Subgroups — C₂×C₃₁×#2326 #21490826

Order 7936.

3566 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2326	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#388	C₂	1	4	yes	maximal, complemented
8	C₂×C₃₁×#385	C₂	1	4	yes	maximal, complemented
9	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#124	C₄	1	16	yes	complemented
15	C₂×C₃₁×#124	C₂²	1	8	yes	—
16	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
17	C₂×C₄×Q₈×C₃₁	C₂²	1	8	yes	complemented
18	C₂×C₄×D₄×C₃₁	—	2	8	no	—
19	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#391	C₂²	1	8	yes	complemented
32	C₃₁×#390	C₂²	1	8	yes	—
33	C₃₁×#388	C₂²	1	32	yes	complemented
34	C₃₁×#385	C₂²	1	32	yes	complemented
35	C₃₁×#384	C₂²	1	8	yes	complemented
36	C₃₁×#376	C₂²	1	8	yes	complemented
37	C₂×C₃₁×#95	C₂²	1	2	yes	—
38	C₂×C₃₁×#95	C₂²	1	2	yes	—
39	C₂×C₃₁×#95	C₂²	1	4	yes	—
40	C₂⁵×C₃₁	C₂³	1	2	yes	—
41	C₂²×Q₈×C₃₁	—	2	4	no	—
42	C₂²×Q₈×C₃₁	—	2	4	no	—
43	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
44	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
45	C₂²×D₄×C₃₁	—	2	4	no	—
46	C₂²×D₄×C₃₁	—	2	4	no	—
47	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
48	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
49	C₂³×C₄×C₃₁	—	2	8	no	—
50	C₂³×C₄×C₃₁	—	4	8	no	—
51	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#124	C₂×C₄	1	64	yes	complemented
61	C₄×Q₈×C₃₁	—	2	32	no	—
62	C₄×D₄×C₃₁	—	2	32	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
66	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
73	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	32	yes	complemented
74	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
76	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
77	C₂×C₄²×C₃₁	—	2	8	no	—
78	C₂×C₄²×C₃₁	—	2	4	no	—
79	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
80	C₂×C₄²×C₃₁	D₄	1	8	yes	—
81	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
82	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
84	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
85	C₃₁×#95	C₂³	1	8	yes	—
86	C₃₁×#95	C₂³	1	8	yes	—
87	C₃₁×#95	C₂³	1	16	yes	—
88	C₂⁴×C₃₁	—	2	4	no	—
89	C₂⁴×C₃₁	—	2	4	no	—
90	C₂⁴×C₃₁	—	2	8	no	—
91	C₂⁴×C₃₁	—	2	8	no	—
92	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
93	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
94	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₂×Q₈×C₃₁	—	2	32	no	—
96	C₂×Q₈×C₃₁	—	2	8	no	—
97	C₂×Q₈×C₃₁	—	2	8	no	—
98	C₂×Q₈×C₃₁	C₂²×C₄	1	16	yes	complemented
99	C₂×D₄×C₃₁	—	2	32	no	—
100	C₂×D₄×C₃₁	—	2	8	no	—
101	C₂×D₄×C₃₁	—	2	8	no	—
102	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	2	4	no	—
105	C₂²×C₄×C₃₁	—	2	4	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	8	no	—
114	C₂²×C₄×C₃₁	—	2	8	no	—
115	C₂²×C₄×C₃₁	—	2	4	no	—
116	C₂²×C₄×C₃₁	—	2	16	no	—
117	C₂²×C₄×C₃₁	—	4	16	no	—
118	C₂²×C₄×C₃₁	—	4	32	no	—
119	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
120	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
121	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₃₁	#41	1	4	yes	—
126	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	8	no	—
135	C₄⋊C₄×C₃₁	—	2	16	no	—
136	C₄⋊C₄×C₃₁	—	2	8	no	—
137	C₄⋊C₄×C₃₁	—	4	16	no	—
138	C₂²⋊C₄×C₃₁	—	2	16	no	—
139	C₂²⋊C₄×C₃₁	—	4	16	no	—
140	C₂²⋊C₄×C₃₁	C₂²×C₄	1	64	yes	complemented
141	C₄²×C₃₁	—	2	16	no	—
142	C₄²×C₃₁	—	2	8	no	—
143	C₄²×C₃₁	C₂²×C₄	1	16	yes	complemented
144	C₄²×C₃₁	C₂×D₄	1	16	yes	—
145	C₂×#2326	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	16	no	—
147	C₂³×C₃₁	—	2	32	no	—
148	C₂³×C₃₁	—	2	4	no	—
149	C₂³×C₃₁	—	2	4	no	—
150	C₂³×C₃₁	—	2	8	no	—
151	C₂³×C₃₁	—	2	8	no	—
152	C₂³×C₃₁	—	4	8	no	—
153	C₂³×C₃₁	—	4	8	no	—
154	C₂³×C₃₁	—	4	16	no	—
155	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
156	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
157	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
158	C₂³×C₃₁	C₂×#41	1	2	yes	—
159	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
160	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
161	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
162	Q₈×C₃₁	—	2	32	no	—
163	D₄×C₃₁	—	2	32	no	—
164	C₂×C₄×C₃₁	—	2	16	no	—
165	C₂×C₄×C₃₁	—	2	16	no	—
166	C₂×C₄×C₃₁	—	2	8	no	—
167	C₂×C₄×C₃₁	—	2	8	no	—
168	C₂×C₄×C₃₁	—	2	4	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	2	4	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	4	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	8	no	—
179	C₂×C₄×C₃₁	—	2	8	no	—
180	C₂×C₄×C₃₁	—	2	4	no	—
181	C₂×C₄×C₃₁	—	4	32	no	—
182	C₂×C₄×C₃₁	—	4	8	no	—
183	C₂×C₄×C₃₁	—	4	8	no	—
184	C₂×C₄×C₃₁	—	4	8	no	—
185	C₂×C₄×C₃₁	C₄×D₄	1	16	yes	—
186	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
187	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
188	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
189	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
190	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
191	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
192	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
193	C₂²×#124	C₂×C₃₁	1	1	yes	characteristic
194	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
195	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
196	#2326	C₂×C₃₁	1	16	yes	direct product factor
197	C₂×#391	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
199	C₂×#388	C₂×C₃₁	1	4	yes	complemented
200	C₂×#385	C₂×C₃₁	1	4	yes	complemented
201	C₂×#384	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂×#376	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	8	no	—
204	C₂²×C₃₁	—	2	16	no	—
205	C₂²×C₃₁	—	4	32	no	—
206	C₂²×C₃₁	—	4	4	no	—
207	C₂²×C₃₁	—	4	4	no	—
208	C₂²×C₃₁	—	4	8	no	—
209	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
210	C₂²×C₃₁	#520	1	8	yes	—
211	C₂²×C₃₁	#535	1	4	yes	—
212	C₂²×C₃₁	#536	1	4	yes	—
213	C₂²×C₃₁	#542	1	8	yes	—
214	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
215	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
216	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
217	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
218	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
219	C₄×C₃₁	—	2	8	no	—
220	C₄×C₃₁	—	2	8	no	—
221	C₄×C₃₁	—	2	4	no	—
222	C₄×C₃₁	—	2	4	no	—
223	C₄×C₃₁	—	4	4	no	—
224	C₄×C₃₁	—	4	4	no	—
225	C₄×C₃₁	—	4	4	no	—
226	C₄×C₃₁	C₂×C₄×D₄	1	8	yes	—
227	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
229	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
230	C₂×#124	C₄×C₃₁	1	16	yes	complemented
231	C₂×#124	C₂²×C₃₁	1	8	yes	—
232	C₂×C₄×Q₈	—	2	8	no	—
233	C₂×C₄×Q₈	C₂²×C₃₁	1	8	yes	complemented
234	C₂×C₄×D₄	—	2	8	no	—
235	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
236	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
237	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
238	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
239	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
240	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
241	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
242	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
243	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
244	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
245	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
246	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
247	#391	C₂²×C₃₁	1	8	yes	complemented
248	#390	C₂²×C₃₁	1	8	yes	—
249	#388	C₂²×C₃₁	1	32	yes	complemented
250	#385	C₂²×C₃₁	1	32	yes	complemented
251	#384	C₂²×C₃₁	1	8	yes	complemented
252	#376	C₂²×C₃₁	1	8	yes	complemented
253	C₂×#95	C₂²×C₃₁	1	2	yes	—
254	C₂×#95	C₂²×C₃₁	1	2	yes	—
255	C₂×#95	C₂²×C₃₁	1	4	yes	—
256	C₂×C₃₁	—	4	8	no	—
257	C₂×C₃₁	#2326	1	8	yes	direct product factor
258	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
259	C₂×C₃₁	C₂×#520	1	2	yes	—
260	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
261	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
262	C₂×C₃₁	C₂×#542	1	2	yes	—
263	C₂⁵	C₂³×C₃₁	1	2	yes	—
264	C₂²×Q₈	—	2	4	no	—
265	C₂²×Q₈	—	2	4	no	—
266	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
267	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
268	C₂²×D₄	—	2	4	no	—
269	C₂²×D₄	—	2	4	no	—
270	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
271	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
272	C₂³×C₄	—	2	8	no	—
273	C₂³×C₄	—	4	8	no	—
274	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
275	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
276	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
277	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
278	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
279	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
280	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
282	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
283	#124	C₂×C₄×C₃₁	1	64	yes	complemented
284	C₄×Q₈	—	2	32	no	—
285	C₄×D₄	—	2	32	no	—
286	C₂×C₄⋊C₄	—	2	4	no	—
287	C₂×C₄⋊C₄	—	2	8	no	—
288	C₂×C₄⋊C₄	—	2	4	no	—
289	C₂×C₄⋊C₄	—	4	8	no	—
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
291	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
292	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
293	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
294	C₂×C₂²⋊C₄	—	2	8	no	—
295	C₂×C₂²⋊C₄	—	4	8	no	—
296	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	32	yes	complemented
297	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
298	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
299	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
300	C₂×C₄²	—	2	8	no	—
301	C₂×C₄²	—	2	4	no	—
302	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
303	C₂×C₄²	D₄×C₃₁	1	8	yes	—
304	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
305	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
306	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
307	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
308	#95	C₂³×C₃₁	1	8	yes	—
309	#95	C₂³×C₃₁	1	8	yes	—
310	#95	C₂³×C₃₁	1	16	yes	—
311	C₃₁	C₂×#2326	1	1	yes	characteristic, direct product factor, Sylow
312	C₂⁴	—	2	4	no	—
313	C₂⁴	—	2	4	no	—
314	C₂⁴	—	2	8	no	—
315	C₂⁴	—	2	8	no	—
316	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
317	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
318	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
319	C₂×Q₈	—	2	32	no	—
320	C₂×Q₈	—	2	8	no	—
321	C₂×Q₈	—	2	8	no	—
322	C₂×Q₈	C₂²×C₄×C₃₁	1	16	yes	complemented
323	C₂×D₄	—	2	32	no	—
324	C₂×D₄	—	2	8	no	—
325	C₂×D₄	—	2	8	no	—
326	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	8	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	16	no	—
341	C₂²×C₄	—	4	16	no	—
342	C₂²×C₄	—	4	32	no	—
343	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
344	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
345	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
346	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
347	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
348	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
349	C₂²×C₄	#41×C₃₁	1	4	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
351	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
352	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
353	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
354	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
355	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
356	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
357	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
358	C₄⋊C₄	—	2	8	no	—
359	C₄⋊C₄	—	2	16	no	—
360	C₄⋊C₄	—	2	8	no	—
361	C₄⋊C₄	—	4	16	no	—
362	C₂²⋊C₄	—	2	16	no	—
363	C₂²⋊C₄	—	4	16	no	—
364	C₂²⋊C₄	C₂²×C₄×C₃₁	1	64	yes	complemented
365	C₄²	—	2	16	no	—
366	C₄²	—	2	8	no	—
367	C₄²	C₂²×C₄×C₃₁	1	16	yes	complemented
368	C₄²	C₂×D₄×C₃₁	1	16	yes	—
369	C₂³	—	2	16	no	—
370	C₂³	—	2	32	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	2	4	no	—
373	C₂³	—	2	8	no	—
374	C₂³	—	2	8	no	—
375	C₂³	—	4	8	no	—
376	C₂³	—	4	8	no	—
377	C₂³	—	4	16	no	—
378	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
379	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
380	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
381	C₂³	C₂×#41×C₃₁	1	2	yes	—
382	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
383	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
384	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
385	Q₈	—	2	32	no	—
386	D₄	—	2	32	no	—
387	C₂×C₄	—	2	16	no	—
388	C₂×C₄	—	2	16	no	—
389	C₂×C₄	—	2	8	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	8	no	—
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	4	32	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	—	4	8	no	—
407	C₂×C₄	—	4	8	no	—
408	C₂×C₄	C₄×D₄×C₃₁	1	16	yes	—
409	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
410	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
411	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
412	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
413	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
414	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
415	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
416	C₂²	—	2	8	no	—
417	C₂²	—	2	16	no	—
418	C₂²	—	4	32	no	—
419	C₂²	—	4	4	no	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	8	no	—
422	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
423	C₂²	C₃₁×#520	1	8	yes	—
424	C₂²	C₃₁×#535	1	4	yes	—
425	C₂²	C₃₁×#536	1	4	yes	—
426	C₂²	C₃₁×#542	1	8	yes	—
427	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
428	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
429	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
430	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
431	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
432	C₄	—	2	8	no	—
433	C₄	—	2	8	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	4	no	—
436	C₄	—	4	4	no	—
437	C₄	—	4	4	no	—
438	C₄	—	4	4	no	—
439	C₄	C₂×C₄×D₄×C₃₁	1	8	yes	—
440	C₂	—	4	8	no	—
441	C₂	C₃₁×#2326	1	8	yes	direct product factor
442	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
443	C₂	C₂×C₃₁×#520	1	2	yes	—
444	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
445	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
446	C₂	C₂×C₃₁×#542	1	2	yes	—

	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. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#2326	C₂	direct (N × H)
5. C₂×C₃₁×#391	C₂	semidirect
7. C₂×C₃₁×#388	C₂	semidirect
8. C₂×C₃₁×#385	C₂	semidirect
9. C₂×C₃₁×#384	C₂	semidirect
10. C₂×C₃₁×#376	C₂	semidirect
14. C₂×C₃₁×#124	C₄	semidirect
17. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
31. C₃₁×#391	C₂²	semidirect
33. C₃₁×#388	C₂²	semidirect
34. C₃₁×#385	C₂²	semidirect
35. C₃₁×#384	C₂²	semidirect
36. C₃₁×#376	C₂²	semidirect
43. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
60. C₃₁×#124	C₂×C₄	semidirect
73. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
79. C₂×C₄²×C₃₁	C₂×C₄	semidirect
98. C₂×Q₈×C₃₁	C₂²×C₄	semidirect
140. C₂²⋊C₄×C₃₁	C₂²×C₄	semidirect
143. C₄²×C₃₁	C₂²×C₄	semidirect
145. C₂×#2326	C₃₁	direct (N × H)
194. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
196. #2326	C₂×C₃₁	direct (N × H)
197. C₂×#391	C₂×C₃₁	semidirect
199. C₂×#388	C₂×C₃₁	semidirect
200. C₂×#385	C₂×C₃₁	semidirect
201. C₂×#384	C₂×C₃₁	semidirect
202. C₂×#376	C₂×C₃₁	semidirect
230. C₂×#124	C₄×C₃₁	semidirect
233. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
247. #391	C₂²×C₃₁	semidirect
249. #388	C₂²×C₃₁	semidirect
250. #385	C₂²×C₃₁	semidirect
251. #384	C₂²×C₃₁	semidirect
252. #376	C₂²×C₃₁	semidirect
257. C₂×C₃₁	#2326	direct (N × H)
266. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
283. #124	C₂×C₄×C₃₁	semidirect
296. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
302. C₂×C₄²	C₂×C₄×C₃₁	semidirect
311. C₃₁	C₂×#2326	direct (N × H)
322. C₂×Q₈	C₂²×C₄×C₃₁	semidirect
364. C₂²⋊C₄	C₂²×C₄×C₃₁	semidirect
367. C₄²	C₂²×C₄×C₃₁	semidirect
441. C₂	C₃₁×#2326	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×#124	2
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2326	2
C₂×C₃₁×#391	2
C₂×C₃₁×#390	2
C₂×C₃₁×#388	2
C₂×C₃₁×#385	2
C₂×C₃₁×#384	2
C₂×C₃₁×#376	2
C₂×#2326	31

Subgroups — C2×C31×#2326 #21490826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2326 #21490826