Subgroups — C₂×C₃₁×#3109 #21517300

Order 7936.

2830 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3109	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#481	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#476	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#459	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#447	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#442	C₂	1	4	yes	maximal, complemented
9	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₃₁	—	2	1	no	—
11	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₃₁	—	2	1	no	—
13	C₂²×D₈×C₃₁	C₂²	1	4	yes	—
14	C₃₁×#532	—	2	4	no	—
15	C₃₁×#532	C₂²	1	8	yes	complemented
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃₁×#124	—	2	1	no	—
19	C₂×C₃₁×#121	—	2	1	no	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₃₁	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	—	2	1	no	—
23	C₂×C₄×D₄×C₃₁	—	2	4	no	—
24	C₂×C₄×D₄×C₃₁	—	2	1	no	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	complemented
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
31	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#481	C₂²	1	16	yes	complemented
33	C₃₁×#476	C₂²	1	16	yes	complemented
34	C₃₁×#459	C₂²	1	16	yes	complemented
35	C₃₁×#447	C₂²	1	16	yes	complemented
36	C₃₁×#442	C₂²	1	32	yes	complemented
37	C₃₁×#432	C₂²	1	8	yes	complemented
38	C₂×C₃₁×#106	C₂²	1	4	yes	complemented
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₂×C₃₁×#102	C₂²	1	4	yes	—
41	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
42	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
44	C₂⁵×C₃₁	—	2	1	no	—
45	C₂×#41×C₃₁	—	2	8	no	—
46	C₂×#41×C₃₁	—	4	8	no	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	8	no	—
49	C₂²×D₄×C₃₁	—	2	2	no	—
50	C₂²×D₄×C₃₁	—	2	2	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	—	2	1	no	—
54	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
55	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
56	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₃₁	—	2	1	no	—
58	C₂³×C₄×C₃₁	—	2	2	no	—
59	C₂³×C₄×C₃₁	—	2	2	no	—
60	C₂³×C₄×C₃₁	—	2	8	no	—
61	C₂³×C₄×C₃₁	—	2	2	no	—
62	C₂³×C₄×C₃₁	—	4	8	no	—
63	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×D₈×C₃₁	—	2	16	no	—
67	C₂×D₈×C₃₁	C₂³	1	16	yes	—
68	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
69	C₃₁×C₄⋊Q₈	—	2	2	no	—
70	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	complemented
71	C₃₁×#127	—	2	2	no	—
72	C₃₁×#127	C₂³	1	4	yes	complemented
73	C₃₁×#124	—	2	8	no	—
74	C₃₁×#121	—	2	8	no	—
75	C₃₁×#121	—	2	4	no	—
76	C₃₁×#121	C₂³	1	8	yes	—
77	C₄×D₄×C₃₁	—	2	4	no	—
78	C₄×D₄×C₃₁	—	2	4	no	—
79	C₄×D₄×C₃₁	—	2	8	no	—
80	C₄×D₄×C₃₁	—	2	8	no	—
81	C₄×D₄×C₃₁	—	2	8	no	—
82	C₄×D₄×C₃₁	—	2	4	no	—
83	C₄×D₄×C₃₁	C₂³	1	8	yes	—
84	C₄×D₄×C₃₁	C₂³	1	8	yes	—
85	C₄×D₄×C₃₁	C₂³	1	8	yes	complemented
86	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
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	4	yes	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
95	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
97	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	4	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
103	C₃₁×#106	C₂³	1	16	yes	complemented
104	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
105	C₃₁×#102	C₂³	1	16	yes	—
106	C₃₁×#102	C₂³	1	16	yes	complemented
107	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
108	C₃₁×#97	C₂³	1	8	yes	complemented
109	C₂⁴×C₃₁	—	2	8	no	—
110	C₂⁴×C₃₁	—	2	4	no	—
111	C₂⁴×C₃₁	—	2	2	no	—
112	C₂⁴×C₃₁	—	2	2	no	—
113	C₂⁴×C₃₁	—	4	4	no	—
114	C₂⁴×C₃₁	—	4	4	no	—
115	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
116	#41×C₃₁	—	4	32	no	—
117	C₂×Q₈×C₃₁	—	2	4	no	—
118	C₂×Q₈×C₃₁	—	4	4	no	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	16	no	—
121	C₂×D₄×C₃₁	—	2	16	no	—
122	C₂×D₄×C₃₁	—	2	8	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	4	no	—
127	C₂×D₄×C₃₁	—	2	4	no	—
128	C₂×D₄×C₃₁	—	2	4	no	—
129	C₂×D₄×C₃₁	—	2	8	no	—
130	C₂×D₄×C₃₁	—	4	4	no	—
131	C₂×D₄×C₃₁	—	4	8	no	—
132	C₂×D₄×C₃₁	—	4	4	no	—
133	C₂×D₄×C₃₁	#41	1	8	yes	complemented
134	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
135	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	8	no	—
143	C₂²×C₄×C₃₁	—	2	8	no	—
144	C₂²×C₄×C₃₁	—	2	8	no	—
145	C₂²×C₄×C₃₁	—	4	8	no	—
146	C₂²×C₄×C₃₁	—	4	32	no	—
147	C₂²×C₄×C₃₁	—	4	8	no	—
148	C₂²×C₄×C₃₁	—	4	4	no	—
149	C₂²×C₄×C₃₁	—	4	8	no	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	D₈×C₃₁	—	2	32	no	—
161	C₂×C₈×C₃₁	—	2	8	no	—
162	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
163	C₄⋊C₄×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	4	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
168	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
169	C₂²⋊C₄×C₃₁	—	2	8	no	—
170	C₂²⋊C₄×C₃₁	—	2	8	no	—
171	C₂²⋊C₄×C₃₁	—	2	8	no	—
172	C₂²⋊C₄×C₃₁	—	2	8	no	—
173	C₄²×C₃₁	C₂×D₄	1	4	yes	—
174	C₄²×C₃₁	C₂×D₄	1	2	yes	—
175	C₄²×C₃₁	C₂×D₄	1	4	yes	—
176	C₄²×C₃₁	C₂×D₄	1	2	yes	—
177	C₄²×C₃₁	C₂⁴	1	2	yes	—
178	C₄²×C₃₁	C₂⁴	1	2	yes	—
179	C₂×#3109	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	2	2	no	—
182	C₂³×C₃₁	—	2	16	no	—
183	C₂³×C₃₁	—	2	8	no	—
184	C₂³×C₃₁	—	2	4	no	—
185	C₂³×C₃₁	—	2	4	no	—
186	C₂³×C₃₁	—	4	8	no	—
187	C₂³×C₃₁	—	4	32	no	—
188	C₂³×C₃₁	—	4	4	no	—
189	C₂³×C₃₁	—	4	8	no	—
190	C₂³×C₃₁	—	4	4	no	—
191	C₂³×C₃₁	—	4	4	no	—
192	C₂³×C₃₁	—	4	4	no	—
193	C₂³×C₃₁	—	4	4	no	—
194	C₂³×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
195	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
196	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	Q₈×C₃₁	—	4	8	no	—
198	D₄×C₃₁	—	2	16	no	—
199	D₄×C₃₁	—	2	8	no	—
200	D₄×C₃₁	—	2	4	no	—
201	D₄×C₃₁	—	2	8	no	—
202	D₄×C₃₁	—	4	8	no	—
203	D₄×C₃₁	—	4	16	no	—
204	D₄×C₃₁	—	4	4	no	—
205	D₄×C₃₁	—	4	8	no	—
206	D₄×C₃₁	C₂×#41	1	8	yes	complemented
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	8	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	8	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	4	16	no	—
218	C₂×C₄×C₃₁	—	4	16	no	—
219	C₂×C₄×C₃₁	—	4	8	no	—
220	C₂×C₄×C₃₁	—	4	4	no	—
221	C₂×C₄×C₃₁	—	4	4	no	—
222	C₂×C₄×C₃₁	—	4	16	no	—
223	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
231	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
232	C₈×C₃₁	—	2	8	no	—
233	C₂×#532	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#3109	C₂×C₃₁	1	16	yes	direct product factor
236	C₂×#481	C₂×C₃₁	1	2	yes	complemented
237	C₂×#476	C₂×C₃₁	1	2	yes	complemented
238	C₂×#459	C₂×C₃₁	1	2	yes	complemented
239	C₂×#447	C₂×C₃₁	1	2	yes	complemented
240	C₂×#442	C₂×C₃₁	1	4	yes	complemented
241	C₂×#432	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂²×C₃₁	—	2	8	no	—
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	2	8	no	—
246	C₂²×C₃₁	—	4	32	no	—
247	C₂²×C₃₁	—	4	16	no	—
248	C₂²×C₃₁	—	4	4	no	—
249	C₂²×C₃₁	—	4	4	no	—
250	C₂²×C₃₁	—	4	4	no	—
251	C₂²×C₃₁	—	4	4	no	—
252	C₂²×C₃₁	—	4	4	no	—
253	C₂²×C₃₁	—	8	8	no	—
254	C₂²×C₃₁	#525	1	2	yes	—
255	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
256	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
257	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
258	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	2	2	no	—
261	C₄×C₃₁	—	2	2	no	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	4	4	no	—
264	C₄×C₃₁	—	4	4	no	—
265	C₄×C₃₁	#525	1	2	yes	—
266	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
267	C₄×C₃₁	C₂²×#41	1	2	yes	—
268	C₂²×#41	—	2	1	no	—
269	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₂⁴×C₄	—	2	1	no	—
271	C₂²×D₈	C₂²×C₃₁	1	4	yes	—
272	#532	—	2	4	no	—
273	#532	C₂²×C₃₁	1	8	yes	complemented
274	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
275	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
276	C₂×#124	—	2	1	no	—
277	C₂×#121	—	2	1	no	—
278	C₂×#121	C₂²×C₃₁	1	2	yes	—
279	C₂×C₄×D₄	—	2	1	no	—
280	C₂×C₄×D₄	—	2	1	no	—
281	C₂×C₄×D₄	—	2	4	no	—
282	C₂×C₄×D₄	—	2	1	no	—
283	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
284	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
285	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	complemented
286	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
287	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
288	C₂²×C₂²⋊C₄	—	2	1	no	—
289	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
290	#481	C₂²×C₃₁	1	16	yes	complemented
291	#476	C₂²×C₃₁	1	16	yes	complemented
292	#459	C₂²×C₃₁	1	16	yes	complemented
293	#447	C₂²×C₃₁	1	16	yes	complemented
294	#442	C₂²×C₃₁	1	32	yes	complemented
295	#432	C₂²×C₃₁	1	8	yes	complemented
296	C₂×#106	C₂²×C₃₁	1	4	yes	complemented
297	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
298	C₂×#102	C₂²×C₃₁	1	4	yes	—
299	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
300	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
301	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
302	C₂×C₃₁	—	2	4	no	—
303	C₂×C₃₁	—	4	8	no	—
304	C₂×C₃₁	—	8	4	no	—
305	C₂×C₃₁	#3109	1	4	yes	direct product factor
306	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
307	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₂⁵	—	2	1	no	—
310	C₂×#41	—	2	8	no	—
311	C₂×#41	—	4	8	no	—
312	C₂²×Q₈	—	2	1	no	—
313	C₂²×D₄	—	2	8	no	—
314	C₂²×D₄	—	2	2	no	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	—	2	2	no	—
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
320	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
321	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
322	C₂³×C₄	—	2	1	no	—
323	C₂³×C₄	—	2	2	no	—
324	C₂³×C₄	—	2	2	no	—
325	C₂³×C₄	—	2	8	no	—
326	C₂³×C₄	—	2	2	no	—
327	C₂³×C₄	—	4	8	no	—
328	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
329	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
330	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×D₈	—	2	16	no	—
332	C₂×D₈	C₂³×C₃₁	1	16	yes	—
333	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
334	C₄⋊Q₈	—	2	2	no	—
335	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	complemented
336	#127	—	2	2	no	—
337	#127	C₂³×C₃₁	1	4	yes	complemented
338	#124	—	2	8	no	—
339	#121	—	2	8	no	—
340	#121	—	2	4	no	—
341	#121	C₂³×C₃₁	1	8	yes	—
342	C₄×D₄	—	2	4	no	—
343	C₄×D₄	—	2	4	no	—
344	C₄×D₄	—	2	8	no	—
345	C₄×D₄	—	2	8	no	—
346	C₄×D₄	—	2	8	no	—
347	C₄×D₄	—	2	4	no	—
348	C₄×D₄	C₂³×C₃₁	1	8	yes	—
349	C₄×D₄	C₂³×C₃₁	1	8	yes	—
350	C₄×D₄	C₂³×C₃₁	1	8	yes	complemented
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
356	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	8	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
364	C₂×C₄²	D₄×C₃₁	1	4	yes	—
365	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
368	#106	C₂³×C₃₁	1	16	yes	complemented
369	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
370	#102	C₂³×C₃₁	1	16	yes	—
371	#102	C₂³×C₃₁	1	16	yes	complemented
372	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
373	#97	C₂³×C₃₁	1	8	yes	complemented
374	C₃₁	C₂×#3109	1	1	yes	characteristic, direct product factor, Sylow
375	C₂⁴	—	2	8	no	—
376	C₂⁴	—	2	4	no	—
377	C₂⁴	—	2	2	no	—
378	C₂⁴	—	2	2	no	—
379	C₂⁴	—	4	4	no	—
380	C₂⁴	—	4	4	no	—
381	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
382	#41	—	4	32	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	4	4	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	16	no	—
387	C₂×D₄	—	2	16	no	—
388	C₂×D₄	—	2	8	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	2	4	no	—
395	C₂×D₄	—	2	8	no	—
396	C₂×D₄	—	4	4	no	—
397	C₂×D₄	—	4	8	no	—
398	C₂×D₄	—	4	4	no	—
399	C₂×D₄	#41×C₃₁	1	8	yes	complemented
400	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
401	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	4	8	no	—
412	C₂²×C₄	—	4	32	no	—
413	C₂²×C₄	—	4	8	no	—
414	C₂²×C₄	—	4	4	no	—
415	C₂²×C₄	—	4	8	no	—
416	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
423	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	D₈	—	2	32	no	—
427	C₂×C₈	—	2	8	no	—
428	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
434	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₄²	C₂×D₄×C₃₁	1	4	yes	—
440	C₄²	C₂×D₄×C₃₁	1	2	yes	—
441	C₄²	C₂×D₄×C₃₁	1	4	yes	—
442	C₄²	C₂×D₄×C₃₁	1	2	yes	—
443	C₄²	C₂⁴×C₃₁	1	2	yes	—
444	C₄²	C₂⁴×C₃₁	1	2	yes	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	16	no	—
448	C₂³	—	2	8	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	4	no	—
451	C₂³	—	4	8	no	—
452	C₂³	—	4	32	no	—
453	C₂³	—	4	4	no	—
454	C₂³	—	4	8	no	—
455	C₂³	—	4	4	no	—
456	C₂³	—	4	4	no	—
457	C₂³	—	4	4	no	—
458	C₂³	—	4	4	no	—
459	C₂³	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
460	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
461	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	Q₈	—	4	8	no	—
463	D₄	—	2	16	no	—
464	D₄	—	2	8	no	—
465	D₄	—	2	4	no	—
466	D₄	—	2	8	no	—
467	D₄	—	4	8	no	—
468	D₄	—	4	16	no	—
469	D₄	—	4	4	no	—
470	D₄	—	4	8	no	—
471	D₄	C₂×#41×C₃₁	1	8	yes	complemented
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	4	16	no	—
483	C₂×C₄	—	4	16	no	—
484	C₂×C₄	—	4	8	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	4	16	no	—
488	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
489	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
490	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
492	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
493	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
495	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
496	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
497	C₈	—	2	8	no	—
498	C₂²	—	2	8	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	4	32	no	—
503	C₂²	—	4	16	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	4	4	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	4	4	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	8	8	no	—
510	C₂²	C₃₁×#525	1	2	yes	—
511	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
512	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
513	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
514	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
515	C₄	—	2	2	no	—
516	C₄	—	2	2	no	—
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	4	no	—
521	C₄	C₃₁×#525	1	2	yes	—
522	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
523	C₄	C₂²×#41×C₃₁	1	2	yes	—
524	C₂	—	2	4	no	—
525	C₂	—	4	8	no	—
526	C₂	—	8	4	no	—
527	C₂	C₃₁×#3109	1	4	yes	direct product factor
528	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
529	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
530	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#532	C₂	semidirect
2. C₂²×C₄×D₄×C₃₁	C₂	semidirect
3. C₃₁×#3109	C₂	direct (N × H)
4. C₂×C₃₁×#481	C₂	semidirect
5. C₂×C₃₁×#476	C₂	semidirect
6. C₂×C₃₁×#459	C₂	semidirect
7. C₂×C₃₁×#447	C₂	semidirect
8. C₂×C₃₁×#442	C₂	semidirect
9. C₂×C₃₁×#432	C₂	semidirect
15. C₃₁×#532	C₂²	semidirect
16. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₃₁×#127	C₂²	semidirect
27. C₂×C₄×D₄×C₃₁	C₂²	semidirect
28. C₂×C₄×D₄×C₃₁	C₂²	semidirect
32. C₃₁×#481	C₂²	semidirect
33. C₃₁×#476	C₂²	semidirect
34. C₃₁×#459	C₂²	semidirect
35. C₃₁×#447	C₂²	semidirect
36. C₃₁×#442	C₂²	semidirect
37. C₃₁×#432	C₂²	semidirect
38. C₂×C₃₁×#106	C₂²	semidirect
39. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
41. C₂×C₃₁×#102	C₂²	semidirect
43. C₂×C₃₁×#97	C₂²	semidirect
70. C₃₁×C₄⋊Q₈	C₂³	semidirect
72. C₃₁×#127	C₂³	semidirect
85. C₄×D₄×C₃₁	C₂³	semidirect
103. C₃₁×#106	C₂³	semidirect
104. C₃₁×C₄⋊C₈	C₂³	semidirect
106. C₃₁×#102	C₂³	semidirect
108. C₃₁×#97	C₂³	semidirect
133. C₂×D₄×C₃₁	#41	semidirect
179. C₂×#3109	C₃₁	direct (N × H)
206. D₄×C₃₁	C₂×#41	semidirect
233. C₂×#532	C₂×C₃₁	semidirect
234. C₂²×C₄×D₄	C₂×C₃₁	semidirect
235. #3109	C₂×C₃₁	direct (N × H)
236. C₂×#481	C₂×C₃₁	semidirect
237. C₂×#476	C₂×C₃₁	semidirect
238. C₂×#459	C₂×C₃₁	semidirect
239. C₂×#447	C₂×C₃₁	semidirect
240. C₂×#442	C₂×C₃₁	semidirect
241. C₂×#432	C₂×C₃₁	semidirect
273. #532	C₂²×C₃₁	semidirect
274. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
275. C₂×#127	C₂²×C₃₁	semidirect
285. C₂×C₄×D₄	C₂²×C₃₁	semidirect
286. C₂×C₄×D₄	C₂²×C₃₁	semidirect
290. #481	C₂²×C₃₁	semidirect
291. #476	C₂²×C₃₁	semidirect
292. #459	C₂²×C₃₁	semidirect
293. #447	C₂²×C₃₁	semidirect
294. #442	C₂²×C₃₁	semidirect
295. #432	C₂²×C₃₁	semidirect
296. C₂×#106	C₂²×C₃₁	semidirect
297. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
299. C₂×#102	C₂²×C₃₁	semidirect
301. C₂×#97	C₂²×C₃₁	semidirect
305. C₂×C₃₁	#3109	direct (N × H)
335. C₄⋊Q₈	C₂³×C₃₁	semidirect
337. #127	C₂³×C₃₁	semidirect
350. C₄×D₄	C₂³×C₃₁	semidirect
368. #106	C₂³×C₃₁	semidirect
369. C₄⋊C₈	C₂³×C₃₁	semidirect
371. #102	C₂³×C₃₁	semidirect
373. #97	C₂³×C₃₁	semidirect
374. C₃₁	C₂×#3109	direct (N × H)
399. C₂×D₄	#41×C₃₁	semidirect
471. D₄	C₂×#41×C₃₁	semidirect
527. C₂	C₃₁×#3109	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×#532	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#3109	2
C₂×C₃₁×#481	2
C₂×C₃₁×#476	2
C₂×C₃₁×#459	2
C₂×C₃₁×#447	2
C₂×C₃₁×#442	2
C₂×C₃₁×#432	2
C₂×#3109	31

Subgroups — C2×C31×#3109 #21517300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3109 #21517300