
360	Q₁₆	—	2	16	no	—
361	Q₁₆	—	2	4	no	—
362	Q₁₆	—	2	4	no	—
363	Q₁₆	C₂×D₄×C₃₁	1	4	yes	—
364	Q₁₆	C₂×D₄×C₃₁	1	4	yes	—
365	SD₁₆	—	2	16	no	—
366	SD₁₆	—	2	16	no	—
367	SD₁₆	—	2	8	no	—
368	SD₁₆	—	2	8	no	—
369	SD₁₆	C₂×D₄×C₃₁	1	8	yes	—
370	SD₁₆	C₂×D₄×C₃₁	1	8	yes	—
371	D₈	—	2	16	no	—
372	D₈	—	2	4	no	—
373	D₈	—	2	4	no	—
374	D₈	C₂×D₄×C₃₁	1	4	yes	—
375	D₈	C₂×D₄×C₃₁	1	4	yes	—
376	M₁₆	—	2	4	no	—
377	M₁₆	—	2	8	no	—
378	M₁₆	—	2	8	no	—
379	M₁₆	C₂⁴×C₃₁	1	8	yes	—
380	C₂×C₈	—	2	4	no	—
381	C₂×C₈	—	2	8	no	—
382	C₂×C₈	—	2	8	no	—
383	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
384	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
385	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
386	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
387	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
388	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
389	C₄⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	4	no	—
391	C₄²	—	2	1	no	—
392	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
393	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
394	C₂³	—	2	4	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	4	8	no	—
397	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
398	Q₈	—	2	4	no	—
399	Q₈	—	2	8	no	—
400	Q₈	—	4	8	no	—
401	Q₈	C₂²×D₄×C₃₁	1	8	yes	—
402	D₄	—	2	8	no	—
403	D₄	—	2	8	no	—
404	D₄	—	2	4	no	—
405	D₄	—	2	8	no	—
406	D₄	—	4	16	no	—
407	D₄	—	4	8	no	—
408	D₄	—	4	8	no	—
409	D₄	—	8	4	no	—
410	D₄	C₂²×D₄×C₃₁	1	8	yes	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	4	8	no	—
418	C₂×C₄	—	4	16	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
423	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
424	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
425	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
426	C₈	—	2	8	no	—
427	C₈	C₂²×D₄×C₃₁	1	2	yes	—
428	C₈	C₂²×D₄×C₃₁	1	2	yes	—
429	C₈	C₂²×D₄×C₃₁	1	2	yes	—
430	C₈	C₂²×D₄×C₃₁	1	2	yes	—
431	C₂²	—	2	8	no	—
432	C₂²	—	4	1	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	8	4	no	—
435	C₂²	—	8	4	no	—
436	C₂²	—	16	8	no	—
437	C₂²	D₄²×C₃₁	1	2	yes	—
438	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
439	C₄	—	2	2	no	—
440	C₄	—	2	8	no	—
441	C₄	—	4	1	no	—
442	C₄	—	4	4	no	—
443	C₄	D₄²×C₃₁	1	1	yes	characteristic
444	C₄	D₄²×C₃₁	1	1	yes	characteristic
445	C₄	C₂³×D₄×C₃₁	1	2	yes	—
446	C₂	—	2	1	no	—
447	C₂	—	2	2	no	—
448	C₂	—	4	8	no	—
449	C₂	—	8	4	no	—
450	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3584	C₂	semidirect
2. C₃₁×#3293	C₂	semidirect
3. C₃₁×#3292	C₂	semidirect
4. C₃₁×#3291	C₂	semidirect
5. C₃₁×#3290	C₂	semidirect
6. C₃₁×#3147	C₂	semidirect
7. C₃₁×#3067	C₂	semidirect
8. C₃₁×#3018	C₂	semidirect
9. C₂×C₃₁×#443	C₂	semidirect
10. C₃₁×#585	C₂²	semidirect
12. C₃₁×#578	C₂²	semidirect
14. C₃₁×#577	C₂²	semidirect
16. C₃₁×#576	C₂²	semidirect
18. C₃₁×#575	C₂²	semidirect
21. C₂×C₃₁×#135	C₂²	semidirect
22. C₂×C₃₁×#131	C₂²	semidirect
23. C₃₁×#532	C₂²	semidirect
24. C₃₁×#495	C₂²	semidirect
25. C₃₁×#494	C₂²	semidirect
26. C₃₁×#493	C₂²	semidirect
27. C₃₁×#492	C₂²	semidirect
28. C₃₁×#473	C₂²	semidirect
29. C₃₁×#472	C₂²	semidirect
30. C₃₁×#471	C₂²	semidirect
31. C₃₁×#456	C₂²	semidirect
32. C₃₁×#455	C₂²	semidirect
33. C₃₁×#454	C₂²	semidirect
34. C₃₁×#453	C₂²	semidirect
35. C₃₁×#443	C₂²	semidirect
36. C₃₁×#443	C₂²	semidirect
37. C₃₁×#443	C₂²	semidirect
38. C₃₁×#443	C₂²	semidirect
39. C₂×C₃₁×#108	C₂²	semidirect
40. C₂×C₃₁×#104	C₂²	semidirect
41. C₃₁×#413	C₂²	semidirect
42. C₂×C₄×C₈×C₃₁	C₂²	semidirect
68. C₃₁×#135	D₄	semidirect
69. C₃₁×#135	D₄	semidirect
73. C₂×Q₁₆×C₃₁	D₄	semidirect
76. C₂×SD₁₆×C₃₁	D₄	semidirect
79. C₂×D₈×C₃₁	D₄	semidirect
168. #61105	C₃₁	direct (N × H)
206. #3584	C₂×C₃₁	semidirect
207. #3293	C₂×C₃₁	semidirect
208. #3292	C₂×C₃₁	semidirect
209. #3291	C₂×C₃₁	semidirect
210. #3290	C₂×C₃₁	semidirect
211. #3147	C₂×C₃₁	semidirect
212. #3067	C₂×C₃₁	semidirect
213. #3018	C₂×C₃₁	semidirect
214. C₂×#443	C₂×C₃₁	semidirect
230. #585	C₂²×C₃₁	semidirect
232. #578	C₂²×C₃₁	semidirect
234. #577	C₂²×C₃₁	semidirect
236. #576	C₂²×C₃₁	semidirect
238. #575	C₂²×C₃₁	semidirect
241. C₂×#135	C₂²×C₃₁	semidirect
242. C₂×#131	C₂²×C₃₁	semidirect
243. #532	C₂²×C₃₁	semidirect
244. #495	C₂²×C₃₁	semidirect
245. #494	C₂²×C₃₁	semidirect
246. #493	C₂²×C₃₁	semidirect
247. #492	C₂²×C₃₁	semidirect
248. #473	C₂²×C₃₁	semidirect
249. #472	C₂²×C₃₁	semidirect
250. #471	C₂²×C₃₁	semidirect
251. #456	C₂²×C₃₁	semidirect
252. #455	C₂²×C₃₁	semidirect
253. #454	C₂²×C₃₁	semidirect
254. #453	C₂²×C₃₁	semidirect
255. #443	C₂²×C₃₁	semidirect
256. #443	C₂²×C₃₁	semidirect
257. #443	C₂²×C₃₁	semidirect
258. #443	C₂²×C₃₁	semidirect
259. C₂×#108	C₂²×C₃₁	semidirect
260. C₂×#104	C₂²×C₃₁	semidirect
261. #413	C₂²×C₃₁	semidirect
262. C₂×C₄×C₈	C₂²×C₃₁	semidirect
293. #135	D₄×C₃₁	semidirect
294. #135	D₄×C₃₁	semidirect
298. C₂×Q₁₆	D₄×C₃₁	semidirect
301. C₂×SD₁₆	D₄×C₃₁	semidirect
304. C₂×D₈	D₄×C₃₁	semidirect
329. C₃₁	#61105	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×#3584	2
C₃₁×#3293	2
C₃₁×#3292	2
C₃₁×#3291	2
C₃₁×#3290	2
C₃₁×#3147	2
C₃₁×#3067	2
C₃₁×#3018	2
C₂×C₃₁×#443	2
#61105	31

Subgroups — C31×#61105 #21517864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61105 #21517864