Subgroups — C₃₁×#60968 #21517727

Order 7936.

2340 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3253	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3227	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3205	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3192	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3184	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3109	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	—	2	2	no	—
10	C₃₁×#532	—	2	2	no	—
11	C₃₁×#532	—	2	2	no	—
12	C₃₁×#532	—	2	2	no	—
13	C₃₁×#532	—	2	2	no	—
14	C₃₁×#532	C₂²	1	4	yes	—
15	C₃₁×#532	C₂²	1	4	yes	—
16	C₃₁×#532	C₂²	1	4	yes	complemented
17	C₃₁×#532	C₂²	1	4	yes	complemented
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
19	C₂×C₃₁×#127	C₂²	1	2	yes	complemented
20	C₂×C₃₁×#124	—	2	2	no	—
21	C₂×C₃₁×#121	—	2	2	no	—
22	C₂×C₃₁×#121	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₃₁	—	2	2	no	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
25	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#498	C₂²	1	4	yes	complemented
27	C₃₁×#493	C₂²	1	4	yes	complemented
28	C₃₁×#486	C₂²	1	8	yes	complemented
29	C₃₁×#481	C₂²	1	16	yes	complemented
30	C₃₁×#476	C₂²	1	16	yes	complemented
31	C₃₁×#469	C₂²	1	16	yes	complemented
32	C₃₁×#459	C₂²	1	16	yes	complemented
33	C₃₁×#447	C₂²	1	16	yes	complemented
34	C₃₁×#442	C₂²	1	16	yes	complemented
35	C₃₁×#432	C₂²	1	4	yes	complemented
36	C₃₁×#431	C₂²	1	4	yes	complemented
37	C₃₁×#423	C₂²	1	2	yes	complemented
38	C₃₁×#407	C₂²	1	2	yes	—
39	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
40	C₂×#41×C₃₁	—	2	4	no	—
41	C₂×#41×C₃₁	—	2	4	no	—
42	C₂×#41×C₃₁	—	2	4	no	—
43	C₂×#41×C₃₁	—	2	4	no	—
44	C₂×#41×C₃₁	—	4	16	no	—
45	C₂²×Q₈×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	4	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂²×D₄×C₃₁	—	2	2	no	—
49	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
50	C₂³×C₄×C₃₁	—	2	4	no	—
51	C₂³×C₄×C₃₁	D₄	1	2	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×D₈×C₃₁	C₂³	1	16	yes	—
54	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
58	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
59	C₃₁×#127	—	2	2	no	—
60	C₃₁×#127	—	2	2	no	—
61	C₃₁×#127	C₂³	1	4	yes	—
62	C₃₁×#127	C₂³	1	4	yes	—
63	C₃₁×#124	—	2	4	no	—
64	C₃₁×#124	—	2	4	no	—
65	C₃₁×#124	—	2	4	no	—
66	C₃₁×#124	—	2	4	no	—
67	C₃₁×#121	—	2	4	no	—
68	C₃₁×#121	—	2	4	no	—
69	C₃₁×#121	—	2	4	no	—
70	C₃₁×#121	—	2	4	no	—
71	C₃₁×#121	—	2	4	no	—
72	C₃₁×#121	—	2	4	no	—
73	C₃₁×#121	C₂³	1	8	yes	—
74	C₃₁×#121	C₂³	1	8	yes	—
75	C₄×D₄×C₃₁	—	2	4	no	—
76	C₄×D₄×C₃₁	—	2	4	no	—
77	C₄×D₄×C₃₁	—	2	4	no	—
78	C₄×D₄×C₃₁	—	2	4	no	—
79	C₄×D₄×C₃₁	—	2	4	no	—
80	C₄×D₄×C₃₁	—	2	4	no	—
81	C₄×D₄×C₃₁	C₂³	1	8	yes	—
82	C₄×D₄×C₃₁	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
87	C₂×C₄²×C₃₁	D₄	1	2	yes	—
88	C₂×C₄²×C₃₁	D₄	1	2	yes	—
89	C₂×C₄²×C₃₁	D₄	1	2	yes	—
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
93	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
94	C₃₁×#106	C₂³	1	16	yes	—
95	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
96	C₃₁×#102	C₂³	1	32	yes	—
97	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
98	C₃₁×#97	C₂³	1	4	yes	—
99	C₄×C₈×C₃₁	C₂³	1	4	yes	—
100	C₂⁴×C₃₁	—	2	4	no	—
101	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
102	#41×C₃₁	—	4	16	no	—
103	#41×C₃₁	—	4	16	no	—
104	#41×C₃₁	—	4	16	no	—
105	#41×C₃₁	—	4	16	no	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	4	no	—
108	C₂×Q₈×C₃₁	—	4	8	no	—
109	C₂×D₄×C₃₁	—	2	8	no	—
110	C₂×D₄×C₃₁	—	2	8	no	—
111	C₂×D₄×C₃₁	—	2	4	no	—
112	C₂×D₄×C₃₁	—	2	4	no	—
113	C₂×D₄×C₃₁	—	2	4	no	—
114	C₂×D₄×C₃₁	—	2	4	no	—
115	C₂×D₄×C₃₁	—	2	4	no	—
116	C₂×D₄×C₃₁	—	4	8	no	—
117	C₂×D₄×C₃₁	—	4	8	no	—
118	C₂×D₄×C₃₁	—	4	16	no	—
119	C₂×D₄×C₃₁	—	4	8	no	—
120	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
121	C₂²×C₄×C₃₁	—	2	8	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	8	no	—
124	C₂²×C₄×C₃₁	—	2	8	no	—
125	C₂²×C₄×C₃₁	—	4	2	no	—
126	C₂²×C₄×C₃₁	—	4	16	no	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
137	D₈×C₃₁	—	4	16	no	—
138	M₁₆×C₃₁	—	4	4	no	—
139	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
140	C₄⋊C₄×C₃₁	—	2	4	no	—
141	C₄⋊C₄×C₃₁	—	2	4	no	—
142	C₄⋊C₄×C₃₁	—	2	4	no	—
143	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
144	C₂²⋊C₄×C₃₁	—	2	8	no	—
145	C₂²⋊C₄×C₃₁	—	2	8	no	—
146	C₂²⋊C₄×C₃₁	—	2	8	no	—
147	C₂²⋊C₄×C₃₁	—	2	8	no	—
148	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₃₁	C₂×D₄	1	2	yes	—
150	C₄²×C₃₁	C₂×D₄	1	2	yes	—
151	C₄²×C₃₁	C₂×D₄	1	2	yes	—
152	C₄²×C₃₁	C₂×D₄	1	2	yes	—
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂×D₄	1	2	yes	—
155	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
156	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
157	C₄²×C₃₁	C₂⁴	1	2	yes	—
158	#60968	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	2	4	no	—
160	C₂³×C₃₁	—	2	8	no	—
161	C₂³×C₃₁	—	4	1	no	—
162	C₂³×C₃₁	—	4	8	no	—
163	C₂³×C₃₁	—	4	8	no	—
164	C₂³×C₃₁	—	4	8	no	—
165	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
167	Q₈×C₃₁	—	4	8	no	—
168	Q₈×C₃₁	—	4	8	no	—
169	D₄×C₃₁	—	2	16	no	—
170	D₄×C₃₁	—	4	8	no	—
171	D₄×C₃₁	—	4	8	no	—
172	D₄×C₃₁	—	4	16	no	—
173	D₄×C₃₁	—	4	16	no	—
174	D₄×C₃₁	—	4	8	no	—
175	D₄×C₃₁	—	4	8	no	—
176	D₄×C₃₁	—	4	8	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	—	2	4	no	—
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	8	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	16	no	—
188	C₂×C₄×C₃₁	—	4	8	no	—
189	C₂×C₄×C₃₁	—	4	16	no	—
190	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	8	yes	—
191	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
195	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
196	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
198	C₈×C₃₁	—	2	8	no	—
199	C₂×#532	C₂×C₃₁	1	2	yes	complemented
200	#3253	C₂×C₃₁	1	4	yes	complemented
201	#3227	C₂×C₃₁	1	4	yes	complemented
202	#3205	C₂×C₃₁	1	4	yes	complemented
203	#3192	C₂×C₃₁	1	4	yes	complemented
204	#3184	C₂×C₃₁	1	4	yes	complemented
205	#3109	C₂×C₃₁	1	8	yes	complemented
206	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	2	2	no	—
209	C₂²×C₃₁	—	2	1	no	—
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	4	4	no	—
212	C₂²×C₃₁	—	4	8	no	—
213	C₂²×C₃₁	—	4	8	no	—
214	C₂²×C₃₁	—	4	8	no	—
215	C₂²×C₃₁	—	8	16	no	—
216	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
217	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	2	2	no	—
221	C₄×C₃₁	—	2	2	no	—
222	C₄×C₃₁	—	4	8	no	—
223	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
224	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
225	C₄×C₃₁	#585	1	2	yes	—
226	C₂²×#41	—	2	2	no	—
227	#532	—	2	2	no	—
228	#532	—	2	2	no	—
229	#532	—	2	2	no	—
230	#532	—	2	2	no	—
231	#532	C₂²×C₃₁	1	4	yes	—
232	#532	C₂²×C₃₁	1	4	yes	—
233	#532	C₂²×C₃₁	1	4	yes	complemented
234	#532	C₂²×C₃₁	1	4	yes	complemented
235	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
236	C₂×#127	C₂²×C₃₁	1	2	yes	complemented
237	C₂×#124	—	2	2	no	—
238	C₂×#121	—	2	2	no	—
239	C₂×#121	C₂²×C₃₁	1	4	yes	—
240	C₂×C₄×D₄	—	2	2	no	—
241	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
242	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
243	#498	C₂²×C₃₁	1	4	yes	complemented
244	#493	C₂²×C₃₁	1	4	yes	complemented
245	#486	C₂²×C₃₁	1	8	yes	complemented
246	#481	C₂²×C₃₁	1	16	yes	complemented
247	#476	C₂²×C₃₁	1	16	yes	complemented
248	#469	C₂²×C₃₁	1	16	yes	complemented
249	#459	C₂²×C₃₁	1	16	yes	complemented
250	#447	C₂²×C₃₁	1	16	yes	complemented
251	#442	C₂²×C₃₁	1	16	yes	complemented
252	#432	C₂²×C₃₁	1	4	yes	complemented
253	#431	C₂²×C₃₁	1	4	yes	complemented
254	#423	C₂²×C₃₁	1	2	yes	complemented
255	#407	C₂²×C₃₁	1	2	yes	—
256	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
257	C₂×C₃₁	—	2	4	no	—
258	C₂×C₃₁	—	4	1	no	—
259	C₂×C₃₁	—	8	8	no	—
260	C₂×C₃₁	#3449	1	1	yes	characteristic
261	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	2	yes	—
262	C₂×#41	—	2	4	no	—
263	C₂×#41	—	2	4	no	—
264	C₂×#41	—	2	4	no	—
265	C₂×#41	—	2	4	no	—
266	C₂×#41	—	4	16	no	—
267	C₂²×Q₈	—	2	2	no	—
268	C₂²×D₄	—	2	4	no	—
269	C₂²×D₄	—	2	2	no	—
270	C₂²×D₄	—	2	2	no	—
271	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
272	C₂³×C₄	—	2	4	no	—
273	C₂³×C₄	D₄×C₃₁	1	2	yes	—
274	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
275	C₂×D₈	C₂³×C₃₁	1	16	yes	—
276	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
277	C₄⋊Q₈	—	2	2	no	—
278	C₄⋊Q₈	—	2	2	no	—
279	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
280	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
281	#127	—	2	2	no	—
282	#127	—	2	2	no	—
283	#127	C₂³×C₃₁	1	4	yes	—
284	#127	C₂³×C₃₁	1	4	yes	—
285	#124	—	2	4	no	—
286	#124	—	2	4	no	—
287	#124	—	2	4	no	—
288	#124	—	2	4	no	—
289	#121	—	2	4	no	—
290	#121	—	2	4	no	—
291	#121	—	2	4	no	—
292	#121	—	2	4	no	—
293	#121	—	2	4	no	—
294	#121	—	2	4	no	—
295	#121	C₂³×C₃₁	1	8	yes	—
296	#121	C₂³×C₃₁	1	8	yes	—
297	C₄×D₄	—	2	4	no	—
298	C₄×D₄	—	2	4	no	—
299	C₄×D₄	—	2	4	no	—
300	C₄×D₄	—	2	4	no	—
301	C₄×D₄	—	2	4	no	—
302	C₄×D₄	—	2	4	no	—
303	C₄×D₄	C₂³×C₃₁	1	8	yes	—
304	C₄×D₄	C₂³×C₃₁	1	8	yes	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
307	C₂×C₂²⋊C₄	—	2	4	no	—
308	C₂×C₂²⋊C₄	—	2	4	no	—
309	C₂×C₄²	D₄×C₃₁	1	2	yes	—
310	C₂×C₄²	D₄×C₃₁	1	2	yes	—
311	C₂×C₄²	D₄×C₃₁	1	2	yes	—
312	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
313	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
314	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
316	#106	C₂³×C₃₁	1	16	yes	—
317	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
318	#102	C₂³×C₃₁	1	32	yes	—
319	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
320	#97	C₂³×C₃₁	1	4	yes	—
321	C₄×C₈	C₂³×C₃₁	1	4	yes	—
322	C₃₁	#60968	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	—	2	4	no	—
324	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
325	#41	—	4	16	no	—
326	#41	—	4	16	no	—
327	#41	—	4	16	no	—
328	#41	—	4	16	no	—
329	C₂×Q₈	—	2	4	no	—
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	—	4	8	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	4	no	—
338	C₂×D₄	—	2	4	no	—
339	C₂×D₄	—	4	8	no	—
340	C₂×D₄	—	4	8	no	—
341	C₂×D₄	—	4	16	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
344	C₂²×C₄	—	2	8	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	8	no	—
347	C₂²×C₄	—	2	8	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	—	4	16	no	—
350	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
358	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
359	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
360	D₈	—	4	16	no	—
361	M₁₆	—	4	4	no	—
362	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	2	8	no	—
369	C₂²⋊C₄	—	2	8	no	—
370	C₂²⋊C₄	—	2	8	no	—
371	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₄²	C₂×D₄×C₃₁	1	2	yes	—
373	C₄²	C₂×D₄×C₃₁	1	2	yes	—
374	C₄²	C₂×D₄×C₃₁	1	2	yes	—
375	C₄²	C₂×D₄×C₃₁	1	2	yes	—
376	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₄²	C₂×D₄×C₃₁	1	2	yes	—
378	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
379	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
380	C₄²	C₂⁴×C₃₁	1	2	yes	—
381	C₂³	—	2	4	no	—
382	C₂³	—	2	8	no	—
383	C₂³	—	4	1	no	—
384	C₂³	—	4	8	no	—
385	C₂³	—	4	8	no	—
386	C₂³	—	4	8	no	—
387	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
389	Q₈	—	4	8	no	—
390	Q₈	—	4	8	no	—
391	D₄	—	2	16	no	—
392	D₄	—	4	8	no	—
393	D₄	—	4	8	no	—
394	D₄	—	4	16	no	—
395	D₄	—	4	16	no	—
396	D₄	—	4	8	no	—
397	D₄	—	4	8	no	—
398	D₄	—	4	8	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	8	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	16	no	—
410	C₂×C₄	—	4	8	no	—
411	C₂×C₄	—	4	16	no	—
412	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	8	yes	—
413	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
415	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
416	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
417	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
418	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
419	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
420	C₈	—	2	8	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	1	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	4	4	no	—
426	C₂²	—	4	8	no	—
427	C₂²	—	4	8	no	—
428	C₂²	—	4	8	no	—
429	C₂²	—	8	16	no	—
430	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
431	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
432	C₄	—	2	2	no	—
433	C₄	—	2	1	no	—
434	C₄	—	2	2	no	—
435	C₄	—	2	2	no	—
436	C₄	—	4	8	no	—
437	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
438	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
439	C₄	C₃₁×#585	1	2	yes	—
440	C₂	—	2	4	no	—
441	C₂	—	4	1	no	—
442	C₂	—	8	8	no	—
443	C₂	C₃₁×#3449	1	1	yes	characteristic
444	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#532	C₂	semidirect
2. C₃₁×#3253	C₂	semidirect
3. C₃₁×#3227	C₂	semidirect
4. C₃₁×#3205	C₂	semidirect
5. C₃₁×#3192	C₂	semidirect
6. C₃₁×#3184	C₂	semidirect
7. C₃₁×#3109	C₂	semidirect
8. C₃₁×#2920	C₂	semidirect
16. C₃₁×#532	C₂²	semidirect
17. C₃₁×#532	C₂²	semidirect
18. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
19. C₂×C₃₁×#127	C₂²	semidirect
24. C₂×C₄×D₄×C₃₁	C₂²	semidirect
26. C₃₁×#498	C₂²	semidirect
27. C₃₁×#493	C₂²	semidirect
28. C₃₁×#486	C₂²	semidirect
29. C₃₁×#481	C₂²	semidirect
30. C₃₁×#476	C₂²	semidirect
31. C₃₁×#469	C₂²	semidirect
32. C₃₁×#459	C₂²	semidirect
33. C₃₁×#447	C₂²	semidirect
34. C₃₁×#442	C₂²	semidirect
35. C₃₁×#432	C₂²	semidirect
36. C₃₁×#431	C₂²	semidirect
37. C₃₁×#423	C₂²	semidirect
39. C₄×M₁₆×C₃₁	C₂²	semidirect
158. #60968	C₃₁	direct (N × H)
199. C₂×#532	C₂×C₃₁	semidirect
200. #3253	C₂×C₃₁	semidirect
201. #3227	C₂×C₃₁	semidirect
202. #3205	C₂×C₃₁	semidirect
203. #3192	C₂×C₃₁	semidirect
204. #3184	C₂×C₃₁	semidirect
205. #3109	C₂×C₃₁	semidirect
206. #2920	C₂×C₃₁	semidirect
233. #532	C₂²×C₃₁	semidirect
234. #532	C₂²×C₃₁	semidirect
235. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
236. C₂×#127	C₂²×C₃₁	semidirect
241. C₂×C₄×D₄	C₂²×C₃₁	semidirect
243. #498	C₂²×C₃₁	semidirect
244. #493	C₂²×C₃₁	semidirect
245. #486	C₂²×C₃₁	semidirect
246. #481	C₂²×C₃₁	semidirect
247. #476	C₂²×C₃₁	semidirect
248. #469	C₂²×C₃₁	semidirect
249. #459	C₂²×C₃₁	semidirect
250. #447	C₂²×C₃₁	semidirect
251. #442	C₂²×C₃₁	semidirect
252. #432	C₂²×C₃₁	semidirect
253. #431	C₂²×C₃₁	semidirect
254. #423	C₂²×C₃₁	semidirect
256. C₄×M₁₆	C₂²×C₃₁	semidirect
322. C₃₁	#60968	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#532	2
C₃₁×#3253	2
C₃₁×#3227	2
C₃₁×#3205	2
C₃₁×#3192	2
C₃₁×#3184	2
C₃₁×#3109	2
C₃₁×#2920	2
#60968	31

Subgroups — C31×#60968 #21517727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60968 #21517727