Subgroups — C₂×C₃₁×#3331 #21517910

Order 7936.

2158 conjugacy classes of subgroups.

The subgroup lattice has 560 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, complemented
2	C₂×C₃₁×#547	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3331	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#483	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#470	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#452	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#417	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₃₁	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#554	—	2	4	no	—
18	C₃₁×#554	C₂²	1	8	yes	complemented
19	C₃₁×#547	—	2	4	no	—
20	C₃₁×#547	C₂²	1	8	yes	—
21	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#125	—	2	1	no	—
24	C₂×C₃₁×#123	—	2	2	no	—
25	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₂×C₃₁×#121	—	2	1	no	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₃₁	—	2	1	no	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₃₁×#487	C₂²	1	8	yes	complemented
36	C₃₁×#483	C₂²	1	16	yes	complemented
37	C₃₁×#476	C₂²	1	8	yes	complemented
38	C₃₁×#470	C₂²	1	16	yes	complemented
39	C₃₁×#462	C₂²	1	8	yes	complemented
40	C₃₁×#452	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
42	C₃₁×#436	C₂²	1	8	yes	complemented
43	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#106	C₂²	1	2	yes	—
45	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₃₁×#417	C₂²	1	16	yes	complemented
47	C₂×C₃₁×#103	C₂²	1	2	yes	—
48	C₂×C₃₁×#103	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	2	yes	—
50	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₂×#41×C₃₁	—	2	4	no	—
56	C₂×#41×C₃₁	—	2	8	no	—
57	C₂×#41×C₃₁	—	4	4	no	—
58	C₂×#41×C₃₁	C₂³	1	8	yes	—
59	C₂²×Q₈×C₃₁	—	2	1	no	—
60	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₃₁	—	2	2	no	—
63	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	—	2	2	no	—
66	C₂³×C₄×C₃₁	—	2	1	no	—
67	C₂³×C₄×C₃₁	—	2	2	no	—
68	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₃₁	—	4	4	no	—
70	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
71	C₂×SD₁₆×C₃₁	—	2	4	no	—
72	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
73	C₂×M₁₆×C₃₁	—	4	4	no	—
74	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
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×#123	—	2	16	no	—
84	C₃₁×C₂²⋊Q₈	—	2	4	no	—
85	C₃₁×C₂²⋊Q₈	—	2	4	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	C₃₁×#121	—	2	8	no	—
90	C₄×Q₈×C₃₁	—	2	2	no	—
91	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
92	C₄×D₄×C₃₁	—	2	8	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	D₄	1	4	yes	—
95	C₄×D₄×C₃₁	C₂³	1	4	yes	—
96	C₄×D₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	8	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	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
115	C₂×C₄²×C₃₁	—	2	1	no	—
116	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
117	C₃₁×#106	C₂³	1	4	yes	—
118	C₃₁×#106	C₂³	1	8	yes	—
119	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
120	C₃₁×#103	C₂³	1	8	yes	—
121	C₃₁×#103	C₂³	1	8	yes	—
122	C₃₁×#103	C₂³	1	4	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#102	C₂³	1	4	yes	—
125	C₃₁×#102	C₂³	1	8	yes	—
126	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
127	C₄×C₈×C₃₁	C₂³	1	4	yes	—
128	C₂⁴×C₃₁	—	2	1	no	—
129	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
130	#41×C₃₁	—	2	16	no	—
131	#41×C₃₁	—	4	8	no	—
132	#41×C₃₁	—	4	16	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	2	2	no	—
135	C₂×Q₈×C₃₁	—	2	4	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	—	8	2	no	—
141	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
142	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
143	C₂×D₄×C₃₁	—	2	8	no	—
144	C₂×D₄×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	4	8	no	—
148	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
149	C₂×D₄×C₃₁	#41	1	4	yes	—
150	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	2	1	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	—	2	1	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	8	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	8	no	—
161	C₂²×C₄×C₃₁	—	4	8	no	—
162	C₂²×C₄×C₃₁	—	4	4	no	—
163	C₂²×C₄×C₃₁	—	8	4	no	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	Q₁₆×C₃₁	—	4	8	no	—
171	SD₁₆×C₃₁	—	2	8	no	—
172	M₁₆×C₃₁	—	4	8	no	—
173	C₂×C₈×C₃₁	—	2	2	no	—
174	C₂×C₈×C₃₁	—	2	2	no	—
175	C₂×C₈×C₃₁	—	2	4	no	—
176	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
177	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
178	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	8	no	—
182	C₄⋊C₄×C₃₁	—	2	8	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	2	4	no	—
185	C₄⋊C₄×C₃₁	—	2	4	no	—
186	C₄⋊C₄×C₃₁	—	2	8	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
190	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
191	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
192	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
193	C₂²⋊C₄×C₃₁	—	2	8	no	—
194	C₂²⋊C₄×C₃₁	—	2	8	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
197	C₄²×C₃₁	—	2	4	no	—
198	C₄²×C₃₁	C₂×D₄	1	2	yes	—
199	C₄²×C₃₁	C₂⁴	1	2	yes	—
200	C₂×#3331	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂³×C₃₁	—	2	4	no	—
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	2	4	no	—
204	C₂³×C₃₁	—	2	4	no	—
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	4	2	no	—
207	C₂³×C₃₁	—	4	2	no	—
208	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
209	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₃₁	—	2	2	no	—
211	Q₈×C₃₁	—	2	4	no	—
212	Q₈×C₃₁	—	2	2	no	—
213	Q₈×C₃₁	—	2	4	no	—
214	Q₈×C₃₁	—	4	2	no	—
215	Q₈×C₃₁	—	4	4	no	—
216	Q₈×C₃₁	—	8	4	no	—
217	D₄×C₃₁	—	2	4	no	—
218	D₄×C₃₁	—	2	8	no	—
219	D₄×C₃₁	—	2	2	no	—
220	D₄×C₃₁	—	4	16	no	—
221	D₄×C₃₁	C₂×#41	1	4	yes	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	8	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	4	16	no	—
233	C₂×C₄×C₃₁	—	4	4	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	4	2	no	—
236	C₂×C₄×C₃₁	—	4	2	no	—
237	C₂×C₄×C₃₁	—	4	4	no	—
238	C₂×C₄×C₃₁	—	4	8	no	—
239	C₂×C₄×C₃₁	—	8	8	no	—
240	C₂×C₄×C₃₁	#137	1	4	yes	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
246	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
247	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
248	C₈×C₃₁	—	2	2	no	—
249	C₈×C₃₁	—	2	2	no	—
250	C₈×C₃₁	—	2	4	no	—
251	C₂×#554	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂×#547	C₂×C₃₁	1	1	yes	characteristic
253	#3331	C₂×C₃₁	1	16	yes	direct product factor
254	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂×#483	C₂×C₃₁	1	2	yes	complemented
256	C₂×#476	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂×#470	C₂×C₃₁	1	2	yes	complemented
258	C₂×#462	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂×#452	C₂×C₃₁	1	2	yes	complemented
260	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
261	C₂×#436	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂×#417	C₂×C₃₁	1	2	yes	complemented
263	C₂²×C₃₁	—	2	4	no	—
264	C₂²×C₃₁	—	2	4	no	—
265	C₂²×C₃₁	—	2	4	no	—
266	C₂²×C₃₁	—	2	4	no	—
267	C₂²×C₃₁	—	4	8	no	—
268	C₂²×C₃₁	—	4	8	no	—
269	C₂²×C₃₁	—	4	2	no	—
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	#546	1	2	yes	—
272	C₂²×C₃₁	C₂×#137	1	2	yes	—
273	C₂²×C₃₁	#577	1	2	yes	—
274	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
275	C₄×C₃₁	—	2	4	no	—
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	4	4	no	—
278	C₄×C₃₁	—	4	2	no	—
279	C₄×C₃₁	—	4	2	no	—
280	C₄×C₃₁	—	4	2	no	—
281	C₄×C₃₁	—	4	4	no	—
282	C₄×C₃₁	#546	1	2	yes	—
283	C₄×C₃₁	C₂×#137	1	4	yes	—
284	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
285	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂²×#41	C₂²×C₃₁	1	2	yes	—
287	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
288	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
289	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
290	#554	—	2	4	no	—
291	#554	C₂²×C₃₁	1	8	yes	complemented
292	#547	—	2	4	no	—
293	#547	C₂²×C₃₁	1	8	yes	—
294	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#125	—	2	1	no	—
297	C₂×#123	—	2	2	no	—
298	C₂×C₂²⋊Q₈	—	2	2	no	—
299	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
300	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
301	C₂×#121	—	2	1	no	—
302	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₄×D₄	—	2	1	no	—
304	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
306	C₂²×C₄⋊C₄	—	2	1	no	—
307	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
308	#487	C₂²×C₃₁	1	8	yes	complemented
309	#483	C₂²×C₃₁	1	16	yes	complemented
310	#476	C₂²×C₃₁	1	8	yes	complemented
311	#470	C₂²×C₃₁	1	16	yes	complemented
312	#462	C₂²×C₃₁	1	8	yes	complemented
313	#452	C₂²×C₃₁	1	16	yes	complemented
314	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
315	#436	C₂²×C₃₁	1	8	yes	complemented
316	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×#106	C₂²×C₃₁	1	2	yes	—
318	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
319	#417	C₂²×C₃₁	1	16	yes	complemented
320	C₂×#103	C₂²×C₃₁	1	2	yes	—
321	C₂×#103	C₂²×C₃₁	1	2	yes	—
322	C₂×#103	C₂²×C₃₁	1	2	yes	—
323	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×#102	C₂²×C₃₁	1	2	yes	—
326	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
327	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×C₃₁	—	4	4	no	—
329	C₂×C₃₁	—	4	4	no	—
330	C₂×C₃₁	#3331	1	4	yes	direct product factor
331	C₂×C₃₁	C₂×#546	1	1	yes	characteristic
332	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
333	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
334	C₂×#41	—	2	4	no	—
335	C₂×#41	—	2	8	no	—
336	C₂×#41	—	4	4	no	—
337	C₂×#41	C₂³×C₃₁	1	8	yes	—
338	C₂²×Q₈	—	2	1	no	—
339	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
340	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
341	C₂²×D₄	—	2	2	no	—
342	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
343	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂³×C₄	—	2	2	no	—
345	C₂³×C₄	—	2	1	no	—
346	C₂³×C₄	—	2	2	no	—
347	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
348	C₂×Q₁₆	—	4	4	no	—
349	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
350	C₂×SD₁₆	—	2	4	no	—
351	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
352	C₂×M₁₆	—	4	4	no	—
353	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
354	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
355	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
356	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₄⋊Q₈	—	2	2	no	—
358	C₄⋊Q₈	—	2	2	no	—
359	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
360	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
361	#125	—	2	8	no	—
362	#123	—	2	16	no	—
363	C₂²⋊Q₈	—	2	4	no	—
364	C₂²⋊Q₈	—	2	4	no	—
365	C₂²⋊Q₈	—	2	16	no	—
366	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
367	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
368	#121	—	2	8	no	—
369	C₄×Q₈	—	2	2	no	—
370	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
371	C₄×D₄	—	2	8	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	D₄×C₃₁	1	4	yes	—
374	C₄×D₄	C₂³×C₃₁	1	4	yes	—
375	C₄×D₄	C₂³×C₃₁	1	4	yes	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	4	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	8	no	—
385	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
389	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
394	C₂×C₄²	—	2	1	no	—
395	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
396	#106	C₂³×C₃₁	1	4	yes	—
397	#106	C₂³×C₃₁	1	8	yes	—
398	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
399	#103	C₂³×C₃₁	1	8	yes	—
400	#103	C₂³×C₃₁	1	8	yes	—
401	#103	C₂³×C₃₁	1	4	yes	—
402	#103	C₂³×C₃₁	1	8	yes	—
403	#102	C₂³×C₃₁	1	4	yes	—
404	#102	C₂³×C₃₁	1	8	yes	—
405	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
406	C₄×C₈	C₂³×C₃₁	1	4	yes	—
407	C₃₁	C₂×#3331	1	1	yes	characteristic, direct product factor, Sylow
408	C₂⁴	—	2	1	no	—
409	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
410	#41	—	2	16	no	—
411	#41	—	4	8	no	—
412	#41	—	4	16	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	—	8	2	no	—
421	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
422	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
429	C₂×D₄	#41×C₃₁	1	4	yes	—
430	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	8	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	8	no	—
441	C₂²×C₄	—	4	8	no	—
442	C₂²×C₄	—	4	4	no	—
443	C₂²×C₄	—	8	4	no	—
444	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
446	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
447	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
450	Q₁₆	—	4	8	no	—
451	SD₁₆	—	2	8	no	—
452	M₁₆	—	4	8	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	4	no	—
456	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
457	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
458	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
470	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
471	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
472	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
477	C₄²	—	2	4	no	—
478	C₄²	C₂×D₄×C₃₁	1	2	yes	—
479	C₄²	C₂⁴×C₃₁	1	2	yes	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	—	4	2	no	—
487	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
488	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
489	Q₈	—	2	2	no	—
490	Q₈	—	2	4	no	—
491	Q₈	—	2	2	no	—
492	Q₈	—	2	4	no	—
493	Q₈	—	4	2	no	—
494	Q₈	—	4	4	no	—
495	Q₈	—	8	4	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	8	no	—
498	D₄	—	2	2	no	—
499	D₄	—	4	16	no	—
500	D₄	C₂×#41×C₃₁	1	4	yes	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	4	16	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	8	8	no	—
519	C₂×C₄	C₃₁×#137	1	4	yes	—
520	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
521	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
522	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
524	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
525	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
526	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
527	C₈	—	2	2	no	—
528	C₈	—	2	2	no	—
529	C₈	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	4	8	no	—
535	C₂²	—	4	8	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	C₃₁×#546	1	2	yes	—
539	C₂²	C₂×C₃₁×#137	1	2	yes	—
540	C₂²	C₃₁×#577	1	2	yes	—
541	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
542	C₄	—	2	4	no	—
543	C₄	—	2	2	no	—
544	C₄	—	4	4	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	4	no	—
549	C₄	C₃₁×#546	1	2	yes	—
550	C₄	C₂×C₃₁×#137	1	4	yes	—
551	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
552	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
553	C₂	—	4	4	no	—
554	C₂	—	4	4	no	—
555	C₂	C₃₁×#3331	1	4	yes	direct product factor
556	C₂	C₂×C₃₁×#546	1	1	yes	characteristic
557	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic
558	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	C₂×C₃₁×#547	C₂
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
1. C₂×C₃₁×#554	C₂	semidirect
3. C₃₁×#3331	C₂	direct (N × H)
4. C₂×C₃₁×#487	C₂	semidirect
5. C₂×C₃₁×#483	C₂	semidirect
6. C₂×C₃₁×#476	C₂	semidirect
7. C₂×C₃₁×#470	C₂	semidirect
8. C₂×C₃₁×#462	C₂	semidirect
9. C₂×C₃₁×#452	C₂	semidirect
10. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
11. C₂×C₃₁×#436	C₂	semidirect
12. C₂×C₃₁×#417	C₂	semidirect
18. C₃₁×#554	C₂²	semidirect
35. C₃₁×#487	C₂²	semidirect
36. C₃₁×#483	C₂²	semidirect
37. C₃₁×#476	C₂²	semidirect
38. C₃₁×#470	C₂²	semidirect
39. C₃₁×#462	C₂²	semidirect
40. C₃₁×#452	C₂²	semidirect
41. C₄×SD₁₆×C₃₁	C₂²	semidirect
42. C₃₁×#436	C₂²	semidirect
46. C₃₁×#417	C₂²	semidirect
200. C₂×#3331	C₃₁	direct (N × H)
251. C₂×#554	C₂×C₃₁	semidirect
253. #3331	C₂×C₃₁	direct (N × H)
254. C₂×#487	C₂×C₃₁	semidirect
255. C₂×#483	C₂×C₃₁	semidirect
256. C₂×#476	C₂×C₃₁	semidirect
257. C₂×#470	C₂×C₃₁	semidirect
258. C₂×#462	C₂×C₃₁	semidirect
259. C₂×#452	C₂×C₃₁	semidirect
260. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
261. C₂×#436	C₂×C₃₁	semidirect
262. C₂×#417	C₂×C₃₁	semidirect
291. #554	C₂²×C₃₁	semidirect
308. #487	C₂²×C₃₁	semidirect
309. #483	C₂²×C₃₁	semidirect
310. #476	C₂²×C₃₁	semidirect
311. #470	C₂²×C₃₁	semidirect
312. #462	C₂²×C₃₁	semidirect
313. #452	C₂²×C₃₁	semidirect
314. C₄×SD₁₆	C₂²×C₃₁	semidirect
315. #436	C₂²×C₃₁	semidirect
319. #417	C₂²×C₃₁	semidirect
330. C₂×C₃₁	#3331	direct (N × H)
407. C₃₁	C₂×#3331	direct (N × H)
555. C₂	C₃₁×#3331	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×C₃₁×#547	2
C₃₁×#3331	2
C₂×C₃₁×#487	2
C₂×C₃₁×#483	2
C₂×C₃₁×#476	2
C₂×C₃₁×#470	2
C₂×C₃₁×#462	2
C₂×C₃₁×#452	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×C₃₁×#436	2
C₂×C₃₁×#417	2
C₂×#3331	31

Subgroups — C2×C31×#3331 #21517910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3331 #21517910