Subgroups — C₃₁×#34199 #21490958

Order 7936.

3190 conjugacy classes of subgroups.

The subgroup lattice has 594 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	C₂×C₃₁×#518	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2359	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2344	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2342	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2322	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2315	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2308	C₂	1	1	yes	maximal, characteristic
10	C₄×C₃₁×#123	C₂	1	4	yes	maximal, complemented
11	C₂×C₃₁×#381	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2273	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#123	C₄	1	16	yes	complemented
16	C₂×C₃₁×#123	C₂²	1	2	yes	—
17	C₂×C₃₁×#123	C₂²	1	4	yes	—
18	C₂×C₃₁×#123	C₂²	1	2	yes	—
19	C₃₁×#518	—	2	8	no	—
20	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
22	C₂×C₃₁×#117	—	2	8	no	—
23	C₂×C₃₁×#117	C₂²	1	4	yes	—
24	C₂×C₃₁×#117	C₂²	1	2	yes	—
25	C₂×C₃₁×#117	C₂²	1	4	yes	complemented
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
36	C₃₁×#389	C₂²	1	8	yes	complemented
37	C₃₁×#388	C₂²	1	8	yes	complemented
38	C₃₁×#388	C₂²	1	8	yes	complemented
39	C₃₁×#387	C₂²	1	8	yes	complemented
40	C₃₁×#387	C₂²	1	8	yes	complemented
41	C₃₁×#386	C₂²	1	4	yes	—
42	C₃₁×#386	C₂²	1	4	yes	—
43	C₃₁×#385	C₂²	1	4	yes	complemented
44	C₃₁×#385	C₂²	1	4	yes	complemented
45	C₃₁×#385	C₂²	1	8	yes	complemented
46	C₃₁×#381	C₂²	1	2	yes	—
47	C₃₁×#381	C₂²	1	4	yes	—
48	C₃₁×#381	C₂²	1	2	yes	complemented
49	C₃₁×#381	C₂²	1	2	yes	complemented
50	C₃₁×#380	C₂²	1	4	yes	—
51	C₃₁×#379	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
53	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
54	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
55	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
56	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
57	C₂×C₃₁×#95	C₂²	1	4	yes	complemented
58	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
59	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₃₁	—	2	4	no	—
61	C₂²×D₄×C₃₁	—	2	4	no	—
62	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
63	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
64	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	—	2	4	no	—
66	C₂³×C₄×C₃₁	—	2	4	no	—
67	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
68	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
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	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
79	C₃₁×#123	C₂×C₄	1	16	yes	complemented
80	C₃₁×#123	C₂×C₄	1	32	yes	complemented
81	C₃₁×#123	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₃₁	—	2	16	no	—
83	C₄×D₄×C₃₁	—	2	16	no	—
84	C₃₁×#117	—	2	16	no	—
85	C₃₁×#117	—	2	16	no	—
86	C₃₁×#117	—	2	8	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
88	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
95	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
98	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
99	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
100	C₂×C₂²⋊C₄×C₃₁	D₄	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×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄²×C₃₁	—	2	8	no	—
116	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
118	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
119	C₃₁×#95	C₂³	1	8	yes	—
120	C₃₁×#95	C₂³	1	8	yes	—
121	C₃₁×#95	C₂³	1	8	yes	—
122	C₃₁×#95	C₂³	1	8	yes	—
123	C₃₁×#95	C₂³	1	8	yes	—
124	C₂⁴×C₃₁	—	2	2	no	—
125	C₂⁴×C₃₁	—	2	2	no	—
126	C₂⁴×C₃₁	—	4	4	no	—
127	C₂⁴×C₃₁	—	4	4	no	—
128	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
129	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
130	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
131	C₂⁴×C₃₁	#41	1	4	yes	—
132	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
133	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
134	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
135	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
136	C₂×D₄×C₃₁	—	2	32	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
142	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
143	C₂²×C₄×C₃₁	—	2	32	no	—
144	C₂²×C₄×C₃₁	—	2	2	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	2	4	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	4	4	no	—
159	C₂²×C₄×C₃₁	—	4	4	no	—
160	C₂²×C₄×C₃₁	—	4	4	no	—
161	C₂²×C₄×C₃₁	—	4	8	no	—
162	C₂²×C₄×C₃₁	—	4	4	no	—
163	C₂²×C₄×C₃₁	—	4	4	no	—
164	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
165	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
169	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
179	C₄⋊C₄×C₃₁	—	2	16	no	—
180	C₄⋊C₄×C₃₁	—	2	8	no	—
181	C₄⋊C₄×C₃₁	—	4	8	no	—
182	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
183	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
184	C₂²⋊C₄×C₃₁	—	2	4	no	—
185	C₂²⋊C₄×C₃₁	—	2	4	no	—
186	C₂²⋊C₄×C₃₁	—	2	4	no	—
187	C₂²⋊C₄×C₃₁	—	2	4	no	—
188	C₂²⋊C₄×C₃₁	—	2	4	no	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	—	2	8	no	—
191	C₂²⋊C₄×C₃₁	—	4	8	no	—
192	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
193	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
194	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
195	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
196	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
197	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
198	C₄²×C₃₁	—	2	16	no	—
199	C₄²×C₃₁	—	2	8	no	—
200	C₄²×C₃₁	—	4	8	no	—
201	#34199	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	2	4	no	—
204	C₂³×C₃₁	—	2	4	no	—
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	8	no	—
208	C₂³×C₃₁	—	2	8	no	—
209	C₂³×C₃₁	—	4	8	no	—
210	C₂³×C₃₁	—	4	8	no	—
211	C₂³×C₃₁	—	4	8	no	—
212	C₂³×C₃₁	—	4	8	no	—
213	C₂³×C₃₁	—	8	4	no	—
214	C₂³×C₃₁	—	8	4	no	—
215	C₂³×C₃₁	C₄×D₄	1	16	yes	—
216	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
217	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
218	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
219	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
220	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
221	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
222	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
223	C₂³×C₃₁	C₂×#41	1	2	yes	—
224	C₂³×C₃₁	C₂×#41	1	2	yes	—
225	C₂³×C₃₁	C₂×#41	1	2	yes	—
226	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
227	D₄×C₃₁	—	2	16	no	—
228	D₄×C₃₁	—	2	16	no	—
229	C₂×C₄×C₃₁	—	2	16	no	—
230	C₂×C₄×C₃₁	—	2	8	no	—
231	C₂×C₄×C₃₁	—	2	8	no	—
232	C₂×C₄×C₃₁	—	2	16	no	—
233	C₂×C₄×C₃₁	—	2	8	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	8	no	—
241	C₂×C₄×C₃₁	—	2	8	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	8	no	—
246	C₂×C₄×C₃₁	—	2	8	no	—
247	C₂×C₄×C₃₁	—	2	8	no	—
248	C₂×C₄×C₃₁	—	4	8	no	—
249	C₂×C₄×C₃₁	—	4	8	no	—
250	C₂×C₄×C₃₁	—	4	8	no	—
251	C₂×C₄×C₃₁	—	8	8	no	—
252	C₂×C₄×C₃₁	—	8	4	no	—
253	C₂×C₄×C₃₁	—	8	4	no	—
254	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
255	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
256	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
257	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
259	C₂²×#123	C₂×C₃₁	1	1	yes	characteristic
260	C₂×#518	C₂×C₃₁	1	1	yes	characteristic
261	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#2359	C₂×C₃₁	1	8	yes	complemented
263	#2344	C₂×C₃₁	1	4	yes	complemented
264	#2342	C₂×C₃₁	1	4	yes	complemented
265	#2322	C₂×C₃₁	1	4	yes	complemented
266	#2315	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#2308	C₂×C₃₁	1	1	yes	characteristic
268	C₄×#123	C₂×C₃₁	1	4	yes	complemented
269	C₂×#381	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#2273	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂²×C₃₁	—	2	4	no	—
272	C₂²×C₃₁	—	2	4	no	—
273	C₂²×C₃₁	—	2	4	no	—
274	C₂²×C₃₁	—	2	4	no	—
275	C₂²×C₃₁	—	2	4	no	—
276	C₂²×C₃₁	—	2	4	no	—
277	C₂²×C₃₁	—	2	4	no	—
278	C₂²×C₃₁	—	2	4	no	—
279	C₂²×C₃₁	—	2	4	no	—
280	C₂²×C₃₁	—	2	4	no	—
281	C₂²×C₃₁	—	2	8	no	—
282	C₂²×C₃₁	—	4	16	no	—
283	C₂²×C₃₁	—	4	4	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	—	8	8	no	—
286	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
287	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
288	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
289	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
290	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
291	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
292	C₂²×C₃₁	#585	1	4	yes	—
293	C₄×C₃₁	—	2	4	no	—
294	C₄×C₃₁	—	2	4	no	—
295	C₄×C₃₁	—	2	4	no	—
296	C₄×C₃₁	—	2	8	no	—
297	C₄×C₃₁	—	2	8	no	—
298	C₄×C₃₁	—	4	4	no	—
299	C₄×C₃₁	—	4	4	no	—
300	C₄×C₃₁	—	4	4	no	—
301	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
302	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#123	C₄×C₃₁	1	16	yes	complemented
304	C₂×#123	C₂²×C₃₁	1	2	yes	—
305	C₂×#123	C₂²×C₃₁	1	4	yes	—
306	C₂×#123	C₂²×C₃₁	1	2	yes	—
307	#518	—	2	8	no	—
308	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
309	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
310	C₂×#117	—	2	8	no	—
311	C₂×#117	C₂²×C₃₁	1	4	yes	—
312	C₂×#117	C₂²×C₃₁	1	2	yes	—
313	C₂×#117	C₂²×C₃₁	1	4	yes	complemented
314	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
315	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
316	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
317	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
318	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
319	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
320	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
321	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
322	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
323	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
324	#389	C₂²×C₃₁	1	8	yes	complemented
325	#388	C₂²×C₃₁	1	8	yes	complemented
326	#388	C₂²×C₃₁	1	8	yes	complemented
327	#387	C₂²×C₃₁	1	8	yes	complemented
328	#387	C₂²×C₃₁	1	8	yes	complemented
329	#386	C₂²×C₃₁	1	4	yes	—
330	#386	C₂²×C₃₁	1	4	yes	—
331	#385	C₂²×C₃₁	1	4	yes	complemented
332	#385	C₂²×C₃₁	1	4	yes	complemented
333	#385	C₂²×C₃₁	1	8	yes	complemented
334	#381	C₂²×C₃₁	1	2	yes	—
335	#381	C₂²×C₃₁	1	4	yes	—
336	#381	C₂²×C₃₁	1	2	yes	complemented
337	#381	C₂²×C₃₁	1	2	yes	complemented
338	#380	C₂²×C₃₁	1	4	yes	—
339	#379	C₂²×C₃₁	1	2	yes	—
340	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
341	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
342	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
343	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
344	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
345	C₂×#95	C₂²×C₃₁	1	4	yes	complemented
346	C₂×C₃₁	—	2	4	no	—
347	C₂×C₃₁	—	2	4	no	—
348	C₂×C₃₁	—	2	4	no	—
349	C₂×C₃₁	—	4	4	no	—
350	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
351	C₂×C₃₁	#3428	1	2	yes	—
352	C₂×C₃₁	#3485	1	2	yes	—
353	C₂×C₃₁	#3491	1	2	yes	—
354	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
355	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
356	C₂²×D₄	—	2	4	no	—
357	C₂²×D₄	—	2	4	no	—
358	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
359	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
360	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
361	C₂³×C₄	—	2	4	no	—
362	C₂³×C₄	—	2	4	no	—
363	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	complemented
364	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
365	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
368	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
370	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
371	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
372	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
373	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
374	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
375	#123	C₂×C₄×C₃₁	1	16	yes	complemented
376	#123	C₂×C₄×C₃₁	1	32	yes	complemented
377	#123	C₂×C₄×C₃₁	1	16	yes	complemented
378	C₄×D₄	—	2	16	no	—
379	C₄×D₄	—	2	16	no	—
380	#117	—	2	16	no	—
381	#117	—	2	16	no	—
382	#117	—	2	8	no	—
383	C₂×C₄⋊C₄	—	2	8	no	—
384	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
385	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
386	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
387	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
388	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
390	C₂×C₂²⋊C₄	—	2	4	no	—
391	C₂×C₂²⋊C₄	—	2	4	no	—
392	C₂×C₂²⋊C₄	—	2	4	no	—
393	C₂×C₂²⋊C₄	—	2	8	no	—
394	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
395	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
396	C₂×C₂²⋊C₄	D₄×C₃₁	1	8	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
411	C₂×C₄²	—	2	8	no	—
412	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
413	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
414	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
415	#95	C₂³×C₃₁	1	8	yes	—
416	#95	C₂³×C₃₁	1	8	yes	—
417	#95	C₂³×C₃₁	1	8	yes	—
418	#95	C₂³×C₃₁	1	8	yes	—
419	#95	C₂³×C₃₁	1	8	yes	—
420	C₃₁	#34199	1	1	yes	characteristic, direct product factor, Sylow
421	C₂⁴	—	2	2	no	—
422	C₂⁴	—	2	2	no	—
423	C₂⁴	—	4	4	no	—
424	C₂⁴	—	4	4	no	—
425	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
426	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
427	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
428	C₂⁴	#41×C₃₁	1	4	yes	—
429	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
430	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
431	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
432	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
433	C₂×D₄	—	2	32	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
439	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
440	C₂²×C₄	—	2	32	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	8	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	4	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	—	4	8	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
462	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
463	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
464	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
465	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
466	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
467	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
468	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
469	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
470	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
471	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
476	C₄⋊C₄	—	2	16	no	—
477	C₄⋊C₄	—	2	8	no	—
478	C₄⋊C₄	—	4	8	no	—
479	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
480	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	4	8	no	—
489	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
490	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
491	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
492	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
493	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
494	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
495	C₄²	—	2	16	no	—
496	C₄²	—	2	8	no	—
497	C₄²	—	4	8	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	8	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	4	8	no	—
506	C₂³	—	4	8	no	—
507	C₂³	—	4	8	no	—
508	C₂³	—	4	8	no	—
509	C₂³	—	8	4	no	—
510	C₂³	—	8	4	no	—
511	C₂³	C₄×D₄×C₃₁	1	16	yes	—
512	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
513	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
514	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
515	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
516	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
517	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
518	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
519	C₂³	C₂×#41×C₃₁	1	2	yes	—
520	C₂³	C₂×#41×C₃₁	1	2	yes	—
521	C₂³	C₂×#41×C₃₁	1	2	yes	—
522	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
523	D₄	—	2	16	no	—
524	D₄	—	2	16	no	—
525	C₂×C₄	—	2	16	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	8	no	—
528	C₂×C₄	—	2	16	no	—
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	8	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	8	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	8	no	—
546	C₂×C₄	—	4	8	no	—
547	C₂×C₄	—	8	8	no	—
548	C₂×C₄	—	8	4	no	—
549	C₂×C₄	—	8	4	no	—
550	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
551	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
552	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
553	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
554	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
555	C₂²	—	2	4	no	—
556	C₂²	—	2	4	no	—
557	C₂²	—	2	4	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	2	8	no	—
566	C₂²	—	4	16	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	8	8	no	—
570	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
571	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
572	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
573	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
574	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
575	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
576	C₂²	C₃₁×#585	1	4	yes	—
577	C₄	—	2	4	no	—
578	C₄	—	2	4	no	—
579	C₄	—	2	4	no	—
580	C₄	—	2	8	no	—
581	C₄	—	2	8	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	4	no	—
584	C₄	—	4	4	no	—
585	C₂	—	2	4	no	—
586	C₂	—	2	4	no	—
587	C₂	—	2	4	no	—
588	C₂	—	4	4	no	—
589	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
590	C₂	C₃₁×#3428	1	2	yes	—
591	C₂	C₃₁×#3485	1	2	yes	—
592	C₂	C₃₁×#3491	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂²×C₃₁×#117	C₂	semidirect
4. C₃₁×#2359	C₂	semidirect
5. C₃₁×#2344	C₂	semidirect
6. C₃₁×#2342	C₂	semidirect
7. C₃₁×#2322	C₂	semidirect
8. C₃₁×#2315	C₂	semidirect
10. C₄×C₃₁×#123	C₂	semidirect
11. C₂×C₃₁×#381	C₂	semidirect
12. C₃₁×#2273	C₂	semidirect
15. C₂×C₃₁×#123	C₄	semidirect
25. C₂×C₃₁×#117	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
35. C₂²×C₄²×C₃₁	C₂²	semidirect
36. C₃₁×#389	C₂²	semidirect
37. C₃₁×#388	C₂²	semidirect
38. C₃₁×#388	C₂²	semidirect
39. C₃₁×#387	C₂²	semidirect
40. C₃₁×#387	C₂²	semidirect
43. C₃₁×#385	C₂²	semidirect
44. C₃₁×#385	C₂²	semidirect
45. C₃₁×#385	C₂²	semidirect
48. C₃₁×#381	C₂²	semidirect
49. C₃₁×#381	C₂²	semidirect
52. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
54. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
55. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
56. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
57. C₂×C₃₁×#95	C₂²	semidirect
67. C₂³×C₄×C₃₁	C₂×C₄	semidirect
79. C₃₁×#123	C₂×C₄	semidirect
80. C₃₁×#123	C₂×C₄	semidirect
81. C₃₁×#123	C₂×C₄	semidirect
88. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
98. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
99. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
201. #34199	C₃₁	direct (N × H)
261. C₂²×#117	C₂×C₃₁	semidirect
262. #2359	C₂×C₃₁	semidirect
263. #2344	C₂×C₃₁	semidirect
264. #2342	C₂×C₃₁	semidirect
265. #2322	C₂×C₃₁	semidirect
266. #2315	C₂×C₃₁	semidirect
268. C₄×#123	C₂×C₃₁	semidirect
269. C₂×#381	C₂×C₃₁	semidirect
270. #2273	C₂×C₃₁	semidirect
303. C₂×#123	C₄×C₃₁	semidirect
313. C₂×#117	C₂²×C₃₁	semidirect
315. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
323. C₂²×C₄²	C₂²×C₃₁	semidirect
324. #389	C₂²×C₃₁	semidirect
325. #388	C₂²×C₃₁	semidirect
326. #388	C₂²×C₃₁	semidirect
327. #387	C₂²×C₃₁	semidirect
328. #387	C₂²×C₃₁	semidirect
331. #385	C₂²×C₃₁	semidirect
332. #385	C₂²×C₃₁	semidirect
333. #385	C₂²×C₃₁	semidirect
336. #381	C₂²×C₃₁	semidirect
337. #381	C₂²×C₃₁	semidirect
340. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
342. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
343. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
344. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
345. C₂×#95	C₂²×C₃₁	semidirect
363. C₂³×C₄	C₂×C₄×C₃₁	semidirect
375. #123	C₂×C₄×C₃₁	semidirect
376. #123	C₂×C₄×C₃₁	semidirect
377. #123	C₂×C₄×C₃₁	semidirect
384. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
394. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
395. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
420. C₃₁	#34199	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
C₂×C₃₁×#518	2
C₂²×C₃₁×#117	2
C₃₁×#2359	2
C₃₁×#2344	2
C₃₁×#2342	2
C₃₁×#2322	2
C₃₁×#2315	2
C₃₁×#2308	2
C₄×C₃₁×#123	2
C₂×C₃₁×#381	2
C₃₁×#2273	2
#34199	31

Subgroups — C31×#34199 #21490958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34199 #21490958