Subgroups — C₂×C₃₁×#2374 #21490909

Order 7936.

3094 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#123	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2374	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#388	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#388	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#387	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#386	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
11	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#123	C₄	1	16	yes	complemented
15	C₂×C₃₁×#123	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#390	C₂²	1	8	yes	—
30	C₃₁×#388	C₂²	1	16	yes	complemented
31	C₃₁×#388	C₂²	1	16	yes	complemented
32	C₃₁×#387	C₂²	1	16	yes	complemented
33	C₃₁×#386	C₂²	1	8	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	16	yes	complemented
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	16	yes	complemented
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	16	yes	complemented
37	C₂×C₃₁×#95	C₂²	1	2	yes	—
38	C₂×C₃₁×#95	C₂²	1	2	yes	—
39	C₂×C₃₁×#95	C₂²	1	2	yes	—
40	C₂×C₃₁×#95	C₂²	1	2	yes	—
41	C₂×C₃₁×#95	C₂²	1	2	yes	—
42	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
43	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	—	4	2	no	—
45	C₂²×D₄×C₃₁	—	4	2	no	—
46	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
48	C₂³×C₄×C₃₁	—	2	8	no	—
49	C₂³×C₄×C₃₁	—	4	2	no	—
50	C₂³×C₄×C₃₁	—	4	2	no	—
51	C₂³×C₄×C₃₁	—	4	2	no	—
52	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
53	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
62	C₃₁×#123	C₂×C₄	1	64	yes	complemented
63	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
64	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
74	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
75	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
77	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄²×C₃₁	—	2	8	no	—
82	C₂×C₄²×C₃₁	—	4	2	no	—
83	C₂×C₄²×C₃₁	—	4	4	no	—
84	C₂×C₄²×C₃₁	—	4	4	no	—
85	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
87	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
88	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
89	C₃₁×#95	C₂³	1	8	yes	—
90	C₃₁×#95	C₂³	1	8	yes	—
91	C₃₁×#95	C₂³	1	8	yes	—
92	C₃₁×#95	C₂³	1	8	yes	—
93	C₃₁×#95	C₂³	1	8	yes	—
94	C₂⁴×C₃₁	—	2	4	no	—
95	C₂⁴×C₃₁	—	2	4	no	—
96	C₂⁴×C₃₁	—	2	4	no	—
97	C₂⁴×C₃₁	—	2	4	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂⁴×C₃₁	—	2	2	no	—
100	C₂⁴×C₃₁	—	2	2	no	—
101	C₂⁴×C₃₁	—	2	2	no	—
102	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
103	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
104	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
105	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
106	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₂×D₄×C₃₁	—	4	16	no	—
108	C₂×D₄×C₃₁	—	4	4	no	—
109	C₂×D₄×C₃₁	—	4	4	no	—
110	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	4	no	—
114	C₂²×C₄×C₃₁	—	2	4	no	—
115	C₂²×C₄×C₃₁	—	2	4	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	2	4	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	16	no	—
128	C₂²×C₄×C₃₁	—	4	16	no	—
129	C₂²×C₄×C₃₁	—	4	4	no	—
130	C₂²×C₄×C₃₁	—	4	4	no	—
131	C₂²×C₄×C₃₁	—	4	4	no	—
132	C₂²×C₄×C₃₁	—	4	16	no	—
133	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
134	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
137	C₂²×C₄×C₃₁	#41	1	8	yes	—
138	C₂²×C₄×C₃₁	#41	1	8	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
146	C₄⋊C₄×C₃₁	—	2	16	no	—
147	C₄⋊C₄×C₃₁	—	4	4	no	—
148	C₄⋊C₄×C₃₁	C₂²×C₄	1	32	yes	complemented
149	C₂²⋊C₄×C₃₁	—	2	16	no	—
150	C₂²⋊C₄×C₃₁	—	2	16	no	—
151	C₂²⋊C₄×C₃₁	—	2	16	no	—
152	C₂²⋊C₄×C₃₁	—	4	4	no	—
153	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	—
154	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
155	C₄²×C₃₁	—	2	16	no	—
156	C₄²×C₃₁	—	4	4	no	—
157	C₄²×C₃₁	—	4	8	no	—
158	C₄²×C₃₁	—	4	8	no	—
159	C₂×#2374	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	16	no	—
161	C₂³×C₃₁	—	2	16	no	—
162	C₂³×C₃₁	—	2	8	no	—
163	C₂³×C₃₁	—	2	8	no	—
164	C₂³×C₃₁	—	2	4	no	—
165	C₂³×C₃₁	—	2	4	no	—
166	C₂³×C₃₁	—	2	4	no	—
167	C₂³×C₃₁	—	2	4	no	—
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	4	4	no	—
173	C₂³×C₃₁	—	4	4	no	—
174	C₂³×C₃₁	—	4	8	no	—
175	C₂³×C₃₁	—	4	8	no	—
176	C₂³×C₃₁	—	4	4	no	—
177	C₂³×C₃₁	—	4	4	no	—
178	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
179	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
180	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
181	C₂³×C₃₁	C₂×#41	1	4	yes	—
182	C₂³×C₃₁	C₂×#41	1	4	yes	—
183	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
184	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
185	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
186	D₄×C₃₁	—	4	16	no	—
187	C₂×C₄×C₃₁	—	2	16	no	—
188	C₂×C₄×C₃₁	—	2	16	no	—
189	C₂×C₄×C₃₁	—	2	8	no	—
190	C₂×C₄×C₃₁	—	2	16	no	—
191	C₂×C₄×C₃₁	—	2	16	no	—
192	C₂×C₄×C₃₁	—	2	4	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	4	16	no	—
209	C₂×C₄×C₃₁	—	4	16	no	—
210	C₂×C₄×C₃₁	—	4	8	no	—
211	C₂×C₄×C₃₁	—	4	8	no	—
212	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
213	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
214	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
215	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
216	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
217	C₂²×#123	C₂×C₃₁	1	1	yes	characteristic
218	#2374	C₂×C₃₁	1	16	yes	direct product factor
219	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
220	C₂×#388	C₂×C₃₁	1	2	yes	complemented
221	C₂×#388	C₂×C₃₁	1	2	yes	complemented
222	C₂×#387	C₂×C₃₁	1	2	yes	complemented
223	C₂×#386	C₂×C₃₁	1	1	yes	characteristic
224	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
225	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
226	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
227	C₂²×C₃₁	—	2	8	no	—
228	C₂²×C₃₁	—	2	8	no	—
229	C₂²×C₃₁	—	2	4	no	—
230	C₂²×C₃₁	—	2	4	no	—
231	C₂²×C₃₁	—	4	16	no	—
232	C₂²×C₃₁	—	4	16	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	4	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	C₄×#41	1	8	yes	—
240	C₂²×C₃₁	#518	1	4	yes	—
241	C₂²×C₃₁	#551	1	4	yes	—
242	C₂²×C₃₁	#553	1	8	yes	—
243	C₂²×C₃₁	#559	1	4	yes	—
244	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
245	C₂²×C₃₁	C₂²×#41	1	2	yes	—
246	C₂²×C₃₁	C₂²×#41	1	2	yes	—
247	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
249	C₄×C₃₁	—	2	8	no	—
250	C₄×C₃₁	—	2	8	no	—
251	C₄×C₃₁	—	2	4	no	—
252	C₄×C₃₁	—	2	8	no	—
253	C₄×C₃₁	—	2	8	no	—
254	C₄×C₃₁	—	4	4	no	—
255	C₄×C₃₁	—	4	4	no	—
256	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
257	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#123	C₄×C₃₁	1	16	yes	complemented
260	C₂×#123	C₂²×C₃₁	1	8	yes	—
261	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
262	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
263	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
264	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
265	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
266	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
267	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
269	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
271	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
272	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
273	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
274	#390	C₂²×C₃₁	1	8	yes	—
275	#388	C₂²×C₃₁	1	16	yes	complemented
276	#388	C₂²×C₃₁	1	16	yes	complemented
277	#387	C₂²×C₃₁	1	16	yes	complemented
278	#386	C₂²×C₃₁	1	8	yes	—
279	C₄×C₄⋊C₄	C₂²×C₃₁	1	16	yes	complemented
280	C₄×C₂²⋊C₄	C₂²×C₃₁	1	16	yes	complemented
281	C₄×C₂²⋊C₄	C₂²×C₃₁	1	16	yes	complemented
282	C₂×#95	C₂²×C₃₁	1	2	yes	—
283	C₂×#95	C₂²×C₃₁	1	2	yes	—
284	C₂×#95	C₂²×C₃₁	1	2	yes	—
285	C₂×#95	C₂²×C₃₁	1	2	yes	—
286	C₂×#95	C₂²×C₃₁	1	2	yes	—
287	C₂×C₃₁	—	4	4	no	—
288	C₂×C₃₁	—	4	4	no	—
289	C₂×C₃₁	#2374	1	8	yes	direct product factor
290	C₂×C₃₁	C₂×C₄×#41	1	2	yes	—
291	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
292	C₂×C₃₁	C₂×#551	1	1	yes	characteristic
293	C₂×C₃₁	C₂×#553	1	2	yes	—
294	C₂×C₃₁	C₂×#559	1	1	yes	characteristic
295	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
296	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×D₄	—	4	2	no	—
298	C₂²×D₄	—	4	2	no	—
299	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
300	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
301	C₂³×C₄	—	2	8	no	—
302	C₂³×C₄	—	4	2	no	—
303	C₂³×C₄	—	4	2	no	—
304	C₂³×C₄	—	4	2	no	—
305	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	complemented
306	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
307	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
308	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
309	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
310	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
311	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
313	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
314	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
315	#123	C₂×C₄×C₃₁	1	64	yes	complemented
316	C₂×C₄⋊C₄	—	2	8	no	—
317	C₂×C₄⋊C₄	—	4	2	no	—
318	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
319	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
320	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
321	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
322	C₂×C₂²⋊C₄	—	2	8	no	—
323	C₂×C₂²⋊C₄	—	2	8	no	—
324	C₂×C₂²⋊C₄	—	2	8	no	—
325	C₂×C₂²⋊C₄	—	4	2	no	—
326	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	—
327	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
328	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
334	C₂×C₄²	—	2	8	no	—
335	C₂×C₄²	—	4	2	no	—
336	C₂×C₄²	—	4	4	no	—
337	C₂×C₄²	—	4	4	no	—
338	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
339	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
340	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
341	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
342	#95	C₂³×C₃₁	1	8	yes	—
343	#95	C₂³×C₃₁	1	8	yes	—
344	#95	C₂³×C₃₁	1	8	yes	—
345	#95	C₂³×C₃₁	1	8	yes	—
346	#95	C₂³×C₃₁	1	8	yes	—
347	C₃₁	C₂×#2374	1	1	yes	characteristic, direct product factor, Sylow
348	C₂⁴	—	2	4	no	—
349	C₂⁴	—	2	4	no	—
350	C₂⁴	—	2	4	no	—
351	C₂⁴	—	2	4	no	—
352	C₂⁴	—	2	2	no	—
353	C₂⁴	—	2	2	no	—
354	C₂⁴	—	2	2	no	—
355	C₂⁴	—	2	2	no	—
356	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
357	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
358	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
359	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
360	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
361	C₂×D₄	—	4	16	no	—
362	C₂×D₄	—	4	4	no	—
363	C₂×D₄	—	4	4	no	—
364	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
365	C₂²×C₄	—	2	4	no	—
366	C₂²×C₄	—	2	4	no	—
367	C₂²×C₄	—	2	4	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	4	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	4	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	16	no	—
382	C₂²×C₄	—	4	16	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	4	16	no	—
387	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
388	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
389	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
390	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	complemented
391	C₂²×C₄	#41×C₃₁	1	8	yes	—
392	C₂²×C₄	#41×C₃₁	1	8	yes	—
393	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
394	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
397	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
400	C₄⋊C₄	—	2	16	no	—
401	C₄⋊C₄	—	4	4	no	—
402	C₄⋊C₄	C₂²×C₄×C₃₁	1	32	yes	complemented
403	C₂²⋊C₄	—	2	16	no	—
404	C₂²⋊C₄	—	2	16	no	—
405	C₂²⋊C₄	—	2	16	no	—
406	C₂²⋊C₄	—	4	4	no	—
407	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	—
408	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	complemented
409	C₄²	—	2	16	no	—
410	C₄²	—	4	4	no	—
411	C₄²	—	4	8	no	—
412	C₄²	—	4	8	no	—
413	C₂³	—	2	16	no	—
414	C₂³	—	2	16	no	—
415	C₂³	—	2	8	no	—
416	C₂³	—	2	8	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	4	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	4	8	no	—
428	C₂³	—	4	8	no	—
429	C₂³	—	4	4	no	—
430	C₂³	—	4	4	no	—
431	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
432	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
433	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
434	C₂³	C₂×#41×C₃₁	1	4	yes	—
435	C₂³	C₂×#41×C₃₁	1	4	yes	—
436	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
437	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
438	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
439	D₄	—	4	16	no	—
440	C₂×C₄	—	2	16	no	—
441	C₂×C₄	—	2	16	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	16	no	—
444	C₂×C₄	—	2	16	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	4	16	no	—
462	C₂×C₄	—	4	16	no	—
463	C₂×C₄	—	4	8	no	—
464	C₂×C₄	—	4	8	no	—
465	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
466	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
467	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
468	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
469	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
470	C₂²	—	2	8	no	—
471	C₂²	—	2	8	no	—
472	C₂²	—	2	4	no	—
473	C₂²	—	2	4	no	—
474	C₂²	—	4	16	no	—
475	C₂²	—	4	16	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	C₄×#41×C₃₁	1	8	yes	—
483	C₂²	C₃₁×#518	1	4	yes	—
484	C₂²	C₃₁×#551	1	4	yes	—
485	C₂²	C₃₁×#553	1	8	yes	—
486	C₂²	C₃₁×#559	1	4	yes	—
487	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
488	C₂²	C₂²×#41×C₃₁	1	2	yes	—
489	C₂²	C₂²×#41×C₃₁	1	2	yes	—
490	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
491	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
492	C₄	—	2	8	no	—
493	C₄	—	2	8	no	—
494	C₄	—	2	4	no	—
495	C₄	—	2	8	no	—
496	C₄	—	2	8	no	—
497	C₄	—	4	4	no	—
498	C₄	—	4	4	no	—
499	C₂	—	4	4	no	—
500	C₂	—	4	4	no	—
501	C₂	C₃₁×#2374	1	8	yes	direct product factor
502	C₂	C₂×C₄×#41×C₃₁	1	2	yes	—
503	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
504	C₂	C₂×C₃₁×#551	1	1	yes	characteristic
505	C₂	C₂×C₃₁×#553	1	2	yes	—
506	C₂	C₂×C₃₁×#559	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2374	C₂	direct (N × H)
4. C₂×C₃₁×#388	C₂	semidirect
5. C₂×C₃₁×#388	C₂	semidirect
6. C₂×C₃₁×#387	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
10. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
14. C₂×C₃₁×#123	C₄	semidirect
30. C₃₁×#388	C₂²	semidirect
31. C₃₁×#388	C₂²	semidirect
32. C₃₁×#387	C₂²	semidirect
34. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
35. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
36. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
52. C₂³×C₄×C₃₁	C₂×C₄	semidirect
62. C₃₁×#123	C₂×C₄	semidirect
65. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
74. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
136. C₂²×C₄×C₃₁	C₂²×C₄	semidirect
148. C₄⋊C₄×C₃₁	C₂²×C₄	semidirect
154. C₂²⋊C₄×C₃₁	C₂²×C₄	semidirect
159. C₂×#2374	C₃₁	direct (N × H)
218. #2374	C₂×C₃₁	direct (N × H)
220. C₂×#388	C₂×C₃₁	semidirect
221. C₂×#388	C₂×C₃₁	semidirect
222. C₂×#387	C₂×C₃₁	semidirect
224. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
225. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
226. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
259. C₂×#123	C₄×C₃₁	semidirect
275. #388	C₂²×C₃₁	semidirect
276. #388	C₂²×C₃₁	semidirect
277. #387	C₂²×C₃₁	semidirect
279. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
280. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
281. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
289. C₂×C₃₁	#2374	direct (N × H)
305. C₂³×C₄	C₂×C₄×C₃₁	semidirect
315. #123	C₂×C₄×C₃₁	semidirect
318. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
327. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
347. C₃₁	C₂×#2374	direct (N × H)
390. C₂²×C₄	C₂²×C₄×C₃₁	semidirect
402. C₄⋊C₄	C₂²×C₄×C₃₁	semidirect
408. C₂²⋊C₄	C₂²×C₄×C₃₁	semidirect
501. C₂	C₃₁×#2374	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#123	2
C₃₁×#2374	2
C₂×C₃₁×#390	2
C₂×C₃₁×#388	2
C₂×C₃₁×#387	2
C₂×C₃₁×#386	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂×#2374	31

Subgroups — C2×C31×#2374 #21490909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2374 #21490909