Subgroups — C₂×C₃₁×#2439 #21491471

Order 7936.

2854 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2439	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#396	C₂	1	2	yes	maximal
5	C₂×C₃₁×#393	C₂	1	2	yes	maximal
6	C₂×C₃₁×#391	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#387	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#380	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
13	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	—	2	8	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	8	yes	complemented
17	C₂×C₃₁×#117	—	2	8	no	—
18	C₂×C₃₁×#117	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#396	C₂²	1	16	yes	—
34	C₃₁×#393	C₂²	1	16	yes	—
35	C₃₁×#391	C₂²	1	16	yes	complemented
36	C₃₁×#389	C₂²	1	8	yes	complemented
37	C₃₁×#389	C₂²	1	8	yes	complemented
38	C₃₁×#387	C₂²	1	16	yes	complemented
39	C₃₁×#382	C₂²	1	8	yes	complemented
40	C₃₁×#380	C₂²	1	8	yes	—
41	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
42	C₂×C₃₁×#95	C₂²	1	2	yes	—
43	C₂×C₃₁×#95	C₂²	1	2	yes	—
44	C₂×C₃₁×#95	C₂²	1	2	yes	—
45	C₂×C₃₁×#95	C₂²	1	2	yes	—
46	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#95	C₂²	1	2	yes	—
49	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	—	2	8	no	—
51	C₂²×Q₈×C₃₁	—	2	8	no	—
52	C₂²×Q₈×C₃₁	—	4	4	no	—
53	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
54	C₂³×C₄×C₃₁	—	2	4	no	—
55	C₂³×C₄×C₃₁	—	2	4	no	—
56	C₂³×C₄×C₃₁	Q₈	1	8	yes	complemented
57	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	—	2	32	no	—
69	C₃₁×#117	—	2	32	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
75	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
76	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₄²×C₃₁	—	2	8	no	—
93	C₂×C₄²×C₃₁	—	2	4	no	—
94	C₂×C₄²×C₃₁	—	2	4	no	—
95	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
96	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
98	C₃₁×#95	C₂³	1	8	yes	—
99	C₃₁×#95	C₂³	1	4	yes	—
100	C₃₁×#95	C₂³	1	4	yes	—
101	C₃₁×#95	C₂³	1	8	yes	—
102	C₃₁×#95	C₂³	1	8	yes	—
103	C₃₁×#95	C₂³	1	8	yes	—
104	C₃₁×#95	C₂³	1	8	yes	—
105	C₂⁴×C₃₁	—	2	2	no	—
106	C₂⁴×C₃₁	—	2	2	no	—
107	C₂⁴×C₃₁	—	2	2	no	—
108	C₂⁴×C₃₁	—	2	2	no	—
109	C₂⁴×C₃₁	—	2	2	no	—
110	C₂⁴×C₃₁	—	2	2	no	—
111	C₂⁴×C₃₁	C₂×Q₈	1	4	yes	—
112	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
113	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
114	C₂×Q₈×C₃₁	—	2	16	no	—
115	C₂×Q₈×C₃₁	—	2	16	no	—
116	C₂×Q₈×C₃₁	—	4	8	no	—
117	C₂×Q₈×C₃₁	—	4	32	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	32	no	—
127	C₂²×C₄×C₃₁	—	2	8	no	—
128	C₂²×C₄×C₃₁	—	2	8	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	2	no	—
135	C₂²×C₄×C₃₁	—	2	4	no	—
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	4	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂×Q₈	1	16	yes	complemented
149	C₂²×C₄×C₃₁	#41	1	4	yes	—
150	C₂²×C₄×C₃₁	#41	1	4	yes	—
151	C₂²×C₄×C₃₁	#41	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₃₁	—	2	8	no	—
163	C₄⋊C₄×C₃₁	—	2	16	no	—
164	C₄⋊C₄×C₃₁	—	2	8	no	—
165	C₄⋊C₄×C₃₁	—	2	16	no	—
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	4	8	no	—
168	C₄⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
169	C₂²⋊C₄×C₃₁	—	2	16	no	—
170	C₂²⋊C₄×C₃₁	—	2	8	no	—
171	C₂²⋊C₄×C₃₁	—	2	8	no	—
172	C₂²⋊C₄×C₃₁	—	4	8	no	—
173	C₄²×C₃₁	—	2	16	no	—
174	C₄²×C₃₁	—	2	8	no	—
175	C₄²×C₃₁	—	2	8	no	—
176	C₂×#2439	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₃₁	—	2	8	no	—
178	C₂³×C₃₁	—	2	8	no	—
179	C₂³×C₃₁	—	2	8	no	—
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	2	2	no	—
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	2	2	no	—
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	4	4	no	—
187	C₂³×C₃₁	—	4	4	no	—
188	C₂³×C₃₁	—	4	4	no	—
189	C₂³×C₃₁	—	4	4	no	—
190	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
191	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
192	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
193	C₂³×C₃₁	C₂×#41	1	2	yes	—
194	C₂³×C₃₁	C₂×#41	1	2	yes	—
195	C₂³×C₃₁	C₂×#41	1	2	yes	—
196	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
197	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
198	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
199	Q₈×C₃₁	—	4	32	no	—
200	C₂×C₄×C₃₁	—	2	8	no	—
201	C₂×C₄×C₃₁	—	2	16	no	—
202	C₂×C₄×C₃₁	—	2	8	no	—
203	C₂×C₄×C₃₁	—	2	8	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	32	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	4	8	no	—
229	C₂×C₄×C₃₁	—	4	8	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	8	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	C₂²⋊Q₈	1	16	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
236	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
237	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
238	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
239	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
240	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
241	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic
243	#2439	C₂×C₃₁	1	16	yes	direct product factor
244	C₂×#396	C₂×C₃₁	1	2	yes	—
245	C₂×#393	C₂×C₃₁	1	2	yes	—
246	C₂×#391	C₂×C₃₁	1	2	yes	complemented
247	C₂×#389	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂×#389	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂×#387	C₂×C₃₁	1	2	yes	complemented
250	C₂×#382	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂×#380	C₂×C₃₁	1	1	yes	characteristic
252	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
253	C₂²×C₃₁	—	2	4	no	—
254	C₂²×C₃₁	—	2	4	no	—
255	C₂²×C₃₁	—	2	4	no	—
256	C₂²×C₃₁	—	4	16	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
262	C₂²×C₃₁	#533	1	4	yes	—
263	C₂²×C₃₁	#544	1	4	yes	—
264	C₂²×C₃₁	#546	1	4	yes	—
265	C₂²×C₃₁	#548	1	4	yes	—
266	C₂²×C₃₁	#555	1	4	yes	—
267	C₂²×C₃₁	#556	1	4	yes	—
268	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
269	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
270	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
271	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
272	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
273	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
275	C₄×C₃₁	—	2	4	no	—
276	C₄×C₃₁	—	2	8	no	—
277	C₄×C₃₁	—	2	4	no	—
278	C₄×C₃₁	—	2	4	no	—
279	C₄×C₃₁	—	4	4	no	—
280	C₄×C₃₁	—	4	4	no	—
281	C₄×C₃₁	—	4	2	no	—
282	C₄×C₃₁	—	4	4	no	—
283	C₄×C₃₁	—	4	2	no	—
284	C₄×C₃₁	C₂×C₂²⋊Q₈	1	8	yes	—
285	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
286	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄⋊Q₈	—	2	8	no	—
288	C₂×C₄⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
289	C₂×#117	—	2	8	no	—
290	C₂×#117	C₂²×C₃₁	1	8	yes	—
291	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
292	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
293	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
294	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
295	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
296	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
297	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
298	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
299	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
300	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
301	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
302	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
303	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
304	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
305	#396	C₂²×C₃₁	1	16	yes	—
306	#393	C₂²×C₃₁	1	16	yes	—
307	#391	C₂²×C₃₁	1	16	yes	complemented
308	#389	C₂²×C₃₁	1	8	yes	complemented
309	#389	C₂²×C₃₁	1	8	yes	complemented
310	#387	C₂²×C₃₁	1	16	yes	complemented
311	#382	C₂²×C₃₁	1	8	yes	complemented
312	#380	C₂²×C₃₁	1	8	yes	—
313	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
314	C₂×#95	C₂²×C₃₁	1	2	yes	—
315	C₂×#95	C₂²×C₃₁	1	2	yes	—
316	C₂×#95	C₂²×C₃₁	1	2	yes	—
317	C₂×#95	C₂²×C₃₁	1	2	yes	—
318	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×#95	C₂²×C₃₁	1	2	yes	—
321	C₂×C₃₁	—	4	4	no	—
322	C₂×C₃₁	#2439	1	8	yes	direct product factor
323	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
324	C₂×C₃₁	C₂×#533	1	1	yes	characteristic
325	C₂×C₃₁	C₂×#544	1	1	yes	characteristic
326	C₂×C₃₁	C₂×#546	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
328	C₂×C₃₁	C₂×#555	1	1	yes	characteristic
329	C₂×C₃₁	C₂×#556	1	1	yes	characteristic
330	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
331	C₂²×Q₈	—	2	8	no	—
332	C₂²×Q₈	—	2	8	no	—
333	C₂²×Q₈	—	4	4	no	—
334	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
335	C₂³×C₄	—	2	4	no	—
336	C₂³×C₄	—	2	4	no	—
337	C₂³×C₄	Q₈×C₃₁	1	8	yes	complemented
338	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
339	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
340	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
341	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
342	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
344	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
346	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
348	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
349	C₄⋊Q₈	—	2	32	no	—
350	#117	—	2	32	no	—
351	C₂×C₄⋊C₄	—	2	4	no	—
352	C₂×C₄⋊C₄	—	2	8	no	—
353	C₂×C₄⋊C₄	—	2	4	no	—
354	C₂×C₄⋊C₄	—	2	8	no	—
355	C₂×C₄⋊C₄	—	2	4	no	—
356	C₂×C₄⋊C₄	—	4	4	no	—
357	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	—
358	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
359	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
360	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
365	C₂×C₂²⋊C₄	—	2	8	no	—
366	C₂×C₂²⋊C₄	—	2	4	no	—
367	C₂×C₂²⋊C₄	—	2	4	no	—
368	C₂×C₂²⋊C₄	—	4	4	no	—
369	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
370	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
373	C₂×C₄²	—	2	8	no	—
374	C₂×C₄²	—	2	4	no	—
375	C₂×C₄²	—	2	4	no	—
376	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
377	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
378	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
379	#95	C₂³×C₃₁	1	4	yes	—
380	#95	C₂³×C₃₁	1	4	yes	—
381	#95	C₂³×C₃₁	1	8	yes	—
382	#95	C₂³×C₃₁	1	8	yes	—
383	#95	C₂³×C₃₁	1	8	yes	—
384	#95	C₂³×C₃₁	1	8	yes	—
385	#95	C₂³×C₃₁	1	8	yes	—
386	C₃₁	C₂×#2439	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	—	2	2	no	—
388	C₂⁴	—	2	2	no	—
389	C₂⁴	—	2	2	no	—
390	C₂⁴	—	2	2	no	—
391	C₂⁴	—	2	2	no	—
392	C₂⁴	—	2	2	no	—
393	C₂⁴	C₂×Q₈×C₃₁	1	4	yes	—
394	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
395	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
396	C₂×Q₈	—	2	16	no	—
397	C₂×Q₈	—	2	16	no	—
398	C₂×Q₈	—	4	8	no	—
399	C₂×Q₈	—	4	32	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	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	32	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
427	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
428	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
429	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
430	C₂²×C₄	C₂×Q₈×C₃₁	1	16	yes	complemented
431	C₂²×C₄	#41×C₃₁	1	4	yes	—
432	C₂²×C₄	#41×C₃₁	1	4	yes	—
433	C₂²×C₄	#41×C₃₁	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
436	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
437	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	16	no	—
446	C₄⋊C₄	—	2	8	no	—
447	C₄⋊C₄	—	2	16	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	4	8	no	—
450	C₄⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
451	C₂²⋊C₄	—	2	16	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	4	8	no	—
455	C₄²	—	2	16	no	—
456	C₄²	—	2	8	no	—
457	C₄²	—	2	8	no	—
458	C₂³	—	2	8	no	—
459	C₂³	—	2	8	no	—
460	C₂³	—	2	8	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	4	4	no	—
470	C₂³	—	4	4	no	—
471	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
472	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
473	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
474	C₂³	C₂×#41×C₃₁	1	2	yes	—
475	C₂³	C₂×#41×C₃₁	1	2	yes	—
476	C₂³	C₂×#41×C₃₁	1	2	yes	—
477	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
478	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
479	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
480	Q₈	—	4	32	no	—
481	C₂×C₄	—	2	8	no	—
482	C₂×C₄	—	2	16	no	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	8	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	32	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	4	8	no	—
510	C₂×C₄	—	4	8	no	—
511	C₂×C₄	—	4	4	no	—
512	C₂×C₄	—	4	8	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	C₃₁×C₂²⋊Q₈	1	16	yes	—
515	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
516	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
517	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
518	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
519	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
520	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
521	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	16	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
531	C₂²	C₃₁×#533	1	4	yes	—
532	C₂²	C₃₁×#544	1	4	yes	—
533	C₂²	C₃₁×#546	1	4	yes	—
534	C₂²	C₃₁×#548	1	4	yes	—
535	C₂²	C₃₁×#555	1	4	yes	—
536	C₂²	C₃₁×#556	1	4	yes	—
537	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
538	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
539	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
540	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
541	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
542	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
543	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
544	C₄	—	2	4	no	—
545	C₄	—	2	8	no	—
546	C₄	—	2	4	no	—
547	C₄	—	2	4	no	—
548	C₄	—	4	4	no	—
549	C₄	—	4	4	no	—
550	C₄	—	4	2	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	2	no	—
553	C₄	C₂×C₃₁×C₂²⋊Q₈	1	8	yes	—
554	C₂	—	4	4	no	—
555	C₂	C₃₁×#2439	1	8	yes	direct product factor
556	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
557	C₂	C₂×C₃₁×#533	1	1	yes	characteristic
558	C₂	C₂×C₃₁×#544	1	1	yes	characteristic
559	C₂	C₂×C₃₁×#546	1	1	yes	characteristic
560	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
561	C₂	C₂×C₃₁×#555	1	1	yes	characteristic
562	C₂	C₂×C₃₁×#556	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₃₁×#2439	C₂	direct (N × H)
6. C₂×C₃₁×#391	C₂	semidirect
7. C₂×C₃₁×#389	C₂	semidirect
8. C₂×C₃₁×#389	C₂	semidirect
9. C₂×C₃₁×#387	C₂	semidirect
10. C₂×C₃₁×#382	C₂	semidirect
16. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
35. C₃₁×#391	C₂²	semidirect
36. C₃₁×#389	C₂²	semidirect
37. C₃₁×#389	C₂²	semidirect
38. C₃₁×#387	C₂²	semidirect
39. C₃₁×#382	C₂²	semidirect
56. C₂³×C₄×C₃₁	Q₈	semidirect
148. C₂²×C₄×C₃₁	C₂×Q₈	semidirect
176. C₂×#2439	C₃₁	direct (N × H)
241. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
243. #2439	C₂×C₃₁	direct (N × H)
246. C₂×#391	C₂×C₃₁	semidirect
247. C₂×#389	C₂×C₃₁	semidirect
248. C₂×#389	C₂×C₃₁	semidirect
249. C₂×#387	C₂×C₃₁	semidirect
250. C₂×#382	C₂×C₃₁	semidirect
288. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
307. #391	C₂²×C₃₁	semidirect
308. #389	C₂²×C₃₁	semidirect
309. #389	C₂²×C₃₁	semidirect
310. #387	C₂²×C₃₁	semidirect
311. #382	C₂²×C₃₁	semidirect
322. C₂×C₃₁	#2439	direct (N × H)
337. C₂³×C₄	Q₈×C₃₁	semidirect
386. C₃₁	C₂×#2439	direct (N × H)
430. C₂²×C₄	C₂×Q₈×C₃₁	semidirect
555. C₂	C₃₁×#2439	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×C₄⋊Q₈	2
C₂²×C₃₁×#117	2
C₃₁×#2439	2
C₂×C₃₁×#396	2
C₂×C₃₁×#393	2
C₂×C₃₁×#391	2
C₂×C₃₁×#389	2
C₂×C₃₁×#387	2
C₂×C₃₁×#382	2
C₂×C₃₁×#380	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂×#2439	31

Subgroups — C2×C31×#2439 #21491471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2439 #21491471