Subgroups — C₃₁×#34060 #21490819

Order 7936.

2458 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2320	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2319	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2318	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2317	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2314	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2311	C₂	1	2	yes	maximal
8	C₃₁×#2295	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2293	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2290	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2285	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2283	C₂	1	2	yes	maximal, complemented
13	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×#41×C₃₁	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₂×C₃₁×#117	C₂²	1	4	yes	—
25	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#117	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#391	C₂²	1	4	yes	—
32	C₃₁×#390	C₂²	1	4	yes	—
33	C₃₁×#389	C₂²	1	4	yes	—
34	C₃₁×#389	C₂²	1	4	yes	—
35	C₃₁×#388	C₂²	1	8	yes	—
36	C₃₁×#388	C₂²	1	8	yes	—
37	C₃₁×#387	C₂²	1	8	yes	—
38	C₃₁×#387	C₂²	1	8	yes	—
39	C₃₁×#386	C₂²	1	8	yes	—
40	C₃₁×#385	C₂²	1	8	yes	—
41	C₃₁×#384	C₂²	1	4	yes	—
42	C₃₁×#383	C₂²	1	4	yes	—
43	C₃₁×#382	C₂²	1	4	yes	—
44	C₃₁×#382	C₂²	1	4	yes	—
45	C₃₁×#381	C₂²	1	4	yes	—
46	C₃₁×#380	C₂²	1	4	yes	—
47	C₃₁×#380	C₂²	1	4	yes	—
48	C₃₁×#380	C₂²	1	4	yes	—
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
50	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
57	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
58	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
59	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
60	C₃₁×#376	C₂²	1	2	yes	—
61	C₃₁×#376	C₂²	1	4	yes	—
62	C₂×C₃₁×#95	C₂²	1	2	yes	—
63	C₄³×C₃₁	C₂²	1	2	yes	—
64	C₂×#41×C₃₁	—	2	8	no	—
65	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	—
79	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	—
80	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
81	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
82	C₃₁×#117	—	2	8	no	—
83	C₃₁×#117	—	4	8	no	—
84	C₃₁×#117	C₂×C₄	1	8	yes	—
85	C₃₁×#117	C₂×C₄	1	16	yes	complemented
86	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄²×C₃₁	—	4	4	no	—
104	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
105	C₂×C₄²×C₃₁	C₂×C₄	1	16	yes	complemented
106	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
111	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
119	C₃₁×#95	C₂³	1	4	yes	—
120	C₃₁×#95	C₂³	1	4	yes	—
121	C₃₁×#95	C₂³	1	2	yes	—
122	C₃₁×#95	C₂³	1	2	yes	—
123	C₃₁×#95	C₂³	1	4	yes	—
124	C₃₁×#95	C₂³	1	8	yes	—
125	C₃₁×#95	C₂³	1	8	yes	—
126	C₃₁×#95	C₂³	1	8	yes	—
127	C₃₁×#95	C₂³	1	8	yes	—
128	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
129	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
130	#41×C₃₁	—	2	32	no	—
131	C₂×Q₈×C₃₁	—	2	4	no	—
132	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
133	C₂×D₄×C₃₁	—	2	8	no	—
134	C₂×D₄×C₃₁	—	2	4	no	—
135	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
136	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₃₁	—	2	8	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	2	8	no	—
143	C₂²×C₄×C₃₁	—	4	4	no	—
144	C₂²×C₄×C₃₁	—	4	2	no	—
145	C₂²×C₄×C₃₁	—	4	4	no	—
146	C₂²×C₄×C₃₁	—	4	4	no	—
147	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
149	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
150	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
152	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₃₁	#41	1	8	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	2	8	no	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	4	8	no	—
171	C₄⋊C₄×C₃₁	—	4	8	no	—
172	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
173	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
174	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
175	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
176	C₂²⋊C₄×C₃₁	—	2	8	no	—
177	C₂²⋊C₄×C₃₁	—	4	8	no	—
178	C₂²⋊C₄×C₃₁	—	4	8	no	—
179	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
180	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
181	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
182	C₄²×C₃₁	—	2	8	no	—
183	C₄²×C₃₁	—	2	8	no	—
184	C₄²×C₃₁	—	2	4	no	—
185	C₄²×C₃₁	—	2	4	no	—
186	C₄²×C₃₁	—	4	4	no	—
187	C₄²×C₃₁	—	4	4	no	—
188	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
189	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
190	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
191	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
192	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
193	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
194	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
195	#34060	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	1	no	—
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	4	no	—
200	C₂³×C₃₁	—	2	2	no	—
201	C₂³×C₃₁	—	2	2	no	—
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
204	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
205	C₂³×C₃₁	C₂×#41	1	4	yes	—
206	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
207	Q₈×C₃₁	—	2	8	no	—
208	D₄×C₃₁	—	2	16	no	—
209	D₄×C₃₁	—	2	8	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	16	no	—
219	C₂×C₄×C₃₁	—	2	8	no	—
220	C₂×C₄×C₃₁	—	2	16	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	8	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	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	8	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	8	no	—
232	C₂×C₄×C₃₁	—	4	16	no	—
233	C₂×C₄×C₃₁	#117	1	16	yes	complemented
234	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
235	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
237	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
238	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
239	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
240	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
243	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
244	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
245	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
246	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
247	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
248	#2320	C₂×C₃₁	1	2	yes	complemented
249	#2319	C₂×C₃₁	1	2	yes	complemented
250	#2318	C₂×C₃₁	1	4	yes	complemented
251	#2317	C₂×C₃₁	1	2	yes	complemented
252	#2314	C₂×C₃₁	1	4	yes	complemented
253	#2311	C₂×C₃₁	1	2	yes	—
254	#2295	C₂×C₃₁	1	2	yes	complemented
255	#2293	C₂×C₃₁	1	2	yes	complemented
256	#2290	C₂×C₃₁	1	4	yes	complemented
257	#2285	C₂×C₃₁	1	2	yes	complemented
258	#2283	C₂×C₃₁	1	2	yes	complemented
259	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	2	2	no	—
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	—	2	4	no	—
268	C₂²×C₃₁	—	4	4	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	C₂×#117	1	4	yes	—
271	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
272	C₂²×C₃₁	C₂²×#41	1	2	yes	—
273	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
275	C₂²×C₃₁	#585	1	2	yes	—
276	C₄×C₃₁	—	2	4	no	—
277	C₄×C₃₁	—	2	2	no	—
278	C₄×C₃₁	—	2	4	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	2	4	no	—
282	C₄×C₃₁	—	4	4	no	—
283	C₄×C₃₁	—	4	8	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	C₂×#117	1	4	yes	—
286	C₄×C₃₁	C₂×#117	1	4	yes	—
287	C₄×C₃₁	#520	1	4	yes	—
288	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
289	C₄×#41	C₄×C₃₁	1	16	yes	complemented
290	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
293	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
294	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#117	C₂²×C₃₁	1	2	yes	—
297	C₂×#117	C₂²×C₃₁	1	4	yes	—
298	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×#117	C₂²×C₃₁	1	2	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
301	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
302	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
303	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
304	#391	C₂²×C₃₁	1	4	yes	—
305	#390	C₂²×C₃₁	1	4	yes	—
306	#389	C₂²×C₃₁	1	4	yes	—
307	#389	C₂²×C₃₁	1	4	yes	—
308	#388	C₂²×C₃₁	1	8	yes	—
309	#388	C₂²×C₃₁	1	8	yes	—
310	#387	C₂²×C₃₁	1	8	yes	—
311	#387	C₂²×C₃₁	1	8	yes	—
312	#386	C₂²×C₃₁	1	8	yes	—
313	#385	C₂²×C₃₁	1	8	yes	—
314	#384	C₂²×C₃₁	1	4	yes	—
315	#383	C₂²×C₃₁	1	4	yes	—
316	#382	C₂²×C₃₁	1	4	yes	—
317	#382	C₂²×C₃₁	1	4	yes	—
318	#381	C₂²×C₃₁	1	4	yes	—
319	#380	C₂²×C₃₁	1	4	yes	—
320	#380	C₂²×C₃₁	1	4	yes	—
321	#380	C₂²×C₃₁	1	4	yes	—
322	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
323	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
324	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
325	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
326	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
327	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
328	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
329	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
330	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
331	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
332	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
333	#376	C₂²×C₃₁	1	2	yes	—
334	#376	C₂²×C₃₁	1	4	yes	—
335	C₂×#95	C₂²×C₃₁	1	2	yes	—
336	C₄³	C₂²×C₃₁	1	2	yes	—
337	C₂×C₃₁	—	2	4	no	—
338	C₂×C₃₁	—	4	4	no	—
339	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
340	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
341	C₂×C₃₁	#3428	1	1	yes	characteristic
342	C₂×C₃₁	#3460	1	2	yes	—
343	C₂×C₃₁	#3461	1	2	yes	—
344	C₂×#41	—	2	8	no	—
345	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
346	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
347	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
348	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
350	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
351	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
353	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
357	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
358	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	—
359	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	—
360	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
361	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
362	#117	—	2	8	no	—
363	#117	—	4	8	no	—
364	#117	C₂×C₄×C₃₁	1	8	yes	—
365	#117	C₂×C₄×C₃₁	1	16	yes	complemented
366	C₂×C₄⋊C₄	—	2	8	no	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
383	C₂×C₄²	—	4	4	no	—
384	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
385	C₂×C₄²	C₂×C₄×C₃₁	1	16	yes	complemented
386	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
387	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
388	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
389	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
390	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
391	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
392	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
394	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
397	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
398	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
399	#95	C₂³×C₃₁	1	4	yes	—
400	#95	C₂³×C₃₁	1	4	yes	—
401	#95	C₂³×C₃₁	1	2	yes	—
402	#95	C₂³×C₃₁	1	2	yes	—
403	#95	C₂³×C₃₁	1	4	yes	—
404	#95	C₂³×C₃₁	1	8	yes	—
405	#95	C₂³×C₃₁	1	8	yes	—
406	#95	C₂³×C₃₁	1	8	yes	—
407	#95	C₂³×C₃₁	1	8	yes	—
408	C₃₁	#34060	1	1	yes	characteristic, direct product factor, Sylow
409	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
410	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
411	#41	—	2	32	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
417	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	8	no	—
424	C₂²×C₄	—	4	4	no	—
425	C₂²×C₄	—	4	2	no	—
426	C₂²×C₄	—	4	4	no	—
427	C₂²×C₄	—	4	4	no	—
428	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
429	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
430	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
431	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
432	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
433	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
434	C₂²×C₄	#41×C₃₁	1	8	yes	—
435	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
436	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
437	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	4	8	no	—
452	C₄⋊C₄	—	4	8	no	—
453	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
454	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
455	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
456	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	4	8	no	—
459	C₂²⋊C₄	—	4	8	no	—
460	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
461	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
462	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
463	C₄²	—	2	8	no	—
464	C₄²	—	2	8	no	—
465	C₄²	—	2	4	no	—
466	C₄²	—	2	4	no	—
467	C₄²	—	4	4	no	—
468	C₄²	—	4	4	no	—
469	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
470	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
471	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
472	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
473	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
474	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
475	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	4	no	—
483	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
484	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
485	C₂³	C₂×#41×C₃₁	1	4	yes	—
486	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
487	Q₈	—	2	8	no	—
488	D₄	—	2	16	no	—
489	D₄	—	2	8	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	16	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	16	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	4	8	no	—
512	C₂×C₄	—	4	16	no	—
513	C₂×C₄	C₃₁×#117	1	16	yes	complemented
514	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
515	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
516	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
517	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
518	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
519	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
520	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
521	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
522	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
523	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
524	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
525	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
526	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	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	4	no	—
535	C₂²	—	4	4	no	—
536	C₂²	C₂×C₃₁×#117	1	4	yes	—
537	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
538	C₂²	C₂²×#41×C₃₁	1	2	yes	—
539	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
540	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
541	C₂²	C₃₁×#585	1	2	yes	—
542	C₄	—	2	4	no	—
543	C₄	—	2	2	no	—
544	C₄	—	2	4	no	—
545	C₄	—	2	2	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	4	no	—
548	C₄	—	4	8	no	—
549	C₄	—	4	4	no	—
550	C₄	—	4	4	no	—
551	C₄	C₂×C₃₁×#117	1	4	yes	—
552	C₄	C₂×C₃₁×#117	1	4	yes	—
553	C₄	C₃₁×#520	1	4	yes	—
554	C₂	—	2	4	no	—
555	C₂	—	4	4	no	—
556	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
557	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
558	C₂	C₃₁×#3428	1	1	yes	characteristic
559	C₂	C₃₁×#3460	1	2	yes	—
560	C₂	C₃₁×#3461	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂²×#41×C₃₁	C₂²
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
2. C₃₁×#2320	C₂	semidirect
3. C₃₁×#2319	C₂	semidirect
4. C₃₁×#2318	C₂	semidirect
5. C₃₁×#2317	C₂	semidirect
6. C₃₁×#2314	C₂	semidirect
8. C₃₁×#2295	C₂	semidirect
9. C₃₁×#2293	C₂	semidirect
10. C₃₁×#2290	C₂	semidirect
11. C₃₁×#2285	C₂	semidirect
12. C₃₁×#2283	C₂	semidirect
13. C₄×C₃₁×#117	C₂	semidirect
14. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
16. C₄×#41×C₃₁	C₄	semidirect
76. C₄×Q₈×C₃₁	C₂×C₄	semidirect
77. C₄×Q₈×C₃₁	C₂×C₄	semidirect
80. C₄×D₄×C₃₁	C₂×C₄	semidirect
81. C₄×D₄×C₃₁	C₂×C₄	semidirect
85. C₃₁×#117	C₂×C₄	semidirect
105. C₂×C₄²×C₃₁	C₂×C₄	semidirect
195. #34060	C₃₁	direct (N × H)
233. C₂×C₄×C₃₁	#117	semidirect
248. #2320	C₂×C₃₁	semidirect
249. #2319	C₂×C₃₁	semidirect
250. #2318	C₂×C₃₁	semidirect
251. #2317	C₂×C₃₁	semidirect
252. #2314	C₂×C₃₁	semidirect
254. #2295	C₂×C₃₁	semidirect
255. #2293	C₂×C₃₁	semidirect
256. #2290	C₂×C₃₁	semidirect
257. #2285	C₂×C₃₁	semidirect
258. #2283	C₂×C₃₁	semidirect
259. C₄×#117	C₂×C₃₁	semidirect
260. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
289. C₄×#41	C₄×C₃₁	semidirect
356. C₄×Q₈	C₂×C₄×C₃₁	semidirect
357. C₄×Q₈	C₂×C₄×C₃₁	semidirect
360. C₄×D₄	C₂×C₄×C₃₁	semidirect
361. C₄×D₄	C₂×C₄×C₃₁	semidirect
365. #117	C₂×C₄×C₃₁	semidirect
385. C₂×C₄²	C₂×C₄×C₃₁	semidirect
408. C₃₁	#34060	direct (N × H)
513. C₂×C₄	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₃₁	2
C₃₁×#2320	2
C₃₁×#2319	2
C₃₁×#2318	2
C₃₁×#2317	2
C₃₁×#2314	2
C₃₁×#2311	2
C₃₁×#2295	2
C₃₁×#2293	2
C₃₁×#2290	2
C₃₁×#2285	2
C₃₁×#2283	2
C₄×C₃₁×#117	2
C₂×C₄×C₄⋊C₄×C₃₁	2
#34060	31

Subgroups — C31×#34060 #21490819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34060 #21490819