Subgroups — C₂×C₃₁×#2549 #21491579

Order 7936.

3094 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal
2	C₃₁×#2549	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#398	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#395	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#391	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#389	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#385	C₂	1	2	yes	maximal
8	C₂×C₃₁×#380	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₃₁×#95	C₂	1	1	yes	maximal, characteristic
11	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	—	2	16	no	—
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	16	yes	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#398	C₂²	1	16	yes	complemented
27	C₃₁×#395	C₂²	1	16	yes	complemented
28	C₃₁×#391	C₂²	1	16	yes	complemented
29	C₃₁×#389	C₂²	1	16	yes	complemented
30	C₃₁×#385	C₂²	1	16	yes	—
31	C₃₁×#380	C₂²	1	8	yes	—
32	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
33	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#95	C₂²	1	2	yes	—
35	C₂×C₃₁×#95	C₂²	1	2	yes	—
36	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#95	C₂²	1	2	yes	—
38	C₂×C₃₁×#95	C₂²	1	2	yes	—
39	C₂×C₃₁×#95	C₂²	1	8	yes	—
40	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
42	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×Q₈×C₃₁	—	2	8	no	—
44	C₂²×Q₈×C₃₁	—	2	8	no	—
45	C₂²×Q₈×C₃₁	—	4	4	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
47	C₂³×C₄×C₃₁	—	2	8	no	—
48	C₂³×C₄×C₃₁	—	2	8	no	—
49	C₂³×C₄×C₃₁	—	4	4	no	—
50	C₂³×C₄×C₃₁	—	4	4	no	—
51	C₂³×C₄×C₃₁	Q₈	1	8	yes	complemented
52	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
62	C₃₁×C₂²⋊Q₈	—	2	64	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
67	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
68	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₃₁	D₄	1	8	yes	—
77	C₂×C₂²⋊C₄×C₃₁	Q₈	1	8	yes	complemented
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄²×C₃₁	—	4	4	no	—
82	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
83	C₃₁×#95	C₂³	1	4	yes	—
84	C₃₁×#95	C₂³	1	8	yes	—
85	C₃₁×#95	C₂³	1	8	yes	—
86	C₃₁×#95	C₂³	1	4	yes	—
87	C₃₁×#95	C₂³	1	8	yes	—
88	C₃₁×#95	C₂³	1	8	yes	—
89	C₃₁×#95	C₂³	1	4	yes	—
90	C₃₁×#95	C₂³	1	4	yes	—
91	C₂⁴×C₃₁	—	2	4	no	—
92	C₂⁴×C₃₁	—	2	4	no	—
93	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
94	C₂⁴×C₃₁	C₂×Q₈	1	4	yes	—
95	C₂⁴×C₃₁	C₂×Q₈	1	4	yes	—
96	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
97	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
98	C₂×Q₈×C₃₁	—	2	16	no	—
99	C₂×Q₈×C₃₁	—	2	16	no	—
100	C₂×Q₈×C₃₁	—	4	8	no	—
101	C₂×Q₈×C₃₁	—	4	32	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	2	4	no	—
105	C₂²×C₄×C₃₁	—	2	4	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	16	no	—
114	C₂²×C₄×C₃₁	—	2	4	no	—
115	C₂²×C₄×C₃₁	—	2	4	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	2	16	no	—
121	C₂²×C₄×C₃₁	—	4	16	no	—
122	C₂²×C₄×C₃₁	—	4	8	no	—
123	C₂²×C₄×C₃₁	—	4	8	no	—
124	C₂²×C₄×C₃₁	—	4	32	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
126	C₂²×C₄×C₃₁	C₂×Q₈	1	8	yes	—
127	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
128	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂×Q₈	1	16	yes	complemented
130	C₂²×C₄×C₃₁	#41	1	8	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
139	C₄⋊C₄×C₃₁	—	2	16	no	—
140	C₄⋊C₄×C₃₁	—	2	16	no	—
141	C₄⋊C₄×C₃₁	—	2	16	no	—
142	C₄⋊C₄×C₃₁	—	2	16	no	—
143	C₄⋊C₄×C₃₁	—	4	8	no	—
144	C₄⋊C₄×C₃₁	—	4	8	no	—
145	C₂²⋊C₄×C₃₁	—	2	16	no	—
146	C₂²⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
147	C₂²⋊C₄×C₃₁	C₂×Q₈	1	16	yes	complemented
148	C₄²×C₃₁	—	4	8	no	—
149	C₂×#2549	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	8	no	—
151	C₂³×C₃₁	—	2	8	no	—
152	C₂³×C₃₁	—	2	16	no	—
153	C₂³×C₃₁	—	2	16	no	—
154	C₂³×C₃₁	—	2	4	no	—
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	C₄⋊Q₈	1	16	yes	—
157	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
158	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
159	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
160	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
161	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
162	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
163	C₂³×C₃₁	C₂×#41	1	4	yes	—
164	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
165	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
166	Q₈×C₃₁	—	4	32	no	—
167	C₂×C₄×C₃₁	—	2	8	no	—
168	C₂×C₄×C₃₁	—	2	8	no	—
169	C₂×C₄×C₃₁	—	2	16	no	—
170	C₂×C₄×C₃₁	—	2	16	no	—
171	C₂×C₄×C₃₁	—	2	16	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	4	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	—	2	4	no	—
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	4	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	4	no	—
186	C₂×C₄×C₃₁	—	2	4	no	—
187	C₂×C₄×C₃₁	—	2	4	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	4	16	no	—
190	C₂×C₄×C₃₁	—	4	32	no	—
191	C₂×C₄×C₃₁	—	4	8	no	—
192	C₂×C₄×C₃₁	—	4	8	no	—
193	C₂×C₄×C₃₁	—	4	8	no	—
194	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
195	C₂×C₄×C₃₁	C₂²×Q₈	1	8	yes	—
196	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
197	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
198	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
199	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	—
200	#2549	C₂×C₃₁	1	16	yes	direct product factor
201	C₂×#398	C₂×C₃₁	1	2	yes	complemented
202	C₂×#395	C₂×C₃₁	1	2	yes	complemented
203	C₂×#391	C₂×C₃₁	1	2	yes	complemented
204	C₂×#389	C₂×C₃₁	1	2	yes	complemented
205	C₂×#385	C₂×C₃₁	1	2	yes	—
206	C₂×#380	C₂×C₃₁	1	1	yes	characteristic
207	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
208	C₂²×#95	C₂×C₃₁	1	1	yes	characteristic
209	C₂²×C₃₁	—	2	4	no	—
210	C₂²×C₃₁	—	2	4	no	—
211	C₂²×C₃₁	—	2	8	no	—
212	C₂²×C₃₁	—	2	8	no	—
213	C₂²×C₃₁	—	2	32	no	—
214	C₂²×C₃₁	C₂×C₄⋊Q₈	1	8	yes	—
215	C₂²×C₃₁	C₂×C₄⋊Q₈	1	4	yes	—
216	C₂²×C₃₁	#533	1	4	yes	—
217	C₂²×C₃₁	#543	1	4	yes	—
218	C₂²×C₃₁	#546	1	8	yes	—
219	C₂²×C₃₁	#554	1	8	yes	—
220	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
221	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
222	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
223	C₂²×C₃₁	C₂²×#41	1	2	yes	—
224	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
225	C₄×C₃₁	—	2	4	no	—
226	C₄×C₃₁	—	2	4	no	—
227	C₄×C₃₁	—	2	8	no	—
228	C₄×C₃₁	—	2	8	no	—
229	C₄×C₃₁	—	2	8	no	—
230	C₄×C₃₁	—	4	4	no	—
231	C₄×C₃₁	—	4	4	no	—
232	C₄×C₃₁	—	4	4	no	—
233	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
234	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
235	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×C₂²⋊Q₈	—	2	16	no	—
237	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	16	yes	—
238	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
239	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
240	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
241	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
242	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
243	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
244	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
245	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
246	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
247	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
248	#398	C₂²×C₃₁	1	16	yes	complemented
249	#395	C₂²×C₃₁	1	16	yes	complemented
250	#391	C₂²×C₃₁	1	16	yes	complemented
251	#389	C₂²×C₃₁	1	16	yes	complemented
252	#385	C₂²×C₃₁	1	16	yes	—
253	#380	C₂²×C₃₁	1	8	yes	—
254	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
255	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#95	C₂²×C₃₁	1	2	yes	—
257	C₂×#95	C₂²×C₃₁	1	2	yes	—
258	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#95	C₂²×C₃₁	1	2	yes	—
260	C₂×#95	C₂²×C₃₁	1	2	yes	—
261	C₂×#95	C₂²×C₃₁	1	8	yes	—
262	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₃₁	—	2	8	no	—
265	C₂×C₃₁	#2549	1	8	yes	direct product factor
266	C₂×C₃₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic
267	C₂×C₃₁	C₂×#533	1	1	yes	characteristic
268	C₂×C₃₁	C₂×#543	1	1	yes	characteristic
269	C₂×C₃₁	C₂×#546	1	2	yes	—
270	C₂×C₃₁	C₂×#554	1	2	yes	—
271	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
272	C₂²×Q₈	—	2	8	no	—
273	C₂²×Q₈	—	2	8	no	—
274	C₂²×Q₈	—	4	4	no	—
275	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
276	C₂³×C₄	—	2	8	no	—
277	C₂³×C₄	—	2	8	no	—
278	C₂³×C₄	—	4	4	no	—
279	C₂³×C₄	—	4	4	no	—
280	C₂³×C₄	Q₈×C₃₁	1	8	yes	complemented
281	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
282	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
283	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
284	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
285	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
286	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
287	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
290	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
291	C₂²⋊Q₈	—	2	64	no	—
292	C₂×C₄⋊C₄	—	2	8	no	—
293	C₂×C₄⋊C₄	—	2	8	no	—
294	C₂×C₄⋊C₄	—	2	8	no	—
295	C₂×C₄⋊C₄	—	2	8	no	—
296	C₂×C₄⋊C₄	—	4	4	no	—
297	C₂×C₄⋊C₄	—	4	4	no	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
304	C₂×C₂²⋊C₄	—	2	8	no	—
305	C₂×C₂²⋊C₄	D₄×C₃₁	1	8	yes	—
306	C₂×C₂²⋊C₄	Q₈×C₃₁	1	8	yes	complemented
307	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
310	C₂×C₄²	—	4	4	no	—
311	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
312	#95	C₂³×C₃₁	1	8	yes	—
313	#95	C₂³×C₃₁	1	4	yes	—
314	#95	C₂³×C₃₁	1	4	yes	—
315	#95	C₂³×C₃₁	1	8	yes	—
316	#95	C₂³×C₃₁	1	4	yes	—
317	#95	C₂³×C₃₁	1	4	yes	—
318	#95	C₂³×C₃₁	1	8	yes	—
319	#95	C₂³×C₃₁	1	8	yes	—
320	C₃₁	C₂×#2549	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	2	4	no	—
322	C₂⁴	—	2	4	no	—
323	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
324	C₂⁴	C₂×Q₈×C₃₁	1	4	yes	—
325	C₂⁴	C₂×Q₈×C₃₁	1	4	yes	—
326	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
327	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
328	C₂×Q₈	—	2	16	no	—
329	C₂×Q₈	—	2	16	no	—
330	C₂×Q₈	—	4	8	no	—
331	C₂×Q₈	—	4	32	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	16	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	16	no	—
351	C₂²×C₄	—	4	16	no	—
352	C₂²×C₄	—	4	8	no	—
353	C₂²×C₄	—	4	8	no	—
354	C₂²×C₄	—	4	32	no	—
355	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
356	C₂²×C₄	C₂×Q₈×C₃₁	1	8	yes	—
357	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
358	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
359	C₂²×C₄	C₂×Q₈×C₃₁	1	16	yes	complemented
360	C₂²×C₄	#41×C₃₁	1	8	yes	—
361	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
362	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
363	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
364	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
366	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
367	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
368	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
369	C₄⋊C₄	—	2	16	no	—
370	C₄⋊C₄	—	2	16	no	—
371	C₄⋊C₄	—	2	16	no	—
372	C₄⋊C₄	—	2	16	no	—
373	C₄⋊C₄	—	4	8	no	—
374	C₄⋊C₄	—	4	8	no	—
375	C₂²⋊C₄	—	2	16	no	—
376	C₂²⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
377	C₂²⋊C₄	C₂×Q₈×C₃₁	1	16	yes	complemented
378	C₄²	—	4	8	no	—
379	C₂³	—	2	8	no	—
380	C₂³	—	2	8	no	—
381	C₂³	—	2	16	no	—
382	C₂³	—	2	16	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	2	4	no	—
385	C₂³	C₃₁×C₄⋊Q₈	1	16	yes	—
386	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
387	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
388	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
389	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
390	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
391	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
392	C₂³	C₂×#41×C₃₁	1	4	yes	—
393	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
394	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
395	Q₈	—	4	32	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	8	no	—
398	C₂×C₄	—	2	16	no	—
399	C₂×C₄	—	2	16	no	—
400	C₂×C₄	—	2	16	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	4	16	no	—
419	C₂×C₄	—	4	32	no	—
420	C₂×C₄	—	4	8	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	—	4	8	no	—
423	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
424	C₂×C₄	C₂²×Q₈×C₃₁	1	8	yes	—
425	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
426	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
427	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
428	C₂²	—	2	4	no	—
429	C₂²	—	2	4	no	—
430	C₂²	—	2	8	no	—
431	C₂²	—	2	8	no	—
432	C₂²	—	2	32	no	—
433	C₂²	C₂×C₃₁×C₄⋊Q₈	1	8	yes	—
434	C₂²	C₂×C₃₁×C₄⋊Q₈	1	4	yes	—
435	C₂²	C₃₁×#533	1	4	yes	—
436	C₂²	C₃₁×#543	1	4	yes	—
437	C₂²	C₃₁×#546	1	8	yes	—
438	C₂²	C₃₁×#554	1	8	yes	—
439	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
440	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
441	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
442	C₂²	C₂²×#41×C₃₁	1	2	yes	—
443	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
444	C₄	—	2	4	no	—
445	C₄	—	2	4	no	—
446	C₄	—	2	8	no	—
447	C₄	—	2	8	no	—
448	C₄	—	2	8	no	—
449	C₄	—	4	4	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	4	no	—
452	C₂	—	2	8	no	—
453	C₂	C₃₁×#2549	1	8	yes	direct product factor
454	C₂	C₂²×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
455	C₂	C₂×C₃₁×#533	1	1	yes	characteristic
456	C₂	C₂×C₃₁×#543	1	1	yes	characteristic
457	C₂	C₂×C₃₁×#546	1	2	yes	—
458	C₂	C₂×C₃₁×#554	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2549	C₂	direct (N × H)
3. C₂×C₃₁×#398	C₂	semidirect
4. C₂×C₃₁×#395	C₂	semidirect
5. C₂×C₃₁×#391	C₂	semidirect
6. C₂×C₃₁×#389	C₂	semidirect
26. C₃₁×#398	C₂²	semidirect
27. C₃₁×#395	C₂²	semidirect
28. C₃₁×#391	C₂²	semidirect
29. C₃₁×#389	C₂²	semidirect
51. C₂³×C₄×C₃₁	Q₈	semidirect
77. C₂×C₂²⋊C₄×C₃₁	Q₈	semidirect
129. C₂²×C₄×C₃₁	C₂×Q₈	semidirect
147. C₂²⋊C₄×C₃₁	C₂×Q₈	semidirect
149. C₂×#2549	C₃₁	direct (N × H)
200. #2549	C₂×C₃₁	direct (N × H)
201. C₂×#398	C₂×C₃₁	semidirect
202. C₂×#395	C₂×C₃₁	semidirect
203. C₂×#391	C₂×C₃₁	semidirect
204. C₂×#389	C₂×C₃₁	semidirect
248. #398	C₂²×C₃₁	semidirect
249. #395	C₂²×C₃₁	semidirect
250. #391	C₂²×C₃₁	semidirect
251. #389	C₂²×C₃₁	semidirect
265. C₂×C₃₁	#2549	direct (N × H)
280. C₂³×C₄	Q₈×C₃₁	semidirect
306. C₂×C₂²⋊C₄	Q₈×C₃₁	semidirect
320. C₃₁	C₂×#2549	direct (N × H)
359. C₂²×C₄	C₂×Q₈×C₃₁	semidirect
377. C₂²⋊C₄	C₂×Q₈×C₃₁	semidirect
453. C₂	C₃₁×#2549	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×C₂²⋊Q₈	2
C₃₁×#2549	2
C₂×C₃₁×#398	2
C₂×C₃₁×#395	2
C₂×C₃₁×#391	2
C₂×C₃₁×#389	2
C₂×C₃₁×#385	2
C₂×C₃₁×#380	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂²×C₃₁×#95	2
C₂×#2549	31

Subgroups — C2×C31×#2549 #21491579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2549 #21491579