Subgroups — C₃₁×#23351 #21480110

Order 7936.

696 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2398	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1924	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1870	C₂	1	2	yes	maximal
7	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1832	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#400	C₄	1	1	yes	characteristic
23	C₃₁×#400	C₄	1	1	yes	characteristic
24	C₃₁×#397	C₄	1	2	yes	—
25	C₃₁×#394	C₄	1	1	yes	characteristic
26	C₃₁×#394	C₄	1	1	yes	characteristic
27	C₃₁×#393	C₄	1	2	yes	—
28	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#376	C₂²	1	1	yes	characteristic
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×M₁₆×C₃₁	—	4	2	no	—
41	C₂×M₁₆×C₃₁	—	4	2	no	—
42	C₂×M₁₆×C₃₁	—	4	2	no	—
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₄×Q₈×C₃₁	—	4	1	no	—
48	C₄×Q₈×C₃₁	—	4	1	no	—
49	C₄×D₄×C₃₁	—	4	1	no	—
50	C₄×D₄×C₃₁	—	4	1	no	—
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊C₈	—	4	2	no	—
65	C₃₁×C₄⋊C₈	—	4	2	no	—
66	C₃₁×C₄⋊C₈	—	4	1	no	—
67	C₃₁×C₄⋊C₈	—	4	1	no	—
68	C₃₁×C₂²⋊C₈	—	4	2	no	—
69	C₃₁×C₂²⋊C₈	—	4	2	no	—
70	C₃₁×C₂²⋊C₈	—	4	2	no	—
71	C₃₁×#97	—	4	1	no	—
72	C₄×C₈×C₃₁	—	4	1	no	—
73	C₃₁×#95	C₂×C₄	1	2	yes	—
74	C₃₁×#95	C₂×C₄	1	2	yes	—
75	C₃₁×#95	C₂×C₄	1	2	yes	—
76	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
77	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
78	C₃₁×#95	C₂×C₄	1	2	yes	—
79	C₃₁×#95	C₂×C₄	1	2	yes	—
80	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	4	1	no	—
83	C₂×Q₈×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	—	4	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	M₁₆×C₃₁	—	4	8	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	4	1	no	—
116	C₄⋊C₄×C₃₁	—	4	1	no	—
117	C₄⋊C₄×C₃₁	—	4	1	no	—
118	C₄⋊C₄×C₃₁	—	4	1	no	—
119	C₂²⋊C₄×C₃₁	—	2	4	no	—
120	C₂²⋊C₄×C₃₁	—	4	2	no	—
121	C₂²⋊C₄×C₃₁	—	4	2	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₂²⋊C₄×C₃₁	—	4	2	no	—
124	C₄²×C₃₁	—	2	1	no	—
125	C₄²×C₃₁	—	2	1	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	1	no	—
128	C₄²×C₃₁	—	2	1	no	—
129	C₄²×C₃₁	—	2	1	no	—
130	C₄²×C₃₁	—	2	1	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	#23351	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	2	2	no	—
135	C₂³×C₃₁	—	2	2	no	—
136	C₂³×C₃₁	—	2	2	no	—
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
141	Q₈×C₃₁	—	4	4	no	—
142	D₄×C₃₁	—	4	4	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	#131	1	2	yes	—
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
175	C₈×C₃₁	—	4	2	no	—
176	C₈×C₃₁	—	4	2	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	4	2	no	—
179	#2398	C₂×C₃₁	1	1	yes	characteristic
180	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1924	C₂×C₃₁	1	2	yes	complemented
183	#1923	C₂×C₃₁	1	2	yes	complemented
184	#1870	C₂×C₃₁	1	2	yes	—
185	#1851	C₂×C₃₁	1	2	yes	complemented
186	#1851	C₂×C₃₁	1	2	yes	complemented
187	#1832	C₂×C₃₁	1	2	yes	complemented
188	C₂²×C₃₁	—	4	2	no	—
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	—	4	2	no	—
191	C₂²×C₃₁	—	4	2	no	—
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	#518	1	1	yes	characteristic
196	C₂²×C₃₁	#520	1	2	yes	—
197	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
198	C₂²×C₃₁	#568	1	1	yes	characteristic
199	C₂²×C₃₁	#568	1	2	yes	—
200	C₄×C₃₁	—	2	2	no	—
201	C₄×C₃₁	—	4	2	no	—
202	C₄×C₃₁	—	4	1	no	—
203	C₄×C₃₁	—	4	1	no	—
204	C₄×C₃₁	—	4	1	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	4	1	no	—
207	C₄×C₃₁	—	4	2	no	—
208	C₄×C₃₁	—	4	2	no	—
209	C₄×C₃₁	H(3,𝔽₄)	1	1	yes	characteristic
210	C₄×C₃₁	#562	1	2	yes	—
211	C₄×C₃₁	#563	1	1	yes	characteristic
212	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
213	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
214	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
216	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
217	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
220	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
221	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
222	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
224	#400	C₄×C₃₁	1	1	yes	characteristic
225	#400	C₄×C₃₁	1	1	yes	characteristic
226	#397	C₄×C₃₁	1	2	yes	—
227	#394	C₄×C₃₁	1	1	yes	characteristic
228	#394	C₄×C₃₁	1	1	yes	characteristic
229	#393	C₄×C₃₁	1	2	yes	—
230	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
231	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
232	#376	C₂²×C₃₁	1	1	yes	characteristic
233	#336	C₂²×C₃₁	1	2	yes	—
234	#336	C₂²×C₃₁	1	2	yes	—
235	#336	C₂²×C₃₁	1	2	yes	—
236	#336	C₂²×C₃₁	1	2	yes	—
237	#336	C₂²×C₃₁	1	2	yes	—
238	#336	C₂²×C₃₁	1	2	yes	—
239	C₂×C₃₁	—	8	2	no	—
240	C₂×C₃₁	#2379	1	1	yes	characteristic
241	C₂×C₃₁	#2976	1	1	yes	characteristic
242	C₂×C₃₁	#2978	1	1	yes	characteristic
243	C₂×C₃₁	#2993	1	2	yes	—
244	C₂×C₃₁	#2995	1	2	yes	—
245	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
246	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
247	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
248	C₂×M₁₆	—	4	2	no	—
249	C₂×M₁₆	—	4	2	no	—
250	C₂×M₁₆	—	4	2	no	—
251	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
252	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
253	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
254	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
255	C₄×Q₈	—	4	1	no	—
256	C₄×Q₈	—	4	1	no	—
257	C₄×D₄	—	4	1	no	—
258	C₄×D₄	—	4	1	no	—
259	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
263	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
264	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
265	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
266	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
267	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
268	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
269	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
270	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
272	C₄⋊C₈	—	4	2	no	—
273	C₄⋊C₈	—	4	2	no	—
274	C₄⋊C₈	—	4	1	no	—
275	C₄⋊C₈	—	4	1	no	—
276	C₂²⋊C₈	—	4	2	no	—
277	C₂²⋊C₈	—	4	2	no	—
278	C₂²⋊C₈	—	4	2	no	—
279	#97	—	4	1	no	—
280	C₄×C₈	—	4	1	no	—
281	#95	C₂×C₄×C₃₁	1	2	yes	—
282	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
283	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
284	#95	C₂×C₄×C₃₁	1	2	yes	—
285	#95	C₂×C₄×C₃₁	1	2	yes	—
286	#95	C₂×C₄×C₃₁	1	2	yes	—
287	#95	C₂×C₄×C₃₁	1	2	yes	—
288	C₃₁	#23351	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	4	1	no	—
292	C₂×Q₈	—	4	1	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	4	2	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
302	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
306	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
311	M₁₆	—	4	8	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₂²⋊C₄	—	2	4	no	—
329	C₂²⋊C₄	—	4	2	no	—
330	C₂²⋊C₄	—	4	2	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₄²	—	2	1	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	1	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	2	no	—
341	C₂³	—	2	2	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
349	Q₈	—	4	4	no	—
350	D₄	—	4	4	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	C₃₁×#131	1	2	yes	—
379	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
380	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
381	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
382	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
383	C₈	—	4	2	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	4	2	no	—
394	C₂²	C₃₁×#518	1	1	yes	characteristic
395	C₂²	C₃₁×#520	1	2	yes	—
396	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
397	C₂²	C₃₁×#568	1	1	yes	characteristic
398	C₂²	C₃₁×#568	1	2	yes	—
399	C₄	—	2	2	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	2	no	—
408	C₄	C₃₁×H(3,𝔽₄)	1	1	yes	characteristic
409	C₄	C₃₁×#562	1	2	yes	—
410	C₄	C₃₁×#563	1	1	yes	characteristic
411	C₂	—	8	2	no	—
412	C₂	C₃₁×#2379	1	1	yes	characteristic
413	C₂	C₃₁×#2976	1	1	yes	characteristic
414	C₂	C₃₁×#2978	1	1	yes	characteristic
415	C₂	C₃₁×#2993	1	2	yes	—
416	C₂	C₃₁×#2995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂³ ▷ 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×#1985	C₂	semidirect
3. C₃₁×#1984	C₂	semidirect
4. C₃₁×#1924	C₂	semidirect
5. C₃₁×#1923	C₂	semidirect
7. C₃₁×#1851	C₂	semidirect
8. C₃₁×#1851	C₂	semidirect
9. C₃₁×#1832	C₂	semidirect
132. #23351	C₃₁	direct (N × H)
180. #1985	C₂×C₃₁	semidirect
181. #1984	C₂×C₃₁	semidirect
182. #1924	C₂×C₃₁	semidirect
183. #1923	C₂×C₃₁	semidirect
185. #1851	C₂×C₃₁	semidirect
186. #1851	C₂×C₃₁	semidirect
187. #1832	C₂×C₃₁	semidirect
288. C₃₁	#23351	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#2398	2
C₃₁×#1985	2
C₃₁×#1984	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1870	2
C₃₁×#1851	2
C₃₁×#1832	2
#23351	31

Subgroups — C31×#23351 #21480110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23351 #21480110