Subgroups — C₂×C₃₁×#3299 #21517876

Order 7936.

2158 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3299	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#484	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#467	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#451	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₃₁×#103	C₂	1	2	yes	maximal, complemented
13	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#554	—	2	4	no	—
18	C₃₁×#554	C₂²	1	8	yes	—
19	D₄×Q₈×C₃₁	—	2	4	no	—
20	D₄×Q₈×C₃₁	C₂²	1	8	yes	—
21	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
22	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#125	—	2	1	no	—
25	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
26	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
27	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₃₁	—	2	1	no	—
32	C₂×C₄×D₄×C₃₁	—	2	1	no	—
33	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₃₁×#487	C₂²	1	8	yes	complemented
36	C₃₁×#484	C₂²	1	16	yes	complemented
37	C₃₁×#477	C₂²	1	8	yes	complemented
38	C₃₁×#467	C₂²	1	16	yes	complemented
39	C₃₁×#462	C₂²	1	8	yes	complemented
40	C₃₁×#451	C₂²	1	16	yes	complemented
41	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
42	C₈×D₄×C₃₁	C₂²	1	8	yes	—
43	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#107	C₂²	1	2	yes	—
45	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×#103	C₂²	1	2	yes	—
47	C₂×C₃₁×#103	C₂²	1	2	yes	—
48	C₂×C₃₁×#103	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#103	C₂²	1	2	yes	—
51	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#103	C₂²	1	16	yes	complemented
53	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₃₁	—	2	2	no	—
56	C₂²×Q₈×C₃₁	—	2	4	no	—
57	C₂²×Q₈×C₃₁	—	2	1	no	—
58	C₂²×Q₈×C₃₁	—	2	8	no	—
59	C₂²×Q₈×C₃₁	—	4	4	no	—
60	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
63	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	—	2	2	no	—
66	C₂³×C₄×C₃₁	—	2	2	no	—
67	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×Q₁₆×C₃₁	—	2	4	no	—
69	C₂×Q₁₆×C₃₁	—	4	4	no	—
70	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
71	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
72	C₂²×C₈×C₃₁	—	4	4	no	—
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
77	C₃₁×C₄⋊Q₈	—	2	8	no	—
78	C₃₁×C₄⋊Q₈	—	2	2	no	—
79	C₃₁×C₄⋊Q₈	—	2	2	no	—
80	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
81	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
82	C₃₁×#125	—	2	8	no	—
83	C₃₁×C₂²⋊Q₈	—	2	4	no	—
84	C₃₁×C₂²⋊Q₈	—	2	4	no	—
85	C₃₁×C₂²⋊Q₈	—	2	16	no	—
86	C₃₁×C₂²⋊Q₈	—	2	16	no	—
87	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
88	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
89	C₄×Q₈×C₃₁	—	2	2	no	—
90	C₄×Q₈×C₃₁	—	2	2	no	—
91	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
93	C₄×D₄×C₃₁	—	2	8	no	—
94	C₄×D₄×C₃₁	—	2	8	no	—
95	C₄×D₄×C₃₁	D₄	1	4	yes	—
96	C₄×D₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
114	C₂×C₄²×C₃₁	—	2	1	no	—
115	C₂×C₄²×C₃₁	—	2	1	no	—
116	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
117	C₃₁×#107	C₂³	1	4	yes	—
118	C₃₁×#107	C₂³	1	8	yes	—
119	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
120	C₃₁×#103	C₂³	1	4	yes	—
121	C₃₁×#103	C₂³	1	8	yes	—
122	C₃₁×#103	C₂³	1	8	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#103	C₂³	1	4	yes	—
125	C₃₁×#103	C₂³	1	8	yes	—
126	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
127	C₄×C₈×C₃₁	C₂³	1	4	yes	—
128	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	8	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	2	4	no	—
134	C₂×Q₈×C₃₁	—	2	2	no	—
135	C₂×Q₈×C₃₁	—	2	16	no	—
136	C₂×Q₈×C₃₁	—	2	2	no	—
137	C₂×Q₈×C₃₁	—	2	4	no	—
138	C₂×Q₈×C₃₁	—	2	2	no	—
139	C₂×Q₈×C₃₁	—	2	4	no	—
140	C₂×Q₈×C₃₁	—	4	8	no	—
141	C₂×Q₈×C₃₁	—	4	8	no	—
142	C₂×Q₈×C₃₁	—	4	16	no	—
143	C₂×Q₈×C₃₁	—	8	2	no	—
144	C₂×Q₈×C₃₁	#41	1	4	yes	—
145	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
146	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
147	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
148	C₂×D₄×C₃₁	Q₁₆	1	8	yes	complemented
149	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	2	8	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	—	2	1	no	—
156	C₂²×C₄×C₃₁	—	2	1	no	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	8	no	—
160	C₂²×C₄×C₃₁	—	4	8	no	—
161	C₂²×C₄×C₃₁	—	8	4	no	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₃₁	—	2	8	no	—
169	Q₁₆×C₃₁	—	4	8	no	—
170	C₂×C₈×C₃₁	—	2	2	no	—
171	C₂×C₈×C₃₁	—	2	2	no	—
172	C₂×C₈×C₃₁	—	2	4	no	—
173	C₂×C₈×C₃₁	—	4	8	no	—
174	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
175	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
176	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	8	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	2	4	no	—
185	C₄⋊C₄×C₃₁	—	2	8	no	—
186	C₄⋊C₄×C₃₁	—	2	8	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
190	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
191	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
192	C₂²⋊C₄×C₃₁	—	2	8	no	—
193	C₂²⋊C₄×C₃₁	—	2	8	no	—
194	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
195	C₄²×C₃₁	—	2	4	no	—
196	C₄²×C₃₁	—	2	4	no	—
197	C₄²×C₃₁	C₂×D₄	1	2	yes	—
198	C₄²×C₃₁	C₂⁴	1	2	yes	—
199	C₂×#3299	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	2	4	no	—
202	C₂³×C₃₁	C₂×Q₁₆	1	8	yes	—
203	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
204	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	4	no	—
206	Q₈×C₃₁	—	2	2	no	—
207	Q₈×C₃₁	—	2	2	no	—
208	Q₈×C₃₁	—	2	4	no	—
209	Q₈×C₃₁	—	2	2	no	—
210	Q₈×C₃₁	—	2	4	no	—
211	Q₈×C₃₁	—	4	16	no	—
212	Q₈×C₃₁	—	4	2	no	—
213	Q₈×C₃₁	—	4	4	no	—
214	Q₈×C₃₁	—	8	4	no	—
215	Q₈×C₃₁	C₂×#41	1	4	yes	—
216	D₄×C₃₁	C₂×Q₁₆	1	16	yes	complemented
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	8	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	4	16	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	4	2	no	—
233	C₂×C₄×C₃₁	—	4	2	no	—
234	C₂×C₄×C₃₁	—	4	4	no	—
235	C₂×C₄×C₃₁	—	8	8	no	—
236	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
237	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
242	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
243	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
244	C₈×C₃₁	—	2	2	no	—
245	C₈×C₃₁	—	2	2	no	—
246	C₈×C₃₁	—	2	4	no	—
247	C₂×#554	C₂×C₃₁	1	1	yes	characteristic
248	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
249	#3299	C₂×C₃₁	1	16	yes	direct product factor
250	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂×#484	C₂×C₃₁	1	2	yes	complemented
252	C₂×#477	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₂×#467	C₂×C₃₁	1	2	yes	complemented
254	C₂×#462	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂×#451	C₂×C₃₁	1	2	yes	complemented
256	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
258	C₂²×#103	C₂×C₃₁	1	2	yes	complemented
259	C₂²×C₃₁	—	2	4	no	—
260	C₂²×C₃₁	—	2	16	no	—
261	C₂²×C₃₁	—	2	4	no	—
262	C₂²×C₃₁	#546	1	2	yes	—
263	C₂²×C₃₁	C₂²×Q₁₆	1	8	yes	—
264	C₂²×C₃₁	C₂²×Q₁₆	1	2	yes	—
265	C₂²×C₃₁	#577	1	2	yes	—
266	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
267	C₄×C₃₁	—	2	4	no	—
268	C₄×C₃₁	—	2	4	no	—
269	C₄×C₃₁	—	2	2	no	—
270	C₄×C₃₁	—	4	4	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	4	4	no	—
275	C₄×C₃₁	#546	1	2	yes	—
276	C₄×C₃₁	C₂²×Q₁₆	1	4	yes	—
277	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
278	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
279	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
280	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
281	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
282	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
283	#554	—	2	4	no	—
284	#554	C₂²×C₃₁	1	8	yes	—
285	D₄×Q₈	—	2	4	no	—
286	D₄×Q₈	C₂²×C₃₁	1	8	yes	—
287	C₂×C₄⋊Q₈	—	2	1	no	—
288	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#125	—	2	1	no	—
291	C₂×C₂²⋊Q₈	—	2	2	no	—
292	C₂×C₂²⋊Q₈	—	2	2	no	—
293	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
294	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
295	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×C₄×D₄	—	2	1	no	—
298	C₂×C₄×D₄	—	2	1	no	—
299	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
300	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
301	#487	C₂²×C₃₁	1	8	yes	complemented
302	#484	C₂²×C₃₁	1	16	yes	complemented
303	#477	C₂²×C₃₁	1	8	yes	complemented
304	#467	C₂²×C₃₁	1	16	yes	complemented
305	#462	C₂²×C₃₁	1	8	yes	complemented
306	#451	C₂²×C₃₁	1	16	yes	complemented
307	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
308	C₈×D₄	C₂²×C₃₁	1	8	yes	—
309	C₂×#107	C₂²×C₃₁	1	2	yes	—
310	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×#103	C₂²×C₃₁	1	2	yes	—
313	C₂×#103	C₂²×C₃₁	1	2	yes	—
314	C₂×#103	C₂²×C₃₁	1	2	yes	—
315	C₂×#103	C₂²×C₃₁	1	2	yes	—
316	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×#103	C₂²×C₃₁	1	16	yes	complemented
319	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
320	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×C₃₁	—	2	8	no	—
322	C₂×C₃₁	#3299	1	4	yes	direct product factor
323	C₂×C₃₁	C₂×#546	1	1	yes	characteristic
324	C₂×C₃₁	C₂³×Q₁₆	1	1	yes	characteristic
325	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
326	C₂²×Q₈	—	2	2	no	—
327	C₂²×Q₈	—	2	4	no	—
328	C₂²×Q₈	—	2	1	no	—
329	C₂²×Q₈	—	2	8	no	—
330	C₂²×Q₈	—	4	4	no	—
331	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
333	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
334	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
335	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
336	C₂³×C₄	—	2	2	no	—
337	C₂³×C₄	—	2	2	no	—
338	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
339	C₂×Q₁₆	—	2	4	no	—
340	C₂×Q₁₆	—	4	4	no	—
341	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
342	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
343	C₂²×C₈	—	4	4	no	—
344	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
346	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
347	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
348	C₄⋊Q₈	—	2	8	no	—
349	C₄⋊Q₈	—	2	2	no	—
350	C₄⋊Q₈	—	2	2	no	—
351	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
352	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
353	#125	—	2	8	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	—	2	16	no	—
357	C₂²⋊Q₈	—	2	16	no	—
358	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
359	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
360	C₄×Q₈	—	2	2	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
363	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
364	C₄×D₄	—	2	8	no	—
365	C₄×D₄	—	2	8	no	—
366	C₄×D₄	D₄×C₃₁	1	4	yes	—
367	C₄×D₄	C₂³×C₃₁	1	4	yes	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	4	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
380	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
381	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×C₂²⋊C₄	—	2	2	no	—
383	C₂×C₂²⋊C₄	—	2	2	no	—
384	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
388	#107	C₂³×C₃₁	1	4	yes	—
389	#107	C₂³×C₃₁	1	8	yes	—
390	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
391	#103	C₂³×C₃₁	1	4	yes	—
392	#103	C₂³×C₃₁	1	8	yes	—
393	#103	C₂³×C₃₁	1	8	yes	—
394	#103	C₂³×C₃₁	1	8	yes	—
395	#103	C₂³×C₃₁	1	4	yes	—
396	#103	C₂³×C₃₁	1	8	yes	—
397	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
398	C₄×C₈	C₂³×C₃₁	1	4	yes	—
399	C₃₁	C₂×#3299	1	1	yes	characteristic, direct product factor, Sylow
400	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	2	8	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	16	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	—	2	4	no	—
412	C₂×Q₈	—	4	8	no	—
413	C₂×Q₈	—	4	8	no	—
414	C₂×Q₈	—	4	16	no	—
415	C₂×Q₈	—	8	2	no	—
416	C₂×Q₈	#41×C₃₁	1	4	yes	—
417	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
418	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
419	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
420	C₂×D₄	Q₁₆×C₃₁	1	8	yes	complemented
421	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	—	4	8	no	—
433	C₂²×C₄	—	8	4	no	—
434	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
437	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
440	Q₁₆	—	2	8	no	—
441	Q₁₆	—	4	8	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	4	no	—
445	C₂×C₈	—	4	8	no	—
446	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
447	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
448	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
461	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
462	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
463	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
464	C₂²⋊C₄	—	2	8	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
467	C₄²	—	2	4	no	—
468	C₄²	—	2	4	no	—
469	C₄²	C₂×D₄×C₃₁	1	2	yes	—
470	C₄²	C₂⁴×C₃₁	1	2	yes	—
471	C₂³	—	2	4	no	—
472	C₂³	—	2	4	no	—
473	C₂³	C₂×Q₁₆×C₃₁	1	8	yes	—
474	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
475	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
476	Q₈	—	2	4	no	—
477	Q₈	—	2	2	no	—
478	Q₈	—	2	2	no	—
479	Q₈	—	2	4	no	—
480	Q₈	—	2	2	no	—
481	Q₈	—	2	4	no	—
482	Q₈	—	4	16	no	—
483	Q₈	—	4	2	no	—
484	Q₈	—	4	4	no	—
485	Q₈	—	8	4	no	—
486	Q₈	C₂×#41×C₃₁	1	4	yes	—
487	D₄	C₂×Q₁₆×C₃₁	1	16	yes	complemented
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	4	16	no	—
501	C₂×C₄	—	4	4	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	8	8	no	—
507	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
509	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
510	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
512	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
513	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
514	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
515	C₈	—	2	2	no	—
516	C₈	—	2	2	no	—
517	C₈	—	2	4	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	16	no	—
520	C₂²	—	2	4	no	—
521	C₂²	C₃₁×#546	1	2	yes	—
522	C₂²	C₂²×Q₁₆×C₃₁	1	8	yes	—
523	C₂²	C₂²×Q₁₆×C₃₁	1	2	yes	—
524	C₂²	C₃₁×#577	1	2	yes	—
525	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
526	C₄	—	2	4	no	—
527	C₄	—	2	4	no	—
528	C₄	—	2	2	no	—
529	C₄	—	4	4	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	2	no	—
534	C₄	C₃₁×#546	1	2	yes	—
535	C₄	C₂²×Q₁₆×C₃₁	1	4	yes	—
536	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
537	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
538	C₂	—	2	8	no	—
539	C₂	C₃₁×#3299	1	4	yes	direct product factor
540	C₂	C₂×C₃₁×#546	1	1	yes	characteristic
541	C₂	C₂³×Q₁₆×C₃₁	1	1	yes	characteristic
542	C₂	C₂×C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂×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
3. C₃₁×#3299	C₂	direct (N × H)
4. C₂×C₃₁×#487	C₂	semidirect
5. C₂×C₃₁×#484	C₂	semidirect
6. C₂×C₃₁×#477	C₂	semidirect
7. C₂×C₃₁×#467	C₂	semidirect
8. C₂×C₃₁×#462	C₂	semidirect
9. C₂×C₃₁×#451	C₂	semidirect
10. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
12. C₂²×C₃₁×#103	C₂	semidirect
35. C₃₁×#487	C₂²	semidirect
36. C₃₁×#484	C₂²	semidirect
37. C₃₁×#477	C₂²	semidirect
38. C₃₁×#467	C₂²	semidirect
39. C₃₁×#462	C₂²	semidirect
40. C₃₁×#451	C₂²	semidirect
41. C₄×Q₁₆×C₃₁	C₂²	semidirect
52. C₂×C₃₁×#103	C₂²	semidirect
148. C₂×D₄×C₃₁	Q₁₆	semidirect
199. C₂×#3299	C₃₁	direct (N × H)
216. D₄×C₃₁	C₂×Q₁₆	semidirect
249. #3299	C₂×C₃₁	direct (N × H)
250. C₂×#487	C₂×C₃₁	semidirect
251. C₂×#484	C₂×C₃₁	semidirect
252. C₂×#477	C₂×C₃₁	semidirect
253. C₂×#467	C₂×C₃₁	semidirect
254. C₂×#462	C₂×C₃₁	semidirect
255. C₂×#451	C₂×C₃₁	semidirect
256. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
258. C₂²×#103	C₂×C₃₁	semidirect
301. #487	C₂²×C₃₁	semidirect
302. #484	C₂²×C₃₁	semidirect
303. #477	C₂²×C₃₁	semidirect
304. #467	C₂²×C₃₁	semidirect
305. #462	C₂²×C₃₁	semidirect
306. #451	C₂²×C₃₁	semidirect
307. C₄×Q₁₆	C₂²×C₃₁	semidirect
318. C₂×#103	C₂²×C₃₁	semidirect
322. C₂×C₃₁	#3299	direct (N × H)
399. C₃₁	C₂×#3299	direct (N × H)
420. C₂×D₄	Q₁₆×C₃₁	semidirect
487. D₄	C₂×Q₁₆×C₃₁	semidirect
539. C₂	C₃₁×#3299	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×#554	2
C₂×D₄×Q₈×C₃₁	2
C₃₁×#3299	2
C₂×C₃₁×#487	2
C₂×C₃₁×#484	2
C₂×C₃₁×#477	2
C₂×C₃₁×#467	2
C₂×C₃₁×#462	2
C₂×C₃₁×#451	2
C₂×C₄×Q₁₆×C₃₁	2
C₂×C₈×D₄×C₃₁	2
C₂²×C₃₁×#103	2
C₂×#3299	31

Subgroups — C2×C31×#3299 #21517876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3299 #21517876