Subgroups — C₃₁×#21241 #21478000

Order 7936.

1498 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#575	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#104	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2879	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1975	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1974	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#575	—	2	4	no	—
10	C₃₁×#575	—	2	4	no	—
11	C₂×C₃₁×#137	C₄	1	4	yes	complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₄	1	4	yes	complemented
13	C₂×C₃₁×#135	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#430	—	2	4	no	—
19	C₂×C₃₁×#108	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#104	—	2	8	no	—
21	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
22	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
23	C₃₁×#408	—	2	4	no	—
24	C₃₁×#408	C₂²	1	8	yes	complemented
25	C₃₁×#407	—	2	4	no	—
26	C₄×M₁₆×C₃₁	—	2	4	no	—
27	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
28	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
29	C₂×#41×C₃₁	—	2	2	no	—
30	C₂×#41×C₃₁	—	2	2	no	—
31	C₂×#41×C₃₁	—	2	4	no	—
32	C₂×#41×C₃₁	—	2	4	no	—
33	C₂×#41×C₃₁	—	2	4	no	—
34	C₂×#41×C₃₁	—	2	4	no	—
35	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
36	C₂×#41×C₃₁	C₂³	1	4	yes	—
37	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₃₁	—	2	2	no	—
39	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₃₁	—	2	1	no	—
41	C₂³×C₄×C₃₁	—	2	2	no	—
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#137	—	2	16	no	—
44	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
45	C₃₁×#135	—	2	8	no	—
46	C₃₁×#135	—	2	8	no	—
47	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
48	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
49	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	4	no	—
54	C₂×M₁₆×C₃₁	—	2	4	no	—
55	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
56	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
57	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
58	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
59	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
60	C₂²×C₈×C₃₁	—	2	2	no	—
61	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄²×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	D₄	1	4	yes	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#108	—	2	8	no	—
69	C₃₁×#104	—	2	8	no	—
70	C₃₁×#104	—	2	8	no	—
71	C₃₁×C₂²⋊C₈	—	2	16	no	—
72	C₃₁×#97	—	2	4	no	—
73	C₄×C₈×C₃₁	—	2	4	no	—
74	C₂⁴×C₃₁	—	2	2	no	—
75	C₂⁴×C₃₁	#41	1	1	yes	characteristic
76	#41×C₃₁	—	2	8	no	—
77	#41×C₃₁	—	2	8	no	—
78	#41×C₃₁	—	2	4	no	—
79	#41×C₃₁	—	2	4	no	—
80	#41×C₃₁	—	2	4	no	—
81	#41×C₃₁	—	2	4	no	—
82	#41×C₃₁	—	4	4	no	—
83	#41×C₃₁	—	4	4	no	—
84	#41×C₃₁	—	4	4	no	—
85	#41×C₃₁	—	4	4	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	4	no	—
88	C₂×Q₈×C₃₁	—	2	4	no	—
89	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	—	2	4	no	—
93	C₂×D₄×C₃₁	—	2	4	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	4	4	no	—
96	C₂×D₄×C₃₁	—	4	4	no	—
97	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	4	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	4	4	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	—	4	4	no	—
107	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	#41	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
115	Q₁₆×C₃₁	—	2	8	no	—
116	SD₁₆×C₃₁	—	2	16	no	—
117	D₈×C₃₁	—	2	8	no	—
118	M₁₆×C₃₁	—	2	8	no	—
119	M₁₆×C₃₁	—	2	4	no	—
120	M₁₆×C₃₁	—	2	4	no	—
121	M₁₆×C₃₁	—	4	4	no	—
122	M₁₆×C₃₁	C₂²⋊C₄	1	16	yes	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	4	no	—
126	C₂×C₈×C₃₁	—	4	4	no	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
128	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
129	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
130	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
131	C₄²×C₃₁	—	2	4	no	—
132	C₄²×C₃₁	—	2	1	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	C₂×D₄	1	2	yes	—
135	C₄²×C₃₁	C₂×D₄	1	2	yes	—
136	#21241	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	4	no	—
138	C₂³×C₃₁	—	4	2	no	—
139	C₂³×C₃₁	—	4	4	no	—
140	C₂³×C₃₁	—	8	4	no	—
141	C₂³×C₃₁	C₄×D₄	1	2	yes	—
142	C₂³×C₃₁	#121	1	2	yes	—
143	C₂³×C₃₁	#124	1	2	yes	—
144	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
145	Q₈×C₃₁	—	2	4	no	—
146	Q₈×C₃₁	—	2	4	no	—
147	Q₈×C₃₁	—	4	4	no	—
148	D₄×C₃₁	—	2	4	no	—
149	D₄×C₃₁	—	2	4	no	—
150	D₄×C₃₁	—	4	4	no	—
151	D₄×C₃₁	—	4	4	no	—
152	D₄×C₃₁	—	4	8	no	—
153	D₄×C₃₁	—	4	4	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	4	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	4	8	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	4	no	—
161	C₂×C₄×C₃₁	—	4	4	no	—
162	C₂×C₄×C₃₁	—	4	4	no	—
163	C₂×C₄×C₃₁	—	4	8	no	—
164	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
165	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
166	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₃₁	#121	1	4	yes	—
168	C₂×C₄×C₃₁	#121	1	2	yes	—
169	C₂×C₄×C₃₁	#124	1	2	yes	—
170	C₂×C₄×C₃₁	#127	1	4	yes	—
171	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
177	C₈×C₃₁	—	2	4	no	—
178	C₈×C₃₁	—	4	4	no	—
179	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
180	C₂×#575	C₂×C₃₁	1	1	yes	characteristic
181	C₂×#430	C₂×C₃₁	1	1	yes	characteristic, complemented
182	C₂²×#104	C₂×C₃₁	1	2	yes	complemented
183	#2879	C₂×C₃₁	1	2	yes	complemented
184	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1975	C₂×C₃₁	1	4	yes	complemented
186	#1974	C₂×C₃₁	1	4	yes	complemented
187	C₂²×C₃₁	—	2	2	no	—
188	C₂²×C₃₁	—	2	4	no	—
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	—	4	2	no	—
191	C₂²×C₃₁	—	4	8	no	—
192	C₂²×C₃₁	—	4	4	no	—
193	C₂²×C₃₁	—	8	8	no	—
194	C₂²×C₃₁	#390	1	4	yes	—
195	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
197	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
198	C₄×C₃₁	—	2	4	no	—
199	C₄×C₃₁	—	4	2	no	—
200	C₄×C₃₁	—	4	4	no	—
201	C₄×C₃₁	#390	1	4	yes	—
202	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
203	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
205	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
206	C₂²×#41	C₂²×C₃₁	1	2	yes	—
207	#575	—	2	4	no	—
208	#575	—	2	4	no	—
209	C₂×#137	C₄×C₃₁	1	4	yes	complemented
210	C₂×AGL(1,ℤ/8)	C₄×C₃₁	1	4	yes	complemented
211	C₂×#135	C₂²×C₃₁	1	4	yes	—
212	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
213	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
214	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
215	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
216	#430	—	2	4	no	—
217	C₂×#108	C₂²×C₃₁	1	4	yes	complemented
218	C₂×#104	—	2	8	no	—
219	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
220	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
221	#408	—	2	4	no	—
222	#408	C₂²×C₃₁	1	8	yes	complemented
223	#407	—	2	4	no	—
224	C₄×M₁₆	—	2	4	no	—
225	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
226	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
227	C₂×C₃₁	—	2	1	no	—
228	C₂×C₃₁	—	2	4	no	—
229	C₂×C₃₁	—	2	1	no	—
230	C₂×C₃₁	—	8	4	no	—
231	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
232	C₂×C₃₁	#2956	1	2	yes	—
233	C₂×#41	—	2	2	no	—
234	C₂×#41	—	2	2	no	—
235	C₂×#41	—	2	4	no	—
236	C₂×#41	—	2	4	no	—
237	C₂×#41	—	2	4	no	—
238	C₂×#41	—	2	4	no	—
239	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
240	C₂×#41	C₂³×C₃₁	1	4	yes	—
241	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
242	C₂²×D₄	—	2	2	no	—
243	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
244	C₂³×C₄	—	2	1	no	—
245	C₂³×C₄	—	2	2	no	—
246	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
247	#137	—	2	16	no	—
248	AGL(1,ℤ/8)	—	2	16	no	—
249	#135	—	2	8	no	—
250	#135	—	2	8	no	—
251	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
252	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
253	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	4	no	—
258	C₂×M₁₆	—	2	4	no	—
259	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
260	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
261	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
262	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
263	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
264	C₂²×C₈	—	2	2	no	—
265	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
266	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
267	C₂×C₄²	—	2	2	no	—
268	C₂×C₄²	—	2	1	no	—
269	C₂×C₄²	D₄×C₃₁	1	4	yes	—
270	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
272	#108	—	2	8	no	—
273	#104	—	2	8	no	—
274	#104	—	2	8	no	—
275	C₂²⋊C₈	—	2	16	no	—
276	#97	—	2	4	no	—
277	C₄×C₈	—	2	4	no	—
278	C₃₁	#21241	1	1	yes	characteristic, direct product factor, Sylow
279	C₂⁴	—	2	2	no	—
280	C₂⁴	#41×C₃₁	1	1	yes	characteristic
281	#41	—	2	8	no	—
282	#41	—	2	8	no	—
283	#41	—	2	4	no	—
284	#41	—	2	4	no	—
285	#41	—	2	4	no	—
286	#41	—	2	4	no	—
287	#41	—	4	4	no	—
288	#41	—	4	4	no	—
289	#41	—	4	4	no	—
290	#41	—	4	4	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	4	no	—
293	C₂×Q₈	—	2	4	no	—
294	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
295	C₂×D₄	—	2	4	no	—
296	C₂×D₄	—	2	4	no	—
297	C₂×D₄	—	2	4	no	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	4	4	no	—
301	C₂×D₄	—	4	4	no	—
302	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	4	4	no	—
310	C₂²×C₄	—	4	4	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
317	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	#41×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
320	Q₁₆	—	2	8	no	—
321	SD₁₆	—	2	16	no	—
322	D₈	—	2	8	no	—
323	M₁₆	—	2	8	no	—
324	M₁₆	—	2	4	no	—
325	M₁₆	—	2	4	no	—
326	M₁₆	—	4	4	no	—
327	M₁₆	C₂²⋊C₄×C₃₁	1	16	yes	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	4	no	—
331	C₂×C₈	—	4	4	no	—
332	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
333	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
334	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
335	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
336	C₄²	—	2	4	no	—
337	C₄²	—	2	1	no	—
338	C₄²	—	2	1	no	—
339	C₄²	C₂×D₄×C₃₁	1	2	yes	—
340	C₄²	C₂×D₄×C₃₁	1	2	yes	—
341	C₂³	—	2	4	no	—
342	C₂³	—	4	2	no	—
343	C₂³	—	4	4	no	—
344	C₂³	—	8	4	no	—
345	C₂³	C₄×D₄×C₃₁	1	2	yes	—
346	C₂³	C₃₁×#121	1	2	yes	—
347	C₂³	C₃₁×#124	1	2	yes	—
348	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
349	Q₈	—	2	4	no	—
350	Q₈	—	2	4	no	—
351	Q₈	—	4	4	no	—
352	D₄	—	2	4	no	—
353	D₄	—	2	4	no	—
354	D₄	—	4	4	no	—
355	D₄	—	4	4	no	—
356	D₄	—	4	8	no	—
357	D₄	—	4	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	8	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	8	no	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
369	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
370	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
371	C₂×C₄	C₃₁×#121	1	4	yes	—
372	C₂×C₄	C₃₁×#121	1	2	yes	—
373	C₂×C₄	C₃₁×#124	1	2	yes	—
374	C₂×C₄	C₃₁×#127	1	4	yes	—
375	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
381	C₈	—	2	4	no	—
382	C₈	—	4	4	no	—
383	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	4	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	8	no	—
389	C₂²	—	4	4	no	—
390	C₂²	—	8	8	no	—
391	C₂²	C₃₁×#390	1	4	yes	—
392	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
393	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
394	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
395	C₄	—	2	4	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	4	no	—
398	C₄	C₃₁×#390	1	4	yes	—
399	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
400	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
401	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
402	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
403	C₂	—	2	1	no	—
404	C₂	—	2	4	no	—
405	C₂	—	2	1	no	—
406	C₂	—	8	4	no	—
407	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
408	C₂	C₃₁×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×C₃₁×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#430	C₂	semidirect
3. C₂²×C₃₁×#104	C₂	semidirect
4. C₃₁×#2879	C₂	semidirect
5. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
6. C₃₁×#1975	C₂	semidirect
7. C₃₁×#1974	C₂	semidirect
11. C₂×C₃₁×#137	C₄	semidirect
12. C₂×C₃₁×AGL(1,ℤ/8)	C₄	semidirect
19. C₂×C₃₁×#108	C₂²	semidirect
21. C₂×C₃₁×#104	C₂²	semidirect
22. C₂×C₃₁×#104	C₂²	semidirect
24. C₃₁×#408	C₂²	semidirect
27. C₂×C₃₁×#97	C₂²	semidirect
28. C₂×C₄×C₈×C₃₁	C₂²	semidirect
47. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
48. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
49. C₂×D₈×C₃₁	C₂×C₄	semidirect
136. #21241	C₃₁	direct (N × H)
181. C₂×#430	C₂×C₃₁	semidirect
182. C₂²×#104	C₂×C₃₁	semidirect
183. #2879	C₂×C₃₁	semidirect
184. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
185. #1975	C₂×C₃₁	semidirect
186. #1974	C₂×C₃₁	semidirect
209. C₂×#137	C₄×C₃₁	semidirect
210. C₂×AGL(1,ℤ/8)	C₄×C₃₁	semidirect
217. C₂×#108	C₂²×C₃₁	semidirect
219. C₂×#104	C₂²×C₃₁	semidirect
220. C₂×#104	C₂²×C₃₁	semidirect
222. #408	C₂²×C₃₁	semidirect
225. C₂×#97	C₂²×C₃₁	semidirect
226. C₂×C₄×C₈	C₂²×C₃₁	semidirect
251. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
252. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
253. C₂×D₈	C₂×C₄×C₃₁	semidirect
278. C₃₁	#21241	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×#575	2
C₂×C₃₁×#430	2
C₂²×C₃₁×#104	2
C₃₁×#2879	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1975	2
C₃₁×#1974	2
#21241	31

Subgroups — C31×#21241 #21478000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21241 #21478000