Subgroups — C₃₁×#33380 #21490139

Order 7936.

1170 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2141	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2140	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2139	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2138	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₃₁	—	2	1	no	—
9	C₂×C₃₁×#135	—	2	1	no	—
10	C₂×C₃₁×#135	—	2	1	no	—
11	C₂×C₃₁×#135	—	2	4	no	—
12	C₂×C₃₁×#135	C₄	1	8	yes	complemented
13	C₂×C₃₁×#135	C₂²	1	2	yes	—
14	C₂×C₃₁×#135	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×SD₁₆×C₃₁	—	2	1	no	—
17	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#117	—	2	1	no	—
20	C₂×C₃₁×M₃₂	—	2	4	no	—
21	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
22	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
23	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
24	C₃₁×#427	—	2	1	no	—
25	C₃₁×#427	—	2	1	no	—
26	C₃₁×#427	C₂²	1	2	yes	—
27	C₃₁×#427	C₂²	1	2	yes	complemented
28	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
29	C₂×C₃₁×#106	—	2	1	no	—
30	C₃₁×#358	C₂²	1	8	yes	complemented
31	C₃₁×#357	C₂²	1	8	yes	complemented
32	C₂×#41×C₃₁	—	2	8	no	—
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₂×#41×C₃₁	—	2	2	no	—
35	C₂×#41×C₃₁	—	4	1	no	—
36	C₂×#41×C₃₁	—	4	1	no	—
37	C₂×#41×C₃₁	—	4	4	no	—
38	C₂²×Q₈×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	4	1	no	—
41	C₂³×C₄×C₃₁	—	4	1	no	—
42	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
43	C₃₁×#135	—	2	8	no	—
44	C₃₁×#135	—	2	8	no	—
45	C₃₁×#135	—	2	4	no	—
46	C₃₁×#135	—	2	4	no	—
47	C₃₁×#135	C₂×C₄	1	8	yes	—
48	C₃₁×#135	C₂×C₄	1	8	yes	complemented
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
52	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
53	C₂×SD₁₆×C₃₁	—	2	8	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₂×D₈×C₃₁	—	2	2	no	—
58	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
59	C₂×D₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
61	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	D₄	1	4	yes	—
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#117	—	2	2	no	—
69	C₃₁×#117	—	2	2	no	—
70	C₃₁×#117	—	4	1	no	—
71	C₃₁×#117	—	4	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
74	C₂×C₄²×C₃₁	—	4	1	no	—
75	C₃₁×M₃₂	—	2	4	no	—
76	C₃₁×M₃₂	—	2	4	no	—
77	C₂×C₁₆×C₃₁	—	2	4	no	—
78	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
79	C₃₁×#107	—	2	2	no	—
80	C₃₁×#107	C₂³	1	4	yes	—
81	C₃₁×#106	—	2	2	no	—
82	C₃₁×#106	—	2	2	no	—
83	C₂⁴×C₃₁	—	4	1	no	—
84	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
85	#41×C₃₁	—	2	8	no	—
86	#41×C₃₁	—	2	8	no	—
87	#41×C₃₁	—	4	8	no	—
88	#41×C₃₁	—	4	8	no	—
89	#41×C₃₁	—	4	4	no	—
90	#41×C₃₁	—	4	4	no	—
91	C₂×Q₈×C₃₁	—	2	4	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	4	1	no	—
94	C₂×Q₈×C₃₁	—	4	2	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	4	8	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	2	no	—
100	C₂×D₄×C₃₁	—	4	1	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂²×C₄×C₃₁	—	4	1	no	—
103	C₂²×C₄×C₃₁	—	4	8	no	—
104	C₂²×C₄×C₃₁	—	4	2	no	—
105	C₂²×C₄×C₃₁	—	4	2	no	—
106	C₂²×C₄×C₃₁	—	4	2	no	—
107	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
108	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	Q₁₆×C₃₁	—	2	4	no	—
116	Q₁₆×C₃₁	—	2	2	no	—
117	Q₁₆×C₃₁	C₂²×C₄	1	4	yes	—
118	SD₁₆×C₃₁	—	2	8	no	—
119	SD₁₆×C₃₁	—	2	8	no	—
120	D₈×C₃₁	—	2	4	no	—
121	D₈×C₃₁	—	2	2	no	—
122	D₈×C₃₁	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
124	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
125	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
126	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
136	C₄⋊C₄×C₃₁	—	2	4	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	4	2	no	—
141	C₄²×C₃₁	—	4	2	no	—
142	C₁₆×C₃₁	—	2	4	no	—
143	#33380	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	—	4	4	no	—
147	C₂³×C₃₁	—	8	4	no	—
148	C₂³×C₃₁	#102	1	4	yes	—
149	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
150	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
151	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
152	Q₈×C₃₁	—	2	4	no	—
153	Q₈×C₃₁	—	4	2	no	—
154	Q₈×C₃₁	—	4	4	no	—
155	D₄×C₃₁	—	2	4	no	—
156	D₄×C₃₁	—	4	8	no	—
157	D₄×C₃₁	—	4	8	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	4	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	4	no	—
162	C₂×C₄×C₃₁	—	4	8	no	—
163	C₂×C₄×C₃₁	—	4	8	no	—
164	C₂×C₄×C₃₁	—	4	4	no	—
165	C₂×C₄×C₃₁	—	8	4	no	—
166	C₂×C₄×C₃₁	#102	1	4	yes	—
167	C₂×C₄×C₃₁	#102	1	4	yes	—
168	C₂×C₄×C₃₁	#102	1	4	yes	—
169	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
171	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
176	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
177	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
180	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
181	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
182	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
183	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic
185	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
186	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#2141	C₂×C₃₁	1	2	yes	complemented
188	#2140	C₂×C₃₁	1	2	yes	complemented
189	#2139	C₂×C₃₁	1	4	yes	complemented
190	#2138	C₂×C₃₁	1	4	yes	complemented
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	4	4	no	—
195	C₂²×C₃₁	—	4	4	no	—
196	C₂²×C₃₁	—	8	8	no	—
197	C₂²×C₃₁	—	8	4	no	—
198	C₂²×C₃₁	C₂×#102	1	2	yes	—
199	C₂²×C₃₁	C₂×#102	1	2	yes	—
200	C₂²×C₃₁	C₂×#102	1	2	yes	—
201	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
202	C₄×C₃₁	—	4	4	no	—
203	C₄×C₃₁	—	8	4	no	—
204	C₄×C₃₁	C₂×#102	1	2	yes	—
205	C₄×C₃₁	C₂×#102	1	2	yes	—
206	C₄×C₃₁	C₂×#102	1	2	yes	—
207	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
208	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
209	C₂²×#41	—	2	1	no	—
210	C₂×#135	—	2	1	no	—
211	C₂×#135	—	2	1	no	—
212	C₂×#135	—	2	4	no	—
213	C₂×#135	C₄×C₃₁	1	8	yes	complemented
214	C₂×#135	C₂²×C₃₁	1	2	yes	—
215	C₂×#135	C₂²×C₃₁	1	2	yes	—
216	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
217	C₂²×SD₁₆	—	2	1	no	—
218	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
219	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×#117	—	2	1	no	—
221	C₂×M₃₂	—	2	4	no	—
222	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
223	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
224	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
225	#427	—	2	1	no	—
226	#427	—	2	1	no	—
227	#427	C₂²×C₃₁	1	2	yes	—
228	#427	C₂²×C₃₁	1	2	yes	complemented
229	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
230	C₂×#106	—	2	1	no	—
231	#358	C₂²×C₃₁	1	8	yes	complemented
232	#357	C₂²×C₃₁	1	8	yes	complemented
233	C₂×C₃₁	—	2	2	no	—
234	C₂×C₃₁	—	2	2	no	—
235	C₂×C₃₁	—	2	2	no	—
236	C₂×C₃₁	—	8	4	no	—
237	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
238	C₂×C₃₁	#3414	1	2	yes	—
239	C₂×#41	—	2	8	no	—
240	C₂×#41	—	2	2	no	—
241	C₂×#41	—	2	2	no	—
242	C₂×#41	—	4	1	no	—
243	C₂×#41	—	4	1	no	—
244	C₂×#41	—	4	4	no	—
245	C₂²×Q₈	—	2	1	no	—
246	C₂²×D₄	—	2	1	no	—
247	C₂²×D₄	—	4	1	no	—
248	C₂³×C₄	—	4	1	no	—
249	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
250	#135	—	2	8	no	—
251	#135	—	2	8	no	—
252	#135	—	2	4	no	—
253	#135	—	2	4	no	—
254	#135	C₂×C₄×C₃₁	1	8	yes	—
255	#135	C₂×C₄×C₃₁	1	8	yes	complemented
256	C₂×Q₁₆	—	2	1	no	—
257	C₂×Q₁₆	—	2	2	no	—
258	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
259	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
260	C₂×SD₁₆	—	2	8	no	—
261	C₂×SD₁₆	—	2	2	no	—
262	C₂×SD₁₆	—	2	2	no	—
263	C₂×D₈	—	2	1	no	—
264	C₂×D₈	—	2	2	no	—
265	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
266	C₂×D₈	C₂³×C₃₁	1	2	yes	—
267	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
268	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₈	D₄×C₃₁	1	4	yes	—
272	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
275	#117	—	2	2	no	—
276	#117	—	2	2	no	—
277	#117	—	4	1	no	—
278	#117	—	4	1	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	4	1	no	—
281	C₂×C₄²	—	4	1	no	—
282	M₃₂	—	2	4	no	—
283	M₃₂	—	2	4	no	—
284	C₂×C₁₆	—	2	4	no	—
285	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
286	#107	—	2	2	no	—
287	#107	C₂³×C₃₁	1	4	yes	—
288	#106	—	2	2	no	—
289	#106	—	2	2	no	—
290	C₃₁	#33380	1	1	yes	characteristic, direct product factor, Sylow
291	C₂⁴	—	4	1	no	—
292	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
293	#41	—	2	8	no	—
294	#41	—	2	8	no	—
295	#41	—	4	8	no	—
296	#41	—	4	8	no	—
297	#41	—	4	4	no	—
298	#41	—	4	4	no	—
299	C₂×Q₈	—	2	4	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	1	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	4	8	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	—	4	1	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	4	8	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	—	4	2	no	—
314	C₂²×C₄	—	4	2	no	—
315	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
316	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	Q₁₆	—	2	4	no	—
324	Q₁₆	—	2	2	no	—
325	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	—
326	SD₁₆	—	2	8	no	—
327	SD₁₆	—	2	8	no	—
328	D₈	—	2	4	no	—
329	D₈	—	2	2	no	—
330	D₈	C₂²×C₄×C₃₁	1	4	yes	—
331	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
332	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
333	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
334	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
336	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
337	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
342	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
343	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₄²	—	4	2	no	—
349	C₄²	—	4	2	no	—
350	C₁₆	—	2	4	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	4	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	8	4	no	—
355	C₂³	C₃₁×#102	1	4	yes	—
356	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
357	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
358	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
359	Q₈	—	2	4	no	—
360	Q₈	—	4	4	no	—
361	Q₈	—	4	2	no	—
362	D₄	—	2	4	no	—
363	D₄	—	4	8	no	—
364	D₄	—	4	8	no	—
365	D₄	—	4	4	no	—
366	D₄	—	4	2	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	8	no	—
370	C₂×C₄	—	4	8	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	8	4	no	—
373	C₂×C₄	C₃₁×#102	1	4	yes	—
374	C₂×C₄	C₃₁×#102	1	4	yes	—
375	C₂×C₄	C₃₁×#102	1	4	yes	—
376	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
377	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
378	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
379	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
383	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
384	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
385	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
386	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
387	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
388	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
389	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
390	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	2	2	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	8	8	no	—
397	C₂²	—	8	4	no	—
398	C₂²	C₂×C₃₁×#102	1	2	yes	—
399	C₂²	C₂×C₃₁×#102	1	2	yes	—
400	C₂²	C₂×C₃₁×#102	1	2	yes	—
401	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
402	C₄	—	4	4	no	—
403	C₄	—	8	4	no	—
404	C₄	C₂×C₃₁×#102	1	2	yes	—
405	C₄	C₂×C₃₁×#102	1	2	yes	—
406	C₄	C₂×C₃₁×#102	1	2	yes	—
407	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
408	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
409	C₂	—	2	2	no	—
410	C₂	—	2	2	no	—
411	C₂	—	2	2	no	—
412	C₂	—	8	4	no	—
413	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic
414	C₂	C₃₁×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₈
prime radical	C₂²×C₃₁×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×M₃₂	C₂	semidirect
3. C₂×C₃₁×#427	C₂	semidirect
4. C₃₁×#2141	C₂	semidirect
5. C₃₁×#2140	C₂	semidirect
6. C₃₁×#2139	C₂	semidirect
7. C₃₁×#2138	C₂	semidirect
12. C₂×C₃₁×#135	C₄	semidirect
22. C₂×C₃₁×M₃₂	C₂²	semidirect
23. C₂²×C₁₆×C₃₁	C₂²	semidirect
27. C₃₁×#427	C₂²	semidirect
28. C₂×C₃₁×#107	C₂²	semidirect
30. C₃₁×#358	C₂²	semidirect
31. C₃₁×#357	C₂²	semidirect
48. C₃₁×#135	C₂×C₄	semidirect
51. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
58. C₂×D₈×C₃₁	C₂×C₄	semidirect
143. #33380	C₃₁	direct (N × H)
185. C₂²×M₃₂	C₂×C₃₁	semidirect
186. C₂×#427	C₂×C₃₁	semidirect
187. #2141	C₂×C₃₁	semidirect
188. #2140	C₂×C₃₁	semidirect
189. #2139	C₂×C₃₁	semidirect
190. #2138	C₂×C₃₁	semidirect
213. C₂×#135	C₄×C₃₁	semidirect
223. C₂×M₃₂	C₂²×C₃₁	semidirect
224. C₂²×C₁₆	C₂²×C₃₁	semidirect
228. #427	C₂²×C₃₁	semidirect
229. C₂×#107	C₂²×C₃₁	semidirect
231. #358	C₂²×C₃₁	semidirect
232. #357	C₂²×C₃₁	semidirect
255. #135	C₂×C₄×C₃₁	semidirect
258. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
265. C₂×D₈	C₂×C₄×C₃₁	semidirect
290. C₃₁	#33380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#135	2
C₂²×C₃₁×M₃₂	2
C₂×C₃₁×#427	2
C₃₁×#2141	2
C₃₁×#2140	2
C₃₁×#2139	2
C₃₁×#2138	2
#33380	31

Subgroups — C31×#33380 #21490139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33380 #21490139