Subgroups — C₃₁×#33483 #21490242

Order 7936.

516 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2169	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2168	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2164	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2108	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#529	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	—
16	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#446	C₄	1	1	yes	characteristic
18	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#436	C₄	1	1	yes	characteristic
20	C₃₁×#435	C₄	1	2	yes	—
21	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₈×D₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#433	C₄	1	2	yes	—
24	C₃₁×#432	C₄	1	1	yes	characteristic
25	C₃₁×#405	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₃₁×C₄⋊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	—
34	C₃₁×#346	C₂²	1	1	yes	characteristic
35	C₃₁×#346	C₂²	1	1	yes	characteristic
36	C₃₁×#346	C₂²	1	2	yes	—
37	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
41	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×#126	C₈	1	2	yes	—
46	C₃₁×#125	C₈	1	1	yes	characteristic
47	C₃₁×#124	C₈	1	1	yes	characteristic
48	C₃₁×#123	C₈	1	2	yes	—
49	C₃₁×C₂²⋊Q₈	C₈	1	1	yes	characteristic
50	C₃₁×#121	C₈	1	1	yes	characteristic
51	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#117	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₃₁×M₃₂	—	2	2	no	—
57	C₃₁×M₃₂	—	2	2	no	—
58	C₃₁×M₃₂	—	2	4	no	—
59	C₂×C₁₆×C₃₁	—	2	4	no	—
60	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
67	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
69	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
70	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
71	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
72	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	C₃₁×#97	C₂×C₄	1	2	yes	—
74	C₃₁×#97	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₃₁	C₂³	1	2	yes	—
80	C₂×Q₈×C₃₁	C₂×C₈	1	1	yes	characteristic
81	C₂×D₄×C₃₁	C₂×C₈	1	1	yes	characteristic
82	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
87	M₁₆×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	—	2	1	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
100	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
101	C₄⋊C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
102	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
103	C₄⋊C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
104	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
109	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
110	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
112	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
115	C₁₆×C₃₁	—	2	2	no	—
116	C₁₆×C₃₁	—	2	1	no	—
117	C₁₆×C₃₁	—	2	1	no	—
118	C₁₆×C₃₁	—	2	2	no	—
119	C₁₆×C₃₁	#41	1	2	yes	—
120	C₁₆×C₃₁	#41	1	2	yes	—
121	#33483	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
123	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
124	Q₈×C₃₁	—	2	2	no	—
125	D₄×C₃₁	—	2	2	no	—
126	D₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₃₁	—	2	2	no	—
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	—	2	1	no	—
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	#131	1	2	yes	—
147	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
149	#2966	C₂×C₃₁	1	1	yes	characteristic
150	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
151	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#2169	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2169	C₂×C₃₁	1	2	yes	complemented
154	#2168	C₂×C₃₁	1	2	yes	complemented
155	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#2164	C₂×C₃₁	1	2	yes	complemented
157	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2108	C₂×C₃₁	1	2	yes	complemented
159	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
160	C₂²×C₃₁	—	2	1	no	—
161	C₂²×C₃₁	—	2	2	no	—
162	C₂²×C₃₁	—	2	1	no	—
163	C₂²×C₃₁	—	2	2	no	—
164	C₂²×C₃₁	—	2	1	no	—
165	C₂²×C₃₁	—	2	2	no	—
166	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	2	2	no	—
169	C₄×C₃₁	—	2	1	no	—
170	C₄×C₃₁	—	2	1	no	—
171	C₄×C₃₁	—	2	2	no	—
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	2	2	no	—
174	C₄×C₃₁	#504	1	2	yes	—
175	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
176	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
177	#529	C₄×C₃₁	1	1	yes	characteristic
178	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
181	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
182	#446	C₄×C₃₁	1	1	yes	characteristic
183	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
184	#436	C₄×C₃₁	1	1	yes	characteristic
185	#435	C₄×C₃₁	1	2	yes	—
186	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
187	C₈×D₄	C₂²×C₃₁	1	2	yes	—
188	#433	C₄×C₃₁	1	2	yes	—
189	#432	C₄×C₃₁	1	1	yes	characteristic
190	#405	C₂²×C₃₁	1	2	yes	—
191	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
192	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
193	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
194	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
195	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
196	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
197	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
198	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
199	#346	C₂²×C₃₁	1	1	yes	characteristic
200	#346	C₂²×C₃₁	1	1	yes	characteristic
201	#346	C₂²×C₃₁	1	2	yes	—
202	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
203	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
204	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
205	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
206	C₂×C₃₁	—	4	1	no	—
207	C₂×C₃₁	—	4	2	no	—
208	C₂×C₃₁	C₈×#41	1	1	yes	characteristic
209	C₂×C₃₁	C₂×#504	1	1	yes	characteristic
210	C₂×C₃₁	#3407	1	1	yes	characteristic
211	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
212	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
215	#126	C₈×C₃₁	1	2	yes	—
216	#125	C₈×C₃₁	1	1	yes	characteristic
217	#124	C₈×C₃₁	1	1	yes	characteristic
218	#123	C₈×C₃₁	1	2	yes	—
219	C₂²⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
220	#121	C₈×C₃₁	1	1	yes	characteristic
221	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
222	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
223	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
224	#117	C₂×C₄×C₃₁	1	2	yes	—
225	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
226	M₃₂	—	2	2	no	—
227	M₃₂	—	2	2	no	—
228	M₃₂	—	2	4	no	—
229	C₂×C₁₆	—	2	4	no	—
230	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
231	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
233	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
234	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
235	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
236	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
237	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
238	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
239	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
240	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
241	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
242	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
243	#97	C₂×C₄×C₃₁	1	2	yes	—
244	#97	C₂×C₄×C₃₁	1	2	yes	—
245	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
246	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₃₁	1	2	yes	—
250	C₃₁	#33483	1	1	yes	characteristic, direct product factor, Sylow
251	C₂×Q₈	C₂×C₈×C₃₁	1	1	yes	characteristic
252	C₂×D₄	C₂×C₈×C₃₁	1	1	yes	characteristic
253	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
254	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
258	M₁₆	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
267	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
268	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
270	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
271	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
272	C₄⋊C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
273	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
274	C₄⋊C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
275	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
276	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
277	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
278	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
279	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
280	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
281	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
282	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
283	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
284	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
285	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	2	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	#41×C₃₁	1	2	yes	—
291	C₁₆	#41×C₃₁	1	2	yes	—
292	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
293	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
308	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
309	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
310	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
311	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	2	no	—
316	C₈	C₃₁×#131	1	2	yes	—
317	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
318	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
319	C₂²	—	2	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	2	no	—
325	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	C₃₁×#504	1	2	yes	—
334	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
335	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	4	2	no	—
338	C₂	C₈×#41×C₃₁	1	1	yes	characteristic
339	C₂	C₂×C₃₁×#504	1	1	yes	characteristic
340	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×#121	C₈
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×#2183	C₂	semidirect
3. Q₈×C₁₆×C₃₁	C₂	semidirect
4. C₃₁×#2169	C₂	semidirect
5. C₃₁×#2169	C₂	semidirect
6. C₃₁×#2168	C₂	semidirect
7. D₄×C₁₆×C₃₁	C₂	semidirect
8. C₃₁×#2164	C₂	semidirect
9. C₃₁×#2162	C₂	semidirect
10. C₃₁×#2108	C₂	semidirect
11. C₄×C₃₁×M₃₂	C₂	semidirect
121. #33483	C₃₁	direct (N × H)
150. #2183	C₂×C₃₁	semidirect
151. Q₈×C₁₆	C₂×C₃₁	semidirect
152. #2169	C₂×C₃₁	semidirect
153. #2169	C₂×C₃₁	semidirect
154. #2168	C₂×C₃₁	semidirect
155. D₄×C₁₆	C₂×C₃₁	semidirect
156. #2164	C₂×C₃₁	semidirect
157. #2162	C₂×C₃₁	semidirect
158. #2108	C₂×C₃₁	semidirect
159. C₄×M₃₂	C₂×C₃₁	semidirect
250. C₃₁	#33483	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×#2966	2
C₃₁×#2183	2
Q₈×C₁₆×C₃₁	2
C₃₁×#2169	2
C₃₁×#2168	2
D₄×C₁₆×C₃₁	2
C₃₁×#2164	2
C₃₁×#2162	2
C₃₁×#2108	2
C₄×C₃₁×M₃₂	2
#33483	31

Subgroups — C31×#33483 #21490242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33483 #21490242