Subgroups — C₂²×C₃₁×#421 #21516850

Order 7936.

5198 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#421	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃₁×#104	C₂	1	4	yes	maximal, complemented
6	C₂²×#41×C₃₁	—	2	1	no	—
7	C₂²×#41×C₃₁	—	2	6	no	—
8	C₂²×#41×C₃₁	—	2	6	no	—
9	C₂²×#41×C₃₁	—	2	1	no	—
10	C₂²×#41×C₃₁	C₄	1	2	yes	complemented
11	C₂²×#41×C₃₁	C₄	1	12	yes	complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂²×#41×C₃₁	C₂²	1	12	yes	—
14	C₂³×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
15	C₂³×D₄×C₃₁	—	2	1	no	—
16	C₂³×D₄×C₃₁	C₄	1	1	yes	characteristic, complemented
17	C₂⁴×C₄×C₃₁	—	2	1	no	—
18	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₃₁	—	2	6	no	—
20	C₂²×M₁₆×C₃₁	—	2	1	no	—
21	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	complemented
22	C₂²×M₁₆×C₃₁	C₂²	1	12	yes	complemented
23	C₂³×C₈×C₃₁	—	2	1	no	—
24	C₂×C₃₁×#117	—	2	6	no	—
25	C₂×C₃₁×#117	C₂²	1	12	yes	complemented
26	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
28	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
29	C₃₁×#421	C₂²	1	64	yes	direct product factor
30	C₂×C₃₁×#104	C₂²	1	48	yes	complemented
31	C₂⁵×C₃₁	—	2	1	no	—
32	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
33	C₂×#41×C₃₁	—	2	12	no	—
34	C₂×#41×C₃₁	—	2	12	no	—
35	C₂×#41×C₃₁	—	2	12	no	—
36	C₂×#41×C₃₁	—	2	48	no	—
37	C₂×#41×C₃₁	—	2	6	no	—
38	C₂×#41×C₃₁	—	2	12	no	—
39	C₂×#41×C₃₁	—	2	24	no	—
40	C₂×#41×C₃₁	—	2	12	no	—
41	C₂×#41×C₃₁	—	2	6	no	—
42	C₂×#41×C₃₁	—	2	24	no	—
43	C₂×#41×C₃₁	C₂×C₄	1	12	yes	—
44	C₂×#41×C₃₁	C₂×C₄	1	24	yes	complemented
45	C₂×#41×C₃₁	C₂×C₄	1	48	yes	complemented
46	C₂×#41×C₃₁	C₂³	1	12	yes	—
47	C₂×#41×C₃₁	C₂³	1	16	yes	—
48	C₂²×Q₈×C₃₁	—	2	1	no	—
49	C₂²×Q₈×C₃₁	—	2	6	no	—
50	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
51	C₂²×Q₈×C₃₁	C₂×C₄	1	12	yes	complemented
52	C₂²×D₄×C₃₁	—	2	2	no	—
53	C₂²×D₄×C₃₁	—	2	12	no	—
54	C₂²×D₄×C₃₁	—	2	2	no	—
55	C₂²×D₄×C₃₁	—	2	12	no	—
56	C₂²×D₄×C₃₁	—	2	6	no	—
57	C₂²×D₄×C₃₁	—	2	1	no	—
58	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₂²×D₄×C₃₁	C₂×C₄	1	12	yes	complemented
60	C₂³×C₄×C₃₁	—	2	1	no	—
61	C₂³×C₄×C₃₁	—	2	2	no	—
62	C₂³×C₄×C₃₁	—	2	12	no	—
63	C₂³×C₄×C₃₁	—	2	12	no	—
64	C₂³×C₄×C₃₁	—	2	2	no	—
65	C₂³×C₄×C₃₁	—	2	2	no	—
66	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	C₂×C₄	1	6	yes	—
69	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	D₄	1	6	yes	—
71	C₂³×C₄×C₃₁	D₄	1	6	yes	—
72	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₃₁	—	2	12	no	—
77	C₂×M₁₆×C₃₁	—	2	6	no	—
78	C₂×M₁₆×C₃₁	—	2	24	no	—
79	C₂×M₁₆×C₃₁	C₂³	1	12	yes	complemented
80	C₂×M₁₆×C₃₁	C₂³	1	16	yes	complemented
81	C₂²×C₈×C₃₁	—	2	12	no	—
82	C₂²×C₈×C₃₁	—	2	2	no	—
83	C₃₁×#117	—	2	24	no	—
84	C₃₁×#117	C₂³	1	16	yes	complemented
85	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	12	no	—
87	C₂×C₄²×C₃₁	—	2	6	no	—
88	C₂×C₄²×C₃₁	C₂³	1	12	yes	complemented
89	C₃₁×#104	C₂³	1	64	yes	complemented
90	C₂⁴×C₃₁	—	2	2	no	—
91	C₂⁴×C₃₁	—	2	12	no	—
92	C₂⁴×C₃₁	—	4	4	no	—
93	C₂⁴×C₃₁	—	4	8	no	—
94	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂⁴×C₃₁	C₂²⋊C₄	1	6	yes	—
97	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂⁴×C₃₁	C₂×D₄	1	6	yes	—
99	#41×C₃₁	—	2	24	no	—
100	#41×C₃₁	—	2	48	no	—
101	#41×C₃₁	—	2	48	no	—
102	#41×C₃₁	—	2	48	no	—
103	#41×C₃₁	—	2	12	no	—
104	#41×C₃₁	—	2	16	no	—
105	#41×C₃₁	—	2	16	no	—
106	#41×C₃₁	—	2	12	no	—
107	#41×C₃₁	C₂²×C₄	1	24	yes	—
108	#41×C₃₁	C₂²×C₄	1	32	yes	complemented
109	#41×C₃₁	C₂⁴	1	8	yes	—
110	C₂×Q₈×C₃₁	—	2	6	no	—
111	C₂×Q₈×C₃₁	—	2	12	no	—
112	C₂×Q₈×C₃₁	—	2	24	no	—
113	C₂×Q₈×C₃₁	C₂²×C₄	1	12	yes	—
114	C₂×Q₈×C₃₁	C₂²×C₄	1	16	yes	complemented
115	C₂×D₄×C₃₁	—	2	12	no	—
116	C₂×D₄×C₃₁	—	2	24	no	—
117	C₂×D₄×C₃₁	—	2	48	no	—
118	C₂×D₄×C₃₁	—	2	12	no	—
119	C₂×D₄×C₃₁	—	2	16	no	—
120	C₂×D₄×C₃₁	—	2	6	no	—
121	C₂×D₄×C₃₁	—	2	24	no	—
122	C₂×D₄×C₃₁	—	2	12	no	—
123	C₂×D₄×C₃₁	C₂²×C₄	1	12	yes	—
124	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	complemented
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	12	no	—
127	C₂²×C₄×C₃₁	—	2	12	no	—
128	C₂²×C₄×C₃₁	—	2	24	no	—
129	C₂²×C₄×C₃₁	—	2	48	no	—
130	C₂²×C₄×C₃₁	—	2	12	no	—
131	C₂²×C₄×C₃₁	—	2	16	no	—
132	C₂²×C₄×C₃₁	—	2	12	no	—
133	C₂²×C₄×C₃₁	—	4	8	no	—
134	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
136	C₂²×C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
137	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
139	C₂²×C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
140	C₂²×C₄×C₃₁	C₂²⋊C₄	1	24	yes	—
141	C₂²×C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
142	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
144	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
145	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
146	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
147	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
148	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	M₁₆×C₃₁	—	2	16	no	—
158	M₁₆×C₃₁	—	2	12	no	—
159	M₁₆×C₃₁	C₂⁴	1	8	yes	complemented
160	C₂×C₈×C₃₁	—	2	12	no	—
161	C₂×C₈×C₃₁	—	2	16	no	—
162	C₄⋊C₄×C₃₁	—	2	16	no	—
163	C₂²⋊C₄×C₃₁	—	2	16	no	—
164	C₄²×C₃₁	—	2	12	no	—
165	C₄²×C₃₁	C₂⁴	1	8	yes	complemented
166	C₂²×#421	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	4	no	—
168	C₂³×C₃₁	—	2	4	no	—
169	C₂³×C₃₁	—	2	4	no	—
170	C₂³×C₃₁	—	2	12	no	—
171	C₂³×C₃₁	—	2	16	no	—
172	C₂³×C₃₁	—	4	12	no	—
173	C₂³×C₃₁	—	4	12	no	—
174	C₂³×C₃₁	—	4	48	no	—
175	C₂³×C₃₁	—	4	8	no	—
176	C₂³×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
177	C₂³×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
178	C₂³×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
179	C₂³×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
180	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
182	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
183	Q₈×C₃₁	—	2	12	no	—
184	Q₈×C₃₁	—	2	16	no	—
185	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
186	D₄×C₃₁	—	2	24	no	—
187	D₄×C₃₁	—	2	32	no	—
188	D₄×C₃₁	—	2	8	no	—
189	D₄×C₃₁	—	2	12	no	—
190	D₄×C₃₁	—	2	16	no	—
191	D₄×C₃₁	C₂³×C₄	1	8	yes	—
192	C₂×C₄×C₃₁	—	2	12	no	—
193	C₂×C₄×C₃₁	—	2	16	no	—
194	C₂×C₄×C₃₁	—	2	24	no	—
195	C₂×C₄×C₃₁	—	2	32	no	—
196	C₂×C₄×C₃₁	—	2	8	no	—
197	C₂×C₄×C₃₁	—	2	8	no	—
198	C₂×C₄×C₃₁	—	4	24	no	—
199	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
200	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
201	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
202	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
203	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
204	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
205	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
206	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
207	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
208	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
209	C₂×C₄×C₃₁	C₂³×C₄	1	3	yes	—
210	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
211	C₂×C₄×C₃₁	C₂³×C₄	1	3	yes	—
212	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
215	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
219	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
220	C₈×C₃₁	—	2	8	no	—
221	C₂³×#41	C₂×C₃₁	1	1	yes	characteristic
222	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂×#421	C₂×C₃₁	1	24	yes	direct product factor
225	C₂²×#104	C₂×C₃₁	1	4	yes	complemented
226	C₂²×C₃₁	—	2	6	no	—
227	C₂²×C₃₁	—	2	12	no	—
228	C₂²×C₃₁	—	2	6	no	—
229	C₂²×C₃₁	—	2	12	no	—
230	C₂²×C₃₁	—	2	8	no	—
231	C₂²×C₃₁	—	4	12	no	—
232	C₂²×C₃₁	—	4	4	no	—
233	C₂²×C₃₁	—	4	32	no	—
234	C₂²×C₃₁	—	4	24	no	—
235	C₂²×C₃₁	#421	1	4	yes	direct product factor
236	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	4	yes	—
237	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	3	yes	—
238	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	4	yes	—
239	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	3	yes	—
240	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
241	C₄×C₃₁	—	2	8	no	—
242	C₄×C₃₁	—	4	8	no	—
243	C₄×C₃₁	C₂²×C₂²⋊C₄	1	3	yes	—
244	C₄×C₃₁	C₂²×C₂²⋊C₄	1	4	yes	—
245	C₄×C₃₁	C₂²×C₂²⋊C₄	1	3	yes	—
246	C₄×C₃₁	C₂²×C₂²⋊C₄	1	4	yes	—
247	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
248	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
249	C₂²×#41	—	2	1	no	—
250	C₂²×#41	—	2	6	no	—
251	C₂²×#41	—	2	1	no	—
252	C₂²×#41	—	2	6	no	—
253	C₂²×#41	C₄×C₃₁	1	2	yes	complemented
254	C₂²×#41	C₄×C₃₁	1	12	yes	complemented
255	C₂²×#41	C₂²×C₃₁	1	2	yes	—
256	C₂²×#41	C₂²×C₃₁	1	12	yes	—
257	C₂³×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
258	C₂³×D₄	—	2	1	no	—
259	C₂³×D₄	C₄×C₃₁	1	1	yes	characteristic, complemented
260	C₂⁴×C₄	—	2	1	no	—
261	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
262	C₂²×M₁₆	—	2	1	no	—
263	C₂²×M₁₆	—	2	6	no	—
264	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	complemented
265	C₂²×M₁₆	C₂²×C₃₁	1	12	yes	complemented
266	C₂³×C₈	—	2	1	no	—
267	C₂×#117	—	2	6	no	—
268	C₂×#117	C₂²×C₃₁	1	12	yes	complemented
269	C₂²×C₄⋊C₄	—	2	1	no	—
270	C₂²×C₂²⋊C₄	—	2	1	no	—
271	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
272	#421	C₂²×C₃₁	1	64	yes	direct product factor
273	C₂×#104	C₂²×C₃₁	1	48	yes	complemented
274	C₂×C₃₁	—	2	1	no	—
275	C₂×C₃₁	—	2	4	no	—
276	C₂×C₃₁	—	2	3	no	—
277	C₂×C₃₁	—	2	4	no	—
278	C₂×C₃₁	—	4	8	no	—
279	C₂×C₃₁	C₂×#421	1	6	yes	direct product factor
280	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
281	C₂⁵	—	2	1	no	—
282	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
283	C₂×#41	—	2	12	no	—
284	C₂×#41	—	2	12	no	—
285	C₂×#41	—	2	12	no	—
286	C₂×#41	—	2	48	no	—
287	C₂×#41	—	2	6	no	—
288	C₂×#41	—	2	12	no	—
289	C₂×#41	—	2	24	no	—
290	C₂×#41	—	2	12	no	—
291	C₂×#41	—	2	6	no	—
292	C₂×#41	—	2	24	no	—
293	C₂×#41	C₂×C₄×C₃₁	1	12	yes	—
294	C₂×#41	C₂×C₄×C₃₁	1	24	yes	complemented
295	C₂×#41	C₂×C₄×C₃₁	1	48	yes	complemented
296	C₂×#41	C₂³×C₃₁	1	12	yes	—
297	C₂×#41	C₂³×C₃₁	1	16	yes	—
298	C₂²×Q₈	—	2	1	no	—
299	C₂²×Q₈	—	2	6	no	—
300	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
301	C₂²×Q₈	C₂×C₄×C₃₁	1	12	yes	complemented
302	C₂²×D₄	—	2	2	no	—
303	C₂²×D₄	—	2	12	no	—
304	C₂²×D₄	—	2	2	no	—
305	C₂²×D₄	—	2	12	no	—
306	C₂²×D₄	—	2	6	no	—
307	C₂²×D₄	—	2	1	no	—
308	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
309	C₂²×D₄	C₂×C₄×C₃₁	1	12	yes	complemented
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	2	no	—
312	C₂³×C₄	—	2	12	no	—
313	C₂³×C₄	—	2	2	no	—
314	C₂³×C₄	—	2	12	no	—
315	C₂³×C₄	—	2	2	no	—
316	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
317	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
318	C₂³×C₄	C₂×C₄×C₃₁	1	6	yes	—
319	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
320	C₂³×C₄	D₄×C₃₁	1	6	yes	—
321	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
322	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
323	C₂³×C₄	D₄×C₃₁	1	6	yes	—
324	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
325	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×M₁₆	—	2	12	no	—
327	C₂×M₁₆	—	2	6	no	—
328	C₂×M₁₆	—	2	24	no	—
329	C₂×M₁₆	C₂³×C₃₁	1	12	yes	complemented
330	C₂×M₁₆	C₂³×C₃₁	1	16	yes	complemented
331	C₂²×C₈	—	2	12	no	—
332	C₂²×C₈	—	2	2	no	—
333	#117	—	2	24	no	—
334	#117	C₂³×C₃₁	1	16	yes	complemented
335	C₂×C₄⋊C₄	—	2	12	no	—
336	C₂×C₂²⋊C₄	—	2	12	no	—
337	C₂×C₄²	—	2	6	no	—
338	C₂×C₄²	C₂³×C₃₁	1	12	yes	complemented
339	#104	C₂³×C₃₁	1	64	yes	complemented
340	C₃₁	C₂²×#421	1	1	yes	characteristic, direct product factor, Sylow
341	C₂⁴	—	2	2	no	—
342	C₂⁴	—	2	12	no	—
343	C₂⁴	—	4	4	no	—
344	C₂⁴	—	4	8	no	—
345	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
346	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
347	C₂⁴	C₂²⋊C₄×C₃₁	1	6	yes	—
348	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂⁴	C₂×D₄×C₃₁	1	6	yes	—
350	#41	—	2	24	no	—
351	#41	—	2	48	no	—
352	#41	—	2	48	no	—
353	#41	—	2	48	no	—
354	#41	—	2	12	no	—
355	#41	—	2	16	no	—
356	#41	—	2	16	no	—
357	#41	—	2	12	no	—
358	#41	C₂²×C₄×C₃₁	1	24	yes	—
359	#41	C₂²×C₄×C₃₁	1	32	yes	complemented
360	#41	C₂⁴×C₃₁	1	8	yes	—
361	C₂×Q₈	—	2	6	no	—
362	C₂×Q₈	—	2	12	no	—
363	C₂×Q₈	—	2	24	no	—
364	C₂×Q₈	C₂²×C₄×C₃₁	1	12	yes	—
365	C₂×Q₈	C₂²×C₄×C₃₁	1	16	yes	complemented
366	C₂×D₄	—	2	12	no	—
367	C₂×D₄	—	2	24	no	—
368	C₂×D₄	—	2	48	no	—
369	C₂×D₄	—	2	12	no	—
370	C₂×D₄	—	2	16	no	—
371	C₂×D₄	—	2	6	no	—
372	C₂×D₄	—	2	24	no	—
373	C₂×D₄	—	2	12	no	—
374	C₂×D₄	C₂²×C₄×C₃₁	1	12	yes	—
375	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	complemented
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	12	no	—
378	C₂²×C₄	—	2	12	no	—
379	C₂²×C₄	—	2	24	no	—
380	C₂²×C₄	—	2	48	no	—
381	C₂²×C₄	—	2	12	no	—
382	C₂²×C₄	—	2	16	no	—
383	C₂²×C₄	—	2	12	no	—
384	C₂²×C₄	—	4	8	no	—
385	C₂²×C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
386	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
387	C₂²×C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
388	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
390	C₂²×C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
391	C₂²×C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
392	C₂²×C₄	C₂²⋊C₄×C₃₁	1	24	yes	—
393	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
395	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
396	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
397	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
404	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
405	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
407	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
408	M₁₆	—	2	16	no	—
409	M₁₆	—	2	12	no	—
410	M₁₆	C₂⁴×C₃₁	1	8	yes	complemented
411	C₂×C₈	—	2	12	no	—
412	C₂×C₈	—	2	16	no	—
413	C₄⋊C₄	—	2	16	no	—
414	C₂²⋊C₄	—	2	16	no	—
415	C₄²	—	2	12	no	—
416	C₄²	C₂⁴×C₃₁	1	8	yes	complemented
417	C₂³	—	2	4	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	2	12	no	—
421	C₂³	—	2	16	no	—
422	C₂³	—	4	12	no	—
423	C₂³	—	4	12	no	—
424	C₂³	—	4	48	no	—
425	C₂³	—	4	8	no	—
426	C₂³	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
427	C₂³	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
428	C₂³	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
429	C₂³	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
430	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
431	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
432	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
433	Q₈	—	2	12	no	—
434	Q₈	—	2	16	no	—
435	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
436	D₄	—	2	24	no	—
437	D₄	—	2	32	no	—
438	D₄	—	2	8	no	—
439	D₄	—	2	12	no	—
440	D₄	—	2	16	no	—
441	D₄	C₂³×C₄×C₃₁	1	8	yes	—
442	C₂×C₄	—	2	12	no	—
443	C₂×C₄	—	2	16	no	—
444	C₂×C₄	—	2	24	no	—
445	C₂×C₄	—	2	32	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	4	24	no	—
449	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
450	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
451	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
452	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
453	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
455	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
456	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
457	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
458	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
459	C₂×C₄	C₂³×C₄×C₃₁	1	3	yes	—
460	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
461	C₂×C₄	C₂³×C₄×C₃₁	1	3	yes	—
462	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
464	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
465	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
466	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
467	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
468	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
469	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
470	C₈	—	2	8	no	—
471	C₂²	—	2	6	no	—
472	C₂²	—	2	12	no	—
473	C₂²	—	2	6	no	—
474	C₂²	—	2	12	no	—
475	C₂²	—	2	8	no	—
476	C₂²	—	4	12	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	32	no	—
479	C₂²	—	4	24	no	—
480	C₂²	C₃₁×#421	1	4	yes	direct product factor
481	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	4	yes	—
482	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	3	yes	—
483	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	4	yes	—
484	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	3	yes	—
485	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
486	C₄	—	2	8	no	—
487	C₄	—	4	8	no	—
488	C₄	C₂²×C₂²⋊C₄×C₃₁	1	3	yes	—
489	C₄	C₂²×C₂²⋊C₄×C₃₁	1	4	yes	—
490	C₄	C₂²×C₂²⋊C₄×C₃₁	1	3	yes	—
491	C₄	C₂²×C₂²⋊C₄×C₃₁	1	4	yes	—
492	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
493	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
494	C₂	—	2	1	no	—
495	C₂	—	2	4	no	—
496	C₂	—	2	3	no	—
497	C₂	—	2	4	no	—
498	C₂	—	4	8	no	—
499	C₂	C₂×C₃₁×#421	1	6	yes	direct product factor
500	C₂	C₂³×C₂²⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×#41×C₃₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆×C₃₁	C₂	semidirect
3. C₂²×C₃₁×#117	C₂	semidirect
4. C₂×C₃₁×#421	C₂	direct (N × H)
5. C₂²×C₃₁×#104	C₂	semidirect
10. C₂²×#41×C₃₁	C₄	semidirect
11. C₂²×#41×C₃₁	C₄	semidirect
14. C₂³×Q₈×C₃₁	C₄	semidirect
16. C₂³×D₄×C₃₁	C₄	semidirect
21. C₂²×M₁₆×C₃₁	C₂²	semidirect
22. C₂²×M₁₆×C₃₁	C₂²	semidirect
25. C₂×C₃₁×#117	C₂²	semidirect
28. C₂²×C₄²×C₃₁	C₂²	semidirect
29. C₃₁×#421	C₂²	direct (N × H)
30. C₂×C₃₁×#104	C₂²	semidirect
44. C₂×#41×C₃₁	C₂×C₄	semidirect
45. C₂×#41×C₃₁	C₂×C₄	semidirect
51. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
59. C₂²×D₄×C₃₁	C₂×C₄	semidirect
79. C₂×M₁₆×C₃₁	C₂³	semidirect
80. C₂×M₁₆×C₃₁	C₂³	semidirect
84. C₃₁×#117	C₂³	semidirect
88. C₂×C₄²×C₃₁	C₂³	semidirect
89. C₃₁×#104	C₂³	semidirect
108. #41×C₃₁	C₂²×C₄	semidirect
114. C₂×Q₈×C₃₁	C₂²×C₄	semidirect
124. C₂×D₄×C₃₁	C₂²×C₄	semidirect
159. M₁₆×C₃₁	C₂⁴	semidirect
165. C₄²×C₃₁	C₂⁴	semidirect
166. C₂²×#421	C₃₁	direct (N × H)
222. C₂³×M₁₆	C₂×C₃₁	semidirect
223. C₂²×#117	C₂×C₃₁	semidirect
224. C₂×#421	C₂×C₃₁	direct (N × H)
225. C₂²×#104	C₂×C₃₁	semidirect
235. C₂²×C₃₁	#421	direct (N × H)
253. C₂²×#41	C₄×C₃₁	semidirect
254. C₂²×#41	C₄×C₃₁	semidirect
257. C₂³×Q₈	C₄×C₃₁	semidirect
259. C₂³×D₄	C₄×C₃₁	semidirect
264. C₂²×M₁₆	C₂²×C₃₁	semidirect
265. C₂²×M₁₆	C₂²×C₃₁	semidirect
268. C₂×#117	C₂²×C₃₁	semidirect
271. C₂²×C₄²	C₂²×C₃₁	semidirect
272. #421	C₂²×C₃₁	direct (N × H)
273. C₂×#104	C₂²×C₃₁	semidirect
279. C₂×C₃₁	C₂×#421	direct (N × H)
294. C₂×#41	C₂×C₄×C₃₁	semidirect
295. C₂×#41	C₂×C₄×C₃₁	semidirect
301. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
309. C₂²×D₄	C₂×C₄×C₃₁	semidirect
329. C₂×M₁₆	C₂³×C₃₁	semidirect
330. C₂×M₁₆	C₂³×C₃₁	semidirect
334. #117	C₂³×C₃₁	semidirect
338. C₂×C₄²	C₂³×C₃₁	semidirect
339. #104	C₂³×C₃₁	semidirect
340. C₃₁	C₂²×#421	direct (N × H)
359. #41	C₂²×C₄×C₃₁	semidirect
365. C₂×Q₈	C₂²×C₄×C₃₁	semidirect
375. C₂×D₄	C₂²×C₄×C₃₁	semidirect
410. M₁₆	C₂⁴×C₃₁	semidirect
416. C₄²	C₂⁴×C₃₁	semidirect
480. C₂²	C₃₁×#421	direct (N × H)
499. C₂	C₂×C₃₁×#421	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×#41×C₃₁	2
C₂³×M₁₆×C₃₁	2
C₂²×C₃₁×#117	2
C₂×C₃₁×#421	2
C₂²×C₃₁×#104	2
C₂²×#421	31

Subgroups — C22×C31×#421 #21516850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#421 #21516850