Subgroups — C₂×C₃₁×#1853 #21477600

Order 7936.

1966 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#106	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#380	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1853	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#340	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₃₁	—	2	1	no	—
8	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
11	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
12	C₂²×C₄⋊C₄×C₃₁	—	2	8	no	—
13	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
18	C₂×C₃₁×#106	—	2	8	no	—
19	C₂×C₃₁×#106	C₂²	1	8	yes	complemented
20	C₂×C₃₁×C₂²⋊C₈	C₄	1	8	yes	complemented
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
22	C₃₁×#380	—	2	2	no	—
23	C₃₁×#380	C₂²	1	4	yes	—
24	C₂×C₃₁×#95	—	2	1	no	—
25	C₃₁×#340	C₂²	1	8	yes	complemented
26	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	—	2	4	no	—
28	C₂³×C₄×C₃₁	—	2	4	no	—
29	C₂³×C₄×C₃₁	—	2	8	no	—
30	C₂³×C₄×C₃₁	—	2	4	no	—
31	C₂³×C₄×C₃₁	—	2	2	no	—
32	C₂³×C₄×C₃₁	—	2	1	no	—
33	C₂³×C₄×C₃₁	—	2	1	no	—
34	C₂³×C₄×C₃₁	—	4	4	no	—
35	C₂³×C₄×C₃₁	—	4	4	no	—
36	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₃₁	D₄	1	2	yes	—
38	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	Q₈	1	4	yes	—
40	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	—	2	4	no	—
43	C₂²×C₈×C₃₁	—	2	4	no	—
44	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
45	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
62	C₃₁×#106	—	2	16	no	—
63	C₃₁×C₂²⋊C₈	C₂×C₄	1	16	yes	complemented
64	C₃₁×#95	—	2	4	no	—
65	C₂⁴×C₃₁	—	2	2	no	—
66	C₂⁴×C₃₁	—	2	2	no	—
67	C₂⁴×C₃₁	—	2	2	no	—
68	C₂⁴×C₃₁	—	2	2	no	—
69	C₂⁴×C₃₁	C₄⋊C₄	1	4	yes	—
70	C₂⁴×C₃₁	SD₁₆	1	4	yes	—
71	C₂⁴×C₃₁	SD₁₆	1	4	yes	—
72	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
73	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	16	no	—
77	C₂²×C₄×C₃₁	—	2	4	no	—
78	C₂²×C₄×C₃₁	—	2	16	no	—
79	C₂²×C₄×C₃₁	—	2	4	no	—
80	C₂²×C₄×C₃₁	—	2	4	no	—
81	C₂²×C₄×C₃₁	—	2	4	no	—
82	C₂²×C₄×C₃₁	—	2	16	no	—
83	C₂²×C₄×C₃₁	—	2	8	no	—
84	C₂²×C₄×C₃₁	—	2	4	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	4	8	no	—
88	C₂²×C₄×C₃₁	—	4	32	no	—
89	C₂²×C₄×C₃₁	—	4	8	no	—
90	C₂²×C₄×C₃₁	—	4	4	no	—
91	C₂²×C₄×C₃₁	—	4	4	no	—
92	C₂²×C₄×C₃₁	—	4	2	no	—
93	C₂²×C₄×C₃₁	—	4	2	no	—
94	C₂²×C₄×C₃₁	—	4	2	no	—
95	C₂²×C₄×C₃₁	—	4	2	no	—
96	C₂²×C₄×C₃₁	—	4	2	no	—
97	C₂²×C₄×C₃₁	—	4	2	no	—
98	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
99	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
101	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
106	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₃₁	#41	1	4	yes	—
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₃₁	—	2	16	no	—
110	C₂×C₈×C₃₁	—	2	4	no	—
111	C₂×C₈×C₃₁	—	2	4	no	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	complemented
113	C₄⋊C₄×C₃₁	—	2	16	no	—
114	C₄⋊C₄×C₃₁	—	2	8	no	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	4	2	no	—
119	C₄⋊C₄×C₃₁	—	4	2	no	—
120	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
121	C₂²⋊C₄×C₃₁	—	4	8	no	—
122	C₂×#1853	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	—	2	2	no	—
125	C₂³×C₃₁	—	2	2	no	—
126	C₂³×C₃₁	—	2	2	no	—
127	C₂³×C₃₁	—	2	8	no	—
128	C₂³×C₃₁	—	2	8	no	—
129	C₂³×C₃₁	—	2	8	no	—
130	C₂³×C₃₁	—	2	8	no	—
131	C₂³×C₃₁	—	2	8	no	—
132	C₂³×C₃₁	—	2	8	no	—
133	C₂³×C₃₁	#106	1	16	yes	complemented
134	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
135	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
136	C₂³×C₃₁	#120	1	2	yes	—
137	C₂³×C₃₁	#123	1	2	yes	—
138	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
139	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
140	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
141	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
142	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
143	C₂³×C₃₁	#137	1	2	yes	—
144	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	8	no	—
148	C₂×C₄×C₃₁	—	2	8	no	—
149	C₂×C₄×C₃₁	—	2	8	no	—
150	C₂×C₄×C₃₁	—	2	16	no	—
151	C₂×C₄×C₃₁	—	2	8	no	—
152	C₂×C₄×C₃₁	—	4	32	no	—
153	C₂×C₄×C₃₁	—	4	4	no	—
154	C₂×C₄×C₃₁	—	4	4	no	—
155	C₂×C₄×C₃₁	—	4	16	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	8	4	no	—
163	C₂×C₄×C₃₁	—	8	4	no	—
164	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
165	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
167	C₂×C₄×C₃₁	#120	1	2	yes	—
168	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
169	C₂×C₄×C₃₁	#123	1	2	yes	—
170	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
174	C₈×C₃₁	—	2	8	no	—
175	C₂³×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
176	C₂²×#106	C₂×C₃₁	1	2	yes	complemented
177	C₂²×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
178	C₂×#380	C₂×C₃₁	1	1	yes	characteristic
179	#1853	C₂×C₃₁	1	8	yes	direct product factor
180	C₂×#340	C₂×C₃₁	1	2	yes	complemented
181	C₂²×C₃₁	—	2	4	no	—
182	C₂²×C₃₁	—	2	4	no	—
183	C₂²×C₃₁	—	2	32	no	—
184	C₂²×C₃₁	—	2	4	no	—
185	C₂²×C₃₁	—	2	4	no	—
186	C₂²×C₃₁	—	2	4	no	—
187	C₂²×C₃₁	—	2	4	no	—
188	C₂²×C₃₁	#385	1	4	yes	—
189	C₂²×C₃₁	C₂×#106	1	4	yes	—
190	C₂²×C₃₁	C₂×#106	1	8	yes	complemented
191	C₂²×C₃₁	#428	1	4	yes	—
192	C₂²×C₃₁	#448	1	4	yes	—
193	C₂²×C₃₁	#450	1	4	yes	—
194	C₂²×C₃₁	#481	1	4	yes	—
195	C₂²×C₃₁	#483	1	4	yes	—
196	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
198	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
199	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
200	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
201	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
203	C₄×C₃₁	—	2	4	no	—
204	C₄×C₃₁	—	4	8	no	—
205	C₄×C₃₁	—	8	2	no	—
206	C₄×C₃₁	—	8	2	no	—
207	C₄×C₃₁	#385	1	4	yes	—
208	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
209	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
210	C₂⁴×C₄	—	2	1	no	—
211	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
212	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
213	C₂²×C₄⋊C₄	—	2	1	no	—
214	C₂²×C₄⋊C₄	—	2	2	no	—
215	C₂²×C₄⋊C₄	—	2	8	no	—
216	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
217	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
218	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
219	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
220	C₂²×C₂²⋊C₄	—	2	1	no	—
221	C₂×#106	—	2	8	no	—
222	C₂×#106	C₂²×C₃₁	1	8	yes	complemented
223	C₂×C₂²⋊C₈	C₄×C₃₁	1	8	yes	complemented
224	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
225	#380	—	2	2	no	—
226	#380	C₂²×C₃₁	1	4	yes	—
227	C₂×#95	—	2	1	no	—
228	#340	C₂²×C₃₁	1	8	yes	complemented
229	C₂×C₃₁	—	2	8	no	—
230	C₂×C₃₁	#1853	1	8	yes	direct product factor
231	C₂×C₃₁	C₂×#385	1	1	yes	characteristic
232	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
233	C₂×C₃₁	C₂×#428	1	1	yes	characteristic
234	C₂×C₃₁	C₂×#450	1	1	yes	characteristic
235	C₂×C₃₁	C₂×#448	1	1	yes	characteristic
236	C₂×C₃₁	C₂×#483	1	1	yes	characteristic
237	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
238	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
239	C₂³×C₄	—	2	4	no	—
240	C₂³×C₄	—	2	4	no	—
241	C₂³×C₄	—	2	8	no	—
242	C₂³×C₄	—	2	4	no	—
243	C₂³×C₄	—	2	2	no	—
244	C₂³×C₄	—	2	1	no	—
245	C₂³×C₄	—	2	1	no	—
246	C₂³×C₄	—	4	4	no	—
247	C₂³×C₄	—	4	4	no	—
248	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
249	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
250	C₂³×C₄	D₄×C₃₁	1	2	yes	—
251	C₂³×C₄	Q₈×C₃₁	1	4	yes	—
252	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
253	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
254	C₂²×C₈	—	2	4	no	—
255	C₂²×C₈	—	2	4	no	—
256	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
257	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
258	C₂×C₄⋊C₄	—	2	8	no	—
259	C₂×C₄⋊C₄	—	2	16	no	—
260	C₂×C₄⋊C₄	—	2	4	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	4	no	—
263	C₂×C₄⋊C₄	—	2	16	no	—
264	C₂×C₄⋊C₄	—	2	4	no	—
265	C₂×C₄⋊C₄	—	2	4	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	—
269	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
271	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
272	C₂×C₂²⋊C₄	—	2	4	no	—
273	C₂×C₂²⋊C₄	—	4	4	no	—
274	#106	—	2	16	no	—
275	C₂²⋊C₈	C₂×C₄×C₃₁	1	16	yes	complemented
276	#95	—	2	4	no	—
277	C₃₁	C₂×#1853	1	1	yes	characteristic, direct product factor, Sylow
278	C₂⁴	—	2	2	no	—
279	C₂⁴	—	2	2	no	—
280	C₂⁴	—	2	2	no	—
281	C₂⁴	—	2	2	no	—
282	C₂⁴	C₄⋊C₄×C₃₁	1	4	yes	—
283	C₂⁴	SD₁₆×C₃₁	1	4	yes	—
284	C₂⁴	SD₁₆×C₃₁	1	4	yes	—
285	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
286	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	16	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	16	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	16	no	—
296	C₂²×C₄	—	2	8	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	4	32	no	—
301	C₂²×C₄	—	4	8	no	—
302	C₂²×C₄	—	4	8	no	—
303	C₂²×C₄	—	4	4	no	—
304	C₂²×C₄	—	4	4	no	—
305	C₂²×C₄	—	4	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	—	4	2	no	—
310	C₂²×C₄	—	4	2	no	—
311	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
312	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
313	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
314	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
318	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
319	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
320	C₂²×C₄	#41×C₃₁	1	4	yes	—
321	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
322	C₂×C₈	—	2	16	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	—	2	4	no	—
325	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	complemented
326	C₄⋊C₄	—	2	16	no	—
327	C₄⋊C₄	—	2	8	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
334	C₂²⋊C₄	—	4	8	no	—
335	C₂³	—	2	8	no	—
336	C₂³	—	2	8	no	—
337	C₂³	—	2	8	no	—
338	C₂³	—	2	8	no	—
339	C₂³	—	2	8	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	2	2	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	8	no	—
345	C₂³	C₃₁×#106	1	16	yes	complemented
346	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
347	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
348	C₂³	C₃₁×#120	1	2	yes	—
349	C₂³	C₃₁×#123	1	2	yes	—
350	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
351	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
352	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
353	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
354	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
355	C₂³	C₃₁×#137	1	2	yes	—
356	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	8	no	—
360	C₂×C₄	—	2	8	no	—
361	C₂×C₄	—	2	8	no	—
362	C₂×C₄	—	2	16	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	4	32	no	—
365	C₂×C₄	—	4	16	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	—	8	4	no	—
376	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
377	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
378	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
379	C₂×C₄	C₃₁×#120	1	2	yes	—
380	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
381	C₂×C₄	C₃₁×#123	1	2	yes	—
382	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
384	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
386	C₈	—	2	8	no	—
387	C₂²	—	2	4	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	2	32	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	4	no	—
394	C₂²	C₃₁×#385	1	4	yes	—
395	C₂²	C₂×C₃₁×#106	1	4	yes	—
396	C₂²	C₂×C₃₁×#106	1	8	yes	complemented
397	C₂²	C₃₁×#428	1	4	yes	—
398	C₂²	C₃₁×#448	1	4	yes	—
399	C₂²	C₃₁×#450	1	4	yes	—
400	C₂²	C₃₁×#481	1	4	yes	—
401	C₂²	C₃₁×#483	1	4	yes	—
402	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
403	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
404	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
405	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
406	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
407	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
408	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
409	C₄	—	2	4	no	—
410	C₄	—	4	8	no	—
411	C₄	—	8	2	no	—
412	C₄	—	8	2	no	—
413	C₄	C₃₁×#385	1	4	yes	—
414	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
415	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
416	C₂	—	2	8	no	—
417	C₂	C₃₁×#1853	1	8	yes	direct product factor
418	C₂	C₂×C₃₁×#385	1	1	yes	characteristic
419	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic
420	C₂	C₂×C₃₁×#428	1	1	yes	characteristic
421	C₂	C₂×C₃₁×#450	1	1	yes	characteristic
422	C₂	C₂×C₃₁×#448	1	1	yes	characteristic
423	C₂	C₂×C₃₁×#483	1	1	yes	characteristic
424	C₂	C₂×C₃₁×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#106	C₂	semidirect
5. C₃₁×#1853	C₂	direct (N × H)
6. C₂×C₃₁×#340	C₂	semidirect
19. C₂×C₃₁×#106	C₂²	semidirect
20. C₂×C₃₁×C₂²⋊C₈	C₄	semidirect
25. C₃₁×#340	C₂²	semidirect
44. C₂²×C₈×C₃₁	C₂×C₄	semidirect
63. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
112. C₂×C₈×C₃₁	C₂²×C₄	semidirect
122. C₂×#1853	C₃₁	direct (N × H)
133. C₂³×C₃₁	#106	semidirect
176. C₂²×#106	C₂×C₃₁	semidirect
179. #1853	C₂×C₃₁	direct (N × H)
180. C₂×#340	C₂×C₃₁	semidirect
190. C₂²×C₃₁	C₂×#106	semidirect
222. C₂×#106	C₂²×C₃₁	semidirect
223. C₂×C₂²⋊C₈	C₄×C₃₁	semidirect
228. #340	C₂²×C₃₁	semidirect
230. C₂×C₃₁	#1853	direct (N × H)
256. C₂²×C₈	C₂×C₄×C₃₁	semidirect
275. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
277. C₃₁	C₂×#1853	direct (N × H)
325. C₂×C₈	C₂²×C₄×C₃₁	semidirect
345. C₂³	C₃₁×#106	semidirect
396. C₂²	C₂×C₃₁×#106	semidirect
417. C₂	C₃₁×#1853	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊C₄×C₃₁	2
C₂²×C₃₁×#106	2
C₂²×C₃₁×C₂²⋊C₈	2
C₂×C₃₁×#380	2
C₃₁×#1853	2
C₂×C₃₁×#340	2
C₂×#1853	31

Subgroups — C2×C31×#1853 #21477600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1853 #21477600