Subgroups — C₃₁×#34121 #21490880

Order 7936.

2554 conjugacy classes of subgroups.

The subgroup lattice has 570 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×#2357	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2356	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2355	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2354	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2353	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2352	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2298	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2297	C₂	1	2	yes	maximal
10	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2283	C₂	1	2	yes	maximal, complemented
12	C₄×C₃₁×#117	C₂	1	2	yes	maximal, complemented
13	C₄×C₃₁×#117	C₂	1	2	yes	maximal, 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	4	yes	—
24	C₂×C₃₁×#117	C₂²	1	2	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×#388	C₂²	1	8	yes	—
34	C₃₁×#388	C₂²	1	8	yes	—
35	C₃₁×#387	C₂²	1	8	yes	—
36	C₃₁×#387	C₂²	1	8	yes	—
37	C₃₁×#386	C₂²	1	8	yes	—
38	C₃₁×#384	C₂²	1	2	yes	—
39	C₃₁×#383	C₂²	1	4	yes	—
40	C₃₁×#382	C₂²	1	4	yes	—
41	C₃₁×#382	C₂²	1	2	yes	—
42	C₃₁×#381	C₂²	1	4	yes	—
43	C₃₁×#380	C₂²	1	2	yes	—
44	C₃₁×#380	C₂²	1	4	yes	—
45	C₃₁×#380	C₂²	1	2	yes	—
46	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
50	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
53	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
56	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
57	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
58	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
60	C₃₁×#376	C₂²	1	4	yes	—
61	C₃₁×#376	C₂²	1	4	yes	—
62	C₂×C₃₁×#95	C₂²	1	2	yes	—
63	C₄³×C₃₁	C₂²	1	4	yes	—
64	C₂×#41×C₃₁	—	2	4	no	—
65	C₂×#41×C₃₁	—	2	4	no	—
66	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
67	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
78	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
79	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	—
80	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	—
81	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
82	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
83	C₃₁×#117	—	2	8	no	—
84	C₃₁×#117	—	2	16	no	—
85	C₃₁×#117	C₂×C₄	1	8	yes	—
86	C₃₁×#117	C₂×C₄	1	16	yes	complemented
87	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄²×C₃₁	—	2	8	no	—
105	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
106	C₂×C₄²×C₃₁	C₂×C₄	1	16	yes	complemented
107	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
109	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
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	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₃₁×#95	C₂³	1	2	yes	—
121	C₃₁×#95	C₂³	1	2	yes	—
122	C₃₁×#95	C₂³	1	4	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	C₃₁×#95	C₂³	1	4	yes	—
129	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
130	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
131	#41×C₃₁	—	2	32	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₃₁	—	2	4	no	—
136	C₂×D₄×C₃₁	—	2	4	no	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
140	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	2	8	no	—
143	C₂²×C₄×C₃₁	—	2	4	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	8	no	—
146	C₂²×C₄×C₃₁	—	2	8	no	—
147	C₂²×C₄×C₃₁	—	4	4	no	—
148	C₂²×C₄×C₃₁	—	4	4	no	—
149	C₂²×C₄×C₃₁	—	4	2	no	—
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	8	yes	—
153	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
155	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₃₁	#41	1	4	yes	—
157	C₂²×C₄×C₃₁	#41	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
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	2	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	8	no	—
173	C₄⋊C₄×C₃₁	—	2	8	no	—
174	C₄⋊C₄×C₃₁	—	4	4	no	—
175	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
176	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
177	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
178	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
179	C₄⋊C₄×C₃₁	#41	1	8	yes	complemented
180	C₂²⋊C₄×C₃₁	—	2	8	no	—
181	C₂²⋊C₄×C₃₁	—	2	16	no	—
182	C₂²⋊C₄×C₃₁	—	4	8	no	—
183	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
184	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
185	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
186	C₄²×C₃₁	—	2	4	no	—
187	C₄²×C₃₁	—	2	8	no	—
188	C₄²×C₃₁	—	2	4	no	—
189	C₄²×C₃₁	—	2	16	no	—
190	C₄²×C₃₁	—	2	8	no	—
191	C₄²×C₃₁	—	4	8	no	—
192	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
193	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
194	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
195	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
196	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
197	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
198	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
199	#34121	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₃₁	—	2	2	no	—
201	C₂³×C₃₁	—	2	2	no	—
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	2	2	no	—
205	C₂³×C₃₁	—	2	4	no	—
206	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
207	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
208	C₂³×C₃₁	C₂×#41	1	2	yes	—
209	C₂³×C₃₁	C₂×#41	1	2	yes	—
210	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
211	Q₈×C₃₁	—	2	8	no	—
212	D₄×C₃₁	—	2	16	no	—
213	D₄×C₃₁	—	2	8	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	4	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	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	16	no	—
224	C₂×C₄×C₃₁	—	2	8	no	—
225	C₂×C₄×C₃₁	—	2	16	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	8	no	—
228	C₂×C₄×C₃₁	—	2	8	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	8	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	—	4	8	no	—
234	C₂×C₄×C₃₁	—	4	16	no	—
235	C₂×C₄×C₃₁	#117	1	16	yes	complemented
236	C₂×C₄×C₃₁	C₂³×C₄	1	2	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	2	yes	—
240	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
241	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
242	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
243	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
244	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
245	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
246	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
247	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
248	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
249	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
250	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
251	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
252	#2357	C₂×C₃₁	1	2	yes	complemented
253	#2356	C₂×C₃₁	1	2	yes	complemented
254	#2355	C₂×C₃₁	1	2	yes	complemented
255	#2354	C₂×C₃₁	1	2	yes	complemented
256	#2353	C₂×C₃₁	1	4	yes	complemented
257	#2352	C₂×C₃₁	1	4	yes	complemented
258	#2298	C₂×C₃₁	1	4	yes	complemented
259	#2297	C₂×C₃₁	1	2	yes	—
260	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#2283	C₂×C₃₁	1	2	yes	complemented
262	C₄×#117	C₂×C₃₁	1	2	yes	complemented
263	C₄×#117	C₂×C₃₁	1	2	yes	complemented
264	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
265	C₂²×C₃₁	—	2	2	no	—
266	C₂²×C₃₁	—	2	2	no	—
267	C₂²×C₃₁	—	2	4	no	—
268	C₂²×C₃₁	—	2	4	no	—
269	C₂²×C₃₁	—	2	4	no	—
270	C₂²×C₃₁	—	2	4	no	—
271	C₂²×C₃₁	—	2	4	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	4	4	no	—
274	C₂²×C₃₁	C₂×#117	1	4	yes	—
275	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
276	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
277	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
278	C₂²×C₃₁	C₂²×#41	1	2	yes	—
279	C₂²×C₃₁	#585	1	2	yes	—
280	C₄×C₃₁	—	2	4	no	—
281	C₄×C₃₁	—	2	4	no	—
282	C₄×C₃₁	—	2	2	no	—
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	2	4	no	—
285	C₄×C₃₁	—	2	2	no	—
286	C₄×C₃₁	—	2	8	no	—
287	C₄×C₃₁	—	4	4	no	—
288	C₄×C₃₁	—	4	8	no	—
289	C₄×C₃₁	C₂×#117	1	4	yes	—
290	C₄×C₃₁	C₂×#117	1	4	yes	—
291	C₄×C₃₁	C₄×#41	1	4	yes	—
292	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
293	C₄×#41	C₄×C₃₁	1	16	yes	complemented
294	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
297	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×#117	C₂²×C₃₁	1	4	yes	—
301	C₂×#117	C₂²×C₃₁	1	2	yes	—
302	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#117	C₂²×C₃₁	1	2	yes	—
304	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
305	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
306	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
307	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
308	#391	C₂²×C₃₁	1	4	yes	—
309	#390	C₂²×C₃₁	1	4	yes	—
310	#388	C₂²×C₃₁	1	8	yes	—
311	#388	C₂²×C₃₁	1	8	yes	—
312	#387	C₂²×C₃₁	1	8	yes	—
313	#387	C₂²×C₃₁	1	8	yes	—
314	#386	C₂²×C₃₁	1	8	yes	—
315	#384	C₂²×C₃₁	1	2	yes	—
316	#383	C₂²×C₃₁	1	4	yes	—
317	#382	C₂²×C₃₁	1	4	yes	—
318	#382	C₂²×C₃₁	1	2	yes	—
319	#381	C₂²×C₃₁	1	4	yes	—
320	#380	C₂²×C₃₁	1	2	yes	—
321	#380	C₂²×C₃₁	1	4	yes	—
322	#380	C₂²×C₃₁	1	2	yes	—
323	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
324	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
325	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
326	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
327	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
328	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
329	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
330	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
331	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
332	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
333	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
334	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
335	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
336	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
337	#376	C₂²×C₃₁	1	4	yes	—
338	#376	C₂²×C₃₁	1	4	yes	—
339	C₂×#95	C₂²×C₃₁	1	2	yes	—
340	C₄³	C₂²×C₃₁	1	4	yes	—
341	C₂×C₃₁	—	2	4	no	—
342	C₂×C₃₁	—	4	4	no	—
343	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
344	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
345	C₂×C₃₁	#3428	1	1	yes	characteristic
346	C₂×C₃₁	#3475	1	2	yes	—
347	C₂×C₃₁	#3480	1	2	yes	—
348	C₂×#41	—	2	4	no	—
349	C₂×#41	—	2	4	no	—
350	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
351	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
352	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
353	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
357	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
358	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
360	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
361	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
362	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
363	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	—
364	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	—
365	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
366	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
367	#117	—	2	8	no	—
368	#117	—	2	16	no	—
369	#117	C₂×C₄×C₃₁	1	8	yes	—
370	#117	C₂×C₄×C₃₁	1	16	yes	complemented
371	C₂×C₄⋊C₄	—	2	8	no	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
377	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
380	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
388	C₂×C₄²	—	2	8	no	—
389	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
390	C₂×C₄²	C₂×C₄×C₃₁	1	16	yes	complemented
391	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
392	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
393	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
394	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
395	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
396	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
397	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
398	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
400	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
402	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
404	#95	C₂³×C₃₁	1	2	yes	—
405	#95	C₂³×C₃₁	1	2	yes	—
406	#95	C₂³×C₃₁	1	4	yes	—
407	#95	C₂³×C₃₁	1	4	yes	—
408	#95	C₂³×C₃₁	1	8	yes	—
409	#95	C₂³×C₃₁	1	8	yes	—
410	#95	C₂³×C₃₁	1	8	yes	—
411	#95	C₂³×C₃₁	1	8	yes	—
412	#95	C₂³×C₃₁	1	4	yes	—
413	C₃₁	#34121	1	1	yes	characteristic, direct product factor, Sylow
414	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
415	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
416	#41	—	2	32	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
425	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	—	4	4	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	—	4	2	no	—
435	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
436	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
437	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
438	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
440	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
441	C₂²×C₄	#41×C₃₁	1	4	yes	—
442	C₂²×C₄	#41×C₃₁	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
445	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
448	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	4	4	no	—
460	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
461	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
462	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
463	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
464	C₄⋊C₄	#41×C₃₁	1	8	yes	complemented
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	16	no	—
467	C₂²⋊C₄	—	4	8	no	—
468	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
469	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
470	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
471	C₄²	—	2	4	no	—
472	C₄²	—	2	8	no	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	16	no	—
475	C₄²	—	2	8	no	—
476	C₄²	—	4	8	no	—
477	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
478	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
479	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
480	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
481	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
482	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
483	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	4	no	—
490	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
491	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
492	C₂³	C₂×#41×C₃₁	1	2	yes	—
493	C₂³	C₂×#41×C₃₁	1	2	yes	—
494	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
495	Q₈	—	2	8	no	—
496	D₄	—	2	16	no	—
497	D₄	—	2	8	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	16	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	16	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	4	16	no	—
519	C₂×C₄	C₃₁×#117	1	16	yes	complemented
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	4	yes	—
523	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
524	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
525	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
526	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
527	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
528	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
529	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
530	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
531	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
532	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
533	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
534	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	4	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	2	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	C₂×C₃₁×#117	1	4	yes	—
545	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
546	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
547	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
548	C₂²	C₂²×#41×C₃₁	1	2	yes	—
549	C₂²	C₃₁×#585	1	2	yes	—
550	C₄	—	2	4	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	2	no	—
553	C₄	—	2	2	no	—
554	C₄	—	2	4	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	8	no	—
557	C₄	—	4	4	no	—
558	C₄	—	4	8	no	—
559	C₄	C₂×C₃₁×#117	1	4	yes	—
560	C₄	C₂×C₃₁×#117	1	4	yes	—
561	C₄	C₄×#41×C₃₁	1	4	yes	—
562	C₂	—	2	4	no	—
563	C₂	—	4	4	no	—
564	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
565	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
566	C₂	C₃₁×#3428	1	1	yes	characteristic
567	C₂	C₃₁×#3475	1	2	yes	—
568	C₂	C₃₁×#3480	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×#2357	C₂	semidirect
3. C₃₁×#2356	C₂	semidirect
4. C₃₁×#2355	C₂	semidirect
5. C₃₁×#2354	C₂	semidirect
6. C₃₁×#2353	C₂	semidirect
7. C₃₁×#2352	C₂	semidirect
8. C₃₁×#2298	C₂	semidirect
10. C₃₁×#2285	C₂	semidirect
11. C₃₁×#2283	C₂	semidirect
12. C₄×C₃₁×#117	C₂	semidirect
13. C₄×C₃₁×#117	C₂	semidirect
14. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
16. C₄×#41×C₃₁	C₄	semidirect
77. C₄×Q₈×C₃₁	C₂×C₄	semidirect
78. C₄×Q₈×C₃₁	C₂×C₄	semidirect
81. C₄×D₄×C₃₁	C₂×C₄	semidirect
82. C₄×D₄×C₃₁	C₂×C₄	semidirect
86. C₃₁×#117	C₂×C₄	semidirect
106. C₂×C₄²×C₃₁	C₂×C₄	semidirect
179. C₄⋊C₄×C₃₁	#41	semidirect
199. #34121	C₃₁	direct (N × H)
235. C₂×C₄×C₃₁	#117	semidirect
252. #2357	C₂×C₃₁	semidirect
253. #2356	C₂×C₃₁	semidirect
254. #2355	C₂×C₃₁	semidirect
255. #2354	C₂×C₃₁	semidirect
256. #2353	C₂×C₃₁	semidirect
257. #2352	C₂×C₃₁	semidirect
258. #2298	C₂×C₃₁	semidirect
260. #2285	C₂×C₃₁	semidirect
261. #2283	C₂×C₃₁	semidirect
262. C₄×#117	C₂×C₃₁	semidirect
263. C₄×#117	C₂×C₃₁	semidirect
264. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
293. C₄×#41	C₄×C₃₁	semidirect
361. C₄×Q₈	C₂×C₄×C₃₁	semidirect
362. C₄×Q₈	C₂×C₄×C₃₁	semidirect
365. C₄×D₄	C₂×C₄×C₃₁	semidirect
366. C₄×D₄	C₂×C₄×C₃₁	semidirect
370. #117	C₂×C₄×C₃₁	semidirect
390. C₂×C₄²	C₂×C₄×C₃₁	semidirect
413. C₃₁	#34121	direct (N × H)
464. C₄⋊C₄	#41×C₃₁	semidirect
519. 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×#2357	2
C₃₁×#2356	2
C₃₁×#2355	2
C₃₁×#2354	2
C₃₁×#2353	2
C₃₁×#2352	2
C₃₁×#2298	2
C₃₁×#2297	2
C₃₁×#2285	2
C₃₁×#2283	2
C₄×C₃₁×#117	2
C₂×C₄×C₄⋊C₄×C₃₁	2
#34121	31

Subgroups — C31×#34121 #21490880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34121 #21490880