Subgroups — C₂×C₃₁×#3480 #21519470

Order 7936.

7802 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3480	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₃₁×#559	C₂	1	3	yes	maximal, complemented
3	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#556	C₂	1	3	yes	maximal, complemented
5	C₂×C₃₁×#553	C₂	1	6	yes	maximal, complemented
6	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#548	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#532	C₂	1	3	yes	maximal, complemented
9	C₂×C₃₁×#529	C₂	1	6	yes	maximal, complemented
10	C₂×C₄×#41×C₃₁	C₂	1	3	yes	maximal, complemented
11	C₂×C₄×#41×C₃₁	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	3	yes	—
14	C₂²×#41×C₃₁	C₂²	1	2	yes	—
15	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#559	C₂²	1	48	yes	complemented
17	C₃₁×#557	C₂²	1	16	yes	complemented
18	C₃₁×#556	C₂²	1	48	yes	complemented
19	C₃₁×#553	C₂²	1	96	yes	complemented
20	C₃₁×#550	C₂²	1	16	yes	complemented
21	C₃₁×#548	C₂²	1	32	yes	complemented
22	C₃₁×#532	C₂²	1	48	yes	complemented
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
24	C₂×C₃₁×#127	C₂²	1	3	yes	—
25	C₃₁×#529	C₂²	1	96	yes	complemented
26	C₂×C₃₁×#126	C₂²	1	12	yes	—
27	C₂×C₃₁×#125	C₂²	1	6	yes	—
28	C₂×C₃₁×#124	C₂²	1	6	yes	—
29	C₂×C₃₁×#124	C₂²	1	6	yes	—
30	C₂×C₃₁×#123	C₂²	1	12	yes	—
31	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	6	yes	—
32	C₂×C₃₁×#121	C₂²	1	6	yes	—
33	C₂×C₃₁×#121	C₂²	1	6	yes	—
34	C₂×C₃₁×#121	C₂²	1	6	yes	—
35	C₄×#41×C₃₁	C₂²	1	48	yes	complemented
36	C₄×#41×C₃₁	C₂²	1	32	yes	complemented
37	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
39	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
41	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
42	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
43	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
44	C₂×C₄×Q₈×C₃₁	C₂²	1	16	yes	complemented
45	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
46	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
47	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
48	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
49	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
50	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
51	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
52	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#117	C₂²	1	6	yes	—
54	C₂×C₃₁×#117	C₂²	1	6	yes	—
55	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
56	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
57	C₂²×C₄²×C₃₁	C₂²	1	6	yes	—
58	C₂×#41×C₃₁	—	2	16	no	—
59	C₂×#41×C₃₁	—	2	24	no	—
60	C₂×#41×C₃₁	C₂³	1	24	yes	—
61	C₂×#41×C₃₁	C₂³	1	16	yes	—
62	C₂²×Q₈×C₃₁	—	2	8	no	—
63	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
64	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
65	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
66	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊Q₈	C₂³	1	24	yes	—
78	C₃₁×#127	C₂³	1	24	yes	—
79	C₃₁×#126	C₂³	1	96	yes	—
80	C₃₁×#125	C₂³	1	48	yes	—
81	C₃₁×#124	C₂³	1	48	yes	—
82	C₃₁×#124	C₂³	1	48	yes	—
83	C₃₁×#123	C₂³	1	96	yes	—
84	C₃₁×C₂²⋊Q₈	C₂³	1	48	yes	—
85	C₃₁×#121	C₂³	1	48	yes	—
86	C₃₁×#121	C₂³	1	48	yes	—
87	C₃₁×#121	C₂³	1	48	yes	—
88	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
89	C₄×Q₈×C₃₁	C₂³	1	24	yes	—
90	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
91	C₄×Q₈×C₃₁	C₂³	1	24	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	24	yes	—
94	C₄×Q₈×C₃₁	C₂³	1	24	yes	—
95	C₄×D₄×C₃₁	C₂³	1	48	yes	—
96	C₄×D₄×C₃₁	C₂³	1	48	yes	—
97	C₄×D₄×C₃₁	C₂³	1	24	yes	—
98	C₄×D₄×C₃₁	C₂³	1	48	yes	—
99	C₄×D₄×C₃₁	C₂³	1	48	yes	—
100	C₄×D₄×C₃₁	C₂³	1	24	yes	—
101	C₄×D₄×C₃₁	C₂³	1	48	yes	—
102	C₃₁×#117	C₂³	1	8	yes	—
103	C₃₁×#117	C₂³	1	48	yes	—
104	C₃₁×#117	C₂³	1	48	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	24	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	48	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
128	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
129	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
130	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
131	#41×C₃₁	—	2	64	no	—
132	#41×C₃₁	—	2	96	no	—
133	C₂×Q₈×C₃₁	—	2	32	no	—
134	C₂×Q₈×C₃₁	—	2	12	no	—
135	C₂×Q₈×C₃₁	—	2	12	no	—
136	C₂×Q₈×C₃₁	#41	1	8	yes	complemented
137	C₂×Q₈×C₃₁	C₂⁴	1	12	yes	—
138	C₂×Q₈×C₃₁	C₂⁴	1	12	yes	—
139	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
140	C₂×D₄×C₃₁	—	2	24	no	—
141	C₂×D₄×C₃₁	—	2	24	no	—
142	C₂×D₄×C₃₁	—	2	12	no	—
143	C₂×D₄×C₃₁	—	4	12	no	—
144	C₂×D₄×C₃₁	C₂⁴	1	24	yes	—
145	C₂×D₄×C₃₁	C₂⁴	1	24	yes	—
146	C₂×D₄×C₃₁	C₂⁴	1	12	yes	—
147	C₂×D₄×C₃₁	C₂⁴	1	24	yes	—
148	C₂²×C₄×C₃₁	—	2	12	no	—
149	C₂²×C₄×C₃₁	—	2	24	no	—
150	C₂²×C₄×C₃₁	—	2	24	no	—
151	C₂²×C₄×C₃₁	—	4	12	no	—
152	C₂²×C₄×C₃₁	—	4	6	no	—
153	C₂²×C₄×C₃₁	#41	1	8	yes	complemented
154	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₃₁	C₂⁴	1	12	yes	—
170	C₄⋊C₄×C₃₁	C₂⁴	1	12	yes	—
171	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
172	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
173	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
175	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
176	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
177	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
178	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
179	C₂²⋊C₄×C₃₁	C₂⁴	1	24	yes	—
180	C₂²⋊C₄×C₃₁	C₂⁴	1	48	yes	—
181	C₂²⋊C₄×C₃₁	C₂⁴	1	24	yes	—
182	C₂²⋊C₄×C₃₁	C₂⁴	1	48	yes	—
183	C₄²×C₃₁	C₂⁴	1	8	yes	—
184	C₄²×C₃₁	C₂⁴	1	12	yes	—
185	C₄²×C₃₁	C₂⁴	1	12	yes	—
186	C₄²×C₃₁	C₂⁴	1	24	yes	—
187	C₄²×C₃₁	C₂⁴	1	24	yes	—
188	C₄²×C₃₁	C₂⁴	1	12	yes	—
189	C₄²×C₃₁	C₂⁴	1	12	yes	—
190	C₂×#3480	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	12	no	—
194	C₂³×C₃₁	—	2	12	no	—
195	C₂³×C₃₁	—	2	12	no	—
196	C₂³×C₃₁	C₂×#41	1	4	yes	—
197	C₂³×C₃₁	C₂⁵	1	12	yes	—
198	C₂³×C₃₁	C₂⁵	1	2	yes	—
199	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
200	Q₈×C₃₁	—	2	24	no	—
201	Q₈×C₃₁	—	2	24	no	—
202	Q₈×C₃₁	C₂×#41	1	16	yes	complemented
203	D₄×C₃₁	—	2	48	no	—
204	D₄×C₃₁	—	2	48	no	—
205	D₄×C₃₁	—	2	24	no	—
206	D₄×C₃₁	—	4	24	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	12	no	—
209	C₂×C₄×C₃₁	—	2	6	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	6	no	—
212	C₂×C₄×C₃₁	—	2	6	no	—
213	C₂×C₄×C₃₁	—	2	24	no	—
214	C₂×C₄×C₃₁	—	2	48	no	—
215	C₂×C₄×C₃₁	—	2	48	no	—
216	C₂×C₄×C₃₁	—	4	24	no	—
217	C₂×C₄×C₃₁	—	4	12	no	—
218	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
219	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
220	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
221	C₂×C₄×C₃₁	C₂×#41	1	16	yes	complemented
222	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
223	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
224	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
225	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
226	C₂×C₄×C₃₁	C₂⁵	1	12	yes	—
227	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
228	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
229	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
230	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
231	#3480	C₂×C₃₁	1	32	yes	direct product factor
232	C₂×#559	C₂×C₃₁	1	3	yes	complemented
233	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×#556	C₂×C₃₁	1	3	yes	complemented
235	C₂×#553	C₂×C₃₁	1	6	yes	complemented
236	C₂×#550	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂×#548	C₂×C₃₁	1	2	yes	complemented
238	C₂×#532	C₂×C₃₁	1	3	yes	complemented
239	C₂×#529	C₂×C₃₁	1	6	yes	complemented
240	C₂×C₄×#41	C₂×C₃₁	1	3	yes	complemented
241	C₂×C₄×#41	C₂×C₃₁	1	2	yes	complemented
242	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	2	8	no	—
246	C₂²×C₃₁	—	2	12	no	—
247	C₂²×C₃₁	—	2	12	no	—
248	C₂²×C₃₁	—	2	12	no	—
249	C₂²×C₃₁	—	4	24	no	—
250	C₂²×C₃₁	C₂²×#41	1	4	yes	—
251	C₂²×C₃₁	C₂²×#41	1	2	yes	—
252	C₂²×C₃₁	C₂²×#41	1	2	yes	—
253	C₂²×C₃₁	#585	1	2	yes	—
254	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
255	C₄×C₃₁	—	2	4	no	—
256	C₄×C₃₁	—	2	12	no	—
257	C₄×C₃₁	—	2	6	no	—
258	C₄×C₃₁	—	2	4	no	—
259	C₄×C₃₁	—	2	6	no	—
260	C₄×C₃₁	—	2	6	no	—
261	C₄×C₃₁	C₂²×#41	1	4	yes	—
262	C₄×C₃₁	C₂²×#41	1	12	yes	—
263	C₄×C₃₁	C₂²×#41	1	4	yes	—
264	C₂²×#41	C₂²×C₃₁	1	3	yes	—
265	C₂²×#41	C₂²×C₃₁	1	2	yes	—
266	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
267	#559	C₂²×C₃₁	1	48	yes	complemented
268	#557	C₂²×C₃₁	1	16	yes	complemented
269	#556	C₂²×C₃₁	1	48	yes	complemented
270	#553	C₂²×C₃₁	1	96	yes	complemented
271	#550	C₂²×C₃₁	1	16	yes	complemented
272	#548	C₂²×C₃₁	1	32	yes	complemented
273	#532	C₂²×C₃₁	1	48	yes	complemented
274	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
275	C₂×#127	C₂²×C₃₁	1	3	yes	—
276	#529	C₂²×C₃₁	1	96	yes	complemented
277	C₂×#126	C₂²×C₃₁	1	12	yes	—
278	C₂×#125	C₂²×C₃₁	1	6	yes	—
279	C₂×#124	C₂²×C₃₁	1	6	yes	—
280	C₂×#124	C₂²×C₃₁	1	6	yes	—
281	C₂×#123	C₂²×C₃₁	1	12	yes	—
282	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	6	yes	—
283	C₂×#121	C₂²×C₃₁	1	6	yes	—
284	C₂×#121	C₂²×C₃₁	1	6	yes	—
285	C₂×#121	C₂²×C₃₁	1	6	yes	—
286	C₄×#41	C₂²×C₃₁	1	48	yes	complemented
287	C₄×#41	C₂²×C₃₁	1	32	yes	complemented
288	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
290	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
292	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
293	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
294	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
295	C₂×C₄×Q₈	C₂²×C₃₁	1	16	yes	complemented
296	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
297	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
298	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
299	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
300	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
301	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
302	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
303	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#117	C₂²×C₃₁	1	6	yes	—
305	C₂×#117	C₂²×C₃₁	1	6	yes	—
306	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
307	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
308	C₂²×C₄²	C₂²×C₃₁	1	6	yes	—
309	C₂×C₃₁	—	2	4	no	—
310	C₂×C₃₁	—	4	12	no	—
311	C₂×C₃₁	#3480	1	4	yes	direct product factor
312	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
313	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
314	C₂×C₃₁	C₂×#585	1	1	yes	characteristic
315	C₂×#41	—	2	16	no	—
316	C₂×#41	—	2	24	no	—
317	C₂×#41	C₂³×C₃₁	1	24	yes	—
318	C₂×#41	C₂³×C₃₁	1	16	yes	—
319	C₂²×Q₈	—	2	8	no	—
320	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
321	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
322	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
323	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
324	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
325	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
326	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
327	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
328	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
329	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
330	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
331	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
332	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
333	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₄⋊Q₈	C₂³×C₃₁	1	24	yes	—
335	#127	C₂³×C₃₁	1	24	yes	—
336	#126	C₂³×C₃₁	1	96	yes	—
337	#125	C₂³×C₃₁	1	48	yes	—
338	#124	C₂³×C₃₁	1	48	yes	—
339	#124	C₂³×C₃₁	1	48	yes	—
340	#123	C₂³×C₃₁	1	96	yes	—
341	C₂²⋊Q₈	C₂³×C₃₁	1	48	yes	—
342	#121	C₂³×C₃₁	1	48	yes	—
343	#121	C₂³×C₃₁	1	48	yes	—
344	#121	C₂³×C₃₁	1	48	yes	—
345	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
346	C₄×Q₈	C₂³×C₃₁	1	24	yes	—
347	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
348	C₄×Q₈	C₂³×C₃₁	1	24	yes	—
349	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
350	C₄×Q₈	C₂³×C₃₁	1	24	yes	—
351	C₄×Q₈	C₂³×C₃₁	1	24	yes	—
352	C₄×D₄	C₂³×C₃₁	1	48	yes	—
353	C₄×D₄	C₂³×C₃₁	1	48	yes	—
354	C₄×D₄	C₂³×C₃₁	1	24	yes	—
355	C₄×D₄	C₂³×C₃₁	1	48	yes	—
356	C₄×D₄	C₂³×C₃₁	1	48	yes	—
357	C₄×D₄	C₂³×C₃₁	1	24	yes	—
358	C₄×D₄	C₂³×C₃₁	1	48	yes	—
359	#117	C₂³×C₃₁	1	8	yes	—
360	#117	C₂³×C₃₁	1	48	yes	—
361	#117	C₂³×C₃₁	1	48	yes	—
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
377	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
378	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
379	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
380	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
381	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
382	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
383	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
384	C₂×C₄²	C₂³×C₃₁	1	24	yes	—
385	C₂×C₄²	C₂³×C₃₁	1	48	yes	—
386	C₃₁	C₂×#3480	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
388	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
389	#41	—	2	64	no	—
390	#41	—	2	96	no	—
391	C₂×Q₈	—	2	32	no	—
392	C₂×Q₈	—	2	12	no	—
393	C₂×Q₈	—	2	12	no	—
394	C₂×Q₈	#41×C₃₁	1	8	yes	complemented
395	C₂×Q₈	C₂⁴×C₃₁	1	12	yes	—
396	C₂×Q₈	C₂⁴×C₃₁	1	12	yes	—
397	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
398	C₂×D₄	—	2	24	no	—
399	C₂×D₄	—	2	24	no	—
400	C₂×D₄	—	2	12	no	—
401	C₂×D₄	—	4	12	no	—
402	C₂×D₄	C₂⁴×C₃₁	1	24	yes	—
403	C₂×D₄	C₂⁴×C₃₁	1	24	yes	—
404	C₂×D₄	C₂⁴×C₃₁	1	12	yes	—
405	C₂×D₄	C₂⁴×C₃₁	1	24	yes	—
406	C₂²×C₄	—	2	12	no	—
407	C₂²×C₄	—	2	24	no	—
408	C₂²×C₄	—	2	24	no	—
409	C₂²×C₄	—	4	12	no	—
410	C₂²×C₄	—	4	6	no	—
411	C₂²×C₄	#41×C₃₁	1	8	yes	complemented
412	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
414	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
415	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
416	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
417	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
418	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
419	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
420	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
421	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
427	C₄⋊C₄	C₂⁴×C₃₁	1	12	yes	—
428	C₄⋊C₄	C₂⁴×C₃₁	1	12	yes	—
429	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
430	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
431	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
432	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
433	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
434	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
435	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
436	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
437	C₂²⋊C₄	C₂⁴×C₃₁	1	24	yes	—
438	C₂²⋊C₄	C₂⁴×C₃₁	1	48	yes	—
439	C₂²⋊C₄	C₂⁴×C₃₁	1	24	yes	—
440	C₂²⋊C₄	C₂⁴×C₃₁	1	48	yes	—
441	C₄²	C₂⁴×C₃₁	1	8	yes	—
442	C₄²	C₂⁴×C₃₁	1	12	yes	—
443	C₄²	C₂⁴×C₃₁	1	12	yes	—
444	C₄²	C₂⁴×C₃₁	1	24	yes	—
445	C₄²	C₂⁴×C₃₁	1	24	yes	—
446	C₄²	C₂⁴×C₃₁	1	12	yes	—
447	C₄²	C₂⁴×C₃₁	1	12	yes	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	12	no	—
451	C₂³	—	2	12	no	—
452	C₂³	—	2	12	no	—
453	C₂³	C₂×#41×C₃₁	1	4	yes	—
454	C₂³	C₂⁵×C₃₁	1	12	yes	—
455	C₂³	C₂⁵×C₃₁	1	2	yes	—
456	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
457	Q₈	—	2	24	no	—
458	Q₈	—	2	24	no	—
459	Q₈	C₂×#41×C₃₁	1	16	yes	complemented
460	D₄	—	2	48	no	—
461	D₄	—	2	48	no	—
462	D₄	—	2	24	no	—
463	D₄	—	4	24	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	12	no	—
466	C₂×C₄	—	2	6	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	6	no	—
469	C₂×C₄	—	2	6	no	—
470	C₂×C₄	—	2	24	no	—
471	C₂×C₄	—	2	48	no	—
472	C₂×C₄	—	2	48	no	—
473	C₂×C₄	—	4	24	no	—
474	C₂×C₄	—	4	12	no	—
475	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
476	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
477	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
478	C₂×C₄	C₂×#41×C₃₁	1	16	yes	complemented
479	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
480	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
481	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
482	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
483	C₂×C₄	C₂⁵×C₃₁	1	12	yes	—
484	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
485	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
486	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
487	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	8	no	—
491	C₂²	—	2	12	no	—
492	C₂²	—	2	12	no	—
493	C₂²	—	2	12	no	—
494	C₂²	—	4	24	no	—
495	C₂²	C₂²×#41×C₃₁	1	4	yes	—
496	C₂²	C₂²×#41×C₃₁	1	2	yes	—
497	C₂²	C₂²×#41×C₃₁	1	2	yes	—
498	C₂²	C₃₁×#585	1	2	yes	—
499	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
500	C₄	—	2	4	no	—
501	C₄	—	2	12	no	—
502	C₄	—	2	6	no	—
503	C₄	—	2	4	no	—
504	C₄	—	2	6	no	—
505	C₄	—	2	6	no	—
506	C₄	C₂²×#41×C₃₁	1	4	yes	—
507	C₄	C₂²×#41×C₃₁	1	12	yes	—
508	C₄	C₂²×#41×C₃₁	1	4	yes	—
509	C₂	—	2	4	no	—
510	C₂	—	4	12	no	—
511	C₂	C₃₁×#3480	1	4	yes	direct product factor
512	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
513	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
514	C₂	C₂×C₃₁×#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
socle	C₂³×C₃₁	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3480	C₂	direct (N × H)
2. C₂×C₃₁×#559	C₂	semidirect
3. C₂×C₃₁×#557	C₂	semidirect
4. C₂×C₃₁×#556	C₂	semidirect
5. C₂×C₃₁×#553	C₂	semidirect
6. C₂×C₃₁×#550	C₂	semidirect
7. C₂×C₃₁×#548	C₂	semidirect
8. C₂×C₃₁×#532	C₂	semidirect
9. C₂×C₃₁×#529	C₂	semidirect
10. C₂×C₄×#41×C₃₁	C₂	semidirect
11. C₂×C₄×#41×C₃₁	C₂	semidirect
12. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
16. C₃₁×#559	C₂²	semidirect
17. C₃₁×#557	C₂²	semidirect
18. C₃₁×#556	C₂²	semidirect
19. C₃₁×#553	C₂²	semidirect
20. C₃₁×#550	C₂²	semidirect
21. C₃₁×#548	C₂²	semidirect
22. C₃₁×#532	C₂²	semidirect
25. C₃₁×#529	C₂²	semidirect
35. C₄×#41×C₃₁	C₂²	semidirect
36. C₄×#41×C₃₁	C₂²	semidirect
44. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
136. C₂×Q₈×C₃₁	#41	semidirect
153. C₂²×C₄×C₃₁	#41	semidirect
190. C₂×#3480	C₃₁	direct (N × H)
202. Q₈×C₃₁	C₂×#41	semidirect
221. C₂×C₄×C₃₁	C₂×#41	semidirect
231. #3480	C₂×C₃₁	direct (N × H)
232. C₂×#559	C₂×C₃₁	semidirect
233. C₂×#557	C₂×C₃₁	semidirect
234. C₂×#556	C₂×C₃₁	semidirect
235. C₂×#553	C₂×C₃₁	semidirect
236. C₂×#550	C₂×C₃₁	semidirect
237. C₂×#548	C₂×C₃₁	semidirect
238. C₂×#532	C₂×C₃₁	semidirect
239. C₂×#529	C₂×C₃₁	semidirect
240. C₂×C₄×#41	C₂×C₃₁	semidirect
241. C₂×C₄×#41	C₂×C₃₁	semidirect
242. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
267. #559	C₂²×C₃₁	semidirect
268. #557	C₂²×C₃₁	semidirect
269. #556	C₂²×C₃₁	semidirect
270. #553	C₂²×C₃₁	semidirect
271. #550	C₂²×C₃₁	semidirect
272. #548	C₂²×C₃₁	semidirect
273. #532	C₂²×C₃₁	semidirect
276. #529	C₂²×C₃₁	semidirect
286. C₄×#41	C₂²×C₃₁	semidirect
287. C₄×#41	C₂²×C₃₁	semidirect
295. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
311. C₂×C₃₁	#3480	direct (N × H)
386. C₃₁	C₂×#3480	direct (N × H)
394. C₂×Q₈	#41×C₃₁	semidirect
411. C₂²×C₄	#41×C₃₁	semidirect
459. Q₈	C₂×#41×C₃₁	semidirect
478. C₂×C₄	C₂×#41×C₃₁	semidirect
511. C₂	C₃₁×#3480	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3480	2
C₂×C₃₁×#559	2
C₂×C₃₁×#557	2
C₂×C₃₁×#556	2
C₂×C₃₁×#553	2
C₂×C₃₁×#550	2
C₂×C₃₁×#548	2
C₂×C₃₁×#532	2
C₂×C₃₁×#529	2
C₂×C₄×#41×C₃₁	2
C₂²×C₄×Q₈×C₃₁	2
C₂×#3480	31

Subgroups — C2×C31×#3480 #21519470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3480 #21519470