Subgroups — C₃₁×#33490 #21490249

Order 7936.

574 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2169	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2168	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2168	C₂	1	2	yes	maximal, complemented
5	D₄×C₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2164	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2113	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#525	C₄	1	1	yes	characteristic
13	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
16	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
19	C₃₁×#435	C₄	1	2	yes	—
20	C₃₁×#435	C₄	1	2	yes	—
21	C₈×D₄×C₃₁	C₂²	1	2	yes	—
22	C₈×D₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#424	C₄	1	1	yes	characteristic
24	C₃₁×#408	C₄	1	1	yes	characteristic
25	C₃₁×#407	C₄	1	1	yes	characteristic
26	C₃₁×#405	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
28	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
30	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
31	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
34	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
35	C₃₁×#346	C₂²	1	2	yes	complemented
36	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
37	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₃₁×#131	—	4	1	no	—
40	C₃₁×#131	—	4	1	no	—
41	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₃₁	—	4	1	no	—
45	C₂²×C₈×C₃₁	—	4	1	no	—
46	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#123	C₈	1	2	yes	—
54	C₃₁×C₂²⋊Q₈	C₈	1	1	yes	characteristic
55	C₃₁×C₂²⋊Q₈	C₈	1	1	yes	characteristic
56	C₃₁×#121	C₈	1	1	yes	characteristic
57	C₃₁×#121	C₈	1	1	yes	characteristic
58	C₃₁×#120	C₈	1	2	yes	—
59	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
62	C₃₁×M₃₂	—	4	1	no	—
63	C₃₁×M₃₂	—	4	1	no	—
64	C₃₁×M₃₂	—	4	2	no	—
65	C₂×C₁₆×C₃₁	—	4	1	no	—
66	C₂×C₁₆×C₃₁	—	4	1	no	—
67	C₂×C₁₆×C₃₁	—	4	2	no	—
68	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
72	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
73	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
74	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
75	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
76	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
77	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
78	C₃₁×#97	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₃₁	C₂³	1	2	yes	—
80	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
81	#41×C₃₁	—	4	1	no	—
82	#41×C₃₁	—	4	1	no	—
83	C₂×Q₈×C₃₁	C₂×C₈	1	1	yes	characteristic
84	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
85	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
86	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
87	C₂×D₄×C₃₁	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	—	4	1	no	—
90	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	M₁₆×C₃₁	—	4	1	no	—
98	M₁₆×C₃₁	—	4	2	no	—
99	M₁₆×C₃₁	—	4	1	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	1	no	—
104	C₂×C₈×C₃₁	—	2	1	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	1	no	—
108	C₂×C₈×C₃₁	—	4	1	no	—
109	C₂×C₈×C₃₁	—	4	1	no	—
110	C₂×C₈×C₃₁	—	4	1	no	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
118	C₄⋊C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
119	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
120	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
121	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
122	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
123	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
124	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
125	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
126	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
127	C₁₆×C₃₁	—	2	2	no	—
128	C₁₆×C₃₁	—	2	2	no	—
129	C₁₆×C₃₁	—	2	2	no	—
130	C₁₆×C₃₁	—	2	2	no	—
131	#33490	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	—	4	1	no	—
134	C₂³×C₃₁	—	4	1	no	—
135	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
136	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
137	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
138	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
139	Q₈×C₃₁	—	4	1	no	—
140	D₄×C₃₁	—	2	4	no	—
141	D₄×C₃₁	—	4	2	no	—
142	D₄×C₃₁	—	4	1	no	—
143	D₄×C₃₁	—	4	2	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	2	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	4	1	no	—
155	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
156	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₃₁	—	2	1	no	—
163	C₈×C₃₁	—	2	1	no	—
164	C₈×C₃₁	—	2	1	no	—
165	C₈×C₃₁	—	2	2	no	—
166	C₈×C₃₁	—	2	2	no	—
167	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
168	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
169	#2929	C₂×C₃₁	1	1	yes	characteristic
170	#2169	C₂×C₃₁	1	2	yes	complemented
171	#2168	C₂×C₃₁	1	2	yes	complemented
172	#2168	C₂×C₃₁	1	2	yes	complemented
173	D₄×C₁₆	C₂×C₃₁	1	2	yes	complemented
174	#2164	C₂×C₃₁	1	2	yes	complemented
175	#2113	C₂×C₃₁	1	2	yes	complemented
176	#2112	C₂×C₃₁	1	1	yes	characteristic, complemented
177	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
178	C₂²×C₃₁	—	2	1	no	—
179	C₂²×C₃₁	—	2	1	no	—
180	C₂²×C₃₁	—	2	2	no	—
181	C₂²×C₃₁	—	2	1	no	—
182	C₂²×C₃₁	—	2	1	no	—
183	C₂²×C₃₁	—	2	1	no	—
184	C₂²×C₃₁	—	2	2	no	—
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	2	2	no	—
189	C₂²×C₃₁	—	4	4	no	—
190	C₂²×C₃₁	#504	1	2	yes	—
191	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
192	C₄×C₃₁	—	2	2	no	—
193	C₄×C₃₁	—	2	2	no	—
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	2	1	no	—
196	C₄×C₃₁	—	2	1	no	—
197	C₄×C₃₁	—	2	2	no	—
198	C₄×C₃₁	#518	1	1	yes	characteristic
199	C₄×C₃₁	#568	1	1	yes	characteristic
200	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
201	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
202	#525	C₄×C₃₁	1	1	yes	characteristic
203	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
206	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
207	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
208	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
209	#435	C₄×C₃₁	1	2	yes	—
210	#435	C₄×C₃₁	1	2	yes	—
211	C₈×D₄	C₂²×C₃₁	1	2	yes	—
212	C₈×D₄	C₂²×C₃₁	1	2	yes	—
213	#424	C₄×C₃₁	1	1	yes	characteristic
214	#408	C₄×C₃₁	1	1	yes	characteristic
215	#407	C₄×C₃₁	1	1	yes	characteristic
216	#405	C₂²×C₃₁	1	1	yes	characteristic
217	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
218	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
219	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
220	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
221	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
222	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
223	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
224	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
225	#346	C₂²×C₃₁	1	2	yes	complemented
226	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
227	C₂×C₃₁	—	2	2	no	—
228	C₂×C₃₁	—	4	2	no	—
229	C₂×C₃₁	—	4	1	no	—
230	C₂×C₃₁	—	4	1	no	—
231	C₂×C₃₁	#2972	1	1	yes	characteristic
232	C₂×C₃₁	C₂×#504	1	1	yes	characteristic
233	C₂×C₃₁	#3407	1	1	yes	characteristic
234	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
235	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
236	#131	—	4	1	no	—
237	#131	—	4	1	no	—
238	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
239	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
240	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
241	C₂²×C₈	—	4	1	no	—
242	C₂²×C₈	—	4	1	no	—
243	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
244	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
247	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
250	#123	C₈×C₃₁	1	2	yes	—
251	C₂²⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
252	C₂²⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
253	#121	C₈×C₃₁	1	1	yes	characteristic
254	#121	C₈×C₃₁	1	1	yes	characteristic
255	#120	C₈×C₃₁	1	2	yes	—
256	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
257	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
258	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
259	M₃₂	—	4	1	no	—
260	M₃₂	—	4	1	no	—
261	M₃₂	—	4	2	no	—
262	C₂×C₁₆	—	4	1	no	—
263	C₂×C₁₆	—	4	1	no	—
264	C₂×C₁₆	—	4	2	no	—
265	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
266	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
267	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
268	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
269	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
270	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
271	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
272	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
273	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
274	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
275	#97	C₂×C₄×C₃₁	1	2	yes	—
276	C₄×C₈	C₂³×C₃₁	1	2	yes	—
277	C₃₁	#33490	1	1	yes	characteristic, direct product factor, Sylow
278	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
279	#41	—	4	1	no	—
280	#41	—	4	1	no	—
281	C₂×Q₈	C₂×C₈×C₃₁	1	1	yes	characteristic
282	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
283	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
284	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
285	C₂×D₄	C₂×C₈×C₃₁	1	1	yes	characteristic
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
291	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
292	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
293	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
294	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
295	M₁₆	—	4	1	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	4	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
310	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
311	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
312	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
313	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
314	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
315	C₄⋊C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
316	C₄⋊C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
317	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
318	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
319	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
320	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
321	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
322	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
323	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
324	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
325	C₁₆	—	2	2	no	—
326	C₁₆	—	2	2	no	—
327	C₁₆	—	2	2	no	—
328	C₁₆	—	2	2	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	4	1	no	—
331	C₂³	—	4	1	no	—
332	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
333	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
334	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
335	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
336	Q₈	—	4	1	no	—
337	D₄	—	2	4	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	1	no	—
340	D₄	—	4	2	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
353	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
354	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
357	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
358	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	2	no	—
363	C₈	—	2	2	no	—
364	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
365	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	2	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	4	4	no	—
378	C₂²	C₃₁×#504	1	2	yes	—
379	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
380	C₄	—	2	2	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	2	no	—
386	C₄	C₃₁×#518	1	1	yes	characteristic
387	C₄	C₃₁×#568	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	4	1	no	—
390	C₂	—	4	1	no	—
391	C₂	—	4	2	no	—
392	C₂	C₃₁×#2972	1	1	yes	characteristic
393	C₂	C₂×C₃₁×#504	1	1	yes	characteristic
394	C₂	C₃₁×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₈
prime radical	C₃₁×#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2169	C₂	semidirect
3. C₃₁×#2168	C₂	semidirect
4. C₃₁×#2168	C₂	semidirect
5. D₄×C₁₆×C₃₁	C₂	semidirect
6. C₃₁×#2164	C₂	semidirect
7. C₃₁×#2113	C₂	semidirect
8. C₃₁×#2112	C₂	semidirect
9. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
15. C₂×C₃₁×M₃₂	C₂²	semidirect
18. C₂²×C₁₆×C₃₁	C₂²	semidirect
27. C₃₁×C₄⋊C₁₆	C₂²	semidirect
28. C₃₁×C₄⋊C₁₆	C₂²	semidirect
33. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
34. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
35. C₃₁×#346	C₂²	semidirect
36. C₄×C₁₆×C₃₁	C₂²	semidirect
131. #33490	C₃₁	direct (N × H)
170. #2169	C₂×C₃₁	semidirect
171. #2168	C₂×C₃₁	semidirect
172. #2168	C₂×C₃₁	semidirect
173. D₄×C₁₆	C₂×C₃₁	semidirect
174. #2164	C₂×C₃₁	semidirect
175. #2113	C₂×C₃₁	semidirect
176. #2112	C₂×C₃₁	semidirect
177. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
205. C₂×M₃₂	C₂²×C₃₁	semidirect
208. C₂²×C₁₆	C₂²×C₃₁	semidirect
217. C₄⋊C₁₆	C₂²×C₃₁	semidirect
218. C₄⋊C₁₆	C₂²×C₃₁	semidirect
223. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
224. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
225. #346	C₂²×C₃₁	semidirect
226. C₄×C₁₆	C₂²×C₃₁	semidirect
277. C₃₁	#33490	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×#2929	2
C₃₁×#2169	2
C₃₁×#2168	2
D₄×C₁₆×C₃₁	2
C₃₁×#2164	2
C₃₁×#2113	2
C₃₁×#2112	2
C₂×C₃₁×C₂²⋊C₁₆	2
#33490	31

Subgroups — C31×#33490 #21490249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33490 #21490249