Subgroups — C₃₁×#24268 #21481027

Order 7936.

1410 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#467	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2409	C₂	1	2	yes	maximal
5	C₃₁×#2056	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2001	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1969	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1787	C₂	1	2	yes	maximal, complemented
10	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₃₁	—	2	8	no	—
12	C₂²×Q₁₆×C₃₁	—	2	4	no	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#467	—	2	8	no	—
20	C₈×D₄×C₃₁	—	2	4	no	—
21	C₂×C₃₁×#107	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#103	C₂²	1	4	yes	—
24	C₂×C₃₁×#103	C₂²	1	4	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#395	—	2	2	no	—
29	C₃₁×#393	—	2	2	no	—
30	C₃₁×#391	C₂²	1	4	yes	—
31	C₃₁×#386	—	2	2	no	—
32	C₃₁×#385	—	2	2	no	—
33	C₃₁×#380	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	—	2	2	no	—
36	C₂²×Q₈×C₃₁	—	2	8	no	—
37	C₂²×Q₈×C₃₁	—	2	4	no	—
38	C₂²×Q₈×C₃₁	—	2	4	no	—
39	C₂²×Q₈×C₃₁	—	4	4	no	—
40	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
41	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	—	2	2	no	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₃₁	—	2	32	no	—
46	C₂×Q₁₆×C₃₁	—	2	8	no	—
47	C₂×Q₁₆×C₃₁	—	2	4	no	—
48	C₂×Q₁₆×C₃₁	—	2	4	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	D₄	1	8	yes	complemented
51	C₂²×C₈×C₃₁	—	2	4	no	—
52	C₂²×C₈×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	2	4	no	—
54	C₂²×C₈×C₃₁	D₄	1	4	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₂²⋊Q₈	—	2	8	no	—
60	C₃₁×C₂²⋊Q₈	—	4	4	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#107	—	2	4	no	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×#103	—	2	8	no	—
74	C₃₁×#103	—	4	4	no	—
75	C₃₁×C₂²⋊C₈	—	2	2	no	—
76	C₃₁×C₂²⋊C₈	—	2	2	no	—
77	C₄×C₈×C₃₁	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	4	no	—
80	C₃₁×#95	—	2	4	no	—
81	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×Q₈×C₃₁	—	2	4	no	—
84	C₂×Q₈×C₃₁	—	2	16	no	—
85	C₂×Q₈×C₃₁	—	2	4	no	—
86	C₂×Q₈×C₃₁	—	2	4	no	—
87	C₂×Q₈×C₃₁	—	2	4	no	—
88	C₂×Q₈×C₃₁	—	2	4	no	—
89	C₂×Q₈×C₃₁	—	4	8	no	—
90	C₂×Q₈×C₃₁	—	4	16	no	—
91	C₂×Q₈×C₃₁	—	4	8	no	—
92	C₂×Q₈×C₃₁	—	4	4	no	—
93	C₂×Q₈×C₃₁	C₂×D₄	1	8	yes	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	Q₁₆	1	4	yes	complemented
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	4	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	4	no	—
105	C₂²×C₄×C₃₁	—	4	8	no	—
106	C₂²×C₄×C₃₁	—	4	4	no	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	#41	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₃₁	—	2	16	no	—
115	Q₁₆×C₃₁	—	2	16	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	8	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	4	4	no	—
123	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	4	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	4	8	no	—
130	C₂²⋊C₄×C₃₁	—	2	2	no	—
131	C₂²⋊C₄×C₃₁	—	2	2	no	—
132	C₂²⋊C₄×C₃₁	—	4	8	no	—
133	C₂²⋊C₄×C₃₁	—	4	4	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	#24268	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	—	2	1	no	—
140	C₂³×C₃₁	—	2	2	no	—
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	2	no	—
143	C₂³×C₃₁	#121	1	2	yes	—
144	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
145	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	complemented
146	C₂³×C₃₁	#135	1	2	yes	—
147	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₃₁	—	2	8	no	—
149	Q₈×C₃₁	—	2	8	no	—
150	Q₈×C₃₁	—	4	16	no	—
151	Q₈×C₃₁	—	4	4	no	—
152	Q₈×C₃₁	—	4	4	no	—
153	D₄×C₃₁	—	2	8	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	8	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	4	16	no	—
166	C₂×C₄×C₃₁	—	4	4	no	—
167	C₂×C₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	4	no	—
172	C₂×C₄×C₃₁	—	4	4	no	—
173	C₂×C₄×C₃₁	—	4	4	no	—
174	C₂×C₄×C₃₁	#121	1	2	yes	—
175	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	complemented
176	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	—	4	1	no	—
183	C₈×C₃₁	#120	1	4	yes	—
184	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂×#467	C₂×C₃₁	1	2	yes	complemented
186	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
187	#2409	C₂×C₃₁	1	2	yes	—
188	#2056	C₂×C₃₁	1	2	yes	complemented
189	#2001	C₂×C₃₁	1	2	yes	complemented
190	#1969	C₂×C₃₁	1	2	yes	complemented
191	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1787	C₂×C₃₁	1	2	yes	complemented
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	2	4	no	—
195	C₂²×C₃₁	—	2	4	no	—
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	2	4	no	—
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	—	4	1	no	—
202	C₂²×C₃₁	#467	1	4	yes	complemented
203	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
204	C₂²×C₃₁	D₄²	1	2	yes	—
205	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
206	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
207	C₂²×C₃₁	#578	1	2	yes	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	4	1	no	—
211	C₄×C₃₁	—	4	8	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	8	4	no	—
214	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
215	C₄×C₃₁	#535	1	1	yes	characteristic
216	C₄×C₃₁	#546	1	2	yes	—
217	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
218	C₂²×Q₁₆	—	2	8	no	—
219	C₂²×Q₁₆	—	2	4	no	—
220	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
221	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
222	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
225	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
226	#467	—	2	8	no	—
227	C₈×D₄	—	2	4	no	—
228	C₂×#107	C₂²×C₃₁	1	2	yes	—
229	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#103	C₂²×C₃₁	1	4	yes	—
231	C₂×#103	C₂²×C₃₁	1	4	yes	—
232	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
235	#395	—	2	2	no	—
236	#393	—	2	2	no	—
237	#391	C₂²×C₃₁	1	4	yes	—
238	#386	—	2	2	no	—
239	#385	—	2	2	no	—
240	#380	C₂²×C₃₁	1	2	yes	—
241	#340	C₂²×C₃₁	1	2	yes	—
242	C₂×C₃₁	—	2	4	no	—
243	C₂×C₃₁	—	4	2	no	—
244	C₂×C₃₁	#2408	1	1	yes	characteristic
245	C₂×C₃₁	C₂×#467	1	1	yes	characteristic
246	C₂×C₃₁	#3059	1	1	yes	characteristic
247	C₂×C₃₁	#3285	1	2	yes	—
248	C₂×C₃₁	D₄×Q₁₆	1	2	yes	—
249	C₂²×Q₈	—	2	2	no	—
250	C₂²×Q₈	—	2	8	no	—
251	C₂²×Q₈	—	2	4	no	—
252	C₂²×Q₈	—	2	4	no	—
253	C₂²×Q₈	—	4	4	no	—
254	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
255	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
256	C₂³×C₄	—	2	2	no	—
257	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
258	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
259	C₂×Q₁₆	—	2	32	no	—
260	C₂×Q₁₆	—	2	8	no	—
261	C₂×Q₁₆	—	2	4	no	—
262	C₂×Q₁₆	—	2	4	no	—
263	C₂×Q₁₆	—	2	4	no	—
264	C₂×Q₁₆	D₄×C₃₁	1	8	yes	complemented
265	C₂²×C₈	—	2	4	no	—
266	C₂²×C₈	—	2	2	no	—
267	C₂²×C₈	—	2	4	no	—
268	C₂²×C₈	D₄×C₃₁	1	4	yes	—
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
272	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²⋊Q₈	—	2	8	no	—
274	C₂²⋊Q₈	—	4	4	no	—
275	C₄×D₄	—	2	2	no	—
276	C₄×D₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	4	no	—
278	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
280	C₂×C₂²⋊C₄	—	2	4	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
285	#107	—	2	4	no	—
286	C₄⋊C₈	—	2	2	no	—
287	#103	—	2	8	no	—
288	#103	—	4	4	no	—
289	C₂²⋊C₈	—	2	2	no	—
290	C₂²⋊C₈	—	2	2	no	—
291	C₄×C₈	—	2	2	no	—
292	#95	—	2	2	no	—
293	#95	—	2	4	no	—
294	#95	—	2	4	no	—
295	C₃₁	#24268	1	1	yes	characteristic, direct product factor, Sylow
296	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₂×Q₈	—	2	4	no	—
299	C₂×Q₈	—	2	16	no	—
300	C₂×Q₈	—	2	4	no	—
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	—	2	4	no	—
303	C₂×Q₈	—	2	4	no	—
304	C₂×Q₈	—	4	8	no	—
305	C₂×Q₈	—	4	16	no	—
306	C₂×Q₈	—	4	8	no	—
307	C₂×Q₈	—	4	4	no	—
308	C₂×Q₈	C₂×D₄×C₃₁	1	8	yes	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	Q₁₆×C₃₁	1	4	yes	complemented
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	4	8	no	—
321	C₂²×C₄	—	4	4	no	—
322	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	#41×C₃₁	1	2	yes	—
328	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
329	Q₁₆	—	2	16	no	—
330	Q₁₆	—	2	16	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	8	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	4	no	—
338	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
339	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
340	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₄⋊C₄	—	2	4	no	—
344	C₄⋊C₄	—	4	8	no	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	2	2	no	—
347	C₂²⋊C₄	—	4	8	no	—
348	C₂²⋊C₄	—	4	4	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	2	1	no	—
351	C₂³	—	2	2	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	2	no	—
357	C₂³	C₃₁×#121	1	2	yes	—
358	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
359	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	complemented
360	C₂³	C₃₁×#135	1	2	yes	—
361	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
362	Q₈	—	2	8	no	—
363	Q₈	—	2	8	no	—
364	Q₈	—	4	16	no	—
365	Q₈	—	4	4	no	—
366	Q₈	—	4	4	no	—
367	D₄	—	2	8	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	8	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	16	no	—
380	C₂×C₄	—	4	4	no	—
381	C₂×C₄	—	4	4	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	4	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	C₃₁×#121	1	2	yes	—
389	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	complemented
390	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
392	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
394	C₈	—	2	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₈	C₃₁×#120	1	4	yes	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	4	no	—
400	C₂²	—	2	4	no	—
401	C₂²	—	2	2	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	4	1	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	1	no	—
407	C₂²	C₃₁×#467	1	4	yes	complemented
408	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
409	C₂²	D₄²×C₃₁	1	2	yes	—
410	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
411	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
412	C₂²	C₃₁×#578	1	2	yes	—
413	C₄	—	2	2	no	—
414	C₄	—	2	2	no	—
415	C₄	—	4	1	no	—
416	C₄	—	4	8	no	—
417	C₄	—	4	1	no	—
418	C₄	—	8	4	no	—
419	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
420	C₄	C₃₁×#535	1	1	yes	characteristic
421	C₄	C₃₁×#546	1	2	yes	—
422	C₂	—	2	4	no	—
423	C₂	—	4	2	no	—
424	C₂	C₃₁×#2408	1	1	yes	characteristic
425	C₂	C₂×C₃₁×#467	1	1	yes	characteristic
426	C₂	C₃₁×#3059	1	1	yes	characteristic
427	C₂	C₃₁×#3285	1	2	yes	—
428	C₂	D₄×Q₁₆×C₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₃₁	C₂	semidirect
2. C₂×C₃₁×#467	C₂	semidirect
5. C₃₁×#2056	C₂	semidirect
6. C₃₁×#2001	C₂	semidirect
7. C₃₁×#1969	C₂	semidirect
8. C₃₁×#1808	C₂	semidirect
9. C₃₁×#1787	C₂	semidirect
50. C₂×Q₁₆×C₃₁	D₄	semidirect
96. C₂×D₄×C₃₁	Q₁₆	semidirect
136. #24268	C₃₁	direct (N × H)
145. C₂³×C₃₁	C₂×Q₁₆	semidirect
175. C₂×C₄×C₃₁	C₂×Q₁₆	semidirect
184. C₂³×Q₁₆	C₂×C₃₁	semidirect
185. C₂×#467	C₂×C₃₁	semidirect
188. #2056	C₂×C₃₁	semidirect
189. #2001	C₂×C₃₁	semidirect
190. #1969	C₂×C₃₁	semidirect
191. #1808	C₂×C₃₁	semidirect
192. #1787	C₂×C₃₁	semidirect
202. C₂²×C₃₁	#467	semidirect
264. C₂×Q₁₆	D₄×C₃₁	semidirect
295. C₃₁	#24268	direct (N × H)
311. C₂×D₄	Q₁₆×C₃₁	semidirect
359. C₂³	C₂×Q₁₆×C₃₁	semidirect
389. C₂×C₄	C₂×Q₁₆×C₃₁	semidirect
407. C₂²	C₃₁×#467	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₃₁	2
C₂×C₃₁×#467	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#2409	2
C₃₁×#2056	2
C₃₁×#2001	2
C₃₁×#1969	2
C₃₁×#1808	2
C₃₁×#1787	2
#24268	31

Subgroups — C31×#24268 #21481027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24268 #21481027