Subgroups — C₃₁×#24988 #21481747

Order 7936.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2081	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2048	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1904	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1833	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1829	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1814	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
15	C₃₁×#518	C₂²	1	2	yes	—
16	C₃₁×#472	—	2	2	no	—
17	C₃₁×#472	—	2	2	no	—
18	C₃₁×#442	—	2	2	no	—
19	C₄×D₈×C₃₁	—	2	2	no	—
20	C₃₁×#430	C₂²	1	2	yes	—
21	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
23	C₃₁×#427	C₂²	1	1	yes	characteristic
24	C₃₁×#421	C₂²	1	2	yes	—
25	C₃₁×#418	C₂²	1	2	yes	—
26	C₃₁×#418	C₂²	1	2	yes	—
27	C₃₁×#413	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#100	C₂²	1	2	yes	—
29	C₂×C₃₁×#100	C₂²	1	2	yes	—
30	C₃₁×#410	C₂²	1	1	yes	characteristic
31	C₃₁×#405	C₂²	1	1	yes	characteristic
32	C₃₁×#405	C₂²	1	1	yes	characteristic
33	C₃₁×#344	C₂²	1	2	yes	—
34	C₃₁×#344	C₂²	1	1	yes	characteristic
35	C₃₁×#344	C₂²	1	1	yes	characteristic
36	C₃₁×#344	C₂²	1	2	yes	—
37	C₃₁×#344	C₂²	1	2	yes	—
38	C₃₁×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
41	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
42	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
43	C₂×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×D₈×C₃₁	—	2	2	no	—
45	C₂×D₈×C₃₁	—	2	1	no	—
46	C₂×D₈×C₃₁	—	2	1	no	—
47	C₂×D₈×C₃₁	—	2	1	no	—
48	C₂×D₈×C₃₁	—	2	1	no	—
49	C₂×D₈×C₃₁	D₄	1	2	yes	—
50	C₂×D₈×C₃₁	D₄	1	2	yes	—
51	C₃₁×#131	—	2	1	no	—
52	C₃₁×#131	—	2	1	no	—
53	C₃₁×#131	—	2	1	no	—
54	C₃₁×#131	—	2	1	no	—
55	C₂×M₁₆×C₃₁	—	2	1	no	—
56	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	—	2	1	no	—
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₃₁	—	2	4	no	—
64	C₄×D₄×C₃₁	—	4	2	no	—
65	C₃₁×#117	C₂³	1	1	yes	characteristic
66	C₃₁×#117	C₂³	1	2	yes	—
67	C₃₁×#117	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
69	C₃₁×#108	—	2	1	no	—
70	C₃₁×#108	—	2	1	no	—
71	C₃₁×#108	—	2	1	no	—
72	C₃₁×#108	—	2	1	no	—
73	C₃₁×#108	—	4	2	no	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#106	—	2	1	no	—
76	C₃₁×#104	—	4	2	no	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#102	—	4	2	no	—
79	C₃₁×#101	—	2	1	no	—
80	C₃₁×#100	—	2	4	no	—
81	C₃₁×#100	—	2	1	no	—
82	C₃₁×#100	—	4	2	no	—
83	C₃₁×#99	—	2	2	no	—
84	C₃₁×#97	—	2	1	no	—
85	C₃₁×#97	—	2	1	no	—
86	C₄×C₈×C₃₁	—	2	1	no	—
87	C₄×C₈×C₃₁	—	2	1	no	—
88	C₂⁴×C₃₁	—	2	2	no	—
89	#41×C₃₁	—	2	1	no	—
90	#41×C₃₁	—	2	1	no	—
91	#41×C₃₁	—	2	1	no	—
92	#41×C₃₁	—	2	1	no	—
93	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
101	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	4	4	no	—
104	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
105	SD₁₆×C₃₁	—	4	4	no	—
106	D₈×C₃₁	—	2	2	no	—
107	D₈×C₃₁	—	2	2	no	—
108	D₈×C₃₁	—	2	2	no	—
109	D₈×C₃₁	—	2	2	no	—
110	D₈×C₃₁	—	4	4	no	—
111	M₁₆×C₃₁	—	2	1	no	—
112	M₁₆×C₃₁	—	2	2	no	—
113	M₁₆×C₃₁	—	2	1	no	—
114	M₁₆×C₃₁	—	2	1	no	—
115	M₁₆×C₃₁	—	2	2	no	—
116	M₁₆×C₃₁	—	2	1	no	—
117	M₁₆×C₃₁	—	2	1	no	—
118	M₁₆×C₃₁	—	2	2	no	—
119	M₁₆×C₃₁	—	2	1	no	—
120	M₁₆×C₃₁	—	2	2	no	—
121	M₁₆×C₃₁	—	4	1	no	—
122	M₁₆×C₃₁	—	4	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	4	1	no	—
129	C₂×C₈×C₃₁	—	4	1	no	—
130	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
133	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
134	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
135	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₂²⋊C₄×C₃₁	—	2	1	no	—
140	C₂²⋊C₄×C₃₁	—	2	2	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₂²⋊C₄×C₃₁	#41	1	2	yes	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	#24988	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂³×C₃₁	—	2	1	no	—
148	C₂³×C₃₁	—	2	4	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	16	2	no	—
152	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
153	Q₈×C₃₁	—	2	1	no	—
154	Q₈×C₃₁	—	2	1	no	—
155	D₄×C₃₁	—	2	4	no	—
156	D₄×C₃₁	—	2	1	no	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	2	1	no	—
159	D₄×C₃₁	—	4	1	no	—
160	D₄×C₃₁	—	4	1	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	16	4	no	—
170	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
176	C₈×C₃₁	—	4	1	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	—	4	2	no	—
180	C₈×C₃₁	—	4	1	no	—
181	C₈×C₃₁	#123	1	1	yes	characteristic
182	C₈×C₃₁	#123	1	1	yes	characteristic
183	C₈×C₃₁	#123	1	1	yes	characteristic
184	C₈×C₃₁	#123	1	1	yes	characteristic
185	C₂×#472	C₂×C₃₁	1	1	yes	characteristic
186	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#2081	C₂×C₃₁	1	2	yes	complemented
188	#2048	C₂×C₃₁	1	2	yes	complemented
189	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1904	C₂×C₃₁	1	2	yes	complemented
192	#1833	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1829	C₂×C₃₁	1	2	yes	complemented
194	#1814	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1757	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₂²×C₃₁	—	4	1	no	—
197	C₂²×C₃₁	—	4	1	no	—
198	C₂²×C₃₁	—	4	4	no	—
199	C₂²×C₃₁	—	8	4	no	—
200	C₂²×C₃₁	—	8	1	no	—
201	C₂²×C₃₁	—	16	2	no	—
202	C₂²×C₃₁	—	16	2	no	—
203	C₂²×C₃₁	#529	1	1	yes	characteristic
204	C₂²×C₃₁	#546	1	1	yes	characteristic
205	C₂²×C₃₁	#555	1	1	yes	characteristic
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	—	4	1	no	—
208	C₄×C₃₁	—	8	1	no	—
209	C₄×C₃₁	—	8	2	no	—
210	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
211	C₄×C₃₁	#540	1	1	yes	characteristic
212	C₄×C₃₁	#546	1	1	yes	characteristic
213	C₄×C₃₁	#553	1	1	yes	characteristic
214	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
215	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
217	#518	C₂²×C₃₁	1	2	yes	—
218	#472	—	2	2	no	—
219	#472	—	2	2	no	—
220	#442	—	2	2	no	—
221	C₄×D₈	—	2	2	no	—
222	#430	C₂²×C₃₁	1	2	yes	—
223	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
225	#427	C₂²×C₃₁	1	1	yes	characteristic
226	#421	C₂²×C₃₁	1	2	yes	—
227	#418	C₂²×C₃₁	1	2	yes	—
228	#418	C₂²×C₃₁	1	2	yes	—
229	#413	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#100	C₂²×C₃₁	1	2	yes	—
231	C₂×#100	C₂²×C₃₁	1	2	yes	—
232	#410	C₂²×C₃₁	1	1	yes	characteristic
233	#405	C₂²×C₃₁	1	1	yes	characteristic
234	#405	C₂²×C₃₁	1	1	yes	characteristic
235	#344	C₂²×C₃₁	1	2	yes	—
236	#344	C₂²×C₃₁	1	2	yes	—
237	#344	C₂²×C₃₁	1	1	yes	characteristic
238	#344	C₂²×C₃₁	1	1	yes	characteristic
239	#344	C₂²×C₃₁	1	2	yes	—
240	H(3,ℤ/4)	C₂²×C₃₁	1	2	yes	—
241	C₂×C₃₁	—	2	1	no	—
242	C₂×C₃₁	—	2	1	no	—
243	C₂×C₃₁	—	2	1	no	—
244	C₂×C₃₁	—	8	1	no	—
245	C₂×C₃₁	—	8	4	no	—
246	C₂×C₃₁	#2420	1	1	yes	characteristic
247	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
248	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
249	AGL(1,ℤ/8)	—	2	4	no	—
250	AGL(1,ℤ/8)	—	4	2	no	—
251	C₂×SD₁₆	—	2	2	no	—
252	C₂×D₈	—	2	2	no	—
253	C₂×D₈	—	2	1	no	—
254	C₂×D₈	—	2	1	no	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	D₄×C₃₁	1	2	yes	—
258	C₂×D₈	D₄×C₃₁	1	2	yes	—
259	#131	—	2	1	no	—
260	#131	—	2	1	no	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	C₂×M₁₆	—	2	1	no	—
264	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
265	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
266	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
268	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	—	2	1	no	—
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₄×D₄	—	2	4	no	—
272	C₄×D₄	—	4	2	no	—
273	#117	C₂³×C₃₁	1	2	yes	—
274	#117	C₂³×C₃₁	1	1	yes	characteristic
275	#117	C₂³×C₃₁	1	1	yes	characteristic
276	C₂×C₂²⋊C₄	—	2	4	no	—
277	#108	—	2	1	no	—
278	#108	—	2	1	no	—
279	#108	—	2	1	no	—
280	#108	—	2	1	no	—
281	#108	—	4	2	no	—
282	#107	—	2	1	no	—
283	#106	—	2	1	no	—
284	#104	—	4	2	no	—
285	#102	—	2	4	no	—
286	#102	—	4	2	no	—
287	#101	—	2	1	no	—
288	#100	—	2	4	no	—
289	#100	—	2	1	no	—
290	#100	—	4	2	no	—
291	#99	—	2	2	no	—
292	#97	—	2	1	no	—
293	#97	—	2	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₃₁	#24988	1	1	yes	characteristic, direct product factor, Sylow
297	C₂⁴	—	2	2	no	—
298	#41	—	2	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	—	2	1	no	—
302	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
310	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	4	4	no	—
313	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
314	SD₁₆	—	4	4	no	—
315	D₈	—	2	2	no	—
316	D₈	—	2	2	no	—
317	D₈	—	2	2	no	—
318	D₈	—	2	2	no	—
319	D₈	—	4	4	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	2	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	1	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	2	1	no	—
326	M₁₆	—	2	1	no	—
327	M₁₆	—	2	2	no	—
328	M₁₆	—	2	1	no	—
329	M₁₆	—	2	2	no	—
330	M₁₆	—	4	1	no	—
331	M₁₆	—	4	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	4	1	no	—
338	C₂×C₈	—	4	1	no	—
339	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
342	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
343	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
344	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₂²⋊C₄	#41×C₃₁	1	2	yes	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	16	2	no	—
360	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	1	no	—
365	D₄	—	2	4	no	—
366	D₄	—	2	1	no	—
367	D₄	—	4	1	no	—
368	D₄	—	4	1	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	16	4	no	—
378	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
383	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	1	no	—
387	C₈	—	4	2	no	—
388	C₈	—	4	1	no	—
389	C₈	C₃₁×#123	1	1	yes	characteristic
390	C₈	C₃₁×#123	1	1	yes	characteristic
391	C₈	C₃₁×#123	1	1	yes	characteristic
392	C₈	C₃₁×#123	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	8	4	no	—
397	C₂²	—	8	1	no	—
398	C₂²	—	16	2	no	—
399	C₂²	—	16	2	no	—
400	C₂²	C₃₁×#529	1	1	yes	characteristic
401	C₂²	C₃₁×#546	1	1	yes	characteristic
402	C₂²	C₃₁×#555	1	1	yes	characteristic
403	C₄	—	4	2	no	—
404	C₄	—	4	1	no	—
405	C₄	—	8	2	no	—
406	C₄	—	8	1	no	—
407	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
408	C₄	C₃₁×#540	1	1	yes	characteristic
409	C₄	C₃₁×#546	1	1	yes	characteristic
410	C₄	C₃₁×#553	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	4	no	—
415	C₂	—	8	1	no	—
416	C₂	C₃₁×#2420	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2420
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#472	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2948	C₂	semidirect
3. C₃₁×#2081	C₂	semidirect
4. C₃₁×#2048	C₂	semidirect
5. C₃₁×#1978	C₂	semidirect
6. C₃₁×#1953	C₂	semidirect
7. C₃₁×#1904	C₂	semidirect
8. C₃₁×#1833	C₂	semidirect
9. C₃₁×#1829	C₂	semidirect
10. C₃₁×#1814	C₂	semidirect
11. C₃₁×#1757	C₂	semidirect
146. #24988	C₃₁	direct (N × H)
186. #2948	C₂×C₃₁	semidirect
187. #2081	C₂×C₃₁	semidirect
188. #2048	C₂×C₃₁	semidirect
189. #1978	C₂×C₃₁	semidirect
190. #1953	C₂×C₃₁	semidirect
191. #1904	C₂×C₃₁	semidirect
192. #1833	C₂×C₃₁	semidirect
193. #1829	C₂×C₃₁	semidirect
194. #1814	C₂×C₃₁	semidirect
195. #1757	C₂×C₃₁	semidirect
296. C₃₁	#24988	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×#472	2
C₃₁×#2948	2
C₃₁×#2081	2
C₃₁×#2048	2
C₃₁×#1978	2
C₃₁×#1953	2
C₃₁×#1904	2
C₃₁×#1833	2
C₃₁×#1829	2
C₃₁×#1814	2
C₃₁×#1757	2
#24988	31

Subgroups — C31×#24988 #21481747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24988 #21481747