Subgroups — C₃₁×#19274 #21476033

Order 7936.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic
4	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1678	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1677	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#486	—	2	1	no	—
15	C₃₁×#486	—	2	1	no	—
16	C₃₁×#486	C₂²	1	4	yes	—
17	C₃₁×#477	C₂²	1	2	yes	complemented
18	C₃₁×#476	C₂²	1	2	yes	complemented
19	C₃₁×#475	—	2	1	no	—
20	C₃₁×#474	—	2	1	no	—
21	C₃₁×#460	—	2	1	no	—
22	C₃₁×#460	C₂²	1	2	yes	—
23	C₃₁×#459	—	2	1	no	—
24	C₃₁×#459	C₂²	1	2	yes	—
25	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
26	C₈×D₄×C₃₁	—	2	2	no	—
27	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#431	C₂²	1	1	yes	characteristic
30	C₃₁×#428	—	2	1	no	—
31	C₃₁×#423	C₂²	1	1	yes	characteristic
32	C₃₁×#423	C₂²	1	1	yes	characteristic
33	C₃₁×#418	—	2	1	no	—
34	C₃₁×#417	—	2	1	no	—
35	C₂×C₃₁×#102	—	2	2	no	—
36	C₂×C₄×C₈×C₃₁	—	2	1	no	—
37	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#334	C₂²	1	4	yes	complemented
39	C₃₁×#333	C₂²	1	4	yes	complemented
40	C₃₁×#331	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	1	no	—
50	C₂×M₁₆×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₂²×C₈×C₃₁	D₄	1	2	yes	—
53	C₃₁×C₄⋊Q₈	—	2	1	no	—
54	C₃₁×C₄⋊Q₈	—	2	1	no	—
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₃₁×#127	—	2	1	no	—
58	C₃₁×#127	—	2	1	no	—
59	C₃₁×#127	—	2	1	no	—
60	C₃₁×#127	C₂³	1	2	yes	—
61	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₃₁	—	2	1	no	—
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
66	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#117	—	2	1	no	—
68	C₃₁×#117	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#107	—	2	2	no	—
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×C₄⋊C₈	—	2	2	no	—
76	C₃₁×C₄⋊C₈	—	2	1	no	—
77	C₃₁×C₄⋊C₈	—	2	1	no	—
78	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	4	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×C₂²⋊C₈	—	2	1	no	—
88	C₃₁×C₂²⋊C₈	—	2	2	no	—
89	C₄×C₈×C₃₁	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	1	no	—
91	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
92	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	2	yes	—
94	C₂⁴×C₃₁	—	4	1	no	—
95	#41×C₃₁	—	2	2	no	—
96	#41×C₃₁	SD₁₆	1	4	yes	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	4	1	no	—
99	C₂×Q₈×C₃₁	—	8	2	no	—
100	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	4	4	no	—
106	C₂×D₄×C₃₁	—	4	1	no	—
107	C₂×D₄×C₃₁	—	4	1	no	—
108	C₂×D₄×C₃₁	—	4	1	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	—	8	2	no	—
112	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	4	1	no	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
120	SD₁₆×C₃₁	—	4	4	no	—
121	D₈×C₃₁	—	4	4	no	—
122	M₁₆×C₃₁	—	4	2	no	—
123	M₁₆×C₃₁	—	4	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	4	2	no	—
128	C₂×C₈×C₃₁	—	4	4	no	—
129	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	4	1	no	—
136	C₄⋊C₄×C₃₁	—	4	1	no	—
137	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₃₁	—	2	1	no	—
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₃₁	#41	1	1	yes	characteristic
148	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
149	#19274	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	—	8	2	no	—
153	C₂³×C₃₁	—	8	2	no	—
154	C₂³×C₃₁	—	8	2	no	—
155	C₂³×C₃₁	#121	1	1	yes	characteristic
156	Q₈×C₃₁	—	2	1	no	—
157	Q₈×C₃₁	—	8	2	no	—
158	Q₈×C₃₁	—	8	2	no	—
159	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
160	D₄×C₃₁	—	2	4	no	—
161	D₄×C₃₁	—	2	1	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	4	no	—
165	D₄×C₃₁	—	8	2	no	—
166	D₄×C₃₁	—	8	2	no	—
167	D₄×C₃₁	—	8	2	no	—
168	D₄×C₃₁	—	8	2	no	—
169	D₄×C₃₁	C₂×SD₁₆	1	2	yes	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	8	2	no	—
180	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
184	C₂×C₄×C₃₁	#135	1	2	yes	—
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	4	2	no	—
191	C₈×C₃₁	—	4	2	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#3079	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#3038	C₂×C₃₁	1	1	yes	characteristic
196	C₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1678	C₂×C₃₁	1	4	yes	complemented
198	#1677	C₂×C₃₁	1	4	yes	complemented
199	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	—	4	1	no	—
207	C₂²×C₃₁	—	8	2	no	—
208	C₂²×C₃₁	—	8	2	no	—
209	C₂²×C₃₁	—	8	2	no	—
210	C₂²×C₃₁	—	16	4	no	—
211	C₂²×C₃₁	#465	1	2	yes	—
212	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
213	C₄×C₃₁	—	2	1	no	—
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	2	1	no	—
216	C₄×C₃₁	—	4	1	no	—
217	C₄×C₃₁	—	4	1	no	—
218	C₄×C₃₁	—	8	2	no	—
219	C₄×C₃₁	#465	1	2	yes	—
220	C₄×C₃₁	#472	1	2	yes	—
221	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
222	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
223	C₄×C₃₁	#575	1	2	yes	—
224	C₂×AGL(1,ℤ/8)	—	2	1	no	—
225	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
227	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
228	#486	—	2	1	no	—
229	#486	—	2	1	no	—
230	#486	C₂²×C₃₁	1	4	yes	—
231	#477	C₂²×C₃₁	1	2	yes	complemented
232	#476	C₂²×C₃₁	1	2	yes	complemented
233	#475	—	2	1	no	—
234	#474	—	2	1	no	—
235	#460	—	2	1	no	—
236	#460	C₂²×C₃₁	1	2	yes	—
237	#459	—	2	1	no	—
238	#459	C₂²×C₃₁	1	2	yes	—
239	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
240	C₈×D₄	—	2	2	no	—
241	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
242	#431	—	2	1	no	—
243	#431	C₂²×C₃₁	1	1	yes	characteristic
244	#428	—	2	1	no	—
245	#423	C₂²×C₃₁	1	1	yes	characteristic
246	#423	C₂²×C₃₁	1	1	yes	characteristic
247	#418	—	2	1	no	—
248	#417	—	2	1	no	—
249	C₂×#102	—	2	2	no	—
250	C₂×C₄×C₈	—	2	1	no	—
251	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
252	#334	C₂²×C₃₁	1	4	yes	complemented
253	#333	C₂²×C₃₁	1	4	yes	complemented
254	#331	C₂²×C₃₁	1	4	yes	—
255	C₂×C₃₁	—	2	2	no	—
256	C₂×C₃₁	—	4	2	no	—
257	C₂×C₃₁	—	16	2	no	—
258	C₂×C₃₁	#3040	1	1	yes	characteristic
259	C₂×C₃₁	C₂×#465	1	1	yes	characteristic
260	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
261	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	—	2	1	no	—
266	AGL(1,ℤ/8)	—	4	4	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	C₂×D₈	—	2	2	no	—
269	C₂×M₁₆	—	2	1	no	—
270	C₂×M₁₆	—	2	1	no	—
271	C₂²×C₈	—	2	2	no	—
272	C₂²×C₈	D₄×C₃₁	1	2	yes	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	—	2	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
277	#127	—	2	1	no	—
278	#127	—	2	1	no	—
279	#127	—	2	1	no	—
280	#127	C₂³×C₃₁	1	2	yes	—
281	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
282	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
283	C₄×D₄	—	2	1	no	—
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
286	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
287	#117	—	2	1	no	—
288	#117	D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
293	#107	—	2	2	no	—
294	#106	—	2	2	no	—
295	C₄⋊C₈	—	2	2	no	—
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	—	2	1	no	—
298	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
299	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
300	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
301	#103	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	—	2	4	no	—
304	#102	—	2	4	no	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	C₂²⋊C₈	—	2	1	no	—
308	C₂²⋊C₈	—	2	2	no	—
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
312	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
313	C₄×C₈	C₂³×C₃₁	1	2	yes	—
314	C₃₁	#19274	1	1	yes	characteristic, direct product factor, Sylow
315	C₂⁴	—	4	1	no	—
316	#41	—	2	2	no	—
317	#41	SD₁₆×C₃₁	1	4	yes	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	4	1	no	—
320	C₂×Q₈	—	8	2	no	—
321	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	4	4	no	—
327	C₂×D₄	—	4	1	no	—
328	C₂×D₄	—	4	1	no	—
329	C₂×D₄	—	4	1	no	—
330	C₂×D₄	—	4	2	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	8	2	no	—
333	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	4	1	no	—
338	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
341	SD₁₆	—	4	4	no	—
342	D₈	—	4	4	no	—
343	M₁₆	—	4	2	no	—
344	M₁₆	—	4	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	4	2	no	—
349	C₂×C₈	—	4	4	no	—
350	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
351	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂²⋊C₄	—	2	1	no	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	2	1	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	1	no	—
366	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₄²	#41×C₃₁	1	1	yes	characteristic
369	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
370	C₂³	—	2	1	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	8	2	no	—
373	C₂³	—	8	2	no	—
374	C₂³	—	8	2	no	—
375	C₂³	C₃₁×#121	1	1	yes	characteristic
376	Q₈	—	2	1	no	—
377	Q₈	—	8	2	no	—
378	Q₈	—	8	2	no	—
379	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
380	D₄	—	2	4	no	—
381	D₄	—	2	1	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	2	no	—
384	D₄	—	4	4	no	—
385	D₄	—	8	2	no	—
386	D₄	—	8	2	no	—
387	D₄	—	8	2	no	—
388	D₄	—	8	2	no	—
389	D₄	C₂×SD₁₆×C₃₁	1	2	yes	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	8	2	no	—
400	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
401	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
402	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
403	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
404	C₂×C₄	C₃₁×#135	1	2	yes	—
405	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
408	C₈	—	2	2	no	—
409	C₈	—	2	2	no	—
410	C₈	—	4	2	no	—
411	C₈	—	4	2	no	—
412	C₈	—	4	2	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	4	1	no	—
417	C₂²	—	4	1	no	—
418	C₂²	—	8	2	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	16	4	no	—
422	C₂²	C₃₁×#465	1	2	yes	—
423	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	2	no	—
426	C₄	—	2	1	no	—
427	C₄	—	4	1	no	—
428	C₄	—	4	1	no	—
429	C₄	—	8	2	no	—
430	C₄	C₃₁×#465	1	2	yes	—
431	C₄	C₃₁×#472	1	2	yes	—
432	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
433	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
434	C₄	C₃₁×#575	1	2	yes	—
435	C₂	—	2	2	no	—
436	C₂	—	4	2	no	—
437	C₂	—	16	2	no	—
438	C₂	C₃₁×#3040	1	1	yes	characteristic
439	C₂	C₂×C₃₁×#465	1	1	yes	characteristic
440	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3038	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#486	C₂	semidirect
2. C₃₁×#3079	C₂	semidirect
4. C₈×#41×C₃₁	C₂	semidirect
5. C₃₁×#1678	C₂	semidirect
6. C₃₁×#1677	C₂	semidirect
7. C₃₁×#1566	C₂	semidirect
8. C₃₁×#1548	C₂	semidirect
9. C₃₁×#1545	C₂	semidirect
17. C₃₁×#477	C₂²	semidirect
18. C₃₁×#476	C₂²	semidirect
25. C₈×Q₈×C₃₁	C₂²	semidirect
27. C₈×D₄×C₃₁	C₂²	semidirect
38. C₃₁×#334	C₂²	semidirect
39. C₃₁×#333	C₂²	semidirect
149. #19274	C₃₁	direct (N × H)
193. C₂×#486	C₂×C₃₁	semidirect
194. #3079	C₂×C₃₁	semidirect
196. C₈×#41	C₂×C₃₁	semidirect
197. #1678	C₂×C₃₁	semidirect
198. #1677	C₂×C₃₁	semidirect
199. #1566	C₂×C₃₁	semidirect
200. #1548	C₂×C₃₁	semidirect
201. #1545	C₂×C₃₁	semidirect
231. #477	C₂²×C₃₁	semidirect
232. #476	C₂²×C₃₁	semidirect
239. C₈×Q₈	C₂²×C₃₁	semidirect
241. C₈×D₄	C₂²×C₃₁	semidirect
252. #334	C₂²×C₃₁	semidirect
253. #333	C₂²×C₃₁	semidirect
314. C₃₁	#19274	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×#486	2
C₃₁×#3079	2
C₃₁×#3038	2
C₈×#41×C₃₁	2
C₃₁×#1678	2
C₃₁×#1677	2
C₃₁×#1566	2
C₃₁×#1548	2
C₃₁×#1545	2
#19274	31

Subgroups — C31×#19274 #21476033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19274 #21476033